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quantum effects, such as particle–antiparticle creation or annihilation. Table 74 shows that the approach distance is near or smaller than the Compton wavelength only in the microscopic world, so that such effects are not observed in everyday life. We do not therefore need quantum field theory to describe common observations. The combined concepts of quantum field theory and general relativity are required in situations in which both conditions are satisfied simultaneously. The necessary approach distance for such situations is calculated by setting r S = 2λ C (the factor 2 is introduced for simplicity). We find that this is the case when lengths or times are (of the order of) l Pl =

¼

ħG~c 3 ¼ t Pl = ħG~c 5

Ref. 1018

= 5.4 ë 10−44 s , the Planck time.

(666)

Whenever we approach objects at these scales, both general relativity and quantum mechanics play a role; at these scales effects of quantum gravity appear. Because the values of the Planck dimensions are extremely small, this level of sophistication is unnecessary in everyday life, in astronomy and even in particle physics. However, to answer the questions posted at the beginning of the book – why do we live in three dimensions and why is the proton 1836.15 times heavier than the electron? – we require a precise and complete description of nature. The contradictions between quantum mechanics and general relativity appear to make the search for answers impossible. However, while the unified theory describing quantum gravity is not yet complete, we can already get a few glimpses at its implications from its present stage of development. Note that the Planck scales specify one of only two domains of nature where quantum mechanics and general relativity apply at the same time. (What is the other?) As Planck scales are the easier of the two to study, they provide the best starting point for the following discussion. When Max Planck discovered the existence of Planck scales or Planck units, he was interested in them mainly as natural units of measurement, and that is what he called them. However, their importance in nature is much more widespread, as we will shall see in the new section. We will discover that they determine what is commonly called quantum geometry. Farewell to instants of time

Ref. 1019, Ref. 1020

The appearance of the quantum of action in the description of motion leads to quantum limits to all measurements. These limits have important consequences at Planck dimensions. Measurement limits appear most clearly when we investigate the properties of clocks and metre rules. Is it possible to construct a clock that is able to measure time intervals shorter than the Planck time? Surprisingly, the answer is no, even though the time–energy indeterminacy relation ∆E∆t E ħ seems to indicate that by making ∆E arbitrary large, we can make ∆t arbitrary small. Every clock is a device with some moving parts. Parts can be mechanical wheels, particles of matter in motion, changing electrodynamic fields, i.e. photons, or decaying

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Time is composed of time atoms ... which in fact are indivisible. Moses Maimonides, 12th century.

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Challenge 1426 n

= 1.6 ë 10−35 m, the Planck length,

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1002

Ref. 1021, Ref. 1022 Ref. 1023

xi general rel ativity versus quantum mechanics

radioactive particles. For each moving component of a clock, such as the two hands, the indeterminacy principle applies. As discussed most clearly by Michael Raymer, the indeterminacy relation for two non-commuting variables describes two different, but related situations: it makes a statement about standard deviations of separate measurements on many identical systems; and it describes the measurement precision for a joint measurement on a single system. Throughout this article, only the second situation is considered. For any clock to work, we need to know both the time and the energy of each hand. Otherwise it would not be a recording device. Put more generally, a clock must be a classical system. We need the combined knowledge of the non-commuting variables for each moving component of the clock. Let us focus on the component with the largest time indeterminacy ∆t. It is evident that the smallest time interval δt that can be measured by a clock is always larger than the quantum limit, i.e. larger than the time indeterminacy ∆t for the most ‘uncertain’ component. Thus we have ħ , ∆E

(667)

where ∆E is the energy indeterminacy of the moving component, and this energy indeterminacy ∆E must be smaller than the total energy E = mc 2 of the component itself.* Furthermore, a clock provides information and thus signals have to be able to leave it. To make this possible, the clock must not be a black hole and its mass m must therefore be smaller than the Schwarzschild mass for its size, i.e. m D c 2 l~G, where l is the size of the clock (neglecting factors of order unity). Finally, for a sensible measurement of the time interval δt, the size l of the clock must be smaller than c δt itself, because otherwise different parts of the clock could not work together to produce the same time display.** If we combine all these conditions, we get

or

δt E δt E

ħG = t Pl . c5

(668)

(669)

In summary, from three simple properties of any clock, namely that there is only a single clock, that we can read its dial and that it gives sensible read-outs, we get the general conclusion that clocks cannot measure time intervals shorter than the Planck time. Note that this argument is independent of the nature of the clock mechanism. Whether the clock is powered by gravitational, electrical, plain mechanical or even nuclear means, the limit still applies.*** The same result can also be found in other ways. For example, any clock small enough to measure small time intervals necessarily has a certain energy indeterminacy due to * Physically, this condition means being sure that there is only one clock; the case ∆E A E would mean that it is impossible to distinguish between a single clock and a clock–anticlock pair created from the vacuum, or a component plus two such pairs, etc. ** It is amusing to explore how a clock larger than c δt would stop working, as a result of the loss of rigidity

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Ref. 1027

¾

ħG c 5 δt

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δt E ∆t E

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Page 410 Ref. 1016

1003

the indeterminacy relation. At the same time, on the basis of general relativity, any energy density induces a deformation of space-time and signals from the deformed region arrive with a certain delay due to that deformation. The energy indeterminacy of the source leads to an indeterminacy in the deformation and thus in the delay. The expression from general relativity for the deformation of the time part of the line element due to a mass m is δt = mG~l c 3 . From the mass–energy relation, an energy spread ∆E produces an indeterminacy ∆t in the delay ∆E G ∆t = . (670) l c5

Farewell to points in space Ref. 1028

Furthermore, the measurement cannot be performed if signals cannot leave the objects; thus they may not be black holes. Therefore their masses must be small enough for their Schwarzschild radius r S = 2Gm~c 2 to be smaller than the distance δl separating them. Challenge 1427 n

Ref. 1025 1024 Ref. 1026

in its components. *** Note that gravitation is essential here. The present argument differs from the well-known study on the limitations of clocks due to their mass and their measuring time which was published by Salecker and Wigner and summarized in pedagogical form by Zimmerman. Here, both quantum mechanics and gravity are included, and therefore a different, lower and much more fundamental limit is found. Note also that the discovery of black hole radiation does not change the argument; black hole radiation notwithstanding, measurement devices cannot exist inside black holes.

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In a similar way, we can deduce that it is impossible to make a metre rule or any other length measuring device that is able to measure lengths shorter than the Planck length. Obviously, we can already deduce this from l Pl = c t Pl , but a separate proof is also possible. The straightforward way to measure the distance between two points is to put an object at rest at each position. In other words, joint measurements of position and momentum are necessary for every length measurement. Now, the minimal length δl that can be measured must be larger than the position indeterminacy of the two objects. From the indeterminacy principle it is known that each object’s position cannot be determined with a precision ∆l better than that given by the indeterminacy relation ∆l ∆p = ħ, where ∆p is the momentum indeterminacy. The requirement that there is only one object at each end, i.e. avoiding pair production from the vacuum, means that ∆p < mc; together, these requirements give ħ . (671) δl E ∆l E mc

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This indeterminacy determines the precision of the clock. Furthermore, the energy indeterminacy of the clock is fixed by the indeterminacy relation for time and energy ∆E E ħ~∆t, in turn fixed by the precision of the clock. Combining all this, we again find the relation δt E t Pl for the minimum measurable time. We are forced to conclude that in nature there is a minimum time interval. In other words, at Planck scales the term ‘instant of time’ has no theoretical or experimental basis. It therefore makes no sense to use the term.

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xi general rel ativity versus quantum mechanics

Again omitting the factor of 2, we get δl E

Ref. 1006, Ref. 1016 Ref. 1028, Ref. 1029, Ref. 1030

Page 349

(672)

Another way to deduce this limit reverses the roles of general relativity and quantum theory. To measure the distance between two objects, we have to localize the first object with respect to the other within a certain interval ∆x. The corresponding energy indeterminacy obeys ∆E = c(c 2 m 2 + (∆p)2 )1~2 E cħ~∆x. However, general relativity shows that a small volume filled with energy changes the curvature of space-time, and thus changes the metric of the surrounding space. For the resulting distance change ∆l, compared to empty space, we find the expression ∆l  G∆E~c 4 . In short, if we localize the first particle in space with a precision ∆x, the distance to a second particle is known only with precision ∆l. The minimum length δl that can be measured is obviously larger than either of these quantities; inserting the expression for ∆E, we find again that the minimum measurable length δl is given by the Planck length. We note that, as the Planck length is the shortest possible length, it follows that there can be no observations of quantum mechanical effects for situations in which the corresponding de Broglie or Compton wavelength is smaller than the Planck length. In proton– proton collisions we observe both pair production and interference effects. In contrast, the Planck limit implies that in everyday, macroscopic situations, such as car–car collisions, we cannot observe embryo–antiembryo pair production and quantum interference effects. In summary, from two simple properties common to all length measuring devices, namely that they can be counted and that they can be read out, we arrive at the conclusion that lengths smaller than the Planck length cannot be found in measurements. Whatever method is used, be it a metre rule or time-of-flight measurement, we cannot overcome this fundamental limit. It follows that the concept of a ‘point in space’ has no experimental basis. In the same way, the term ‘event’, being a combination of a ‘point in space’ and an ‘instant of time’, also loses its meaning for the description of nature. A simple way to deduce the minimum length using the limit statements which structure this ascent is the following. General relativity is based on a maximum force in nature, or alternatively, on a minimum mass change per time; its value is given by dm~dt = c 3 ~4G. Quantum theory is based on a minimum action in nature, given by L = ħ~2. Since a distance d can be expressed like L d2 = , (673) dm~dt one sees directly that a minimum action and a maximum mass rate imply a minimum distance. In other words, quantum theory and general relativity, when put together, imply a minimum distance. These results are often expressed by the so-called generalized indeterminacy principle ∆p∆x E ħ~2 + f

G (∆p)2 c3

(674)

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Ref. 1034

ħG = l Pl . c3

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Ref. 1031, Ref. 1032, Ref. 1033

¾

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d oes matter differ from vacuum?

or

Challenge 1428 e

Ref. 1034 Ref. 1035

Ref. 1037, Ref. 1038, Ref. 1039

Ref. 1040

2 l Pl (∆p)2 , ħ

(675)

where f is a numerical factor of order unity. A similar expression holds for the time– energy indeterminacy relation. The first term on the right hand side is the usual quantum mechanical indeterminacy. The second term, negligible for everyday life energies, plays a role only near Planck energies and is due to the changes in space-time induced by gravity at these high energies. You should be able to show that the generalized principle (674) automatically implies that ∆x can never be smaller than f 1~2 l Pl . The generalized indeterminacy principle is derived in exactly the same way in which Heisenberg derived the original indeterminacy principle ∆p∆x E ħ~2, namely by studying the deflection of light by an object under a microscope. A careful re-evaluation of the process, this time including gravity, yields equation (674). For this reason, all approaches that try to unify quantum mechanics and gravity must yield this relation; indeed, it appears in the theory of canonical quantum gravity, in superstring theory and in the quantum group approach. We remember that quantum mechanics starts when we realize that the classical concept of action makes no sense below the value of ħ~2; similarly, unified theories start when we realize that the classical concepts of time and length make no sense below Planck values. However, the usual description of space-time does contain such small values; the usual description involves the existence of intervals smaller than the smallest measurable one. Therefore, the continuum description of space-time has to be abandoned in favour of a more appropriate description. The new indeterminacy relation appearing at Planck scales shows that continuity cannot be a good description of space-time. Inserting c∆p E ∆E E ħ~∆t into equation (674), we get ∆x∆t E ħG~c 4 = t Pl l Pl , (676)

Farewell to the space-time manifold The consequences of the Planck limits for measurements of time and space can be taken much further. It is commonplace to say that given any two points in space or any two instants of time, there is always a third in between. Physicists sloppily call this property continuity, while mathematicians call it denseness. However, at Planck dimensions this

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which of course has no counterpart in standard quantum mechanics. It shows that spacetime events do not exist. A final way to convince oneself that points have no meaning is that a point is an entity with vanishing volume; however, the minimum volume possible 3 in nature is the Planck volume VPl = l Pl . While space-time points are idealizations of events, this idealization is incorrect. The use of the concept of ‘point’ is similar to the use of the concept of ‘aether’ a century ago: it is impossible to detect and it is only useful for describing observations until a way to describe nature without it has been found. Like ‘aether´, also ‘point´ leads reason astray. In other words, the Planck units do not only provide natural units, they also provide — within a factor of order one — the limit values of space and time intervals.

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Ref. 1036

∆p∆x E ħ~2 + f

1005

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1006

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Ref. 1028, Ref. 1042

property cannot exist, since intervals smaller than the Planck time can never be found. Thus points and instants are not dense, and between two points there is not always a third. This means that space and time are not continuous. Of course, at large scales they are – approximately – continuous, in the same way that a piece of rubber or a liquid seems continuous at everyday dimensions, even though it is not at a small scale. All paradoxes resulting from the infinite divisibility of space and time, such as Zeno’s argument on the impossibility to distinguish motion from rest, or the Banach–Tarski paradox, are now avoided. We can dismiss the paradoxes straight away because of their incorrect premises concerning the nature of space and time. But let us go on. Special relativity, quantum mechanics and general relativity all rely on the idea that time can be defined for all points of a given reference frame. However, two clocks a distance l apart cannot be synchronized with arbitrary precision. Since the distance between two clocks cannot be measured with an error smaller than the Planck length l Pl , and transmission of signals is necessary for synchronization, it is not possible to synchronize two clocks with a better precision than the time l Pl ~c = t Pl , the Planck time. Because it is impossible to synchronize clocks precisely, a single time coordinate for a whole reference frame is only an approximation, and this idea cannot be maintained in a precise description of nature. Moreover, since the time difference between events can only be measured within a Planck time, for two events distant in time by this order of magnitude, it is not possible to say with complete certainty which of the two precedes the other! This is an important result. If events cannot be ordered, the concept of time, which was introduced into physics to describe sequences, cannot be defined at all at Planck scales. In other words, after dropping the idea of a common time coordinate for a complete frame of reference, we are forced to drop the idea of time at a single ‘point’ as well. Therefore, the concept of ‘proper time’ loses its meaning at Planck scales. It is straightforward to use the same arguments to show that length measurements do not allow us to speak of continuous space, but only of approximately continuous space. As a result of the lack of measurement precision at Planck scales, the concepts of spatial order, translation invariance, isotropy of vacuum and global coordinate systems have no experimental basis. But there is more to come. The very existence of a minimum length contradicts special relativity theory, in which it is shown that lengths undergo Lorentz contraction when the frame of reference is changed. A minimum length thus cannot exist in special relativity. But we just deduced that there must be such a minimum distance in nature. There is only one conclusion: special relativity cannot be correct at smallest distances. Thus, space-time is neither Lorentz invariant nor diffeomorphism invariant nor dilatation invariant at Planck dimensions. All symmetries that are at the basis of special and general relativity are thus only approximately valid at Planck scales. As a result of the imprecision of measurement, most familiar concepts used to describe spatial relations become useless. For example, the concept of metric loses its usefulness at Planck scales. Since distances cannot be measured with precision, the metric cannot be determined. We deduce that it is impossible to say precisely whether space is flat or curved. In other words, the impossibility of measuring lengths exactly is equivalent to fluctuations of the curvature, and thus equivalent to fluctuations of gravity. In addition, even the number of spatial dimensions makes no sense at Planck scales.

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Ref. 1041

xi general rel ativity versus quantum mechanics

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d oes matter differ from vacuum?

Let us remind ourselves how to determine this number experimentally. One possible way is to determine how many points we can choose in space such that all the distances between them are equal. If we can find at most n such points, the space has n − 1 dimensions. We can see that if reliable length measurement at Planck scale is not possible, there is no way to determine reliably the number of dimensions of a space with this method. Another way to check for three spatial dimensions is to make a knot in a shoe string and glue the ends together: since it stays knotted we know that space has three dimensions, because there is a mathematical theorem that in spaces with greater or fewer than three dimensions, knots do not exist. Again, at Planck dimensions the errors in measurement do not allow to say whether a string is knotted or not, because measurement limits at crossings make it impossible to say which strand lies above the other; in short, at Planck scales we cannot check whether space has three dimensions or not. There are many other methods for determining the dimensionality of space.* All these methods start from the definition of the concept of dimensionality, which is based on a precise definition of the concept of neighbourhood. However, at Planck scales, as just mentioned, length measurements do not allow us to say whether a given point is inside or outside a given volume. In short, whatever method we use, the lack of reliable length measurements means that at Planck scales, the dimensionality of physical space is not defined. It should therefore not come as a surprise that when we approach these scales, we may get a scale-dependent answer for the number of dimensions, that may be different from three. The reason for the problems with space-time become most evident when we remember Euclid’s well-known definition: ‘A point is that which has no part.’ As Euclid clearly understood, a physical point, and here the stress is on physical, cannot be defined without some measurement method. A physical point is an idealization of position, and as such includes measurement right from the start. In mathematics, however, Euclid’s definition is rejected; mathematical points do not need a metric for their definition. Mathematical points are elements of a set, usually called a space. In mathematics, a measurable or metric space is a set of points equipped afterwards with a measure or a metric. Mathematical points do not need a metric for their definition; they are basic entities. In contrast to the mathematical situation, the case of physical space-time, the concepts of measure and of metric are more fundamental than that of a point. The difficulties distinguishing physical and mathematical space and points arise from the failure to distinguish a mathematical metric from a physical length measurement.**

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* For example, we can determine the dimension using only the topological properties of space. If we draw a so-called covering of a topological space with open sets, there are always points that are elements of several sets of the covering. Let us call p the maximal number of sets of which a point can be an element in a given covering. This number can be determined for all possible coverings. The minimum value of p, minus one, gives the dimension of the space. In fact, if physical space is not a manifold, the various methods may give different answers for the dimensionality. Indeed, for linear spaces without norm, a unique number of dimensions cannot be defined. The value then depends on the specific definition used and is called e.g. fractal dimension, Lyapunov dimension, etc. ** Where does the incorrect idea of continuous space-time have its roots? In everyday life, as well as in physics, space-time is introduced to describe observations. Space-time is a book-keeping device. Its properties are extracted from the properties of observables. Since observables can be added and multiplied, we extrapolate that they can take continuous values. This extrapolation implies that length and time intervals can take continuous values, and, in particular, arbitrary small values. From this result we get the possibil-

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Ref. 1043

1007

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1008

Ref. 1044

xi general rel ativity versus quantum mechanics

There is nothing in the world but matter in motion, and matter in motion cannot move otherwise than in space and time. Lenin

To complete this review of the situation, if space and time are not continuous, no quantities defined as derivatives with respect to space or time are precisely defined. Velocity, acceleration, momentum, energy, etc., are only well-defined under the assumption of continuous space and time. That important tool, the evolution equation, is based on derivatives and thus can no longer be used. Therefore the Schrödinger or the Dirac equation ity of defining points and sets of points. A special field of mathematics, topology, shows how to start from a set of points and construct, with the help of neighbourhood relations and separation properties, first a topological space. Then, with the help of a metric, a metric space can be built. With the appropriate compactness and connectedness relations, a manifold, characterized by its dimension, metric and topology, can be constructed. * A manifold is what locally looks like an Euclidean space. The exact definition can be found in Appendix D.

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Farewell to observables and measurements

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Perhaps the most beautiful way to make this point is the Banach–Tarski theorem, which clearly shows the limits of the concept of volume. The theorem states that a sphere made up of mathematical points can be cut into five pieces in such a way that the pieces can be put together to form two spheres, each of the same volume as the original one. However, the necessary cuts are ‘infinitely’ curved and detailed: they are wildly disconnected. For physical matter such as gold, unfortunately – or fortunately – the existence of a minimum length, namely the atomic distance, makes it impossible to perform such a cut. For vacuum, the puzzle reappears: for example, the energy of zero-point fluctuations is given by the density times the volume; following the Banach–Tarski theorem, the zero point energy content of a single sphere should be equal to the zero point energy of two similar spheres each of the same volume as the original one. The paradox is solved by the Planck length, because it also provides a fundamental length scale for vacuum, thus making infinitely complex cuts impossible. Therefore, the concept of volume is only well defined at Planck scales if a minimum length is introduced. To sum up, physical space-time cannot be a set of mathematical points. But the surprises are not finished. At Planck dimensions, since both temporal and spatial order break down, there is no way to say if the distance between two space-time regions that are close enough together is space-like or time-like. Measurement limits make it impossible to distinguish the two cases. At Planck scales, time and space cannot be distinguished from each other. In addition, it is impossible to state that the topology of space-time is fixed, as general relativity implies. The topology changes – mentioned above – required for particle reactions do become possible. In this way another of the contradictions between general relativity and quantum theory is resolved. In summary, space-time at Planck scales is not continuous, not ordered, not endowed with a metric, not four-dimensional and not made up of points. If we compare this with the definition of the term manifold,* not one of its defining properties is fulfilled. We arrive at the conclusion that the concept of a space-time manifold has no backing at Planck scales. This is a strong result. Even though both general relativity and quantum mechanics use continuous space-time, the combination of both theories does not.

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1009

Ref. 1045 Ref. 1046 Ref. 1047 Ref. 1049

Discretization of space-time has been studied already since 1940s. Recently, the idea that space-time could be described as a lattice has also been explored most notably by David Finkelstein and by Gerard ’t Hooft. The idea of space-time as a lattice is based on the idea that, if a minimum distance exists, then all distances are a multiple of this minimum. It is generally agreed that, in order to get an isotropic and homogeneous situation for large,

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Can space-time be a lattice? – A glimpse of quantum geometry

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lose their basis. Concepts such as ‘derivative’, ‘divergence-free’, ‘source free’ etc., lose their meaning at Planck scales. In fact, all physical observables are defined using length and time measurements. A list of physical units shows that each is a product of powers of length, time (and mass) units. (Even though in the SI system electrical quantities have a separate base quantity, the ampere, the argument still holds; the ampere is itself defined in terms of a force, which is measured using the three base units length, time and mass.) Since time and length are not continuous, observables themselves are not defined, because their value is not fixed. This means that at Planck scales, observables cannot be described by real numbers. In addition, if time and space are not continuous, the usual expression for an observable field A, namely A(t, x), does not make sense: we have to find a more appropriate description. Physical fields cannot exist at Planck scales. The consequences for quantum mechanics are severe. It makes no sense to define multiplication of observables by real numbers, thus by a continuous range of values, but only by a discrete set of numbers. Among other implications, this means that observables do not form a linear algebra. We recognize that, because of measurement errors, we cannot prove that observables do form such an algebra. This means that observables are not described by operators at Planck scales. And, because quantum mechanics is based on the superposition principle, without it, everything comes crumbling down. In particular, the most important observables are the gauge potentials. Since they do not now form an algebra, gauge symmetry is not valid at Planck scales. Even innocuous looking expressions such as [x i , x j ] = 0 for x i x x j , which are at the root of quantum field theory, become meaningless at Planck scales. Since at those scales also the superposition principle cannot be backed up by experiment, even the famous Wheeler–DeWitt equation, often assumed to describe quantum gravity, cannot be valid. Similarly, permutation symmetry is based on the premise that we can distinguish two points by their coordinates, and then exchange particles between those two locations. As we have just seen, this is not possible if the distance between the two particles is small; we conclude that permutation symmetry has no experimental basis at Planck scales. Even discrete symmetries, like charge conjugation, space inversion and time reversal cannot be correct in this domain, because there is no way to verify them exactly by measurement. CPT symmetry is not valid at Planck scales. Finally we note that all types of scaling relations do not work at smallest scales. As a result, renormalization symmetry is also destroyed at Planck scales. All these results are consistent: if there are no symmetries at Planck scales, there are also no observables, since physical observables are representations of symmetry groups. In fact, the limits on time and length measurements imply that the concept of measurement has no significance at Planck scales.

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1010

Ref. 1048

Farewell to particles In every example of motion, some object is involved. One of the important discoveries of the natural sciences was that all objects are composed of small constituents, called elementary particles. Quantum theory shows that all composite, non-elementary objects

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everyday scales, the structure of space-time cannot be periodic, but must be random. In addition, any fixed structure of space-time violates the result that there are no lengths smaller than the Planck length: as a result of the Lorentz contraction, any moving observer would find lattice distances smaller than the Planck value. Worse still, the lattice idea conflicts with general relativity, in particular with the diffeomorphism invariance of vacuum. Finally, where would a particle be during the jump from one lattice point to the next? Thus, in summary, space-time cannot be a lattice. A minimum distance does exist in nature; however, the hope that all other distances are simple multiples of the smallest distance is not correct. We will discover more evidence for this later on. If space-time is not a set of points or events, it must be something else. Three hints already appear at this stage. The first step necessary to improve the description of motion is the recognition that abandoning ‘points’ means abandoning the local description of nature. Both quantum mechanics and general relativity assume that the phrase ‘observable at a point’ has a precise meaning. Because it is impossible to describe space as a manifold, this expression is no longer useful. The unification of general relativity and quantum physics forces the adoption of a non-local description of nature at Planck scales. The existence of a minimum length implies that there is no way to physically distinguish locations that are even closer together. We are tempted to conclude therefore that no pair of locations can be distinguished, even if they are one metre apart, since on any path joining two points, no two locations that are close together can be distinguished. This situation is similar to the question about the size of a cloud or of an atom. If we measure water density or electron density, we find non-vanishing values at any distance from the centre of the cloud or the atom; however, an effective size can still be defined, because it is very unlikely that the effects of the presence of a cloud or of an atom can be seen at distances much larger than this effective size. Similarly, we can guess that two points in space-time at a macroscopic distance from each other can be distinguished because the probability that they will be confused drops rapidly with increasing distance. In short, we are thus led to a probabilistic description of space-time. Space-time becomes a macroscopic observable, a statistical or thermodynamic limit of some microscopic entities. We note that a fluctuating structure for space-time would also avoid the problems of fixed structures with Lorentz invariance. This property is of course compatible with a statistical description. In summary, the experimental observations of special relativity, i.e. Lorentz invariance, isotropy and homogeneity, together with that of a minimum distance, point towards a fluctuating description of space-time. Research efforts in quantum gravity, superstring theory and quantum groups have confirmed independently of each other that a probabilistic and non-local description of space-time at Planck dimensions, resolves the contradictions between general relativity and quantum theory. This is our first result on quantum geometry. To clarify the issue, we have to turn to the concept of the particle.

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Ref. 1050

xi general rel ativity versus quantum mechanics

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have a finite, non-vanishing size. This property allows us to determine whether a particle is elementary or not. If it behaves like a point particle, it is elementary. At present, only the leptons (electron, muon, tau and the neutrinos), the quarks and the radiation quanta of the electromagnetic, weak and strong nuclear interactions (the photon, the W and Z bosons, the gluons) have been found to be elementary. A few more elementary particles are predicted by various refinements of the standard model. Protons, atoms, molecules, cheese, people, galaxies etc., are all composite, as shown in Table 74. Elementary particles are characterized by their vanishing size, their spin and their mass. Even though the definition of ‘elementary particle’ as point particle is all we need in the following argument, it is not complete, because it seems to leave open the possibility that future experiments could show that electrons or quarks are not elementary. This is not so! In fact, any particle smaller than its own Compton wavelength is elementary. If it were composite, there would be a lighter component inside it and this lighter particle would have a larger Compton wavelength than the composite particle. This is impossible, since the size of a composite particle must be larger than the Compton wavelength of its components. (The possibility that all components are heavier than the composite, which would avoid this argument, does not lead to satisfying physical properties; for example, it leads to intrinsically unstable components.) The size of an object, such as those given in Table 74, is defined as the length at which differences from point-like behaviour are observed. This is the way in which, using alpha particle scattering, the radius of the atomic nucleus was determined for the first time in Rutherford’s experiment. In other words, the size d of an object is determined by measuring how it scatters a beam of probe particles. In daily life as well, when we look at objects, we make use of scattered photons. In general, in order to make use of scattering, the effective wavelength λ = ħ~mv of the probe must be smaller than the object size d to be determined. We thus need d A λ = ħ~(mv) E ħ~(mc). In addition, in order to make a scattering experiment possible, the object must not be a black hole, since, if it were, it would simply swallow the approaching particle. This means that its mass m must be smaller than that of a black hole of the same size; in other words, from equation (664) we must have m < dc 2 ~G. Combining this with the previous condition we get ħG = l Pl . c3

(677)

In other words, there is no way to observe that an object is smaller than the Planck length. There is thus no way in principle to deduce from observations that a particle is point-like. In fact, it makes no sense to use the term ‘point particle’ at all! Of course, there is a relation between the existence of a minimum length for empty space and a minimum length for objects. If the term ‘point of space’ is meaningless, then the term ‘point particle’ is also meaningless. As in the case of time, the lower limit on length results from the combination of quantum mechanics and general relativity.* The size d of any elementary particle must by definition be smaller than its own Compton wavelength ħ~(mc). Moreover, the size of a particle is always larger than the * Obviously, the minimum size of a particle has nothing to do with the impossibility, in quantum theory, of localizing a particle to within less than its Compton wavelength.

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dA

¾

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Ref. 1051

1011

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xi general rel ativity versus quantum mechanics

1012

Planck length: d A l Pl . Combining these two requirements and eliminating the size d we get the condition for the mass m of any elementary particle, namely ħ m< = c l Pl

Ref. 1052

Ref. 1054

Ref. 1055

ħc = m Pl = 2.2 ë 10−8 kg = 1.2 ë 1019 GeV~c2 . G

(678)

The limit m Pl , the so-called Planck mass, corresponds roughly to the mass of a human embryo that is ten days old, or equivalently, to that of a small flea. In short, the mass of any elementary particle must be smaller than the Planck mass. This fact was already noted as ‘well-known’ by Andrei Sakharov* in 1968; he explains that these hypothetical particles are sometimes called ‘maximons’. And indeed, the known elementary particles all have masses well below the Planck mass. (In fact, the question why their masses are so incredibly much smaller than the Planck mass is one of the most important questions of high-energy physics. We will come back to it.) There are many other ways to arrive at the mass limit for particles. For example, in order to measure mass by scattering – and that is the only way for very small objects – the Compton wavelength of the scatterer must be larger than the Schwarzschild radius; otherwise the probe will be swallowed. Inserting the definition of the two quantities and neglecting the factor 2, we get again the limit m < m Pl . (In fact it is a general property of descriptions of nature that a minimum spacetime interval leads to an upper limit for elementary particle masses.) The importance of the Planck mass will become clear shortly. Another property connected with the size of a particle is its electric dipole moment. It describes the deviation of its charge distribu- Andrei Sakharov tion from spherical. Some predictions from the standard model of elementary particles give as upper limit for the electron dipole moment d e a value of (679)

Sd e S < 3 ë 10−21 m , e

(680)

where e is the charge of the electron. This value is ten thousand times smaller than the Planck length l Pl e. Since the Planck length is the smallest possible length, we seem to have a potential contradiction here. However, a more recent prediction from the standard model is more careful and only states

which is not in contradiction with a minimal length in nature. The issue is still not settled. We will see below that the experimental limit is expected to allow to test these predictions * Andrei Dmitrievich Sakharov, famous Soviet nuclear physicist (1921–1989). One of the keenest thinkers in physics, Sakharov, among others, invented the Tokamak, directed the construction of nuclear bombs, and explained the matter-antimatter asymmetry of nature. Like many others, he later campaigned against nuclear weapons, a cause for which he was put into jail and exile, together with his wife, Yelena Bonner. He received the Nobel Peace Prize in 1975.

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Sd e S < 10−39 m , e

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Ref. 1053

¾

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Farewell to mass The Planck mass divided by the Planck volume, i.e. the Planck density, is given by ρ Pl =

(681)

and is a useful concept in the following. If we want to measure the (gravitational) mass M enclosed in a sphere of size R and thus (roughly) of volume R 3 , one way to do this is to put a test particle in orbit around it at that same distance R. Universal gravitation then gives for the mass M the expression M = Rv 2 ~G, where v is the speed of the orbiting test particle. From v < c, we thus deduce that M < c 2 R~G; since the minimum value for R is the Planck distance, we get (neglecting again factors of order unity) a limit for the mass density ρ, namely ρ < ρ Pl . (682)

In other words, the Planck density is the maximum possible value for mass density. Unsurprisingly, a volume of Planck dimensions cannot contain a mass larger than the Planck mass. Interesting things happen when we start to determine the error ∆M of a mass measurement in a Planck volume. Let us return to the mass measurement by an orbiting probe.

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Challenge 1429 e

c5 = 5.2 ë 1096 kg~m3 G2ħ

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in the foreseeable future. Planck scales have other strange consequences. In quantum field theory, the difference between a virtual particle and a real particle is that a real particle is ‘on shell’, obeying E 2 = m 2 c 4 + p2 c 2 , whereas a virtual particle is ‘off shell’, obeying E 2 x m 2 c 4 + p2 c 2 . Because of the fundamental limits of measurement precision, at Planck scales we cannot determine whether a particle is real or virtual. However, that is not all. Antimatter can be described as matter moving backwards in time. Since the difference between backwards and forwards cannot be determined at Planck scales, matter and antimatter cannot be distinguished at Planck scales. Particles are also characterized by their spin. Spin describes two properties of a particle: its behaviour under rotations (and if the particle is charged, its behaviour in magnetic fields) and its behaviour under particle exchange. The wave function of particles with spin 1 remains invariant under a rotation of 2π, whereas that of particles with spin 1/2 changes sign. Similarly, the combined wave function of two particles with spin 1 does not change sign under exchange of particles, whereas for two particles with spin 1/2 it does. We see directly that both transformations are impossible to study at Planck scales. Given the limit on position measurements, the position of a rotation axis cannot be well defined, and rotations become impossible to distinguish from translations. Similarly, position imprecision makes impossible the determination of precise separate positions for exchange experiments. In short, spin cannot be defined at Planck scales, and fermions cannot be distinguished from bosons, or, phrased differently, matter cannot be distinguished from radiation at Planck scales. We can thus easily see that supersymmetry, a unifying symmetry between bosons and fermions, somehow becomes natural at Planck dimensions. But let us now move to the main property of elementary particles.

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xi general rel ativity versus quantum mechanics

M

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R

F I G U R E 381 A Gedanken experiment showing that

From the relation GM = rv 2 we deduce by differentiation that G∆M = v 2 ∆r + 2vr∆v A 2vr∆v = 2GM∆v~v. For the error ∆v in the velocity measurement we have the indeterminacy relation ∆v E ħ~(m∆r) + ħ~(MR) E ħ~(MR). Inserting this in the previous inequality, and forgetting again the factor of 2, we find that the mass measurement error ∆M of a mass M enclosed in a volume of size R is subject to the condition ∆M E

ħ . cR

(683)

Copyright © Christoph Schiller November 1997–May 2006

Note that for everyday situations, this error is extremely small, and other errors, such as the technical limits of the balance, are much larger. To check this result, we can explore another situation. We even use relativistic expressions, in order to show that the result does not depend on the details of the situation or the approximations. Imagine having a mass M in a box of size R and weighing the box with a scale. (It is assumed that either the box is massless or that its mass is subtracted by the scale.) The mass error is given by ∆M = ∆E~c 2 , where ∆E is due to the indeterminacy in the kinetic energy of the mass inside the box. Using the expression E 2 = m 2 c 4 + p2 c 2 , we get that ∆M E ∆p~c, which again reduces to equation (683). Now that we are sure of the result, let us continue. From equation (683) we deduce that for a box of Planck dimensions, the mass measurement error is given by the Planck mass. But from above we also know that the mass that can be put inside such a box must not be larger than the Planck mass. Therefore, for a box of Planck dimensions, the mass measurement error is larger than (or at best equal to) the mass contained in it: ∆M E M Pl . In other words, if we build a balance with two boxes of Planck size, one empty and the other full, as shown in Figure 381, nature cannot decide which way the balance should hang! Note that even a repeated or a continuous measurement will not resolve the situation: the balance will only randomly change inclination, staying horizontal on average. The argument can be rephrased as follows. The largest mass that we can put in a box

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at Planck scales, matter and vacuum cannot be distinguished

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of size R is a black hole with a Schwarzschild radius of the same value; the smallest mass present in such a box – corresponding to what we call vacuum – is due to the indeterminacy relation and is given by the mass with a Compton wavelength that matches the size of the box. In other words, inside any box of size R we have a mass m, the limits of which are given by: ħ c2 R AmA (empty box) . (684) (full box) G cR

∆M ∆δλ = . M δλ

(685)

Copyright © Christoph Schiller November 1997–May 2006

In order to determine the mass in a Planck volume, the probe has to have a wavelength of the Planck length. But we know from above that there always is a minimum wavelength indeterminacy, given by the Planck length l Pl . In other words, for a Planck volume the mass error is always as large as the Planck mass itself: ∆M E M Pl . Again, this limit is a direct consequence of the limit on length and space measurements. This result has an astonishing consequence. In these examples, the measurement error is independent of the mass of the scatterer, i.e. independent of whether or not we start with a situation in which there is a particle in the original volume. We thus find that in a volume of Planck size, it is impossible to say whether or not there is something there when we probe it with a beam! In short, all arguments lead to the same conclusion: vacuum, i.e. empty space-time, cannot be distinguished from matter at Planck scales. Another, often used way to express this state of affairs is to say that when a particle of Planck energy travels through space it will be scattered by the fluctuations of space-time itself, thus making it impossible to say whether it was scattered by empty space-time or by matter. These surprising results rely on a simple fact: whatever definition of mass we use, it is always measured via combined length and time measurements. (This is even the case for normal weighing scales: mass is measured by the displacement of some part of the machine.) The error in these measurements makes it impossible to distinguish vacuum from matter. We can put this result in another way. If on one hand, we measure the mass of a piece

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We see directly that for sizes R of the order of the Planck scale, the two limits coincide; in other words, we cannot distinguish a full box from an empty box in that case. To be sure of this strange result, we check whether it also occurs if, instead of measuring the gravitational mass, as we have just done, we measure the inertial mass. The inertial mass for a small object is determined by touching it, i.e. physically speaking, by performing a scattering experiment. To determine the inertial mass inside a region of size R, a probe must have a wavelength smaller than R, and thus a correspondingly high energy. A high energy means that the probe also attracts the particle through gravity. (We thus find the intermediate result that at Planck scales, inertial and gravitational mass cannot be distinguished. Even the balance experiment shown in Figure 381 illustrates this: at Planck scales, the two types of mass are always inextricably linked.) Now, in any scattering experiment, e.g. in a Compton-type experiment, the mass measurement is performed by measuring the wavelength change δλ of the probe before and after the scattering experiment. The mass indeterminacy is given by

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Ref. 1056

of vacuum of size R, the result is always at least ħ~cR; there is no possible way to find a perfect vacuum in an experiment. On the other hand, if we measure the mass of a particle, we find that the result is size dependent; at Planck dimensions it approaches the Planck mass for every type of particle, be it matter or radiation. If we use another image, when two particles approach each other to a separation of the order of the Planck length, the indeterminacy in the length measurements makes it impossible to say whether there is something or nothing between the two objects. In short, matter and vacuum are interchangeable at Planck dimensions. This is an important result: since both mass and empty space-time cannot be differentiated, we have confirmed that they are made of the same ‘fabric’. This approach, already suggested above, is now commonplace in all attempts to find a unified description of nature. This approach is corroborated by the attempts to apply quantum mechanics in highly curved space-time, where a clear distinction between vacuum and particles is impossible. This has already been shown by Fulling–Davies–Unruh radiation. Any accelerated observer and any observer in a gravitational field detects particles hitting him, even if he is in vacuum. The effect shows that for curved space-time the idea of vacuum as a particlefree space does not work. Since at Planck scales it is impossible to say whether space is flat or not, it again follows that it is impossible to say whether it contains particles or not. Curiosities and fun challenges on Planck scales The strange results at Planck scales imply many other consequences. ** Observes are made of matter. Observer are thus biased, because they take a specific standpoint. But at Planck scale, vacuum, radiation and matter cannot me distinguished. Two conclusions result: first, only at those scales would a description be free of any bias in favour of matter; but secondly, observers do not exist at all at Planck energy. **

Challenge 1430 n Challenge 1431 n

The Planck energy is rather large. Imagine that we want to impart this amount of energy to protons using a particle accelerator. How large would that accelerator have to be? In contrast, in everyday life, the Planck energy is rather small. Measured in litres of gasoline, how much fuel does it correspond to?

The usual concepts of matter and of radiation are not applicable at Planck dimensions. Usually, it is assumed that matter and radiation are made up of interacting elementary particles. The concept of an elementary particle is one of an entity that is countable, pointlike, real and not virtual, that has a definite mass and a definite spin, that is distinct from its antiparticle, and, most of all, that is distinct from vacuum, which is assumed to have zero mass. All these properties are found to be incorrect at Planck scales. At Planck dimensions, it does not make sense to use the concepts of ‘mass’, ‘vacuum’, ‘elementary particle’, ‘radiation’ and ‘matter’. **

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**

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d oes matter differ from vacuum?

Challenge 1432 n

1017

Do the large mass measurement errors make it possible to claim that mass can be negative at Planck energy? ** We now have a new answer to the old question: why is there anything rather than nothing? Well, we now see that at Planck scales there is no difference between anything and nothing. In addition, we now can honestly say about ourselves that we are made of nothing.

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** If vacuum and matter or radiation cannot be distinguished, then it is incorrect to claim that the universe appeared from nothing. The impossibility of making this distinction thus shows that naive creation is a logical impossibility. Creation is not a description of reality. The term ‘creation’ turns out to be a result of lack of imagination.

Challenge 1433 r Ref. 1058

Special relativity implies that no length or energy can be invariant. Since we have come to the conclusion that the Planck energy and the Planck length are invariant, there must be deviations from Lorentz invariance at high energy. Can you imagine what the effects would be? In what experiment could they be measured? If you find an answer, publish it; you might get known. First attempts are appearing in the research papers. We return to the issue in the third part, with some interesting insights. **

Ref. 1057 Ref. 1038

Challenge 1434 ny

Quantum mechanics alone gives, via the Heisenberg indeterminacy relation, a lower limit on the spread of measurements, but strangely enough not on their precision, i.e. not on the number of significant digits. Jauch gives the example that atomic lattice constants are known much more precisely than the position indeterminacy of single atoms inside the crystal. It is sometimes claimed that measurement indeterminacies smaller than the Planck values are possible for large enough numbers of particles. Can you show why this is incorrect, at least for space and time? **

**

Challenge 1435 n

A Planck energy particle falling in a gravitational field would gain energy. However, this is impossible, as the Planck energy is the highest energy in nature. What does this imply for this situation? **

Ref. 1027

One way to generalize the results presented here is to assume that, at Planck energy, nature is event symmetric, i.e. nature is symmetric under exchange of any two events. This idea,

Copyright © Christoph Schiller November 1997–May 2006

Of course, the idea that vacuum is not empty is not new. More than two thousand years ago, Aristotle argued for a filled vacuum, even though he used incorrect arguments, as seen from today’s perspective. In the fourteenth century the discussion on whether empty space was composed of indivisible entities was rather common, but died down again later.

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**

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developed by Phil Gibbs, provides an additional formulation of the strange behaviour of nature at extreme scales. ** Page 489

Because there is a minimum length in nature, so-called naked singularities do not exist. The issue, so hotly debated in the twentieth century, becomes uninteresting, thus ending decades of speculation.

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** Page 870

** Page 37

If vacuum and matter cannot be distinguished, we cannot distinguish between objects and their environment. However, this was one the starting points of our journey. Some interesting adventures thus still await us! **

Page 676

We have seen earlier that characterizing nature as made up of particles and vacuum creates problems when interactions are included, since on one hand interactions are the difference between the parts and the whole, while on the other hand, according to quantum theory, interactions are exchanges of particles. This apparent contradiction can be used to show either that vacuum and particles are not the only components of nature, or that something is counted twice. However, since matter and space-time are both made of the same ‘stuff,’ the contradiction is resolved. ** Is there a smallest possible momentum? And a smallest momentum error? **

Challenge 1437 n

There is a maximum acceleration in nature. Can you deduce the value of this so-called Planck acceleration? Does it require quantum theory? ** Given that time becomes an approximation at Planck scales, can we still say whether nature is deterministic? Let us go back to the beginning. We can define time, because in nature change is not random, but gradual. What is the situation now that we know that time is only approximate? Is non-gradual change possible? Is energy conserved? In other words, are surprises possible?

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1436 d

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Ref. 1059

Since mass density and thus energy density are limited, we know that the number of degrees of freedom of any object of finite volume is finite. The entropy of black holes has shown us already that entropy values are always finite. This implies that perfect baths do not exist. Baths play an important role in thermodynamics (which is thus found to be only an approximation) and also in recording and measuring devices: when a device measures, it switches from a neutral state to a state in which it shows the result of the measurement. In order to avoid the device returning to the neutral state, it must be coupled to a bath. Without a bath, a reliable measuring device cannot be made. In short, perfect clocks and length measuring devices do not exist because nature puts a limit on their storage ability.

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Challenge 1438 n

1019

To say that time is not defined at Planck scales and that therefore determinism is an undefinable concept is correct, but not a satisfying answer. What happens at daily life scales? The first answer is that at our everyday scales, the probability of surprises is so small that the world indeed is effectively deterministic. The second answer is that nature is not deterministic, but that the difference is not measurable, since every measurement and observation, by definition, implies a deterministic world. The lack of surprises would be due to the limitations of our human nature, and more precisely to the limitations of our senses and brain. The third answer is that the lack of surprises is only apparent, and that we have not yet experienced them yet. Can you imagine any other possibility? To be honest, it is not possible to answer at this point. But we need to keep the alternatives in mind. We have to continue searching, but with every step we take, we have to consider carefully what we are doing.

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**

**

Ref. 1060

Vacuum has zero mass density at large scales, but Planck mass density at Planck scales. Cosmological measurements show that the cosmos is flat or almost flat at large scales, i.e. its energy density is quite low. In contrast, quantum field theory maintains that vacuum has a high energy density (or mass density) at small scales. Since mass is scale dependent, both viewpoints are right, providing a hint to the solution of what is usually called the cosmological constant problem. The contradiction is only apparent; more about this issue later on. **

Challenge 1439 n

When can matter and vacuum be distinguished? At what energy? **

** Can we distinguish between liquids and gases by looking at a single atom? No, only by looking at many. In the same way, we cannot distinguish between matter and vacuum by looking at one point, but only by looking at many. We must always average. However, even averaging is not completely successful. Distinguishing matter from vacuum is like distinguishing clouds from the clear sky; like clouds, matter also has no defined boundary. ** In our exploration we have found that there is no argument which shows that space and

Copyright Š Christoph Schiller November 1997â&#x20AC;&#x201C;May 2006

If matter and vacuum cannot be distinguished, there is a lack of information, which in turn produces an intrinsic basic entropy associated with any part of the universe. We will come back to this topic shortly, in the discussion of the entropy of black holes.

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If matter and vacuum cannot be distinguished, matter and vacuum each has the properties of the other. For example, since space-time is an extended entity, matter and radiation are also extended entities. Furthermore, as space-time is an entity that reaches the borders of the system under scrutiny, particles must also do so. This is the first hint at the extension of matter; in the following, we will examine this argument in more detail.

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xi general rel ativity versus quantum mechanics

time are either continuous or made up of points. Indeed, in contrast, we have found that the combination of relativity and quantum theory makes this impossible. In order to proceed in our ascent of Motion Mountain, we need to leave behind us the usual concept of space-time. At Planck dimensions, the concept of ‘space-time point’ or ‘mass point’ is not applicable in the description of nature. Farewell to the big bang A minimum length, or equivalently,* a minimum action, both imply that there is a maximum curvature for space-time. Curvature can be measured in several ways; for example, surface curvature is an inverse area. A minimum length thus implies a maximum curvature. Within a factor of order one, we find

Challenge 1440 n

c3 = 0.39 ë 1070 m−2 . Għ

(686)

as limit for surface curvature K in nature. In other words, the universe never has been a point, never had zero age, never had infinite density and never had infinite curvature. It is not difficult to get a similar limit for temperature or any other physical quantity. In short, since events do not exist, also the big bang cannot have been an event. There never was an initial singularity or a beginning of the universe. The baggage left behind

Page 1068

Ref. 1013

* The big bang section was added in summer 2002.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1012

In this rapid journey, we have destroyed all the experimental pillars of quantum theory: the superposition principle, space-time symmetry, gauge symmetry, renormalization symmetry and permutation symmetry. We also have destroyed the foundations of general relativity, namely the existence of the space-time manifold, the field concept, the particle concept and the concept of mass. We have even seen that matter and space-time cannot be distinguished. All these conclusions can be drawn in a simpler manner, by using the minimum action of quantum theory and the maximum force of general relativity. All the mentioned results above are confirmed. It seems that we have lost every concept used for the description of motion, and thus made its description impossible. We naturally ask whether we can save the situation. First of all, since matter is not distinguishable from vacuum, and since this is true for all types of particles, be they matter or radiation, we have an argument which demonstrates that the quest for unification in the description of elementary particles is correct and necessary. Moreover, since the concepts ‘mass’, ‘time’ and ‘space’ cannot be distinguished from each other, we also know that a new, single entity is necessary to define both particles and space-time. To find out more about this new entity, three approaches are being pursued at the beginning of the twenty-first century. The first, quantum gravity, especially the approach using the loop representation and Ashtekar’s new variables, starts by generalizing space-time symmetry. The second, string theory, starts by generalizing gauge symmetries

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K<

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some experimental predictions

Ref. 1014

1021

and interactions, while the third, the algebraic quantum group approach, looks for generalized permutation symmetries. We will describe these approaches in more detail later on. Before we go on however, we should check with experiments what we have deduced so far.

S ome experimental predictions

Ref. 1061

Ref. 1063, Ref. 1064

Ref. 1065, Ref. 1064

Ref. 1066

Ref. 1067

ħ(ω 1 − ω 2 )d . c∆t

(687)

This energy value is 8 ë 1016 GeV for the best measurement to date. This value is not far from the Planck energy; in fact, it is even closer when the missing factors of order unity are included. It is expected that the Planck scale will be reached in a few years, so that tests will become possible on whether the quantum nature of space-time influences the dispersion of light signals. Planck scale effects should produce a minimum dispersion, different from zero. Detecting it would confirm that Lorentz symmetry is not valid at Planck scales. Another candidate experiment is the direct detection of distance fluctuations between bodies. Gravitational wave detectors are sensitive to extremely small noise signals in length measurements. There should be a noise signal due to the distance fluctuations induced near Planck energies. The length indeterminacy with which a length l can be measured is predicted to be l Pl 2~3 δl E‹  (688) l l

The expression is deduced simply by combining the measurement limit of a ruler in quantum theory with the requirement that the ruler cannot be a black hole. We will discuss this result in more detail in the next section. The sensitivity to noise of the detectors might reach the required level in the early twenty-first century. The noise induced by quantum gravity effects is also predicted to lead to detectable quantum decoherence and vacuum fluctuations. * As more candidates appear, they will be added to this section.

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Page 1036

E char =

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Ref. 1062

At present, there is a race going on both in experimental and in theoretical physics: which will be the first experiment that will detect quantum gravity effects, i.e. effects sensitive to the Planck energy?* One might think that the fluctuations of space and time might make images from far away galaxies unsharp or destroy the phase relation between the photons. However, this effect has been shown to be unmeasurable in all possible cases. A better candidate is measurement of the speed of light at different frequencies in far away light flashes. There are natural flashes, called gamma ray bursts, which have an extremely broad spectrum, from 100 GeV down to visible light of about 1 eV. These flashes often originate at cosmological distances d. From the difference in arrival time ∆t for two frequencies we can define a characteristic energy by setting

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1022

Ref. 1064

Ref. 1068

Ref. 1069 Ref. 1067

xi general rel ativity versus quantum mechanics

Copyright © Christoph Schiller November 1997–May 2006

* This subsection, in contrast to the ones so far, is speculative; it was added in February 2001. ** The entropy of a black hole is thus given by the ratio between its horizon and º half the minimum area. Of course, a detailed investigation also shows that the Planck mass (divided by 8 ) is the limit for elementary particles from below and for black holes from above. For everyday systems, there is no limit.

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A third candidate for measurable quantum gravity is the detection of the loss of CPT symmetry at high energies. Especially in the case of the decay of certain elementary particles, such as neutral kaons, the precision of experimental measurement is approaching the detection of Planck scale effects. A fourth candidate is the possibility that quantum gravity effects may change the threshold energy at which certain particle reactions become possible. It may be that extremely high energy photons or cosmic rays will make it possible to prove that Lorentz invariance is indeed broken near the Planck scale. A fifth candidate is the possibility that the phase of light that travels over long distances gets washed out. However, the first tests show that this is not the case; light form extremely distant galaxies still interferes. The precise prediction of the phase washing effect is still in discussion; most probably the effect is too small to be measured. In the domain of atomic physics, it has also been predicted that quantum gravity effects will induce a gravitational Stark effect and a gravitational Lamb shift in atomic transitions. Either effect could be measurable. A few candidates for quantum gravity effects have also been predicted by the author. To get an overview, we summarize and clarify the results found so far.* Special relativity starts with the discovery that observable speeds are limited by the speed of light c. Quantum theory starts with the result that observable actions are limited by ħ~2. Gravitation shows that for every system with length L and mass M, the observable ratio L~M is limited by the constant 4G~c 2 . Combining these results, we have deduced that all physical observables are bound, namely by what are usually called the Planck values, though modified by a factor of square root of 2 (or several of them) to compensate for the numerical factors omitted from the previous sentence. We need to replace ħ by ħ~2 and G by 4G in all the defining expressions for Planck quantities, in order to find º the corresponding measurement limits. In particular, the limit 2 times the Planck value and the limit for energy is the Planck for lengths and times is º value divided by 8 .** Interestingly, the existence of bounds on all observables makes it possible to deduce several experimentally testable predictions for the unification of quantum theory and general relativity. These predictions do not depend on the detailed final theory. However, first we need to correct the argument that we have just presented. The argument is only half the story, because we have cheated. The (corrected) Planck values do not seem to be the actual limits to measurements. The actual measurement limits are even stricter still. First of all, for any measurement, we need certain fundamental conditions to be realized. Take the length measurement of an object. We need to be able to distinguish between matter and radiation, since the object to be measured is made up of matter, and since radiation is the measurement tool that is used to read distances from the ruler. For a measurement process, we need an interaction, which implies the use of radiation. Note that even the use of particle scattering to determine lengths does not invalidate this general requirement.

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some experimental predictions

Page 1104 Ref. 1070

In addition, for the measurement of wavelengths we need to distinguish between matter and radiation, because matter is necessary to compare two wavelengths. In fact, all length measurements require the distinction between matter and radiation.* However, this distinction is impossible at the energy of grand unification, when the electroweak and the strong nuclear interactions are unified. At and above this energy, particles of matter and particles of radiation transform into each other; in practice they cannot be distinguished from each other. If all matter and radiation particles were the same or mixtures of each other, mass could not be defined. Similarly, spin, charge or any other quantum numbers could be defined. To sum up, no measurement can be performed at energies equal to or greater than the GUT unification energy. In other words, the particle concept (and thus the matter concept) does not run into trouble at the Planck scale, it has already done so at the unification scale. Only below the unification scale do our standard particle and space-time concepts apply. Only below the unification scale can particles and vacuum be effectively distinguished. º As a result, the smallest length in nature º is 2 times the Planck length reduced by the ratio between the maximal energy E Pl ~ 8 and the unification energy E GUT . Present estimates give E GUT = 1016 GeV, implying that the smallest accessible length L min is L min =

º E Pl 2 l Pl º  10−32 m  800 l Pl . 8 E GUT

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d A d min = e L min = e 10−32 m = 1.5 ë 10−51 Cm .

Page 1012 Ref. 1072

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We saw that this result is in contradiction with one of the predictions deduced from the standard model, but not with others. More interestingly, the prediction is in the reach of future experiments. This improved limit may be the simplest possible measurement of yet unpredicted quantum gravity effects. Measuring the dipole moment could thus be a way to determine the unification energy (the factor 800) independently of high-energy physics experiments and possibly to a higher precision. * To speak in modern high energy concepts, all measurements require broken supersymmetry.

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It is unlikely that measurements at these dimensions will ever be possible. Anyway, the smallest measurable length is significantly larger than the Planck scale of nature discussed above. The reason for this is that the Planck scale is that length for which particles and vacuum cannot be distinguished, whereas the minimal measurable length is the distance at which particles of matter and particles of radiation cannot be distinguished. The latter happens at lower energy than the former. We thus have to correct our previous statement to: the minimum measurable length cannot be smaller than L min . The experimentally determined factor of about 800 is one of the great riddles of physics. It is the high-energy equivalent of the quest to understand why the electromagnetic coupling constant is about 1~137, or more simply, why all things have the colours they have. Only the final theory of motion will provide the answer. In particular, the minimum length puts a bound on the electric dipole moment d of elementary particles, i.e. on any particles without constituents. We get the limit

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Ref. 1071

1023

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1024 Inverse of coupling

electromagnetism

Dvipsbugw 1/26

weak interaction

105

1010

1015 GUT energy

"1020" Planck energy

Energy in GeV

F I G U R E 382 Coupling constants and their spread as a function of energy

Interestingly, the bound on the measurability of observables also puts a bound on the measurement precision for each observable. This bound is of no importance in everyday life, but it is important at high energy. What is the precision with which a coupling constant can be measured? We can illustrate this by taking the electromagnetic coupling constant as an example. This constant α, also called the fine structure constant, is related to the charge q by ¼ q=

4πε 0 ħcα .

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q ma = . q unit m unit a unit

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Therefore any error in mass and acceleration measurements implies errors in measurements of charge and the coupling constant. We found in the part on quantum theory that the electromagnetic, the weak and the strong interactions are characterized by coupling constants, the inverse of which depend linearly on the logarithm of the energy. It is usually assumed that these three lines meet at the unification energy already mentioned. Measurements put the unification coupling

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Now, any electrical charge itself is defined and measured by comparing, in an electrical field, the acceleration to which the charged object is subjected with the acceleration of some unit charge q unit . In other words, we have

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strong interaction

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some experimental predictions

Ref. 1071

value at about 1/26. We know from the above discussions that the minimum measurement error for any energy measurement at high energies is given by the ratio between the energy to be measured and the limit energy. Inserting this into the graph of the coupling constants ‘running’ with energy – as physicist like to say – we get the result shown in Figure 382. The search for the consequences of this fan-out effect is delightful. One way to put the result is to say that coupling constants are by definition subject to an error. However, all measurement devices, be they clocks, metre rules, scales or any other device, use electromagnetic effects at energies of around 1 eV plus the electron rest energy. This is about 10−16 times the GUT energy. As a consequence, the measurement precision of any observable is limited to about 19 digits. The maximum precision presently achieved is 15 digits, and, for the electromagnetic coupling constant, about 9 digits. It will thus be quite some time before this prediction can be tested. The fun is thus to find a system in which the spreading coupling constant value appears more clearly in the measurements. For example, it may be that high-precision measurements of the д-factor of elementary particles or of high-energy cosmic ray reactions will show some effects of the fan-out. The lifetime of elementary particles could also be affected. Can you find another effect? In summary, the experimental detection of quantum gravity effects should be possible, despite their weakness, at some time during the twenty-first century. The successful detection of any such effect will be one of the highlights of physics, as it will challenge the usual description of space and time even more than general relativity did. We now know that the fundamental entity describing space-time and matter that we are looking for is not point-like. What does it look like? To get to the top of Motion Mountain as rapidly as possible, we will make use of some explosive to blast away a few disturbing obstacles.

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Challenge 1441 n

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Biblio graphy 1003 Steven Weinberg, The cosmological constant problem, Reviews of Modern Physics 61, 1004 1005 1006 1007

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1010

1011 1012

1013

1015 1016 1017

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pp. 1–23, 1989. Cited on page 996. Steven Weinberg, The quantum theory of fields, Cambridge University Press, volume I, 1995, and II, 1996. Cited on page 996. The difficulties are summarised by B.S. de Witt, Quantum field theory in curved spacetime, Physics Reports 19, pp. 295–357, 1975. Cited on page 997. C.W. Misner & K.S. Thorne, J.A. Wheeler, Gravitation, Freeman, 1973. Cited on pages 998, 1000, and 1004. J.A. Wheeler, in Relativity, Groups and Topology, edited by C. DeWitt & B. DeWitt, Gordon and Breach, 1994. See also J.A. Wheeler, Physics at the Planck length, International Journal of Modern Physics A 8, pp. 4013–4018, 1993. Cited on page 998. J.L. Friedman & R.D. Sorkin, Spin 1/2 from gravity, Physical Review Letters 44, pp. 1100–1103, 1980. Cited on page 998. A.P. Bal achandran, G. Bimonte, G. Marmo & A. Simoni, Topology change and quantum physics, Nuclear Physics B 446, pp. 299–314, 1995, http://www.arxiv.org/abs/ hep-th/9503046. Cited on page 998. Many themes discussed here resulted from an intense exchange with Luca Bombelli, now at the Department of Physics and Astronomy of the University of Mississippi. These pages are a reworked version of those published in French as C. Schiller, Le vide diffère-t-il de la matière? in E. Gunzig & S. Diner, editeurs, Le vide – Univers du tout et du rien – Des physiciens et des philosophes s’interrogent, Les Éditions de l’Université de Bruxelles, 1998. An older English version is also available as C. Schiller, Does matter differ from vacuum? http://www.arxiv.org/abs/gr-qc/9610066. Cited on page 998. See for example R ichard P. Feynman, R.B. Leighton & M. Sands, The Feynman Lectures on Physics, Addison Wesley, 1977. Cited on page 998. A. Ashtekar, Recent mathematical developments in quantum general relativity, preprint found at http://www.arxiv.org/abs/gr-qc/9411055. C. Rovelli, Ashtekar Formulation of general relativity and loop-space non-perturbative quantum gravity: a report, Classical and Quantum Gravity 8, pp. 1613–1675, 1991. Cited on pages 999 and 1020. M.B. Green, J.H. Schwarz & E. Witten, Superstring theory, volume 1 and 2 (Cambridge University Press, 1987). G. Veneziano, A stringy nature needs just two constants, Europhysics Letters 2, pp. 199–204, 1986. Cited on pages 999 and 1020. S. Majid, Introduction to Braided Geometry and q-Minkowski Space, preprint downloadable at http://www.arxiv.org/abs/hep-th/9410241. See also S. Majid, Beyond supersymmetry and quantum symmetry (an introduction to braided groups and braided matrices), in Quantum Groups, Integrable Statistical Models and Knot Theory, edited by M.-L. Ge & H.J. de Vega, World Scientific, 1993, pp. 231–282. Cited on pages 999 and 1021. C.J. Isham, Prima facie questions in quantum gravity, http://www.arxiv.org/abs/gr-qc/ 9310031. Cited on page 999. Steven Weinberg, Gravitation and Cosmology, Wiley, 1972. Cited on pages 1000, 1003, and 1004. The observations of black holes at the centre of galaxies and elsewhere are summarised by R. Bl andford & N. Gehrels, Revisiting the black hole, Physics Today 52, June 1999.

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Cited on page 1000. M. Pl anck, Über irreversible Strahlungsvorgänge, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften zu Berlin, pp. 440–480, 1899. Today it is commonplace to use Dirac’s ħ instead of Planck’s h, which Planck originally called b. Cited on page 1001. The argument is given e.g. in E.P. Wigner, Relativistic invariance and quantum phenomena, Reviews of Modern Physics 29, pp. 255–258, 1957. Cited on page 1001. The starting point for the following arguments is taken from M. Schön, Operative time definition and principal indeterminacy, preprint http://www.arxiv.org/abs/gr-qc/9304024, and from T. Padmanabhan, Limitations on the operational definition of space-time events and quantum gravity, Classical and Quantum Gravity 4, pp. L107–L113, 1987; see also Padmanabhan’s earlier papers referenced there. Cited on page 1001. W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik 43, pp. 172–198, 1927. Cited on page 1002. E.H. Kennard, Zur Quantenmechanik einfacher Bewegungstypen, Zeitschrift für Physik 44, pp. 326–352, 1927. Cited on page 1002. M.G. R aymer, Uncertainty principle for joint measurement of noncommuting variables, American Journal of Physics 62, pp. 986–993, 1994. Cited on page 1002. H. Salecker & E.P. Wigner, Quantum limitations of the measurement of space-time distances, Physical Review 109, pp. 571–577, 1958. Cited on page 1003. E.J. Zimmerman, The macroscopic nature of space-time, American Journal of Physics 30, pp. 97–105, 1962. Cited on page 1003. S.W. Hawking, Particle creation by black holes, Communications in Mathematical Physics 43, pp. 199–220, 1975; see also S.W. Hawking, Black hole thermodynamics, Physical Review D13, pp. 191–197, 1976. Cited on page 1003. P. Gibbs, The small structure of space-time: a bibliographical review, unpublished preprint found at http://www.arxiv.org/abs/hep-th/9506171. Cited on pages 1002 and 1017. C.A. Mead, Possible connection between gravitation and fundamental length, Physical Review B 135, pp. 849–862, 1964. Cited on pages 1003, 1004, 1006, and 1027. P.K. Townsend, Small-scale structure of space-time as the origin of the gravitational constant, Physical Review D 15, pp. 2795–2801, 1977. Cited on page 1004. M.-T. Jaekel & S. R enaud, Gravitational quantum limit for length measurement, Physics Letters A 185, pp. 143–148, 1994. Cited on page 1004. D.V. Ahluwalia, Quantum measurement, gravitation and locality, Physics Letters B 339, pp. 301–303, 1994, also preprint http://www.arxiv.org/abs/gr-qc/9308007. Cited on page 1004. L. Garay, Quantum gravity and minimum length, International Journal of Modern Physics A 10, pp. 145–165, 1995, also preprint http://www.arxiv.org/abs/gr-qc/9403008; this paper also includes an extensive bibliography. Cited on page 1004. G. Amelino–Camelia, Limits on the measurability of space-time distances in (the semiclassical approximation of) quantum gravity, Modern Physics Letters A 9, pp. 3415– 3422, 1994, also preprint http://www.arxiv.org/abs/gr-qc/9603014. See also Y.J. Ng & H. Van Dam, Limit to space-time measurement, Modern Physics Letters A 9, pp. 335–340, 1994. Cited on page 1004. The generalised indeterminacy relation is an implicit part of Ref. 1028, especially on page 852, but the issue is explained rather unclearly. Obviously the author found the result too simple to be mentioned explicitly. Cited on pages 1004 and 1005.

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1023

1027

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1035 C. Rovelli & L. Smolin, Discreteness of area and volume in quantum gravity, Nuclear

Physics B 442, pp. 593–619, 1995. R. Loll, The volume operator in discretized quantum gravity, preprint http://www.arxiv.org/abs/gr-qc/9506014. Cited on page 1005. 1036 D. Amati, M. Ciafaloni & G. Veneziano, Superstring collisions at Planckian en-

ergies, Physics Letters B 197, pp. 81–88, 1987. D.J. Gross & P.F. Mende, The high energy behavior of string scattering amplitudes, Physics Letters B 197, pp. 129–134, 1987. K. Konishi, G. Paffu ti & P. Provero, Minimum physical length and the generalised uncertainty principle, Physics Letters B 234, pp. 276–284, 1990. P. Aspinwall, Minimum Distances in Non-Trivial String Target Spaces, Nuclear Physics B 431, pp. 78–96, 1994, also preprint http://www.arxiv.org/abs/hep-th/9404060. Cited on page 1005.

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1037 M. Maggiore, A generalised uncertainty principle in quantum mechanics, Physics Let-

ters B 304, pp. 65–69, 1993. Cited on page 1005. 1038 A simple approach is S. Doplicher, K. Fredenhagen & J.E. Roberts, Space-time

1039 A. Kempf, Uncertainty relation in quantum mechanics with quantum group symmetry,

Journal of Mathematical Physics 35, pp. 4483–4496, 1994. A. Kempf, Quantum Groups and quantum field theory with Nonzero minimal uncertainties in positions and momenta, Czechoslovak Journal of Physics 44, pp. 1041–1048, 1994. Cited on page 1005. 1040 E.J. Hellund & K. Tanaka, Quantized space-time, Physical Review 94, pp. 192–195,

1954. Cited on page 1005. 1041 The impossibility to determine temporal ordering in quantum theory is discussed by J.

Oppenheimer, B. R eznik & W.G. Unruh, Temporal ordering in quantum mechanics, Journal of Physics A 35, pp. 7641–7652, 2001, or http://www.arxiv.org/abs/quant-ph/ 0003130. Cited on page 1006. 1042 A. Peres & N. Rosen, Quantum Limitations on the measurement of gravitational fields,

Physical Review 118, pp. 335–336, 1960. Cited on page 1006. 1043 It is the first definition in Euclid’s Elements, c. 300 bce. For an English translation see T.

Heath, The Thirteen Books of the Elements, Dover, 1969. Cited on page 1007. 1044 A beautiful description of the Banach–Tarski paradox is the one by Ian Stewart, Para-

dox of the spheres, New Scientist, 14 January 1995, pp. 28–31. Cited on page 1008. 1045 H.S. Snyder, Quantized space-time, Physical Review 71, pp. 38–41, 1947. H.S. Snyder,

1046 D. Finkelstein, ‘Superconducting’ causal nets, International Journal of Theoretical Phys-

ics 27, pp. 473–519, 1985. Cited on page 1009. 1047 N.H. Christ, R. Friedberg & T.D. Lee, Random lattice field theory: general formu-

lation, Nuclear Physics B 202, pp. 89–125, 1982. G. ’t Hooft, Quantum field theory for elementary particles – is quantum field theory a theory?, Physics Reports 104, pp. 129–142, 1984. Cited on page 1009. 1048 See for example L. B ombelli, J. Lee, D. Meyer & R.D. Sorkin, Space-time as

a causal set, Physical Review Letters 59, pp. 521–524, 1987. G. Brightwell & R.

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The electromagnetic field in quantised of space-time, Physical Review 72, pp. 68–74, 1947. A. Schild, Discrete space-time and integral Lorentz transformations, Physical Review 73, pp. 414–415, 1948. E.L. Hill, Relativistic theory of discrete momentum space and discrete space-time, Physical Review 100, pp. 1780–1783, 1950. H.T. Flint, The quantization of space-time, Physical Review 74, pp. 209–210, 1948. A. Das, Cellular space-time and quantum field theory, Il Nuovo Cimento 18, pp. 482–504, 1960. Cited on page 1009.

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quantization induced by classical gravity, Physics Letters B 331, pp. 39–44, 1994, Cited on pages 1005 and 1017.

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Gregory, Structure of random space-time, Physical Review Letters 66, pp. 260–263, 1991. Cited on page 1010. For a discussion, see R. Sorabji, Time, Creation and the continuum: theories in antiquity and the early Middle Ages, Duckworth, London 1983. Cited on page 1009. For quantum gravity, see for example A. Ashtekar, Quantum geometry in action: big bang and black holes, http://www.arxiv.org/abs/math-ph/0202008. Cited on page 1010. The false belief that particles like quarks or electrons are composite is slow to die out. See, e.g, S. Fredriksson, Preon prophecies by the standard Model, http://www.arxiv.org/ abs/hep-ph/0309213. Preon models gained popularity in the 1970s and 1980s, in particular through the papers by J.C. Pati & A. Sal am, Physical Review D 10,, p. 275, 1974, H. Harari, Physics Letters B 86,, p. 83, 1979, M.A. Shupe, Physics Letters B 86,, p. 87, 1979, and H. Fritzsch & G. Mandelbaum, Physics Letters B 102,, p. 319, 1981. Cited on page 1011. A.D. Sakharov, Vacuum quantum fluctuations in curved space and the theory of gravitation, Soviet Physics – Doklady, 12, pp. 1040–1041, 1968. Cited on page 1012. C. Wolf, Upper limit for the mass of an elementary particle due to discrete time quantum mechanics, Il Nuovo Cimento B 109, pp. 213–218, 1994. Cited on page 1012. N.F. R amsey & A. Weis, Suche nach permanenten elektrischen Dipolmomenten: ein Test der Zeitumkehrinvarianz, Physikalische Blätter 52, pp. 859–863, 1996. The paper by E.D. Commins, S.B. Ross, D. DeMille & B.C. R egan, Improved experimental limit on the electric dipole moment of the electron, Physical Review A 50, p. 2960, 1994, gives an upper experimental limit to dipole moment of the electron of 3.4 ë 10−29 e m. W. Bernreu ther & M. Suzuki, The electric dipole moment of the electron, Reviews of Modern Physics 63, pp. 313–340, 1991. See also the musings in Hans Dehmelt, Is the electron a composite particle?, Hyperfine Interactions 81, pp. 1–3, 1993. Cited on page 1012. K. Akama, T. Hattori & K. Katsuura, Naturalness bounds on dipole moments from new physics, http://www.arxiv.org/abs/hep-ph/0111238 Cited on page 1012. W.G. Unruh, Notes on black hole evaporation, Physical Review D 14, pp. 870–875, 1976. W.G. Unruh & R.M. Wald, What happens when an accelerating observer detects a Rindler particle, Physical Review D 29, pp. 1047–1056, 1984. Cited on page 1016. W. Jauch, Heisenberg’s uncertainty relation and thermal vibrations in crystals, American Journal of Physics 61, pp. 929–932, 1993. Cited on page 1017. One way out is proposed in J. Magueijo & L. Smolin, Lorentz invariance with an invariant energy scale, Physical Review Letters 88, p. 190403, 26 April 2002. They propose a modification of the mass energy relation of the kind γmc 2 1+

γmc 2 E Pl

and

p=

γmv 1+

γmc 2 E Pl

.

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Another, similar approach of recent years, with a different proposal, is called doubly special relativity. A recent summary is G. Amelino–Camelia, Doubly-special relativity: first results and key open problems, http://www.arxiv.org/abs/gr-qc/0210063. The paper shows how conceptual problems hinder the advance of the field. Cited on page 1017. 1059 H.D. Zeh, On the interpretation of measurement in quantum theory, Foundations of Physics 1, pp. 69–76, 1970. Cited on page 1018. 1060 Steven Weinberg, The cosmological constant problem, Reviews of Modern Physics 61, pp. 1–23, 1989. Cited on page 1019.

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E=

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1053

1029

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1061 See Y.J. Ng, W.A. Christiansen & H. van Dam, Probing Planck-scale physics with

1062 1063

1065

1066

1067

1069

1070

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1068

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extragalactic sources?, Astrophysical Journal 591, pp. L87–L90, 2003, also downloadable electronically at http://www.arxiv.org/abs/astro-ph/0302372; D.H. Coule, Planck scale still safe from stellar images, Classical and Quantum Gravity 20, pp. 3107–3112, 2003, also available as electronic preprint at http://www.arxiv.org/abs/astro-ph/0302333. Negative experimental results (and not always correct calculations) are found in R. Lieu & L. Hillman, The phase coherence of light from extragalactic sources - direct evidence against first order Planck scale fluctuations in time and space, Astrophysical Journal 585, pp. L77–L80, 2003, and R. R agazzoni, M. Turatto & W. Gaessler, The lack of observational evidence for the quantum structure of spacetime at Planck scales, Astrophysical Journal 587, pp. L1–L4, 2003. Cited on page 1021. B.E. Schaefer, Severe limits on variations of the speed of light with frequency, Physical Review Letters 82, pp. 4964–4966, 21 June 1999. Cited on page 1021. G. Amelino–Camelia, J. Ellis, N.E. Mavromatos, D.V. Nanopoulos & S. Sakar, Potential sensitivity of gamma-ray-burster observations to wave dispersion in vacuo, Nature 393, p. 763, 1998, and preprint http://www.arxiv.org/abs/astro-ph/9712103. Cited on page 1021. G. Amelino–Camelia, Phenomenological description of space-time foam, preprint found at http://www.arxiv.org/abs/gr-qc/0104005, 2001. The paper includes a clearly written overview of present experimental approaches to detecting quantum gravity effects. See also his update G. Amelino–Camelia, Quantum-gravity phenomenology: status and prospects, preprint found at http://www.arxiv.org/abs/gr-qc/0204051, 2002. Cited on pages 1021 and 1022. G. Amelino–Camelia, An interferometric gravitational wave detector as a quantum gravity apparatus, Nature 398, p. 216, 1999, and preprint http://www.arxiv.org/abs/gr-qc/ 9808029. Cited on page 1021. F. Karolyhazy, Gravitation and quantum mechanics of macroscopic objects, Il Nuovo Cimento A42, pp. 390–402, 1966, Y.J. Ng & H. van Dam, Limit to space-time measurement, Modern Physics Letters A 9, pp. 335–340, 1994, Y.J. Ng & H. Van Dam, Modern Physics Letters A Remarks on gravitational sources, 10, pp. 2801–2808, 1995. The discussion is neatly summarised in Y.J. Ng & H. van Dam, Comment on ‘Uncertainty in measurements of distance’, http://www.arxiv.org/abs/gr-qc/0209021. See also Y.J. Ng, Spacetime foam, preprint found at http://www.arxiv.org/abs/gr-qc/0201022. Cited on page 1021. L.J. Garay, Spacetime foam as a quantum thermal bath, Physics Review Letters 80, pp. 2508–2511, 1998, also downloadable as http://www.arxiv.org/abs/gr-qc/9801024. Cited on pages 1021 and 1022. G. Amelino–Camelia & T. Piran, Planck scale-deformation of Lorentz symmetry as a solution to the UHECR and the TeV γ paradoxes, preprint http://www.arxiv.org/ astro-ph/0008107, 2000. Cited on page 1022. Experimental tests were described by R. Lieu & L.W. Hillman, The phase coherence of light from extragalactic sources: direct evidence against first-order Planck-scale fluctuations in time and space, Astrophysical Journal 585, pp. L77–L80, 2003, and R. R agazzoni & al., The lack of observational evidence for the quantum structure of spacetime at Planck scales, Astrophysical Journal 587, pp. L1–L4, 2003, Discussions on the predicted phase values are found in http://www.arxiv.org/abs/astro-ph/0302372 and http:// www.arxiv.org/abs/astro-ph/0302333. Cited on page 1022. At the unification scale, supersymmetry is manifest; all particles can transform into each other. See for example ... Cited on page 1023.

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1071 See the paper by U. Amaldi, W. de B oer & H. Fürstenau, Comparison of grand

unified theories with elektroweak and strong coupling constants measured at LEP, Physics Letters 260, pp. 447–455, 1991. This is the standard reference on the topic. Cited on pages 1023 and 1025. 1072 M.V. Romalis, W.C. Griffith, J.P. Jacobs & E.N. Fortson, New limit on the permanent electric dipole moment of 199 Hg, Physical Review Letters 86, pp. 2505–2508, 19 March 2001, or http://www.arxiv.org/abs/hep-ex/0012001. Their upper limit on the dipole moment of mercury atoms is 2.1 ë 1030 e m. Research for better measurement methods is progressing; a modern proposal is by S.K. L amoreaux, Solid state systems for electron electric dipole moment and other fundamental measurements, http://www.arxiv.org/abs/ nucl-ex/0109014. Cited on page 1023.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006

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1032

34.

nature at l arge scales – is the universe something or nothing?

“ Ref. 1073

Die Grenze ist der eigentlich fruchtbare Ort der Erkenntnis.* Paul Tillich, Auf der Grenze.

T

Cosmological scales

Hic sunt leones.*** Antiquity

d  r S = 2Gm~c 2 .

Challenge 1442 n

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It is straightforward to confirm that, with the usually quoted mass m and size d of everything visible in the universe, this condition is indeed fulfilled. We do need general relativity and thus curved space-time when talking about the whole of nature. * ‘The frontier is the really productive place of understanding’. Paul Tillich (1886–1965), German theologian, socialist and philosopher. ** Written between June and December 2000. *** ‘Here are lions.’ Written across unknown and dangerous regions on ancient maps.

Copyright © Christoph Schiller November 1997–May 2006

The description of motion requires the application of general relativity whenever the scale d of the situation are of the order of the Schwarzschild radius, i.e. whenever

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his strange question is the topic of the current leg of our mountain ascent. In he last section we explored nature in the vicinity of Planck dimensions; but he other limit, namely to study the description of motion at large, cosmological scales, is equally fascinating. As we proceed, many incredible results will appear, and at the end we will discover a surprising answer to the question in the section title. This section is not standard textbook material; a large part of it is original** and thus speculative and open to question. Even though this section aims at explaining in simple words the ongoing research in the domains of quantum gravity and superstring theory, be warned. With every sentence of this section you will find at least one physicist who disagrees! We have discussed the universe as a whole several times already. In classical physics we enquired about the initial conditions of the universe and whether it is isolated. In the first intermezzo we asked whether the universe is a set or a concept and indeed, whether it exists at all. In general relativity we gave the classical definition of the term, as the sum of all matter and space-time, we studied the expansion of the universe and we asked about its size and topology. In quantum theory we asked whether the universe has a wave function, whether it is born from a quantum fluctuation, and whether it allows the number of particles to be defined. Here we will settle all these issues by combining general relativity and quantum theory at cosmological scales. That will lead us to some of the strangest results we will encounter in our journey.

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1033

Similarly, quantum theory is required for the description of motion of an object whenever we approach it within a distance d of the order of the Compton wavelength λ C , i.e. whenever h . (695) d  λC = mc

Maximum time

Challenge 1443 n

Does the universe have a definite age? One should never trust a woman who tells one her real age. A woman who would tell one that, would tell one anything. Oscar Wilde

* This conclusion implies that so-called ‘oscillating’ universe models, in which it is claimed that ‘before’ the big bang there were other phenomena, cannot be based on nature or on observations. They are based on beliefs.

Copyright © Christoph Schiller November 1997–May 2006

Is it possible to measure time intervals of any imaginable size? General relativity shows that in nature there is a maximum time interval, with a value of about fourteen thousand million years or 430 Ps, providing an upper limit to the measurement of time. It is called the ‘age’ of the universe and has been deduced from two sets of measurements: the expansion of space-time and the age of matter. We all know of clocks that have been ticking for a long time: the hydrogen atoms in our body. All hydrogen atoms were formed just after the big bang. We can almost say that the electrons in these atoms have been orbiting the nuclei since the dawn of time. In fact, inside the protons in these atoms, the quarks have been moving already a few hundred thousand years longer. Anyway, we thus get a common maximum time limit for any clock made of atoms. Even ‘clocks’ made of radiation (can you describe one?) yield a similar maximum time. Also the study of the spatial expansion of the universe leads to the same maximum. No real or imaginable clock or measurement device was ticking before this maximum time and no clock could provide a record of having done so. In summary, it is not possible to measure time intervals greater than the maximum one, either by using the history of space-time or by using the history of matter or radiation.* The maximum time is thus rightly called the ‘age’ of the universe. Of course, all this is not new, although looking at the issue in more detail does provide some surprises.

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Obviously, for the total mass of the universe this condition is not fulfilled. However, we are not interested in the motion of the universe itself; we are interested in the motion of its components. In the description of these components, quantum theory is required whenever pair production and annihilation play a role. This is especially the case in the early history of the universe and near the horizon, i.e. for the most distant events that we can observe in space and time. We are thus obliged to include quantum theory in any precise description of the universe. Since at cosmological scales we need both quantum theory and general relativity, we start our investigation with the study of time, space and mass, by asking at large scales the same questions that we asked above at Planck scales.

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How precise can age measurements be?

Ref. 1075

Ref. 1074

The first way to measure the age of the universe* is to look at clocks in the usual sense of the term, namely at clocks made of matter. As explained in the part on quantum theory, Salecker and Wigner showed that a clock built to measure a total time T with a precision * Note that the age t 0 is not the same as the Hubble time T = 1~H 0 . The Hubble time is only a computed quantity and (almost) always larger than the age; the relation between the two depends on the value of the cosmological constant, on the density and on other parameters of the universe. For example, for the standard hot big bang scenario, i.e. for the matter-dominated Einstein–de Sitter model, we have the simple relation T = (3~2) t 0 .

Copyright © Christoph Schiller November 1997–May 2006

No woman should ever be quite accurate about her age. It looks so calculating. Oscar Wilde

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Asking about the age of the universe may seem a silly question, because we have just discussed it. Furthermore, the value is found in many books and tables, including that of Appendix B, and its precise determination is actually one of the most important quests in modern astrophysics. But is this quest reasonable? In order to measure the duration of a movement or the age of a system, we need a clock. The clock has to be independent of that movement or system and thus has to be outside the system. However, there are no clocks outside the universe and, inside it, a clock cannot be independent. In fact we have just seen that inside the universe, no clock can run throughout its complete history. Indeed, time can be defined only once it is possible to distinguish between matter and space-time. Once this distinction can be made, only the two possibilities just discussed remain: we can either talk about the age of space-time, as is done in general relativity, by assuming that matter provides suitable and independent clocks; or we can talk about the age of matter, such as stars or galaxies, by assuming that the extension of space-time or some other matter provides a good clock. Both possibilities are being explored experimentally in modern astrophysics; both give the same result of about fourteen thousand million years that was mentioned previously. However, for the universe as a whole, an age cannot be defined. The issue of the starting point of time makes this difficulty even more apparent. We may imagine that going back in time leads to only two possibilities: either the starting instant t = 0 is part of time or it is not. (Mathematically, this means that the segment describing time should be either closed or open.) Both cases assume that it is possible to measure arbitrarily small times, but we know from the combination of general relativity and quantum theory that this is not the case. In other words, neither possibility is incorrect: the beginning cannot be part of time, nor can it not be part of it. To this situation there is only one solution: there was no beginning at all. In other words, the situation is consistently muddled. Neither the age of the universe nor its origin makes sense. What is going wrong? Or, more correctly, how are things going wrong? In other words, what happens if instead of jumping directly to the big bang, we approach it as closely as possible? The best way to clarify the issue is to ask about the measurement error we are making when we say that the universe is fourteen thousand million years old. This turns out to be a fascinating topic.

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1035

∆t has a minimum mass m given by mA

Ref. 1076

ħ T . c 2 (∆t)2

(696)

A simple way to incorporate general relativity into this result was suggested by Ng and van Dam. Any clock of mass m has a minimum resolution ∆t due to the curvature of space that it introduces, given by Gm ∆t A 3 . (697) c

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If m is eliminated, these two results imply that any clock with a precision ∆t can only measure times T up to a certain maximum value, namely

Challenge 1444 e

Challenge 1445 ny

(∆t)3 2 t Pl

,

(698)

» where t Pl = ħG~c 5 = 5.4 ë 10−44 s is the already familiar Planck time. (As usual, we have omitted factors of order one in this and in all the following results of this section.) In other words, the higher the accuracy of a clock, the shorter the time during which the clock works dependably! The precision of a clock is not (only) limited by the budget spent on building it, but by nature itself. Nevertheless, it does not take much to check that for clocks in daily life, this limit is not even remotely approached. For example, you may want to deduce how precisely your own age can be specified. As a consequence of (698), a clock trying to achieve an accuracy of one Planck time can do so for at most one single Planck time! Simply put, a real clock cannot achieve Planck time accuracy. If we try to go beyond limit (698), fluctuations of space-time hinder the working of the clock and prevent higher precision. With every Planck time that passes, the clock accumulates a measuring error of at least one Planck time. Thus, the total measurement error is at least as large as the measurement itself. The conclusion is also valid for clocks based on radiation, for example on background radiation. In short, measuring age with a clock always involves some errors; whenever we try to reduce these errors to Planck level, the clock becomes so imprecise that age measurements become impossible.

“ Page 45

Time is waste of money. Oscar Wilde

From the origins of physics onwards, the concept of ‘time’ has designated what is measured by a clock. Since equation (698) expresses the non-existence of perfect clocks, it also implies that time is only an approximate concept, and that perfect time does not exist. Thus there is no ‘idea’ of time, in the Platonic sense. In fact, all discussion in the previous and present sections can be seen as proof that there are no perfect or ‘ideal’ examples of any classical or everyday concept.

Copyright © Christoph Schiller November 1997–May 2006

Does time exist?

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T<

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Challenge 1446 e

Page 1004 Challenge 1447 e

Time does not exist. Despite this conclusion, time is obviously a useful concept in everyday life. A simple explanation is provided when we focus on the importance of energy. Any clock, in fact any system of nature, is characterized by a simple number, namely the highest ratio of the kinetic energy to the rest energy of its components. In daily life, this fraction is about 1 eV~10 GeV = 10−10 . Such low-energy systems are well suited to building clocks. The more precisely the motion of the main moving part – the pointer of the clock – can be kept constant and can be monitored, the higher the precision of the clock becomes. To achieve the highest possible precision, the highest possible mass of the pointer is required; indeed, both the position and the speed of the pointer must be measured, and the two measurement errors are related by the quantum mechanical indeterminacy relation ∆v ∆x A ħ~m. High mass implies low intrinsic fluctuations. In order to screen the pointer from outside influences, even more mass is needed. This connection might explain why better clocks are usually more expensive than less accurate ones. The usually quoted indeterminacy relation is valid only at everyday energies. Increasing the mass does not allow to reach arbitrary small time errors, since general relativity changes the indeterminacy relation to ∆v ∆x A ħ~m + G(∆v)2 m~c 3 . The additional term on the right hand side, negligible at everyday scales, is proportional to energy. Increasing it by too large an amount limits the achievable precision of the clock. The smallest measurable time interval turns out to be the Planck time. In summary, time exists as a good approximation only for low-energy systems. Any increase in precision beyond a certain limit will require an increase in the energy of the components; at Planck energy, this energy increase will prevent an increase in precision. What is the error in the measurement of the age of the universe?

Challenge 1448 e

Applying the discussion about the measurement of time to the age of the universe is now straightforward. Expression (698) implies that the highest precision possible for a clock is about 10−23 s, or about the time light takes to move across a proton. The finite age of the universe also yields a maximum relative measurement precision. Expression (698) can be written as t Pl ∆t A ( )2~3 (699) T T

and thus decreases with increasing measurement energy. For high energies, however, the error is given by gravitational effects as ∆t E meas  T E Pl

(701)

Copyright © Christoph Schiller November 1997–May 2006

which shows that no time interval can be measured with a precision of more than about 40 decimals. In order to clarify the issue, we can calculate the error in measurement as a function of the observation energy E meas . There are two limit cases. For small energies, the error is given by quantum effects as 1 ∆t  (700) T E meas

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Ref. 1073

xi general rel ativity versus quantum mechanics

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Relative measurement error 1

quantum error

total error quantum gravity error

∆Emin E Energy

Eopt

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EPl

F I G U R E 383 Measurement errors as a function of measurement energy

d . v

(702)

This is in simple words the method used to determine the age of the universe from the expansion of space-time, for galaxies with red-shifts below unity.* Of interest in the fol* At higher red-shifts, the speed of light, as well as the details of the expansion, come into play; if we continue

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T=

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so that the total result is as shown in Figure 383. In particular, energies that are too high do not reduce measurement errors, because any attempt to reduce the measurement error for the age of the universe below 10−23 s would require energies so high that the limits of space-time would be reached, making the measurement itself impossible. We reached this conclusion through an argument based on clocks made of particles. Below we will find out that even by determining the age of the universe using space-time expansion leads to the same limit. Imagine to observe a tree which, as a result of some storm or strong wind, has fallen towards second tree, touching it at the very top, as shown in Figure 384. It is possible to determine the height of both trees by measuring their separation and v the angles at the base. The error in height will depend on the errors in separation and angles. Similarly, the age of the universe follows from the present distance and speed of objects – such as galaxies – observed in the night sky. The present distance d T corresponds to separation of the trees at ground level and the speed v to the angle between the two trees. The Hubble time T of the universe – which, as has already been mentioned, is usud ally assumed to be larger than the age of the universe – then corresponds to the height at which the two trees meet. This time since the universe ‘started’, in a naive sense since the galaxies F I G U R E 384 Trees and galaxies ‘separated’, is then given, within a factor of order one, by

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xi general rel ativity versus quantum mechanics

lowing is the (positive) measurement error ∆T, which becomes ∆T ∆d ∆v = + . T d v

(703)

2~3

∆d l Pl A 2~3 T d

Challenge 1449 e

Challenge 1450 e

Ref. 1074

(704)

with the image of inclined trees, we find that the trees are not straight all the way up to the top and that they grow on a slope, as shown in Figure 385.

Copyright © Christoph Schiller November 1997–May 2006

thus giving the same indeterminacy in the age of the universe as found above in the case of material clocks. We can try to reduce this error in two ways: by choosing objects at either small or large distances. Let us start with the smallest possible distances. In order to get high precision at small distances, we need high observation energies. It is fairly obvious that at observation energies near the Planck value, the value of ∆T~T approaches unity. In fact, both terms on the right-hand side of expression (703) become of order one. At these energies, ∆v approaches c and the maximum value for d approaches the Planck length, for the same reason that at Planck energies the maximum measurable time is the Planck time. In short, at Planck scales it is impossible to say whether the universe is old or young. Let us continue with the other extreme, namely objects extremely far away, say with a red-shift of z Q 1. Relativistic cosmology requires the diagram of Figure 384 to be replaced by the more realistic diagram of Figure 385. The ‘light onion’ replaces the familiar light cone of special relativity: light converges near the big bang. In this case the measurement error for the age of the universe also depends on the distance and velocity errors. At the largest possible distances, the signals an object must send out must be of high energy, because the emitted wavelength must be smaller than the universe itself. Thus, inevitably we reach Planck energies. However, we saw that in such high-energy situations, the emitted radiation, as well as the object itself, are indistinguishable from the space-time background. In other words, the red-shifted signal we would observe today would have a wavelength as large as the size of the universe, with a correspondingly small frequency.

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Exploring this in more detail is worthwhile. For any measurement of T we have to choose the object, i.e. a distance d, as well as an observation time ∆t, or, equivalently, an observation energy ∆E = 2πħ~∆t. We will now investigate the consequences of these choices for expression (703), always taking into account both quantum theory and general relativity. At everyday energies, the result of the determination of the age t 0 is about 14  2 ë 109 Ga. This value is deduced by measuring red-shifts, i.e. velocities, and distances, using stars and galaxies in distance ranges from some hundred thousand light years up to a red-shift of about 1. Measuring red-shifts does not produce large velocity errors. The main source of experimental error is the difficulty in determining the distances of galaxies. What is the smallest possible error in distance? Obviously, equation (699) implies

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nature at l arge scales – is the universe something or nothing?

Another way to describe the situation is the folbig bang space lowing. At Planck energies or near the horizon, the original signal has an error of the same size as the signal itself. When measured at the present time, the red-shifted signal still has an error of the same 4ct0/9 size as the signal. As a result, the error on the holight cone: what we rizon distance becomes as large as the value to be can see measured. time In short, even if space-time expansion and large scales are used, the instant of the so-called beginother galaxies in the night sky ning of the universe cannot be determined with an error smaller than the age of the universe itself, a t0 result we also found at Planck distances. Whenever our galaxy we aim for perfect precision, we find that the universe is 14  14 thousand million years old! In other F I G U R E 385 The speed and distance of words, at both extremal situations it is impossible to remote galaxies say whether the universe has a non-vanishing age. We have to conclude that the anthropomorphic concept of ‘age’ does not make any sense for the universe as a whole. The usual textbook value is useful only for domains in time, space and energy for which matter and space-time are clearly distinguished, namely at everyday, human-scale energies; however, this anthropocentric value has no overall meaning. By the way, you may like to examine the issue of the fate of the universe using the same arguments. In the text, however, we continue on the path outlined at the start of this section; the next topic is the measurement of length. Maximum length

Page 460

˙ the * In cosmology, we need to distinguish between the scale factor R, the Hubble radius c~H = cR~R, horizon distance h and the size d of the universe. The Hubble radius is a computed quantity giving the distance at which objects move away with the speed of light. It is always smaller than the horizon distance, at which in the standard Einstein–de Sitter model, for example, objects move away with twice the speed of light. However, the horizon itself moves away with three times the speed of light.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1074

General relativity shows that in the standard cosmological model, for hyperbolic (open) and parabolic (marginal) evolutions of the universe, the actual size of the universe is infinite. It is only the horizon distance, i.e. the distance of objects with infinite red-shift, which is finite. In a hyperbolic or parabolic universe, even though the size is infinite, the most distant visible events (which form the horizon) are at a finite distance.* For elliptical evolution, the total size is finite and depends on the curvature. However, in this case also the present measurement limit yields a minimum size for the universe many times larger than the horizon distance. At least, this is what general relativity says. On the other hand, quantum field theory is based on flat and infinite space-time. Let us see what happens when the two theories are combined. What can we say about measurements of length in this case? For example, would it be possible to construct and use a metre rule to measure lengths larger than the distance to the horizon? It is true that we would have no time to push it up to there, since in the standard Einstein–de Sitter big

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Challenge 1451 ny

1039

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1040

Ref. 1074

Ref. 1074

xi general rel ativity versus quantum mechanics

bang model the horizon moves away from us faster than the speed of light. We should have started using the metre rule right at the big bang. For fun, let us assume that we have actually managed to do this. How far away can we read off distances? In fact, since the universe was smaller in the past and since every observation of the sky is an observation of the past, Figure 385 shows that the maximum spatial distance an object can be seen away from us is only (4~9)ct 0 . Obviously, for spacetime intervals, the maximum remains ct 0 . Thus, in all cases it turns out to be impossible to measure lengths larger than the horizon distance, even though general relativity predicts such distances. This unsurprising result is in obvious agreement with the existence of a limit for measurements of time intervals. The real surprises come now.

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Is the universe really a big place? Ref. 1077

Astronomers and Hollywood movies answer this question in the affirmative. Indeed, the distance to the horizon of the universe is usually included in tables. Cosmological models specify that the scale factor R, which fixes the distance to the horizon, grows with time t; for the case of the usually assumed mass-dominated Einstein–de Sitter model, i.e. for a vanishing cosmological constant and flat space, we have R(t) = C t 2~3 ,

where the numerical constant C relates the commonly accepted horizon distance to the commonly accepted age. Indeed, observation shows that the universe is large and is still getting larger. But let us investigate what happens if we add to this result from general relativity the limitations of quantum theory. Is it really possible to measure the distance to the horizon? We look first at the situation at high energies. We saw above that space-time and matter are not distinguishable at Planck scales. Therefore, at Planck energies we cannot state whether objects are localized or not. At Planck scales, a basic distinction of our thinking, namely the one between matter and vacuum, becomes obsolete. Equivalently, it is not possible to claim that space-time is extended at Planck scales. Our concept of extension derives from the possibility of measuring distances and time intervals, and from observations such as the ability to align several objects behind one another. Such observations are not possible at Planck scales. In fact, none of the observations in daily life from which we deduce that space is extended are possible at Planck scales. At Planck scales, the basic distinction between vacuum and matter, namely the opposition between extension and localization, disappears. As a consequence, at Planck energies the size of the universe cannot be measured. It cannot even be called larger than a match box. At cosmological distances, the situation is even easier. All the arguments given above on the errors in measurement of the age can be repeated for the distance to the horizon. Essentially, at the largest distances and at Planck energies, the measurement errors are of the same magnitude as the measured value. All this happens because length measurements become impossible at nature’s limits. This is corroborated by the lack of any standard with which to compare the size of the universe. Studying the big bang also produces strange results. At Planck energies, whenever we

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1452 ny

(705)

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Page 1167

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nature at l arge scales – is the universe something or nothing?

Challenge 1453 ny

1041

The boundary of space-time – is the sky a surface?

Challenge 1454 ny

* In addition, the measurement errors imply that no statement can be made about translational symmetry at cosmological scales. Are you able to confirm this? In addition, at the horizon it is impossible to distinguish between spacelike and timelike distances. Even worse, concepts such as ‘mass’ or ‘momentum’ are muddled at the horizon. This means that, as at Planck energies, we are unable to distinguish between objects and the background, and between state and intrinsic properties. We will come back to this important point shortly.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1455 ny

The horizon of the universe, essentially the black part of the night sky, is a fascinating entity. Everybody interested in cosmology wants to know what happens there. In newspapers the horizon is sometimes called the boundary of space-time. Some surprising insights, not yet common in newspapers, appear when the approaches of general relativity and quantum mechanics are combined. We saw above that the errors in measuring the distance of the horizon are substantial. They imply that we cannot pretend that all points of the sky are equally far away from us. Thus we cannot say that the sky is a surface. There is even no way to determine the dimensionality of the horizon or the dimensionality of space-time near the horizon.* Measurements thus do not allow us to determine whether the boundary is a point, a surface, or a line. It may be an arbitrary complex shape, even knotted. In fact, quantum theory tells us that it must be all of these from time to time, in short, that the sky fluctuates in height and shape. In short, it is impossible to determine the topology of the sky. But that is nothing new. As is well known, general relativity is unable to describe pair creation particles with spin 1/2. The reason for this is the change in space-time topology required by the process. On the other hand, the universe is full of such processes, implying that it is impossible to define a topology for the universe and, in particular, to talk of the topology of the horizon itself. Are you able to find at least two other arguments to show this? Worse still, quantum theory shows that space-time is not continuous at a horizon, as

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try to determine the size of the big bang, we cannot claim that the universe was smaller than the present size. At Planck energies, there is no way to distinguish length values. Somehow, Planck dimensions and the size of the universe get confused. There are also other confirmations. Let us go back to the example above. If we had a metre rule spanning the whole universe, even beyond the horizon, with zero at the place where we live, what measurement error would it produce for the horizon? It does not take long to discover that the expansion of space-time from Planck scales to the present also expands the indeterminacy in the Planck size into one of the order of the distance to the horizon. The error is as large as the measurement result. Since this also applies when we try to measure the diameter of the universe instead of its radius, it becomes impossible to state whether the antipodes in the sky really are distant from each other! We can summarize the situation by noting that anything said about the size of the universe is as limited as anything said about its age. The height of the sky depends on the observation energy. At Planck energies, it cannot be distinguished from the Planck length. If we start measuring the sky at standard observation energies, trying to increase the precision of measurement of the distance to the horizon, the measurement error increases beyond all bounds. At Planck energies, the volume of the universe is indistinguishable from the Planck volume!

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Ref. 1073

xi general rel ativity versus quantum mechanics

can easily be deduced by applying the Planck-scale arguments from the previous section. Time and space are not defined there. Finally, there is no way to decide whether the various boundary points are different from each other. The distance between two points in the night sky is undefined. In other words, it is unclear what the diameter of the horizon is. In summary, the horizon has no specific distance or shape. The horizon, and thus the universe, cannot be shown to be manifolds. This leads to the next question:

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Does the universe have initial conditions? Ref. 1078

The number of stars, about 10231 , is included in every book on cosmology, as it is in the table of Appendix B.1167 A subset of this number can be counted on clear nights. If we ask the same question about particles instead of stars, the situation is similar. The commonly quoted number of baryons is 10811 , together with 10901 photons. However, this does not settle the issue. Neither quantum theory nor general relativity alone make predictions

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Does the universe contain particles and stars?

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Ref. 1073

One often reads about the quest for the initial conditions of the universe. But before joining this search, we should ask whether and when such initial conditions make any sense. Obviously, our everyday description of motion requires them. Initial conditions describe the state of a system, i.e. all those aspects that differentiate it from a system with the same intrinsic properties. Initial conditions, like the state of a system, are attributed to a system by an outside observer. More specifically, quantum theory tells us that initial conditions or the state of a system can only be defined by an outside observer with respect to an environment. It is already difficult to be outside the universe. In addition, quite independently of this issue, even inside the universe a state can only be defined if matter can be distinguished from vacuum. However, this is impossible at Planck energies, near the big bang, or at the horizon. Thus the universe has no state. No state also means that there is no wave function of the universe. The limits imposed by the Planck values confirm this conclusion in other ways. First of all, they show that the big bang was not a singularity with infinite curvature, density or temperature, because infinitely large values do not exist in nature. Second, since instants of time do not exist, it is impossible to define the state of any system at a given time. Third, as instants of time do not exist, neither do events exist, and thus the big bang was not an event, so that for this more prosaic reason, neither an initial state nor an initial wave function can be ascribed to the universe. (Note that this also means that the universe cannot have been created.) In short, there are no initial conditions for the universe. Initial conditions make sense only for subsystems and only far away from Planck scales. Thus, for initial conditions to exist, the system must be far away from the horizon and it must have evolved for some time ‘after’ the big bang. Only when these two requirements are fulfilled can objects move in space. Of course, this is always the case in everyday life. At this point in our mountain ascent, where time and length are unclearly defined at cosmological scales, it should come as no surprise that there are similar difficulties concerning the concept of mass.

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Page 773

Challenge 1457 ny

Does the universe contain masses and objects? Page 1167

The average density of the universe, about 10−26 kg~m3 , is frequently cited in texts. Is it different from a vacuum? Quantum theory shows that, as a result of the indeterminacy relation, even an empty volume of size R has a mass. For a zero-energy photon inside such a vacuum, we have E~c = ∆p A ħ~∆x, so that in a volume of size R, we have a minimum mass of at least m mi n (R) = h~cR. For a spherical volume of radius R there is

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about the number of particles, either inside or outside the horizon. What happens if we combine them? In order to define the number of particles in a region, quantum theory first of all requires a vacuum state to be defined. The number of particles is defined by comparing the system with the vacuum. If we neglect or omit general relativity by assuming flat spacetime, this procedure poses no problem. However, if we include general relativity and thus a curved space-time, especially one with such a strangely behaved horizon as the one we have just found, the answer is simple: there is no vacuum state with which we can compare the universe, for two reasons. First, nobody can explain what an empty universe would look like; second, and more importantly, there is no way to define a state of the universe at all. The number of particles in the universe thus becomes undefinable. Only at everyday energies and for finite dimensions are we able to speak of an approximate number of particles. Comparison between a system and the vacuum is also impossible in the case of the universe for purely practical reasons. The requirement for such a comparison effectively translates into the requirement that the particle counter be outside the system. (Can you confirm the reason for this connection?) In addition, it is impossible to remove particles from the universe. The impossibility of defining a vacuum state, and thus the number of particles in the universe, is not surprising. It is an interesting exercise to investigate the measurement errors that appear when we try to determine the number of particles despite this fundamental impossibility. Can we count the stars? In principle, the same conclusion applies as for particles. However, at everyday energies the stars can be counted classically, i.e. without taking them out of the volume in which they are enclosed. For example, this is possible if the stars are differentiated by mass, colour or any other individual property. Only near Planck energies or near the horizon are these methods inapplicable. In short, the number of stars is only defined as long as the observation energy is low, i.e. as long as we stay away from Planck energies and from the horizon. In short, despite what appears to be the case on human scales, there is no definite number of particles in the universe. The universe cannot be distinguished from vacuum by counting particles. Even though particles are necessary for our own existence and functioning, a complete count of them cannot be made. This conclusion is so strange that we should not accept it too easily. Let us try another method of determining the content of matter in the universe: instead of counting particles, let us weigh them.

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thus a minimal mass density given approximately by ρ min 

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For the universe, if the standard horizon distance R 0 of 14 000 million light years is inserted, the value becomes about 10−142 kg~m3 . This describes the density of the vacuum. In other words, the universe, with a density of about 10−26 kg~m3 , seems to be clearly different from vacuum. But are we sure? We have just deduced that the radius of the horizon is undefined: depending on the observation energy, it can be as small as the Planck length. This implies that the density of the universe lies somewhere between the lowest possible value, given by the density of vacuum just mentioned, and the highest possible one, namely the Planck density.* In short, relation (706) does not really provide a clear statement. Another way to measure the mass of the universe would be to apply the original definition of mass, as given by Mach and modified by special relativity. Thus, let us try to collide a standard kilogram with the universe. It is not hard to see that whatever we do, using either low or high energies for the standard kilogram, the mass of the universe cannot be constrained by this method. We would need to produce or to measure a velocity change ∆v for the rest of the universe after the collision. To hit all the mass in the universe at the same time, we need high energy; but then we are hindered by Planck energy effects. In addition, a properly performed collision measurement would require a mass outside the universe, a rather difficult feat to achieve. Yet another way to measure the mass would be to determine the gravitational mass of the universe through straightforward weighing. But the lack of balances outside the universe makes this an impractical solution, to say the least. A way out might be to use the most precise definition of mass provided by general relativity, the so-called ADM mass. However, for definition this requires a specified behaviour at infinity, i.e. a background, which the universe lacks. We are then left with the other general relativistic method: determining the mass of the universe by measuring its average curvature. Let us take the defining expressions for average curvature κ for a region of size R, namely

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1 2 r curvature

=

3 4πR 2 − S 15 4πR 3 ~3 − V = . 4π R4 4π R5

(708)

We have to insert the horizon radius R 0 and either its surface area S 0 or its volume V0 . However, given the error margins on the radius and the volume, especially at Planck energies, for the radius of curvature we again find no reliable result. * In fact, at everyday energies the density of the universe lies almost exactly between the two values, yielding the strange relation 2 2 2 2 4 2 m 0 ~R 0  m Pl ~R Pl = c ~G . (707)

But this is nothing new. The approximate equality can be deduced from equation 16.4.3 (p. 620) of Steven Weinberg, Gravitation and Cosmology, Wiley, 1972, namely Gn b m p = 1~t 02 . The relation is required by several cosmological models.

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Copyright © Christoph Schiller November 1997–May 2006

κ=

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(706)

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m mi n (R) ħ = 4 . 3 R cR

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An equivalent method starts with the usual expression for the indeterminacy ∆κ in the scalar curvature for a region of size R provided by Rosenfeld, namely ∆κ A

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2 16πl Pl . R4

(709)

However, this expression also shows that the error in the radius of curvature behaves like the error in the distance to the horizon. In summary, at Planck energies, the average radius of curvature of nature turns out to lie between infinity and the Planck length. This implies that the density of matter lies between the minimum value and the Planck value. There is thus no method to determine the mass of the universe at Planck energies. (Can you find one?) The concept of mass cannot be applied to the universe as a whole. Thus, the universe has no mass.

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Does the universe have a boundary? It is common to take ‘boundary’ and ‘horizon’ as synonyms in the case of the universe, because they are the same for all practical purposes. To study this concept, knowledge of mathematics does not help us; the properties of mathematical boundaries, e.g. that they themselves have no boundary, are not applicable in the case of nature, since space-time is not continuous. We need other, physical arguments. The boundary of the universe is obviously supposed to represent the boundary between something and nothing. This gives three possibilities:

Copyright © Christoph Schiller November 1997–May 2006

We have already seen that at the horizon, space-time translation symmetry breaks down. Let us have a quick look at the other symmetries. What happens to permutation symmetry? Exchange is an operation on objects in space-time. Exchange thus automatically requires a distinction between matter, space and time. If we cannot distinguish positions, we cannot talk about exchange of particles. However, this is exactly what happens at the horizon. In short, general relativity and quantum theory together make it impossible to define permutation symmetry at the horizon. CPT symmetry suffers the same fate. As a result of measurement errors or of limiting maximum or minimum values, it is impossible to distinguish between the original and the transformed situations. It is therefore impossible to maintain that CPT is a symmetry of nature at horizon scales. In other words, matter and antimatter cannot be distinguished at the horizon. The same happens with gauge symmetry, as you may wish to check in detail yourself. For its definition, the concept of gauge field requires a distinction between time, space and mass; at the horizon this is impossible. We therefore also deduce that at the horizon also concepts such as algebras of observables cannot be used to describe nature. Renormalization also breaks down. All symmetries of nature break down at the horizon. The complete vocabulary we use when we talk about observations, including terms such as such as magnetic field, electric field, potential, spin, charge, or speed, cannot be used at the horizon. And that is not all.

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Do symmetries exist in nature?

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— ‘Nothing’ could mean ‘no matter’. But we have just seen that this distinction cannot be made at Planck scales. As a consequence, the boundary will either not exist at all or it will encompass the horizon as well as the whole universe. — ‘Nothing’ could mean ‘no space-time’. We then have to look for those domains where space and time cease to exist. These occur at Planck scales and at the horizon. Again, the boundary will either not exist or it will encompass the whole universe. — ‘Nothing’ could mean ‘neither space-time nor matter.’ The only possibility is a boundary that encloses domains beyond the Planck scale and beyond the horizon; but again, such a boundary would also encompass all of nature.

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Is the universe a set? – Again

Copyright © Christoph Schiller November 1997–May 2006

We are used to calling the universe the sum of all matter and all space-time. In other words, we imply that the universe is a set of components, all different from each other. This idea was introduced in three situations: it was assumed that matter consists of particles, that space-time consists of events (or points) and that the set of states consists of different initial conditions. However, our discussion so far shows that the universe is not a set of such distinguishable elements. We have encountered several proofs: at the horizon, at the big bang and at Planck scales distinction between events, between particles, between observables and between space-time and matter becomes impossible. In those domains, distinctions of any kind become impossible. We have found that any distinction between two entities, such as between a toothpick and a mountain, is only approximately possible. The approximation is possible because we live at energies much smaller than the Planck energy. Obviously, we are able to distinguish cars from people and from toothpicks; the approximation is so good that we do not notice the error when we perform it. Nevertheless, the discussion of the situation at Planck energies shows that a perfect distinction is impossible in principle. It is impossible to split the universe into separate entities. Another way to reach this result is the following. Distinguishing between two entities requires different measurement results, such as different positions, masses, sizes, etc. Whatever quantity we choose, at Planck energies the distinction becomes impossible. Only at everyday energies is it approximately possible. In short, since the universe contains no distinguishable entities, the universe is not a

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Ref. 1080

This result is puzzling. When combining quantum theory and relativity, we do not seem to be able to find a conceptual definition of the horizon that distinguishes the horizon from what it includes. In fact, if you find one, publish it! A distinction is possible in general relativity; and equally, a distinction is possible in quantum theory. However, as soon as we combine the two, the boundary becomes indistinguishable from its content. The interior of the universe cannot be distinguished from its horizon. There is no boundary. A difficulty to distinguish the horizon and its contents is definitely interesting; it suggests that nature may be symmetric under transformations that exchange interiors and boundaries. This connection is nowadays called holography because it vaguely recalls the working of credit card holograms. It is a busy research field in present-day high-energy physics. However, for the time being, we shall continue with our original theme, which directly leads us to ask:

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* Some people knew this long before physicists; for example, the belief that the universe is or contains information was ridiculed most thoroughly in the popular science fiction parody by Douglas Adams, The Hitchhiker’s Guide to the Galaxy, 1979, and its sequels.

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Copyright © Christoph Schiller November 1997–May 2006

set. We have already envisaged this possibility in the first intermezzo; now it is confirmed. The concept of ‘set’ is already too specialized to describe the universe. The universe must be described by a mathematical concept that does not contain any set. This is a powerful result: it means that the universe cannot be described precisely if any of the concepts used for its description presuppose the use of sets. But all concepts we have used so far to describe nature, such as space-time, phase space, Hilbert space and its generalizations, namely Fock space and particle space, are based on sets. They all must be abandoned at Planck energies, as well as in any precise description. Furthermore, many speculations about unified descriptions do not satisfy the criterion that sets must not be included. In particular, all studies of quantum fluctuations, mathematical categories, posets, complex mathematical spaces, computer programs, Turing machines, Gödel’s theorem, creation of any sort, space-time lattices, quantum lattices and Bohm’s unbroken wholeness fail to satisfy this requirement. In addition, almost none of the speculations about the origin of the universe can be correct. For example, you may wish to check the religious explanations you know against this result. In fact, no approach used by theoretical physics in the year 2000 satisfies the requirement that sets must be abandoned; perhaps a future version of string or M theory will do so. The task is not easy; do you know of a single concept not based on a set? Note that this radical conclusion is deduced from only two statements: the necessity of using quantum theory whenever the dimensions are of the order of the Compton wavelength, and the necessity to use general relativity whenever the dimensions are of the order of the Schwarzschild radius. Together, they mean that any precise description of nature cannot contain sets. We have reached this result after a long and interesting, but in a sense unnecessary, digression. The difficulties in complying with this result may explain why the unification of the two theories has not so far been successful. Not only does unification require that we stop using space, time and mass for the description of nature; it also requires that all distinctions, of any kind, should be only approximate. But all physicists have been educated on the basis of exactly the opposite creed! Note that, because it is not a set, the universe is not a physical system. Specifically, it has no state, no intrinsic properties, no wave function, no initial conditions, no density, no entropy and no cosmological constant. The universe is thus neither thermodynamically closed nor open; and it contains no information. All thermodynamic quantities, such as entropy, temperature and free energy, are defined using ensembles. Ensembles are limits of systems which are thermodynamically either open or closed. As the universe is neither of these two, no thermodynamic quantity can be defined for it.* All physical properties are defined only for parts of nature which are approximated or idealized as sets, and thus are physical systems.

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Curiosities and fun challenges about the universe Insofern sich die Sätze der Mathematik auf die Wirklichkeit beziehen, sind sie nicht sicher, und sofern sie sicher sind, beziehen sie sich nicht auf die Wirklichkeit.* Albert Einstein

The contradictions between the term ‘universe’ and the concept of ‘set’ lead to numerous fascinating issues. Here are a few.

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**

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In mathematics, 2 + 2 = 4. This statement is an idealization of statements such as ‘two apples plus two apples makes four apples.’ However, we now know that at Planck energies this is not a correct statement about nature. At Planck energies, objects cannot be counted. Are you able to confirm this?

Ref. 1081

In 2002, Seth Lloyd estimated how much information the universe can contain, and how many calculations it has performed since the big bang. This estimate is based on two ideas: that the number of particles in the universe is a well-defined quantity, and that the universe is a computer, i.e. a physical system. We now know that neither assumption is correct. This example shows the power of the criteria that we deduced for the final description of motion. **

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People take pictures of the cosmic background radiation and its variations. Is it possible that the photographs will show that the spots in one direction of the sky are exactly the same as those in the diametrically opposite direction? **

Ref. 1082 Challenge 1468 n

In 1714, Leibniz published his Monadologie. In it he explores what he calls a simple substance, which he defined to be a substance that has no parts. He called it a monade and explores some of its properties. However, due mainly to its incorrect deductions, the term has not been taken over by others. Let us forget the strange deductions and focus only on the definition: what is the physical concept most related to that of monade?

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We usually speak of the universe, implying that there is only one of them. Yet there is a simple case to be made that ‘universe’ is an observer-dependent concept, since the idea of ‘all’ is observer-dependent. Does this mean that there are many universes? **

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If all particles would be removed – assuming one would know where to put them – there wouldn’t be much of a universe left. True? * In so far as mathematical statements describe reality, they are not certain, and as far as they are certain, they are not a description of reality.

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**

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**

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** Challenge 1471 ny

At Planck energies, interactions cannot be defined. Therefore, ‘existence’ cannot be defined. In short, at Planck energies we cannot say whether particles exist. True? Hilbert’s sixth problem settled

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Drum hab ich mich der Magie ergeben, [...] Daß ich erkenne, was die Welt Im Innersten zusammenhält.* Goethe, Faust.

Is the universe really the sum of matter–energy and space-time? Or of particles and vacuum? We have heard this so often up to now that we may be lulled into forgetting to * Thus I have devoted myself to magic, [...] that I understand how the innermost world is held together.

Copyright © Christoph Schiller November 1997–May 2006

Does the universe make sense?

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Page 157

In the year 1900, David Hilbert gave a famous lecture in which he listed 23 of the great challenges facing mathematics in the twentieth century. Most of these problems provided challenges to many mathematicians for decades afterwards. A few are still unsolved, among them the sixth, which challenged mathematicians and physicists to find an axiomatic treatment of physics. Since the universe is not even a set, we can deduce that such an axiomatic description of nature is impossible. The reasoning is simple; all mathematical systems, be they algebraic systems, order systems or topological systems, are based on sets. Mathematics does not have axiomatic systems that do not contain sets. The reason for this is simple: any (mathematical) concept contains at least one set. However, nature does not. The impossibility of an axiomatic system for physics is also confirmed in another way. Physics starts with a circular definition: space-time is defined with the help of objects and objects are defined with the help of space-time. Physics thus has never been modelled on the basis of mathematics. Physicists have always had to live with logical problems. The situation is similar to a child’s description of the sky as ‘made of air and clouds’. Looking closely, we discover that clouds are made up of water droplets. However, there is air inside clouds, and there is also water vapour elsewhere in the air. When clouds and air are viewed through a microscope, there is no clear boundary between the two. We cannot define either of the terms ‘cloud’ and ‘air’ without the other. No axiomatic definition is possible. Objects and vacuum also behave in this way. Virtual particles are found in vacuum, and vacuum is found inside objects. At Planck scales there is no clear boundary between the two; we cannot define either of them without the other. Despite the lack of a precise definition and despite the logical problems that ensue, in both cases the description works well at large, everyday scales. We note that, since the universe is not a set and since it contains no information, the paradox of the physics book containing a full description of nature disappears. Such a book can exist, as it does not contradict any property of the universe. But then a question arises naturally:

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TA B L E 75 Physical statements about the universe

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* There is also another well-known, non-physical concept about which nothing can be said. Many scholars have explored it in detail. Can you see what it is?

Copyright © Christoph Schiller November 1997–May 2006

Page 644

check the statement. To find the answer, we do not need magic, as Faust thought; we only need to list what we have found so far, especially in this section, in the section on Planck scales, and in the intermezzo on brain and language. Table 75 shows the result. Not only are we unable to state that the universe is made of space-time and matter; in fact, we are unable to say anything about the universe at all!* It is not even possible to say that it exists, since it is impossible to interact with it. The term ‘universe’ does not allow us to make a single sensible statement. (Can you find one?) We are only able to say which properties it does not have. We are unable to find any property the universe does have. Thus, the universe has no properties! We cannot even say whether the universe is something or nothing. The universe isn’t anything in particular. In other words, the term ‘universe’ is not useful at all for the description of motion. We can obtain a confirmation of this strange conclusion from the first intermezzo. There we found that any concept needs defined content, defined limits and a defined domain of application. In this section, we have found that for the term ‘universe’, not one of these three aspects is defined; there is thus no such concept. If somebody asks: ‘why does the universe exist?’ the answer is: not only does the use of ‘why’ wrongly suggest that something may exist outside the universe, providing a reason for it and thus contradicting the definition of the term ‘universe’ itself; most importantly of all, the universe simply does not exist. In summary, any sentence containing the word universe makes no sense.

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The universe has no age. The universe has no beginning. The universe has no size. The universe has no volume. The universe has no shape. The universe’s particle number is undefined. The universe has no mass. The universe has no energy. The universe has no density. The universe contains no matter. The universe has no cosmological constant. The universe has no initial conditions. The universe has no state. The universe has no wave function. The universe is not a physical system. The universe contains no information. The universe is not isolated. The universe is not open. The universe has no boundaries. The universe does not interact. The universe cannot be measured. The universe cannot be said to exist. The universe cannot be distinguished from The universe cannot be distinguished from a nothing. single event. The universe contains no moments. The universe is not composite. The universe is not a set. The universe is not a concept. The universe cannot be described. There is no plural for ‘universe’. The universe cannot be distinguished from va- The universe was not created. cuum.

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The term ‘universe’ only seems to express something, even if it doesn’t.* The conclusion that the term universe makes no sense may be interesting, even strangely beautiful; but does it help us to understand motion more precisely? Interestingly so, it does. A concept without a set eliminates contradictions

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In the chapter Quantum physics in a nutshell we listed all the unexplained properties of nature left open either by general relativity or by quantum theory. The present conclusions provide us with new connections among them. Indeed, many of the cosmological results of this section sound surprisingly familiar; let us compare them systematically with those of the section on Planck scales. Both sections explored topics – some in more detail than others – from the list of unexplained properties of nature. First, Table 76 shows that none of the unexplained properties makes sense at both limits of nature, the small and the large. All open questions are open at both extremes. Second and more importantly, nature behaves in the same way at horizon scales and at Planck scales. In fact, we have not found any difference between the two cases. (Are you able to discover one?) We are thus led to the hypothesis that nature does not distinguish between the large and the small. Nature seems to be characterized by extremal identity. Is extremal identity a principle of nature? The principle of extremal identity incorporates some rather general points:

* Of course, the term ‘universe’ still makes sense if it is defined more restrictively, such as ‘everything interacting with a particular human or animal observer in everyday life’. But such a definition is not useful for our quest, as it lacks the precision required for any description of motion.

Copyright © Christoph Schiller November 1997–May 2006

— all open questions about nature appear at its two extremes; — a description of nature requires both general relativity and quantum theory; — nature or the universe is not a set; — initial conditions and evolution equations make no sense at nature’s limits; — there is a relation between local and global issues in nature; — the concept of ‘universe’ has no content.

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The discussion about the term ‘universe’ also shows that the term does not contain any set. In other words, this is the first term that will help us on the way to a precise description of nature. We will see later on how this happens. By taking into account the limits on length, time, mass and all other quantities we have encountered, we have deduced a number of almost painful conclusions about nature. However, we also received something in exchange: all the contradictions between general relativity and quantum theory that we mentioned at the beginning of this chapter are now resolved. Although we have had to leave behind us many cherished habits, in exchange we have the promise of a description of nature without contradictions. But we get even more.

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TA B L E 76 Properties of nature at maximal, everyday and minimal scales

At every day scale

At P l a n c k scale

requires quantum theory and relativity intervals can be measured precisely length and time intervals are space-time is not continuous points and events cannot be distinguished space-time is not a manifold space is 3 dimensional space and time are indistinguishable initial conditions make sense space-time fluctuates Lorentz and Poincaré symmetry CPT symmetry renormalization permutation symmetry interactions number of particles algebras of observables matter indistinguishable from vacuum boundaries exist nature is a set

true false limited true true true false true false true does not apply does not apply does not apply does not apply do not exist undefined undefined true false false

false true unlimited false false false true false true false applies applies applies applies exist defined defined false true true

true false limited true true true false true false true does not apply does not apply does not apply does not apply do not exist undefined undefined true false false

Extremal identity thus looks like a good candidate tool for use in the search for a unified description of nature. To be a bit more provocative, it may be the only known principle incorporating the idea that the universe is not a set, and thus might be the only candidate tool for use in the quest of unification. Extremal identity is beautiful in its simplicity, in its unexpectedness and in the richness of its consequences. You might enjoy exploring it by yourself. The study of the consequences of extremal identity is currently the focus of much activity in high energy particle physics, although often under different names. The simplest approach to extremal identity – in fact one that is too simple to be correct – is inversion. It looks as if extremal identity implies a connection such as r

2 l Pl r

or

xµ 

2 l Pl xµ xµ x µ

(710)

relating distances r or coordinates x µ with their inverse values using the Planck length l Pl . Can this mapping, called inversion, be a symmetry of nature? At every point of space? For

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Ref. 1084

At h ori zon scale

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P h ys i c a l p r o p e rt y o f n at u r e

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example, if the horizon distance is inserted, equation (710) implies that lengths smaller than l Pl ~1061  10−96 m never appear in physics. Is this the case? What would inversion imply for the big bang? Numerous fascinating questions are contained in the simple hypothesis of extremal identity. They lead to two main directions for investigation. First, we have to search for some stronger arguments for the validity of extremal identity. We will discover a number of simple arguments, all showing that extremal identity is indeed a property of nature and producing many beautiful insights. The other quest then follows. We need to find the correct version of equation (710). That oversimplified expression is neither sufficient nor correct. It is not sufficient because it does not explain any of the issues left open by general relativity and quantum theory. It only relates some of them, thus reducing their number, but it does not solve any of them. You may wish to check this for yourself. In other words, we need to find the precise description of quantum geometry and of elementary particles. However, inversion is also simply wrong. Inversion is not the correct description of extremal identity because it does not realize a central result discovered above: it does not connect states and intrinsic properties. Inversion keeps them distinct. This means that inversion does not take interactions into account. And most open issues at this point of our mountain ascent are properties of interactions.

Copyright © Christoph Schiller November 1997–May 2006

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Biblio graphy 1073 Available at http://www.dse.nl/motionmountain/C10-QMGR.pdf. Those pages are a re-

worked version of the ones published in French as C. Schiller, Le vide diffère-t-il de la matière? in E. Gunzig & S. Diner, editeurs, Le vide – Univers du tout et du rien – Des physiciens et des philosophes s’interrogent, Les Éditions de l’Université de Bruxelles, 1998. See also C. Schiller, Does matter differ from vacuum? http://www.arxiv.org/abs/gr-qc/ 9610066. Cited on pages 1032, 1036, and 1042.

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1074 See the lucid discussion by G.F.R. Ellis & T. Rothman, Lost horizons, American

Journal of Physics 61, pp. 883–893, 1993. Cited on pages 1034, 1038, 1039, and 1040. 1075 H. Salecker & E.P. Wigner, Quantum limitations of the measurement of space-time

distances, Physical Review 109, pp. 571–577, 1958. See also the popular account of this paper by E.J. Zimmerman, The macroscopic nature of space-time, American Journal of Physics 30, pp. 97–105, 1962. Cited on page 1034. Cimento A42, pp. 390–402, 1966, Y.J. Ng & H. Van Dam, Limit to space-time measurement, Modern Physics Letters A 9, pp. 335–340, 1994, Y.J. Ng & H. Van Dam, Remarks on gravitational sources, Modern Physics Letters A 10, pp. 2801–2808, 1995. Newer summaries are Y.J. Ng & H. van Dam, Comment on “Uncertainty in measurements of distance”, http://www.arxiv.org/abs/gr-qc/0209021 and Y.J. Ng, Spacetime foam, http://www.arxiv. org/abs/gr-qc/0201022. Cited on page 1035. 1077 See, for example, the Hollywood movie Contact, based on the book by Carl Sagan,

Contact, 1985. Cited on page 1040. 1078 See, for example, the international bestseller by S. Hawking, A Brief History of Time –

From the Big Bang to Black Holes, 1988. Cited on page 1042. 1079 L. Rosenfeld, in H.-J. Treder, Entstehung, Entwicklung und Perspektiven der Ein-

steinschen Gravitationstheorie, Springer Verlag, 1966. Cited on page 1045. 1080 Holography in high-energy physics is connected with the work by ’t Hooft and by Susskind.

1081 S. Lloyd, Computational capacity of the universe, Physical Review Letters 88, p. 237901,

2002. Cited on page 1048. 1082 Gottfried Wilhelm Leibniz, La Monadologie, 1714. Written in French, it is avail-

able freely at http://www.uqac.uquebec.ca/zone30/Classiques_des_sciences_sociales and in various other languages on other websites. Cited on page 1048. 1083 See, for example, H. Wussing & P.S. Alexandrov, (eds.), Die Hilbertschen Probleme,

Akademische Verlagsgesellschaft Geest & Portig, 1983, or Ben H. Yandell, The Honours Class: Hilbert’s Problems and their Solvers, A.K. Peters, 2002. Cited on page 1049. 1084 Large part of the study of dualities in string and M theory can be seen as investigations into

the detailed consequences of extremal identity. For a review of duality, see ... A classical version of duality is discussed by M.C.B. Abdall a, A.L. Gadelka & I.V. Vancea, Du-

Copyright © Christoph Schiller November 1997–May 2006

See for example G. ’t Hooft, Dimensional reduction in quantum gravity, pp. 284–296, in A. Ali, J. Ellis & S. R andjbar-Daemi, Salaamfeest, 1993, or the often cited paper by L. Susskind, The world as a hologram, Journal of Mathematical Physics 36, pp. 6377– 6396, 1995, or as http://www.arxiv.org/abs/hep-th/9409089 A good modern overview is R. B ousso, The holographic principle, Review of Modern Physics 74, pp. 825–874, 2002, also available as http://www.arxiv.org/abs/hep-th/0203101, even though it has some argumentation errors, as explained on page 871. Cited on page 1046.

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1076 F. Karolyhazy, Gravitation and quantum mechanics of macroscopic objects, Il Nuovo

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1055 bibliography

ality between coordinates and the Dirac field, http://www.arxiv.org/abs/hep-th/0002217. Cited on page 1052.

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Copyright © Christoph Schiller November 1997–May 2006

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35.

the physics of love – a summary of the first t wo and a half parts

Sex is the physics urge sublimated.

M

Graffito

Love is communication. Love is tiring. Love is an interaction between moving bodies. Love takes time. Love is attractive. Love is repulsive. Love makes noise. In love, size matters. Love is for reproduction. Love can hurt. Love needs memory. Love is Greek. Love uses the sense of sight. Love is animalic. Love is motion. Love is holy. Love is based on touch. Love uses motion again. Love is fun. Love is private. Love makes one dream.

Let us see what these observations imply for the description of nature. **

Ref. 1085

* Here we deduce physics from love. We could also deduce physics from sexuality. The modern habit of saying ‘sex’ instead of ‘sexuality’ mixes up two completely different concepts. In fact, studying the influences of sex on physics is almost fully a waste of time. We avoid it. Maybe one day we shall understand why there do not seem to be any female crackpots proposing pet physical theories.

Copyright © Christoph Schiller November 1997–May 2006

Love is communication. Communication is possible because nature looks similar from different standpoints and because nature shows no surprises. Without similarity we could not understand each other, and a world of surprises would even make thinking impossible; it would not be possible to form concepts to describe observations. But fortunately, the world is regular; it thus allows to use concepts such as time and space for its description.

Dvipsbugw

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aybe you have once met a physicist who has told you, in one of those oments of confidentiality, that studying physics is even more beautiful than aking love. At this statement, many will simply shake their head in disbelief and strongly disapprove. In this section we shall argue that it is possible to learn so much about physics while making love that discussions about their relative beauty can be put aside altogether. Imagine to be with your partner on a beautiful tropical island, just after sunset, and to look together at the evening sky. Imagine as well that you know little of what is taught at school nowadays, e.g. that your knowledge is that of the late Renaissance, which probably is a good description of the average modern education level anyway. Imagine being busy enjoying each other’s company. The most important results of physics can be deduced from the following experimental facts:*

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**

Challenge 1479 n

Love is an interaction between moving bodies. Together with the previous result, this implies that we can and need to describe moving bodies with mass, energy and momentum. That is not a small feat. For example, it implies that the Sun will rise tomorrow if the sea level around the island is the usual one. ** Love is attractive. When feeling attracted to your partner, you may wonder if this attraction is the same which keeps the Moon going around the Earth. You make a quick calculation and find that applying the expression for universal gravity E pot = −

GMm r

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(711)

** Love makes noise. That is no news. However, even after making love, even when everybody and everything is quiet, in a completely silent environment, we do hear something. The noises we hear are produced within the ear, partly by the blood flowing through the head, partly by the electrical noise generated in the nerves. That is strange. If matter were continuous, there would be no noise even for low signal levels. In fact, all proofs for the discreteness of matter, of electric current, of energy, or of light are based on the increase of fluctuations with the smallness of systems under consideration. The persistence of noise thus makes us suspect that matter is made of smallest entities. Making love confirms this suspicion in several ways. **

Copyright © Christoph Schiller November 1997–May 2006

Love is for reproduction. Love is what we owe our life to, as we all are results of reproduction. But the reproduction of a structure is possible only if it can be constructed, in other words if the structure can be built from small standard entities. Thus we again suspect ourselves to be made of smallest, discrete entities. Love is also a complicated method of reproduction. Mathematics provides a much simpler one. If matter objects were not made of particles, but were continuous, it would be possible to perform reproduction by cutting and reassembling. A famous mathematical theorem by Banach and Tarski proves that it is possible to take a continuous solid, cut it into five pieces and rearrange the pieces in such a way that the result are two copies of the same size and volume as the original. In fact, even volume increases can be produced in this way, thus realizing growth without any need for food. Mathematics thus provides

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to both of you, the involved energy is about as much as the energy added by the leg of a fly on the skin. (M and m are your masses, r your distance, and the gravitational constant has the value G = 6.7 ë 10−11 m3 ~kg s2 .) In short, your partner teaches you that in nature there are other attractive interactions apart from gravity; the average modern education is incomplete. Nevertheless, this first equation is important: it allows to predict the position of the planets, the time of the tides, the time of eclipses, the return of comets, etc., to a high accuracy for thousands of years in advance.

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xi general rel ativity versus quantum mechanics

some interesting methods for growth and reproduction. However, they assume that matter is continuous, without a smallest length scale. The observation that these methods do not work in nature is compatible with the idea that matter is not continuous. ** Ref. 1086

Challenge 1480 n

Love needs memory. If you would not recognize your partner among all possible ones, your love life would be quite complicated. A memory is a device which, in order to store information, must have small internal fluctuations. Obviously, fluctuations in systems get smaller as their number of components increase. Since our memory works so well, we can follow that we are made of a large number of small particles. In summary, love shows that we are made of some kind of lego bricks: depending on the level of magnification, these bricks are called molecules, atoms, or elementary particles. It is possible to estimate their size using the sea around the tropical island, as well as a bit of oil. Can you imagine how?

Ref. 1087

Love uses the sense of sight. Seeing each other is only possible because we are cold whereas the Sun is hot. If we and our environment all had the same temperature as the Sun, we would not see each other. This can be checked experimentally by looking into a hot oven: Inside a glowing oven filled with glowing objects it is impossible to discern them against the background. ** Love is motion. Bodies move against each other. Moreover, their speed can be measured. Since measurement is a comparison with a standard, there must be a velocity standard in nature, some special velocity standing out. Such a standard must either be the minimum or the maximum possible value. Now, daily life shows that for velocity a finite minimum value does not exist. We are thus looking for a maximum value. To estimate the value of the maximum, just take your mobile phone and ring home from the island to your family. From the delay in the line and the height of the satellite, you can deduce the telephone speed c and get 3 ë 108 m~s. The existence of a maximum speed c implies that time is different for different observers. Looking into the details, we find that this effect becomes noticeable at energies (712)

where m is the mass of the object. For example, this applies to electrons inside a television tube. **

Challenge 1481 n

Love is based on touching. When we touch our partner, sometimes we get small shocks. The energies involved are larger than than those of touching fly legs. In short, people are electric. In the dark, we observe that discharges emit light. Light is thus related to electricity. In addition, touching proves that light is a wave: simply observe the dark lines between two fingers near your eye in front of a bright background. The lines are due to interference

Copyright © Christoph Schiller November 1997–May 2006

E different time  mc 2 ,

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**

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the physics of love – a summary of the first two and a half parts

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effects. Light thus does not move with infinite speed. In fact, it moves with the same speed as that of telephone calls. **

Page 210

Love is fun. People like to make love in different ways, such as in a dark room. But rooms get dark when the light is switched off only because we live in a space of odd dimensions. In even dimensions, a lamp would not turn off directly after the switch is flipped, but dim only slowly. Love is also fun because with our legs, arms and bodies we can make knots. Knots are possible only in three dimensions. In short, love is real fun only because we live in 3 dimensions.

Dvipsbugw

**

R horizon  Gm~c 2 .

(713)

For example, only due a horizon, albeit one appearing in a different way, the night sky is dark. **

Ref. 1091

Love takes time. It is known that men and women have different opinions on durations. It is also known that love happens between your ears. Indeed, biological research has shown that we have a clock inside the brain, due to circulating electrical currents. This clock provides our normal sense of time. Since such a brain clock can be built, there must be a time standard in nature. Again, such a standard must be either a minimum or a maximum time interval. We shall discover it later on. **

** Love can hurt. For example, it can lead to injuries. Atoms can get ripped apart. That hap* Note that the classical electron radius is not an exception: it contains the elementary charge e, which contains a length scale, as shown on page 492.

Copyright © Christoph Schiller November 1997–May 2006

Love is repulsive. And in love, size matters. Both facts turn out to be the two sides of the same coin. Love is based on touch, and touch needs repulsion. Repulsion needs a length scale, but neither gravity nor classical electrodynamics provide one. Classical physics only allows for the measurement of speed. Classical physics cannot explain that the measurement of length, time, or mass is possible.* Classically, matter cannot be hard; it should be possible to compress it. But love shows us that this is not the case. Love shows us that lengths scales do exist in nature and thus that classical physics is not sufficient for the description of nature.

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Love is tiring. The reason is gravity. But what is gravity? A little thinking shows that since there is a maximum speed, gravity is the curvature of space-time. Curved space also means that a horizon can appear, i.e. a largest possible visible distance. From equations (711) and (712), we deduce that this happens when distances are of the order of

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xi general rel ativity versus quantum mechanics

pens when energies are concentrated on small volumes, such as a few aJ per atom. Investigating such situations more precisely, we finds that strange phenomena appear at distances r if energies exceed the value E

ħc r

;

(714)

r pair production 

ħ . mc

(715)

At such small distances we cannot avoid using the quantum description of nature. ** Love is not only Greek. The Greeks were the first to make theories above love, such as Plato in his Phaedrus. But they also described it in another way. Already before Plato, Democritus said that making love is an example of particles moving and interacting in vacuum. If we change ‘vacuum’ to ‘curved 3+1-dimensional space’ and ‘particle’ to ‘quantum particle’, we do indeed make love in the way Democritus described 2500 years ago. It seems that physics has not made much progress in the mean time. Take the statement made in 1939 by the British astrophysicist Arthur Eddington:

Ref. 1088

I believe there are 15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,914,527,116,709,366,231,425,076,185,631,031,296 protons in the universe and the same number of electrons.

Baryons in the universe: 10811 ; total charge: near zero.

The second is more honest, but which of the two is less sensible? Both sentences show that there are unexplained facts in the Greek description nature, in particular the number of involved particles. ** Love is animalic. We have seen that we can learn a lot about nature from the existence of love. We could be tempted to see this approach of nature as a special case of the so-called

Copyright © Christoph Schiller November 1997–May 2006

Compare it with the version of 2006:

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in particular, energy becomes chunky, things become fuzzy, boxes are not tight, and particles get confused. These are called quantum phenomena. The new constant ħ = 10−34 Js is important: it determines the size of things, because it allows to define distance and time units. In other words, objects tear and break because in nature there is a minimum action, given roughly by ħ. If even more energy is concentrated in small volumes, such as energies of the order of mc 2 per particle, one even observes transformation of energy into matter, or pair production. From equations (712) and (714), we deduce that this happens at distances of

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the physics of love – a summary of the first two and a half parts

1061

** Ref. 1089

Copyright © Christoph Schiller November 1997–May 2006

Love is holy. Following the famous definition by the theologian Rudolf Otto, holiness results from a mixture of a mysterium tremendum and a mysterium fascinans. Tremendum means that it makes one tremble. Indeed, love produces heat and is a dissipative process. All systems in nature which produce heat have a finite lifetime. That is true for machines, stars, animals, lightning, fire, lamps and people. Through heat, love shows us that we are going to die. Physicists call this the second principle of thermodynamics. But love also fascinates. Everything which fascinates has a story. Indeed, this is a principle of nature: every dissipative structure, every structure which appears or is sustained through the release of energy, tells us that it has a story. Take atoms, for example. All the protons we are made of formed during the big bang. Most hydrogen we are made of is also that old. The other elements were formed in stars and then blown into the sky during nova or supernova explosions. They then regrouped during planet formation. We truly are made of stardust. Why do such stories fascinate? If you only think about how you and your partner have met, you will discover that it is through a chain of incredible coincidences. If only one of all these coincidences had not taken place, you and your partner would not be together. And of course, we all owe our existence to such a chain of coincidences, which brought our parents together, our grandparents, and made life appear on Earth. The realization of the importance of coincidences automatically produces two kinds of questions: why? and what if? Physicists have now produced a list of all the answers to repeated why questions and many are working at the list of what-if questions. The first list, the why-list of Table 77, gives all facts still unexplained. It can also be called the complete list of all surprises in nature. (Above, it was said that there are no surprises in nature about what happens. However, so far there still are a handful of surprises on how all these things happen.)

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anthropic principle. However, some care is required here. In fact, we could have learned exactly the same if we had taken as starting point the observation that apes or pigs have love. There is no ‘law’ of nature which distinguishes between them and humans. In fact, there is a simple way to determine whether any ‘anthropic’ statement makes sense: the reasoning must be equally true for humans, apes, and pigs. A famous anthropic deduction was drawn by the British astrophysicist Fred Hoyle. While studying stars, he predicted a resonance in the carbon-12 nucleus. If it did not exist, he argued, stars could not have produced the carbon which afterwards was spread out by explosions into interstellar space and collected on Earth. Also apes or pigs could reason this way; therefore Hoyle’s statement does make sense. On the other hand, claiming that the universe is made especially for people is not sensible: using the same arguments, pigs would say it is made for pigs. The existence of either requires all ‘laws’ of nature. In summary, the anthropic principle is true only in so far as its consequences are indistinguishable from the porcine or the simian principle. In short, the animalic side of love puts limits to the philosophy of physics.

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TA B L E 77 Everything quantum field theory and general relativity do not explain; in other words, a list of

the only experimental data and criteria available for tests of the unified description of motion

O b s e r va b l e P r o p e r t y u n e x p l a i n e d s o f a r Local quantities, from quantum theory the low energy value of the electromagnetic coupling constant the low energy value of the weak coupling constant the low energy value of the strong coupling constant the values of the 6 quark masses the values of 3 lepton masses the values of the independent mass of the W vector boson the value of the Weinberg angle three mixing angles the value of the CP parameter the value of the strong topological angle the number of particle generations the value of the observed vacuum energy density or cosmological constant the number of space and time dimensions

Global quantities, from general relativity 1.2(1) ë 1026 m ? 1082 ? A 1092 ?

the distance of the horizon, i.e. the ‘size’ of the universe (if it makes sense) the number of baryons in the universe, i.e. the average matter density in the universe (if it makes sense) the initial conditions for more than 1092 particle fields in the universe, including those at the origin of galaxies and stars (if they make sense)

Local structures, from quantum theory

Global structures, from general relativity maybe R  S3 ?

the unknown topology of the universe (if it makes sense)

This why-list fascinates through its shortness, which many researchers are still trying

Copyright © Christoph Schiller November 1997–May 2006

the origin of particle identity, i.e. of permutation symmetry the renormalization properties, i.e. the existence of point particles the origin of Lorentz (or Poincaré) symmetry (i.e. of spin, position, energy, momentum) C‡ the origin of the algebra of observables Gauge group the origin of gauge symmetry (and thus of charge, strangeness, beauty, etc.) in particular, for the standard model: U(1) the origin of the electromagnetic gauge group (i.e. of the quantization of electric charge, as well as the vanishing of magnetic charge) SU(2) the origin of weak interaction gauge group SU(3) the origin of strong interaction gauge group S(n) Ren. group SO(3,1)

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α em αw αs mq ml mW θW β1 , β2 , β3 θ CP θ st 3 0.5 nJ~m3 3+1

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1063

to reduce. But it is equally interesting to study what consequences appear if any of the values from Table 77 were only a tiny bit different. It is not a secret that small changes in nature would lead to completely different observations, as shown in Table 78. TA B L E 78 A small selection of the consequences when changing aspects of nature

R e su lt

Moon size

smaller

Moon size

larger

Jupiter

smaller

Jupiter

larger

Oort belt

smaller

small Earth magnetic field; too much cosmic radiation; widespread child cancers. large Earth magnetic field; too little cosmic radiation; no evolution into humans. too many comet impacts on Earth; extinction of animal life. too little comet impacts on Earth; no Moon; no dinosaur extinction. no comets, no irregular asteroids, no Moon; still dinosaurs. irregular planet motion; supernova dangers. proton decay; leucemia.

Star distance smaller Strong coupling smaller constant

The large number of coincidences of life force our mind to realize that we are only a tiny part of nature. We are a small droplet shaken around in the ocean of nature. Even the tiniest changes in nature would prevent the existence of humans, apes and pigs. In other words, making love tells us that the universe is much larger than we are and tells us how much we are dependent and connected to the rest of the universe. **

**

Copyright © Christoph Schiller November 1997–May 2006

We said above that love uses motion. It contains a remarkable mystery, worth a second look: - Motion is the change of position with time of some bodies. - Position is what we measure with a ruler. Time is what we measure with a clock. Both rulers and clocks are bodies. - A body is an entity distinct from its environment by its shape or its mass. Shape is the extension of a body in space (and time). Mass is measured by measuring speed or acceleration, i.e. by measuring space and time. This means that we define space-time with bodies – as done in detail in general relativity – and that we define bodies with space-time – as done in detail in quantum theory. This circular reasoning shows that making love is truly a mystery. The circular reasoning has not yet been eliminated yet; at present, modern theoretical physicists are busy attempting to do so. The most promising approach seems to be M-theory, the modern extension of string theory. But any such attempt has to overcome important difficulties which can also be experienced while making love.

Dvipsbugw

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O b s e r va b l e C h a n g e

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M

Dvipsbugw R

R

F I G U R E 386 A Gedanken experiment showing that

Ref. 1090

Love is private. But is it? Privacy assumes that a person can separate itself from the rest, without important interactions, at least for a given time, and come back later. This is possible if the person puts enough empty space between itself and others. In other words, privacy is based on the idea that objects can be distinguished from vacuum. Let us check whether this is always possible. What is the smallest measurable distance? This question has been almost, but only almost answered by Max Planck in 1899. The distance δl between two objects of mass m is surely larger than their position indeterminacy ħ~∆p; and the momentum indeterminacy must be smaller that the momentum leading to pair production, i.e. ∆p < mc. This means that ħ . (716) δl E ∆l E mc

δl E

¾

ħG = l Pl = 1.6 ë 10−35 m . c3

(717)

This expression defines a minimum length in nature, the so-called Planck length. Every other Gedanken experiment leads to this characteristic length as well. In fact, this minimum distance (and the corresponding minimum time interval) provides the measurement standard we were looking for at the beginning of our musings about length and time measurements. A more detailed discussion shows that the smallest measurable distance is somewhat larger, a multiple of the Planck length, as measurements require the distinction of matter and radiation. This happens at scales about 800 times the Planck length. In other words, privacy has its limits. In fact, the issue is even more muddled when

Copyright © Christoph Schiller November 1997–May 2006

In addition, the measurements require that signals leave the objects; the two masses must not be black holes. Their masses must be so small that the Schwarzschild radius is smaller than the distance to be measured. This means that r S  Gm~c 2 < δl or that

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at Planck scales, matter and vacuum cannot be distinguished

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the physics of love – a summary of the first two and a half parts

1065

we explore the consequences for bodies. A body, also a human one, is something we can touch, throw, hit, carry or weigh. Physicists say that a body is something with energy or momentum. Vacuum has none of it. In addition, vacuum is unbounded, whereas objects are bounded. What happens if we try to weigh objects at Planck scales? Quantum theory makes a simple prediction. If we put an object of mass M in a box of size R onto a scale – as in Figure 386 – equation (714) implies that there is a minimal mass error ∆M given by ∆M 

ħ . cR

(719)

How large is the mass we can put into a box of Planck size? Obviously it is given by the maximum possible mass density. To determine it, imagine a planet and put a satellite in orbit around it, just skimming its surface. The density ρ of the planet with radius r is given by v2 M (720) ρ 3 = 2 . r Gr

Using equation (716) we find that the maximum mass density in nature, within a factor of order one, is the so-called Planck density, given by ρ Pl =

Ref. 1090

(721)

Therefore the maximum mass that can be contained inside a Planck box is the Planck mass. But that was also the measurement error for that situation. This implies that we cannot say whether the original box we measured was empty or full: vacuum cannot be distinguished from matter at Planck scales. This astonishing result is confirmed by every other Gedanken experiment exploring the issue. It is straightforward to deduce with similar arguments that objects are not bound in size at Planck scales, i.e. that they are not localized, and that the vacuum is not necessarily extended at those scales. In addition, the concept of particle number cannot be defined at Planck scales. So, why is there something instead of nothing? Making love shows that there is no difference between the two options! ** Love makes us dream. When we dream, especially at night, we often look at the sky. How far is it away? How many atoms are enclosed by it? How old is it? These questions have an answer for small distances and for large distances; but for the whole of the sky or the whole of nature they cannot have one, as there is no way to be outside of the sky in order to measure it. In fact, each of the impossibilities to measure nature at smallest distances

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1482 n

c5 = 5.2 ë 1096 kg~m3 . G2ħ

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If the box has Planck size, the mass error is the Planck mass » ∆M = M Pl = ħc~G  22 µg .

Dvipsbugw (718)

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1066

Ref. 1092

xi general rel ativity versus quantum mechanics

are found again at the largest scales. There seems to be a fundamental equivalence, or, as physicists say, a duality between the largest and the smallest distances. The coming years will hopefully show how we can translate these results into an even more precise description of motion and of nature. In particular, this description should allow us to reduce the number of unexplained properties of nature. ** In summary, making love is a good physics lesson. Enjoy the rest of your day.

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bibliography

1067

Biblio graphy 1085 An attempt to explain the lack of women in physics is made in Margaret Wertheim,

1086 1087

1088

1089

Pythagoras’ Trousers – God, Physics and the Gender Wars, Fourth Estate, 1997. Cited on page 1056. The consequences of memory loss in this case are already told by Voltaire, Aventure de la mémoire, 1775. Cited on page 1058. A picture of objects in a red hot oven and at room temperature is shown in C.H. Bennett, Demons, Engines and the Second Law, Scientific American 255, pp. 108–117, November 1987. Cited on page 1058. The famous quote is found at the beginning of chapter XI, ‘The Physical Universe’, in Arthur Eddington, The Philosophy of Physical Science, Cambridge, 1939. Cited on page 1060. See the first intermezzo for details and references. Cited on page 1061.

in French as C. Schiller, Le vide diffère-t-il de la matière? in E. Gunzig & S. Diner, editeurs, Le vide – Univers du tout et du rien – Des physiciens et des philosophes s’interrogent, Les Éditions de l’Université de Bruxelles, 1998. Cited on pages 1064 and 1065. 1091 An introduction to the sense of time as result of signals in the ganglia and in the temporal lobe of the brain is found in R.B. Ivry & R. Spencer, The neural representation of time, Current Opinion in Neurobiology 14, pp. 225–232, 2004. Cited on page 1059. 1092 For details of fthe arguments leading to duality, see section 34. It also includes suggestions supporting the notion that the universe is not a even a set, thus proposing a solution for Hilbert’s sixth problem. Cited on page 1066.

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1090 Details can be found in Chapter XI of this text, a reworked version of the pages published

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maximum force and minimum distance – physics in limit statements

Fundamental limits to all observables

Special relativity in one statement The step from everyday or Galilean physics to special relativity can be summarized in a single limit statement on motion. It was popularized by Hendrik Antoon Lorentz: There is a maximum energy speed in nature. For all physical systems and all observers, vDc.

(722)

* These limits were given for the first time in 2003, in this section of the present text. ** Stimulating discussions with Saverio Pascazio, Corrado Massa and Steve Carlip helped shaping this section.

Copyright © Christoph Schiller November 1997–May 2006

At dinner parties physicists are regularly asked to summarize physics in a few sentences. It is useful to be able to present a few simple statements answering the request.** Here we propose such a set of statements. We are already familiar with each statement from what we have found out so far in our exploration of motion. But by putting them all together we will get direct insight into the results of modern research on unification. The main lesson of modern physics is the following: Nature limits the possibilities of motion. These limits are the origin of special relativity, general relativity and quantum theory. In fact, we will find out that nature poses limits to every aspect of motion; but let us put first things first.

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The principle of maximum force or power allows us to summarize special relativity, quantum theory and general relativity in one fundamental limit principle each. The three principles are fully equivalent to the standard formulation of the theories. In particular, we show that the maximum force c 4 ~4G implies and is equivalent to the field equations of general relativity. With the speed limit of special relativity and the action–angular momentum limit of quantum theory, the three fundamental principles imply a bound for every physical observable, from acceleration to size. The new, precise limit values differ from the usual Planck values by numerical prefactors of order unity.* Continuing this approach, we show that every observable in nature has both a lower and an upper limit value. As a result, a maximum force and thus a minimum length imply that the noncontinuity of space-time is an inevitable consequence of the unification of quantum theory and relativity. Furthermore, the limits are shown to imply the maximum entropy bound, including the correct numerical prefactor. The limits also imply a maximum measurement precision possible in nature, thus showing that any description by real numbers is approximate. Finally, the limits show that nature cannot be described by sets. As a result, the limits point to a solution to Hilbert’s sixth problem. The limits also show that vacuum and matter cannot be fully distinguished in detail; they are two limit cases of the same entity. These fascinating results provide the basis for any search for a unified theory of motion.

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fundamental limits to all observables

Page 282

1069

This size limit is induced by the speed of light c; it is also valid for the displacement d of a system, if the acceleration measured by an external observer is used. Finally, the limit implies an ‘indeterminacy’ relation ∆l ∆a D c 2

Challenge 1483 n

(724)

for the length and acceleration indeterminacies. You might want to take a minute to deduce it from the time-frequency indeterminacy. All this is standard knowledge.

Ref. 1093 Page 704

In the same way, the difference between Galilean physics and quantum theory can be summarized in a single statement on motion, due to Niels Bohr: There is a minimum action in nature. For all physical systems and all observers, SE

ħ . 2

(725)

* A physical system is a region of space-time containing mass–energy, the location of which can be followed over time and which interacts incoherently with its environment. With this definition, images, geometrical points or incomplete entangled situations are excluded from the definition of system.

Copyright © Christoph Schiller November 1997–May 2006

Quantum theory in one statement

Dvipsbugw

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

A few well-known remarks set the framework for the discussion that follows. The speed v is smaller than or equal to the speed of light c for all physical systems;* in particular, this limit is valid both for composed systems as well as for elementary particles. The speed limit statement is also valid for all observers; no exception to the statement is known. Indeed, only a maximum speed ensures that cause and effect can be distinguished in nature, or that sequences of observations can be defined. The opposite statement, implying the existence of (real) tachyons, has been explored and tested in great detail; it leads to numerous conflicts with observations. The maximum speed forces us to use the concept of space-time to describe nature. Maximum speed implies that space and time mix. The existence of a maximum speed in nature also implies observer-dependent time and space coordinates, length contraction, time dilation, mass–energy equivalence and all other effects that characterize special relativity. Only the existence of a maximum speed leads to the principle of maximum ageing that governs special relativity; only a maximum speed leads to the principle of least action at low speeds. In addition, only a finite speed limit makes it to define a unit of speed that is valid at all places and at all times. If a global speed limit did not exist, no natural measurement standard for speed, independent of all interactions, would exist in nature; speed would not then be a measurable quantity. Special relativity also limits the size of systems, independently of whether they are composed or elementary. Indeed, the limit speed implies that acceleration a and size l cannot be increased independently without bounds, because the two ends of a system must not interpenetrate. The most important case concerns massive systems, for which we have c2 . (723) lD a

Dvipsbugw


1070

Page 704

dE

Page 708

ħ . 2mc

(726)

In other words, (half) the (reduced) Compton wavelength of quantum theory is recovered as lower limit on the displacement of a system, whenever gravity plays no role. Since the quantum displacement limit applies in particular to an elementary system, the limit is also valid for the size of a composite system. However, the limit is not valid for the size of elementary particles. The action limit of quantum theory also implies Heisenberg’s well-known indeterminacy relation for the displacement d and momentum p of systems:

Dvipsbugw

Copyright © Christoph Schiller November 1997–May 2006

Half the Planck constant ħ is the smallest observable action or angular momentum. This statement is valid for both composite and elementary systems. The action limit is used less frequently than the speed limit. It starts from the usual definition of the action, S = ∫ (T − U)dt, and states that between two observations performed at times t and t + ∆t, even if the evolution of a system is not known, the action is at least ħ~2. Physical action is defined to be the quantity that measures the amount of change in the state of a physical system. In other words, there is always a minimum change of state taking place between two observations of a system. In this way, the quantum of action expresses the well-known fundamental fuzziness of nature at a microscopic scale. It can easily be checked that no observation results in a smaller value of action, irrespective of whether photons, electrons or macroscopic systems are observed. No exception to the statement is known. A minimum action has been observed for fermions, bosons, laser beams and matter systems and for any combination of these. The opposite statement, implying the existence of change that is arbitrary small, has been explored in detail; Einstein’s long discussion with Bohr, for example, can be seen as a repeated attempt by Einstein to find experiments that make it possible to measure arbitrarily small changes in nature. In every case, Bohr found that this aim could not be achieved. The minimum value of action can be used to deduce the indeterminacy relation, the tunnelling effect, entanglement, permutation symmetry, the appearance of probabilities in quantum theory, the information theory aspect of quantum theory and the existence of elementary particle reactions. The minimum value of action implies that, in quantum theory, the three concepts of state, measurement operation and measurement result need to be distinguished from each other; a so-called Hilbert space needs to be introduced. The minimal action is also part of the Einstein–Brillouin–Keller quantization. The details of these connections can be found in the chapter on quantum theory. Obviously, the existence of a quantum of action has been known right from the beginning of quantum theory. The quantum of action is at the basis of all descriptions of quantum theory, including the many-path formulation and the information-theoretic descriptions. The existence of a minimum quantum of action is completely equivalent to all standard developments of quantum theory. We also note that only a finite action limit makes it possible to define a unit of action. If an action limit did not exist, no natural measurement standard for action would exist in nature; action would not then be a measurable quantity. The action bound S D pd D mcd, together with the quantum of action, implies a limit on the displacement d of a system between any two observations:

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Ref. 1094

xi general rel ativity versus quantum mechanics

Dvipsbugw


fundamental limits to all observables

∆d ∆p E

1071

ħ . 2

(727)

This is valid for both massless and massive systems. All this is textbook knowledge, of course. General relativity in one statement Least known of all is the possibility of summarizing the step from Galilean physics to general relativity in a single statement on motion: There is a maximum force or power in nature. For all physical systems and all observers,

Ref. 1095 Page 349

Challenge 1484 e

Page 349

PD

c5 = 9.1 ë 1051 W . 4G

(728)

The limit statements contain both the speed of light c and the constant of gravitation G; they thus indeed qualify as statements concerning relativistic gravitation. Like the previous limit statements, they are valid for all observers. This formulation of general relativity is not common; in fact, it seems that it was only discovered 80 years after the theory of general relativity had first been proposed.* A detailed discussion is given in the chapter on general relativity. The value of the force limit is the energy of a Schwarzschild black hole divided by its diameter; here the ‘diameter’ is defined as the circumference divided by π. The power limit is realized when such a black hole is radiated away in the time that light takes to travel along a length corresponding to the diameter. Force is change of momentum; power is the change of energy. Since momentum and energy are conserved, force or power can be pictured as the flow of momentum or energy through a given surface. The value of the maximum force is the mass–energy of a black hole divided by its diameter. It is also the surface gravity of a black hole times its mass. The force limit thus means that no physical system of a given mass can be concentrated in a region of space-time smaller than a (non-rotating) black hole of that mass. In fact, the mass–energy concentration limit can be easily transformed by simple algebra into the force limit; both are equivalent. It is easily checked that the maximum force is valid for all systems observed in nature, whether they are microscopic, macroscopic or astrophysical. Neither the ‘gravitational force’ (as long as it is operationally defined) nor the electromagnetic or the nuclear interactions are ever found to exceed this limit. The next aspect to check is whether a system can be imagined that does exceed the limit. An extensive discussion shows that this is impossible, if the proper size of observers or test masses is taken into account. Even for a moving observer, when the force value is increased by the (cube of the) relativistic dilation factor, or for an accelerating observer, when the observed acceleration is increased by the acceleration of the observer itself, the force limit must still hold. However, no situations allow the limit to be exceeded, because for high accelerations a, horizons appear at distance a~c 2 ; since a mass m has a minimum * It might be that the present author was the first to point it out, in this very textbook. Also Gary Gibbons found this result independently.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1485 e

c4 = 3.0 ë 1043 N or 4G

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FD

Dvipsbugw

Dvipsbugw


1072

xi general rel ativity versus quantum mechanics

diameter given by l E 4Gm~c 2 , we are again limited by the maximum force. The exploration of the force and power limits shows that they are achieved only at horizons; the limits are not reached in any other situation. The limits are valid for all observers and all interactions. As an alternative to the maximum force and power limits, we can use as basic principle the statement: There is a maximum mass change in nature. For all systems and observers, one has dm c 3 D = 1.0 ë 1035 kg~s . (729) dt 4G

Dvipsbugw

Equivalently, we can state: There is a maximum mass per length ratio in nature. For all systems and observers, one has

Page 349 Page 349

(730)

In detail, both the force and the power limits state that the flow of momentum or of energy through any physical surface – a term defined below – of any size, for any observer, in any coordinate system, never exceeds the limit values. Indeed, as a result to the lack of nearby black holes or horizons, neither limit value is exceeded in any physical system found so far. This is the case at everyday length scales, in the microscopic world and in astrophysical systems. In addition, even Gedanken experiments do not allow to the limits to be exceeded. However, the limits become evident only when in such Gedanken experiments the size of observers or of test masses is taken into account. If this is not done, apparent exceptions can be constructed; however, they are then unphysical. Deducing general relativity*

Ref. 1096

* This section can be skipped at first reading.

Copyright © Christoph Schiller November 1997–May 2006

In order to elevate the force or power limit to a principle of nature, we have to show that, in the same way that special relativity results from the maximum speed, general relativity results from the maximum force. The maximum force and the maximum power are only realized at horizons. Horizons are regions of space-time where the curvature is so high that it limits the possibility of observation. The name ‘horizon’ is due to a certain analogy with the usual horizon of everyday life, which also limits the distance to which one can see. However, in general relativity horizons are surfaces, not lines. In fact, we can define the concept of horizon in general relativity as a region of maximum force; it is then easy to prove that it is always a two-dimensional surface, and that it has all properties usually associated with it. The connection between horizons and the maximum force or power allows a simple deduction of the field equations. We start with a simple connection. All horizons show energy flow at their location. This implies that horizons cannot be planes. An infinitely extended plane would imply an infinite energy flow. To characterize the finite extension of a given horizon, we use its radius R and its total area A. The energy flow through a horizon is characterized by an energy E and a proper length L of the energy pulse. When such an energy pulse flows perpendicularly through a hori-

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dm c 2 D = 3.4 ë 1026 kg~m . dl 4G

Dvipsbugw


fundamental limits to all observables

zon, the momentum change dp~dt = F is given by F=

E . L

1073

(731)

For a horizon, we need to insert the maximum possible values. With the horizon area A and radius R, we can rewrite the limit case as E 1 c4 = 4πR 2 4G A L

Ref. 1097

1 a2 A L . 4πG

(733)

Special relativity shows that horizons limit the product aL between proper length and acceleration to the value c 2 ~2. This leads to the central relation for the energy flow at horizons: c2 E= aA. (734) 8πG

This equation makes three points. First, the energy flowing through a horizon is limited. Second, this energy is proportional to the area of the horizon. Third, the energy flow is proportional to the surface gravity. These results are fundamental statements of general relativity. No other part of physics makes comparable statements. For the differential case the last relation can be rewritten as δE =

(735)

In this way, the result can also be used for general horizons, such as horizons that are curved or time-dependent.* In a well known paper, Jacobson has given a beautiful proof of a simple connection: if energy flow is proportional to horizon area for all observers and all horizons, then * Relation (735) is well known, though with different names for the observables. Since no communication is possible across a horizon, the detailed fate of energy flowing through a horizon is also unknown. Energy whose detailed fate is unknown is often called heat. Relation (735) therefore states that the heat flowing through a horizon is proportional to the horizon area. When quantum theory is introduced into the discussion, the area of a horizon can be called ‘entropy’ and its surface gravity can be called ‘temperature’; relation (735) can then be rewritten as δQ = T δS . (736) However, this translation of relation (735), which requires the quantum of action, is unnecessary here. We only cite it to show the relation between horizon behaviour and quantum gravity.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1098

c2 a δA . 8πG

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where the maximum force and the maximum possible area 4πR 2 of a horizon of (maximum local) radius R were introduced. The fraction E~A is the energy per area flowing through the horizon. Often, horizons are characterized by the so-called surface gravity a instead of the radius R. In the limit case, two are related by a = c 2 ~2R. This leads to E=

Dvipsbugw

(732)

Dvipsbugw


1074

xi general rel ativity versus quantum mechanics

general relativity holds. To see the connection to general relativity, we generalize relation (735) to general coordinate systems and general energy-flow directions. This is achieved by introducing tensor notation. To realize this at horizons, one introduces the general surface element dΣ and the local boost Killing vector field k that generates the horizon (with suitable norm). Jacobson uses them to rewrite the left hand side of relation (735) as δE =

∫T

ab k

a

dΣ b ,

(737)

Dvipsbugw

where Tab is the energy–momentum tensor. This rewrites the energy for arbitrary coordinate systems and arbitrary energy flow directions. Jacobson’s main result is that the essential part of the right hand side of relation (735) can be rewritten, using the (purely geometric) Raychaudhuri equation, as

∫R

ab k

a

dΣ b ,

(738)

where R ab is the Ricci tensor describing space-time curvature. Combining these two steps, we find that the energy–area relation (735) for horizons can be rewritten as c4 Tab k a dΣ b = R ab k a dΣ b . (739) 8πG

Jacobson shows that this equation, together with local conservation of energy (i.e., vanishing divergence of the energy–momentum tensor), can only be satisfied if Tab =

c4 1 ‹R ab − ( R + Λ)дab  , 8πG 2

(740)

Copyright © Christoph Schiller November 1997–May 2006

where Λ is a constant of integration whose value is not specified by the problem. These are the full field equations of general relativity, including the cosmological constant Λ. The field equations are thus shown to be valid at horizons. Since it is possible, by choosing a suitable coordinate transformation, to position a horizon at any desired space-time event, the field equations must be valid over the whole of space-time. It is possible to have a horizon at every event in space-time; therefore, at every event in nature there is the same maximum possible force (or power). This maximum force (or power) is a constant of nature. In other words, we just showed that the field equations of general relativity are a direct consequence of the limited energy flow at horizons, which in turn is due to the existence of a maximum force or power. Maximum force or power implies the field equations. One can thus speak of the maximum force principle. In turn, the field equations imply maximum force. Maximum force and general relativity are equivalent. The bounds on force and power have important consequences. In particular, they imply statements on cosmic censorship, the Penrose inequality, the hoop conjecture, the non-existence of plane gravitational waves, the lack of space-time singularities, new experimental tests of the theory, and on the elimination of competing theories of relativistic

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aδA = c 2

Dvipsbugw


fundamental limits to all observables

Ref. 1099

1075

gravitation. These consequences are presented elsewhere. Deducing universal gravitation

Challenge 1486 e

The size of physical systems in general relativity General relativity, like the other theories of modern physics, provides a limit on the size of systems: there is a limit to the amount of matter that can be concentrated into a small volume. 4Gm lE 2 . (741) c

Ref. 1100

The size limit is only achieved for black holes, those well-known systems which swallow everything that is thrown into them. It is fully equivalent to the force limit. All composite systems in nature comply with the lower size limit. Whether elementary particles fulfil or even achieve this limit remains one of the open issues of modern physics. At present, neither experiment nor theory allow clear statements on their size. More about this issue below. General relativity also implies an ‘indeterminacy relation’: ∆E c4 D . ∆l 4G

(742)

A mechanical analogy for the maximum force The maximum force is central to the theory of general relativity. That is the reason why the value of the force (adorned with a factor 2π) appears in the field equations. The importance of a maximum force becomes clearer when we return to our old image of space-time as a deformable mattress. Like any material body, a mattress is characterized by a material constant that relates the deformation values to the values of applied energy. Similarly, a mattress, like any material, is characterized by the maximum stress it can bear before

Copyright © Christoph Schiller November 1997–May 2006

Since experimental data are available only for composite systems, we cannot say yet whether this inequality also holds for elementary particles. The relation is not as popular as the previous. In fact, testing the relation, for example with binary pulsars, may lead to new tests that would distinguish general relativity from competing theories.

Dvipsbugw

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Universal gravitation can be derived from the force limit in the case where forces and values. The first condition implies º speeds are2 much smaller than the maximum 4GMa  c , the second v  c and al  c 2 . Let us apply this to a specific case. We study a satellite circling a central mass M at distance R with acceleration a. This system, with length l = 2R, has only one characteristic speed. Whenever º this speed v is much 2 smaller than c, v must be proportional both to al = 2aR and to 4GMa . If they are taken together, they imply that a = f GM~R 2 , where the numerical factor f is not yet fixed. A quick check, for example using the observed escape velocity values, shows that f = 1. Forces and speeds much smaller than the limit values thus imply that the inverse square law of gravity describes the interaction between systems. In other words, nature’s limit on force implies the universal law of gravity, as is expected.

Dvipsbugw


xi general rel ativity versus quantum mechanics

1076

breaking. Like mattresses, crystals also have these two values. In fact, for perfect crystals (without dislocations) these two material constants are the same. Empty space-time somehow behaves like a perfect crystal or a perfect mattress: it has a deformation-energy constant that at the same time is the maximum force that can be applied to it. The constant of gravitation thus determines the elasticity of space-time. Now, crystals are not homogeneous, but are made up of atoms, while mattresses are made up of foam bubbles. What is the corresponding structure of space-time? This is a central question in the rest of our adventure. One thing is sure: vacuum has no preferred directions, in complete contrast to crystals. In fact, all these analogies even suggest that the appearance of matter might be nature’s way of preventing space-time from ripping apart. We have to patient for a while, before we can judge this option. A first step towards the answer to the question appears when we put all limits together.

Dvipsbugw

Units and limit values for all physical observables

S E

ħ 2 v Dc

quantum theory on action: special relativity on speed:

F D

general relativity on force:

Challenge 1487 e

c4 . 4G

(743)

The limits are valid for all physical systems, whether composed or elementary, and are valid for all observers. We note that the limit quantities of special relativity, quantum theory and general relativity can also be seen as the right-hand sides of the respective indeterminacy relations. Indeed, the set (724, 727, 742) of indeterminacy relations or the set (723, 726, 741) of length limits is each fully equivalent to the three limit statements (743). Each set of limits can be taken as a (somewhat selective) summary of twentieth century physics. If the three fundamental limits are combined, a limit on a number of physical observables arise. The following limits are valid generally, for both composite and elementary systems:

time distance product: acceleration: angular frequency:

tE

2Għ c5 2Għ td E 4 c ¾ c7 aD 2Għ ¾ c5 ω D 2π 2Għ

= 7.6 ë 10−44 s

= 1.7 ë 10−78 sm

= 4.0 ë 1051 m~s2 = 8.2 ë 1043 ~s

(744) (745) (746) (747)

Copyright © Christoph Schiller November 1997–May 2006

time interval:

¾

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The existence of a maximum force in nature is equivalent to general relativity. As a result, physics can now be seen as making three simple statements on motion that is found in nature:

Dvipsbugw


units and limit values for all physical observables

1077

With the additional knowledge that, in nature, space and time can mix, we get distance: area: volume curvature: mass density:

Ref. 1103

2Għ c3 2Għ AE 3 c 2Għ 3~2 V E‹ 3  c c3 KD 2Għ c5 ρD 8G 2 ħ

= 2.3 ë 10−35 m

= 5.2 ë 10−70 m2

= 1.2 ë 10−104 m3 = 1.9 ë 1069 ~m2

= 6.5 ë 1095 kg~m3

(748) (749) (750)

Dvipsbugw

(751) (752)

Copyright © Christoph Schiller November 1997–May 2006

Of course, speed, action, angular momentum, power and force are also limited, as has already been stated. Except for a small numerical factor, for every physical observable these limits correspond to the Planck value. (The limit values are deduced from the commonly used Planck values simply by substituting 4G for G and ħ~2 for ħ.) These values are the true natural units of nature. In fact, the most aesthetically pleasing solution is to redefine the usual Planck values for every observable to these extremal values by absorbing the numerical factors into the respective definitions. In the following, we call the redefined limits the (corrected) Planck limits and assume that the factors have been properly included. In other words, every natural unit or (corrected) Planck unit is at the same time the limit value of the corresponding physical observable. Most of these limit statements are found throughout the literature, though the numerical factors are often different. Each limit has a string of publications attached to it. The existence of a smallest measurable distance and time interval of the order of the Planck values is discussed in quantum gravity and string theory. The largest curvature has been studied in quantum gravity; it has important consequences for the ‘beginning’ of the universe, where it excludes any infinitely large or small observable. The maximum mass density appears regularly in discussions on the energy of vacuum. With the present deduction of the limits, two results are achieved. First of all, the various arguments used in the literature are reduced to three generally accepted principles. Second, the confusion about the numerical factors is solved. During the history of Planck units, the numerical factors have varied greatly. For example, the fathers of quantum theory forgot the 1/2 in the definition of the quantum of action. Similarly, the specialists of relativity did not emphasize the factor 4. With the present framework, the issue of the correct factors in the Planck units can be considered as settled. We also note that the dimensional independence of the three limits in nature also means that quantum effects cannot be used to overcome the force limit; similarly, the power limit cannot be used to overcome the speed limit. The same is valid for any other combination of limits: they are independent and consistent at the same time.

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Ref. 1101, Ref. 1102

dE

¾

Dvipsbugw


1078

xi general rel ativity versus quantum mechanics

Limits to space and time

Ref. 1104

Ref. 1106 Ref. 1107

Ref. 1102

Mass plays a special role in all these arguments. The set of limits (743) does not make it possible to extract a limit statement on the mass of physical systems. To find one, the aim has to be restricted. The Planck limits mentioned so far apply for all physical systems, whether they are composed or elementary. Additional limits can only be found for elementary systems. In quantum theory, the distance limit is a size limit only for composed systems. A particle is

Copyright © Christoph Schiller November 1997–May 2006

Mass and energy limits

Dvipsbugw

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Ref. 1102, Ref. 1105

The three limits of nature (743) result in a minimum distance and a minimum time interval. These minimum intervals directly result from the unification of quantum theory and relativity. They do not appear if the theories are kept separate. In short, unification implies that there is a smallest length in nature. The result is important: the formulation of physics as a set of limit statements shows that the continuum description of space and time is not correct. Continuity and manifolds are only approximations; they are valid for large values of action, low speeds and low values of force. The reformulation of general relativity and quantum theory with limit statements makes this result especially clear. The result is thus a direct consequence of the unification of quantum theory and general relativity. No other assumptions are needed. In fact, the connection between minimum length and gravity is not new. In 1969, Andrei Sakharov pointed out that a minimum length implies gravity. He showed that regularizing quantum field theory on curved space with a cut-off will induce counter-terms that include to lowest order the cosmological constant and then the Einstein Hilbert action. The existence of limit values for the length observable (and all others) has numerous consequences discussed in detail elsewhere. In particular, the existence of a smallest length – and a corresponding shortest time interval – implies that no surface is physical if any part of it requires a localization in space-time to dimensions smaller that the minimum length. (In addition, a physical surface must not cross any horizon.) Only by stipulation of this condition can unphysical examples that contravene the force and power limits be eliminated. For example, this condition has been overlooked in Bousso’s earlier discussion of Bekenstein’s entropy bound – though not in his more recent ones. The corrected value of the Planck length should also be the expression that appears in the so-called theories of ‘doubly special relativity’. These then try to expand special relativity in such a way that an invariant length appears in the theory. A force limit in nature implies that no physical system can be smaller than a Schwarzschild black hole of the same mass. The force limit thus implies that point particles do not exist. So far, this prediction has not been contradicted by observations, as the predicted sizes are so small that they are outside experimental reach. If quantum theory is taken into account, this bound is sharpened. Because of the minimum length, elementary particles are now predicted to be larger than the corrected Planck length. Detecting the sizes of elementary particles would thus make it possible to check the force limit directly, for example with future electric dipole measurements.

Dvipsbugw


units and limit values for all physical observables

1079

elementary if the system size l is smaller than any conceivable dimension: for elementary particles:

lD

ħ . 2mc

(753)

Ref. 1108

These single-particle limits, corresponding to the corrected Planck mass, energy and momentum, were already discussed in 1968 by Andrei Sakharov, though again with different numerical factors. They are regularly cited in elementary particle theory. Obviously, all known measurements comply with the limits. It is also known that the unification of the electromagnetic and the two nuclear interactions takes place at an energy near, but still clearly below the maximum particle energy. Virtual particles – a new definition

Page 315 Page 759

In fact, there are physical systems that exceed all three limits. Nature does contain systems that move faster than light, that show action values below half the quantum of action and that experience forces larger than the force limit. The systems in question are called virtual particles. We know from special relativity that the virtual particles exchanged in collisions move faster than light. We know from quantum theory that virtual particle exchange implies action values below the minimum action. Virtual particles also imply an instantaneous change of momentum; they thus exceed the force limit. Virtual particles are thus those particles that exceed all the limits that hold for (real) physical systems.

Thermodynamics can also be summarized in a single statement on motion: There is a smallest entropy in nature. k SE . (755) 2

Ref. 1109

The entropy S is limited by half the Boltzmann constant k. The result is almost 100 years old; it was stated most clearly by Leo Szilard, though with a different numerical factor. In the same way as in the other fields of physics, this result can also be phrased as a

Copyright © Christoph Schiller November 1997–May 2006

Limits in thermodynamics

Dvipsbugw

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By using this new limit, valid only for elementary particles, the well-known mass, energy and momentum limits are found: ¾ ħc = 7.7 ë 10−9 kg = 0.42 ë 1019 GeV~c2 for elementary particles: m D 8G ¾ ħc 5 for elementary particles: E D = 6.9 ë 108 J = 0.42 ë 1019 GeV 8G ¾ ħc 3 for elementary particles: p D = 2.3 kg m~s = 0.42 ë 1019 GeV~c (754) 8G

Dvipsbugw


xi general rel ativity versus quantum mechanics

1080

indeterminacy relation: ∆ Ref. 1110

k 1 ∆U E . T 2

(756)

This relation was given by Bohr and discussed by Heisenberg and many others (though with k instead of k~2). It is mentioned here in order to complete the list of indeterminacy relations and fundamental constants. With the single-particle limits, the entropy limit leads to an upper limit for temperature: TD

¾

Dvipsbugw 5

ħc = 1.0 ë 1032 K . 2Gk 2

(757)

Electromagnetic limits and units The discussion of limits can be extended to include electromagnetism. Using the (lowenergy) electromagnetic coupling constant α, one gets the following limits for physical systems interacting electromagnetically: electric charge: electric field: magnetic field: voltage: inductance:

¼

(758) (759) (760) (761) (762)

With the additional assumption that in nature at most one particle can occupy one Planck volume, one gets

charge density: capacitance:

¾ πε o α c 5 c9 = e = 1.3 ë 1085 C~m3 ρe D 2G 3 ħ 8G 3 ħ 3 ¾ ¾ 2ħG 8G 2 =e = 1.0 ë 10−46 F C E 8πε 0 α 3 c c5 ħ ½

(763) (764)

Copyright © Christoph Schiller November 1997–May 2006

4πε 0 αcħ = e = 0.16 aC ¾ c7 c4 = ED = 2.4 ë 1061 V~m 64πε o αħG 2 4Ge ¾ c3 c5 = = 7.9 ë 1052 T BD 64πε 0 αħG 2 4Ge ¾ ¾ c4 ħc 5 1 UD = = 1.5 ë 1027 V 32πε 0 αG e 8G ¾ ¾ 2ħG ħ3 G 1 1 = = 4.4 ë 10−40 H LE 8πε o α c7 e2 2c 5 qE

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This corresponds to the temperature at which the energy per degree of freedom is given by the (corrected) Planck energy. A more realistic value would have to take into account the number of degrees of freedom of a particle at Planck energy. This would change the numerical factor.

Dvipsbugw


units and limit values for all physical observables

1081

For the case of a single conduction channel, one gets electric resistance: electric conductivity: electric current:

Ref. 1111

(765) (766) (767)

Many of these limits have been studied already. The magnetic field limit plays a role in the discussion of extreme stars and black holes. The maximum electric field plays a role in the theory of gamma ray bursters. The studies of limit values for current, conductivity and resistance in single channels are well known; the values and their effects have been studied extensively in the 1980s and 1990s. They will probably win a Nobel prize in the future. The observation of quarks and of collective excitations in semiconductors with charge e~3 does not necessarily invalidate the charge limit for physical systems. In neither case is there is a physical system – defined as localized mass–energy interacting incoherently with the environment – with charge e~3. Vacuum and mass–energy – two sides of the same coin

Page 996 Page 1032

Page 349

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1102, Ref. 1105

In this discussion we have found that there is a fixed limit to every physical observable. Many consequences have been discussed already in previous sections. There we saw that the existence of a limit to all observables implies that at Planck scales no physical observable can be described by real numbers and that no low-energy symmetry is valid. One conclusion is especially important for the rest of our adventure. We saw that there is a limit for the precision of length measurements in nature. The limit is valid both for the length measurements of empty space and for the length measurements of matter (or radiation). Now let us recall what we do when we measure the length of a table with a ruler. To find the ends of the table, we must be able to distinguish the table from the surrounding air. In more precise terms, we must be able to distinguish matter from vacuum. But we have no way to perform this distinction at Planck energy. In these domains, the intrinsic measurement limitations of nature do not allow us to say whether we are measuring vacuum or matter. There is no way, at Planck scales, to distinguish the two. We have explored this conclusion in detail above and have shown that it is the only consistent description at Planck scales. The limitations in length measurement precision, in mass measurement precision and in the precision of any other observable do not allow to tell whether a box at Planck scale is full or empty. You can pick any observable you want to distinguish vacuum from matter. Use colour, mass, size, charge, speed, angular momentum, or anything you want. At Planck scales, the limits to observables also lead to limits in measurement precision. At Planck scales, the measurement limits are of the same size as the observable to be measured. As a result, it is impossible to distinguish matter and vacuum at Planck scales. We put the conclusion in the sharpest terms possible: Vacuum and matter do not differ

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Ref. 1112

ħ 1 = 2 = 2.1 kΩ 8πε 0 αc 2e 2e 2 G D 8πε 0 αc = = 0.49 mS ħ ¾ ¾ 2πε 0 αc 6 c5 ID =e = 7.4 ë 1023 A G 2ħG RE

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1082

Ref. 1113

xi general rel ativity versus quantum mechanics

at Planck scales. This counter-intuitive result is one of the charms of theoretical high energy physics. This result alone inspired many researchers in the field and induced them to write best-sellers. Brian Greene was especially successful in presenting this side of quantum geometry to the wider public. However, at this point of our adventure, most issues are still open. The precise manner in which a minimum distance leads to a homogeneous and isotropic vacuum is unclear. The way to describe matter and vacuum with the same concepts has to be found. And of course, the list of open questions in physics, given above, still waits. However, the conceptual results give hope; there are interesting issues awaiting us.

Dvipsbugw

Curiosities and fun challenges about Planck limits

** The minimum angular momentum may surprise at first, especially when one thinks about particles with spin zero. However, the angular momentum of the statement is total angular momentum, including the orbital part with respect to the observer. The total angular momentum is never smaller than ħ~2. ** If any interaction is stronger than gravity, how can the maximum force be determined by gravity alone, which is the weakest interaction? It turns out that in situations near the maximum force, the other interactions are negligible. This is the reason that gravity must be included in a unified description of nature. **

Challenge 1488 ny

At first sight, it seems that electric charge can be used in such a way that the acceleration of a charged body towards a charged black hole is increased to a value exceeding the force limit. However, the changes in the horizon for charged black holes prevent this. **

Ref. 1114

where e is the positron charge, C capacity and U potential difference, and one between electric current I and time t ∆I ∆t E e (769) and both relations are found in the literature.

Copyright © Christoph Schiller November 1997–May 2006

The general connection that to every limit value in nature there is a corresponding indeterminacy relation is also valid for electricity. Indeed, there is an indeterminacy relation for capacitors of the form ∆C ∆U E e (768)

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The (corrected) Planck limits are statements about properties of nature. There is no way to measure values exceeding these limits, whatever experiment is performed. As can be expected, such a claim provokes the search for counter-examples and leads to many paradoxes.

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units and limit values for all physical observables

1083

** The gravitational attraction between two masses never yields force values sufficiently high to exceed the force limit. Why? First of all, masses m and M cannot come closer than the sum of their horizon radii. Using F = GmM~r 2 with the distance r given by the (naive) sum of the two black hole radii as r = 2G(M + m)~c 2 , one gets FD

Mm c4 , 4G (M + m)2

(770)

Dvipsbugw

which is never larger than the force limit. Even two attracting black holes thus do not exceed the force limit – in the inverse square approximation of universal gravity. The minimum size of masses does not allow to exceed the a maximum force. **

F=

GMm . ¼ M d 2 1 − 2G d c2

(771)

** An absolute power limit implies a limit on the energy that can be transported per time unit through any imaginable surface. At first sight, it may seem that the combined power emitted by two radiation sources that each emit 3/4 of the maximum value should give 3/2 times the upper value. However, the combination forms a black hole or at least prevents part of the radiation to be emitted by swallowing some of it between the sources. **

Copyright © Christoph Schiller November 1997–May 2006

The expression diverges at d = 0, the location of the horizon. However, even a test mass cannot be smaller than its own gravitational radius. If we want to reach the horizon with a realistic test mass, we need to chose a small test mass m; only a small – and thus light – mass can get near the horizon. For vanishingly small masses however, the resulting force tends to zero. Indeed, letting the distance tend to the smallest possible value by letting d = 2G(m + M)~c 2 d = 2GM~c 2 requires m 0, which makes the force F(m, d) vanish. If on the other hand, we remain away from the horizon and look for the maximum force by using a mass as large as can possibly fit into the available distance (the calculation is straightforward algebra) again the force limit is never exceeded. In other words, for realistic test masses, expression (771) is never larger than c 4 ~4G. Taking into account the minimal size of test masses thus prevents that the maximum force is exceeded in gravitational systems.

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Ref. 1115

It is well known that gravity bends space. To be fully convincing, the calculation needs to be repeated taking space curvature into account. The simplest way is to study the force generated by a black hole on a test mass hanging from a wire that is lowered towards a black hole horizon. For an unrealistic point mass, the force would diverge on the horizon. Indeed, for a point mass m lowered towards a black hole of mass M at (conventionally defined radial) distance d, the force would be

Dvipsbugw


1084

xi general rel ativity versus quantum mechanics

One possible system that actually achieves the power limit is the final stage of black hole evaporation. But even in this case the power limit is not exceeded. ** Ref. 1095

The maximum force limit states that the stress–energy tensor, when integrated over any physical surface, does not exceed the limit value. No such integral, over any physical surface whatsoever, of any tensor component in any coordinate system, can exceed the force limit, provided that it is measured by a nearby observer or a test body with a realistic proper size. The maximum force limit thus applies to any component of any force vector, as well as to its magnitude. It applies to gravitational, electromagnetic, and nuclear forces. It applies to all realistic observers. Whether the forces are real or fictitious is not important. It also plays no role whether we discuss 3-forces of Galilean physics or 4-forces of special relativity. Indeed, the force limit applied to the 0-th component of the 4-force is the power limit.

Translated to mass flows, the power limit implies that flow of water through a tube is limited in throughput. Also this limit seems unknown in the literature. ** The force limit cannot be overcome with Lorentz boosts. A Lorentz boost of any nonvanishing force value seems to allow exceeding the force limit at high speeds. However, such a transformation would create a horizon that makes any point with a potentially higher force value inaccessible. ** The power limits is of interest if applied to the universe as a whole. Indeed, it can be used to explain Olber’s paradox. The sky is dark at night because the combined luminosity of all light sources in the universe cannot be brighter than the maximum value. **

One notes that the negative energy volume density −Λc 4 ~4πG introduced by the positive cosmological constant Λ corresponds to a negative pressure (both quantities have the same dimensions). When multiplied with the minimum area it yields a force value Λħc = 4.8 ë 10−79 N . 2π

(772)

This is also the gravitational force between two corrected Planck masses located at the » cosmological distance π~4Λ . If we make the (wishful) assumption that this is the smallest possible force in nature (the numerical prefactor is not finalized yet), we get the fascinating conjecture that the full theory of general relativity, including the cosmological constant, is defined by the combination of a maximum and a minimum force in nature. Another consequence of the limits merits a dedicated section.

Copyright © Christoph Schiller November 1997–May 2006

F=

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**

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upper and lower limits to observables

1085

Upper and lower limits to observables

Page 349

Page 1068

Size and energy dependence Looking for additional limits in nature, we directly note a fundamental property. Any upper limit for angular momentum or any lower limit for power must be system dependent. Such limits will not be absolute, but will depend on properties of the system. Now, any physical system is a part of nature characterized by a boundary and its content.** The simplest properties all systems share are thus their size (characterized in the following by the diameter) L and energy E. With these characteristics we can enjoy deducing systemdependent limits for every physical observable. The general method is straightforward. We take the known inequalities for speed, action, power, charge and entropy and then extract a limit for any observable, by inserting length and energy as required. We then have to select the strictest of the limits we find. Angular momentum, action and speed It only takes a moment to note that the ratio of angular momentum D to mass times length has the dimension of speed. Since speeds are limited by the speed of light, we get

Ref. 1116

Page 800

1 LE . c

(773)

Indeed, there do not seem to be any exceptions to this limit in nature. No known system has a larger angular momentum value, from atoms to molecules, from ice skaters to galaxies. For example, the most violently rotating objects, the so-called extremal black holes, are also limited in angular momentum by D D LE~c. (In fact, this limit is correct only if the energy is taken as the irreducible mass times c 2 ; if the usual mass is used, the limit is too large by a factor 4.) One remarks that the limit deduced from general relativity, given * This section was added in June 2004. ** We mention here that quantum theory narrows down this definition as a part of nature that in addition interacts incoherently with its environment. We assume that this condition is realized in the following.

Copyright © Christoph Schiller November 1997–May 2006

DD

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In our quest to understand motion, we have focused our attention to the limitations it is subjected to. Special relativity poses a limit to speed, namely the speed of light c. General relativity limits force and power respectively by c 4~4G and c 5 ~4G, and quantum theory introduces a smallest value ħ~2 for angular momentum or action. We saw that nature poses a limit on entropy and a limit on electrical charge. Together, all these limits provide an extreme value for every physical observable. The extremum is given by the corresponding (corrected) Planck value. We have explored the list of all Planck limits in the previous section. The question may now arise whether nature provides a limit for physical observables also on the opposite end of the measurement scale. For example, there is a highest force and a highest power in nature; is there also a lowest force and a lowest power? Is there also a lowest speed? We show in the following that there indeed are such limits, for all observables. We give the general method to generate such bounds and explore several examples.* The exploration will lead us along an interesting scan across modern physics.

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1086

Challenge 1489 ny

xi general rel ativity versus quantum mechanics

by D D L 2 c 3 ~4G is not stricter than the one just given. In addition, no system-dependent lower limit for angular momentum can be deduced. The maximum angular momentum value is also interesting when it is seen as action limit. Action is the time integral of the difference between kinetic and potential energy. In fact, since nature always minimizes action W, we are not used to search for systems which maximize its value. You might check by yourself that the action limit W D LE~c is not exceeded in any physical process. Similarly, speed times mass times length is an action. Since action values in nature are limited from below by ħ~2, we get vE

Challenge 1491 e

(774)

This relation is a rewritten form of the indeterminacy relation of quantum theory and is no news. No system of energy E and diameter L has a smaller speed than this limit. Even the slowest imaginable processes show this speed value. For example, the extremely slow radius change of a black hole by evaporation just realizes this minimal speed. Continuing with the method just outlined, one also finds that the limit deduced from general relativity, v D (c 2 ~4G)(L~E), gives no new information. Therefore, no system-dependent upper speed limit exists. Incidentally, the limits are not unique. Additional limits can be found in a systematic way. Upper limits can be multiplied, for example, by factors of (L~E)(c 4 ~4G) or (LE)(2~ħc) yielding additional, but less strict upper limits. A similar rule can be given for lower limits.* With the same approach we can now systematically deduce all size and energy dependent limits for physical observables. We have a tour of the most important ones. Force, power and luminosity We saw that force and power are central to general relativity. Due to the connection W = FLT between action W, force, distance and time, we can deduce ħ 1 . 2c T 2

(775)

Experiments do not reach this limit. The smallest forces measured in nature are those in atomic force microscopes, where values as small as 1 aN are observed. However, the values are all above the lower force limit. The power P emitted by a system of size L and mass M is limited by c3

M M E P E 2ħG 3 L L

(776)

The left, upper limit gives the upper limit for any engine or lamp deduced from relativity; not even the universe exceeds it. The right, lower limit gives the minimum power emit* The strictest upper limits are thus those with the smallest possible exponent for length, and the strictest lower limits are those with the largest sensible exponent of length.

Copyright © Christoph Schiller November 1997–May 2006

FE

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Challenge 1490 ny

ħc 2 1 . 2 LE

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upper and lower limits to observables

1087

ted by any system due to quantum gravity effects. Indeed, no system is completely tight. Even black holes, the systems with the best ability in nature to keep components inside their enclosure, nevertheless radiate. The power radiated by black holes should just saturate this limit, provided the length L is taken to be the circumference of the black hole. (However, present literature values of the numerical factors in the black hole power are not yet consistent). The claim of the quantum gravity limit is thus that the power emitted by a black hole is the smallest power that is emitted by any composed system of the same surface gravity.

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Acceleration When the acceleration of a system is measured by a nearby inertial observer, the acceleration a of a system of size L and mass M is limited by

Ref. 1117

1 M E a E 4G 2 L L

The lower, right limit gives the acceleration due to universal gravity. Indeed, the relative acceleration between a system and an observer has at least this value. The left, upper limit to acceleration is the value due to special relativity. No exception to either of these limits has ever been observed. Using the limit to the size of masses, the upper limit can be transformed into the equivalent acceleration limit 2c 3 MEa ħ

Challenge 1492 ny

(777)

(778)

which has never been approached either, despite many attempts. The upper limit to acceleration is thus a quantum limit, the lower one a gravitational limit. The acceleration of the radius of a black hole due to evaporation can be much slower than the limit a E 4GM~L 2 . Why is this not a counter-example? Momentum

The momentum p of a system of size L is limited by (779)

The lower limit is obviously due to quantum theory; experiments confirmed it for all radiation and matter. The upper limit for momentum is due to general relativity. It has never been exceeded.

Copyright © Christoph Schiller November 1997–May 2006

ħ 1 c3 LEpE 4G 2 L

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c2

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1088

xi general rel ativity versus quantum mechanics

Lifetime, distance and curvature Time is something special. What are the limits to time measurements? Like before, we find that any measured time interval t in a system in thermal equilibrium is limited by 4G 2 ML 2 E t E 3 M ħ c

(780)

2c 4G ML 2 E d E 2 M ħ c

(781)

Since curvature is an inverse square distance, curvature of space-time is also limited. Mass change The mass change dM~dt of a system of size L and mass M is limited by c 5 L dM ħ 1 E E 16G 2 M dt 2 L2

(782)

The limits apply to systems in thermal equilibrium. The left, upper limit is due to general relativity; it is never exceeded. The right, lower limit is due to quantum theory. Again, all experiments are consistent with the limit values.

Limits for rest mass make only sense if the system is characterized by a size L only. We then have ħ 1 c2 LEME (783) 4G 2c L

The upper limit for mass was discussed in general relativity. Adding mass or energy to a black hole always increases its size. No system can show higher values than this value, and indeed, no such system is known or even imaginable. The lower limit on mass is obviously due to quantum theory; it follows directly from the quantum of action.

Copyright © Christoph Schiller November 1997–May 2006

Mass and density

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The lower time limit is the gravitational time. No clock can measure a smaller time than this value. Similarly, no system can produce signals shorter than this duration. Black holes, for example, emit radiation with a frequency given by this minimum value. The upper time limit is expected to be the exact lifetime of a black hole. (There is no consensus in the literature on the numerical factor yet.) The upper limit to time measurements is due to quantum theory. It leads to a question: What happens to a single atom in space after the limit time has passed by? Obviously, an atom is not a composed system comparable with a black hole. The lifetime expression assumes that decay can take place in the most tiny energy steps. As long as there is no decay mechanism, the life-time formula does not apply. The expression (780) thus does not apply to atoms. Distance limits are straightforward.

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upper and lower limits to observables

1089

The mass density a system of size L is limited by ħ 1 c2 1 EρE 4G L 2 2c L 4

Page 1086

(784)

The upper limit for mass density, due to general relativity, is only achieved for black holes. The lower limit is the smallest density of a system due to quantum theory. It also applies to the vacuum, if a piece of vacuum of site L is taken as a physical system. We note again that many equivalent (but less strict) limits can be formulated by using the transformations rules mentioned above.

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The strange charm of the entropy bound Ref. 1118

S S limit which gives SD

Ref. 1119 Page 245

A

(785)

A limit

kc 3 A, 4Għ

(786)

where A is the surface of the system. Equality is realized only for black holes. The old question of the origin of the factor 4 in the entropy of black holes is thus answered here in he following way: it is due to the factor 4 in the force or power bound in nature (provided that the factors from the Planck entropy and the Planck action cancel). The future will tell whether this explanation will stand the winds of time. Stay tuned. We can also derive a more general relation if we use a mysterious assumption that we discuss afterwards. We assume that the limits for vacuum are opposite to those for matter. We can then write c 2 ~4G D M~L for the vacuum. This gives

S S corr. Planck

SD

D

πkc 2πkc ML = MR . ħ ħ M

A

M corr. Planck A corr. Planck

L corr. Planck . L

(787)

(788)

Expression (787) is called Bekenstein’s entropy bound. Up to today, no exception has been found or constructed, despite many attempts. Again, the limit value itself is only realized for black holes. We still need to explain the strange assumption used above. We are exploring the entropy of a horizon. Horizons are not matter, but limits to empty space. The entropy of

Copyright © Christoph Schiller November 1997–May 2006

In other words, we used

Ref. 1106

D

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In 1973, Bekenstein discovered a famous limit that connects the entropy S of a physical system with its size and mass. No system has a larger entropy than one bounded by a horizon. The larger the horizon surface, the larger the entropy. We write

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1090

Ref. 1120

xi general rel ativity versus quantum mechanics

horizons is due to the large amount of virtual particles found there. In order to deduce the maximum entropy of expression (788) one therefore has to use the properties of the vacuum. In other words, either (1) we use a mass to length ratio for vacuum above the Planck limit or (2) we use the Planck entropy as maximum value for vacuum. Other, equivalent limits for entropy can be found if other variables are introduced. For example, since the ratio of shear viscosity η to the volume density of entropy (times k) has the dimensions of an action, we can directly write SD

Challenge 1494 e

Dvipsbugw (789)

Again, equality is only reached in the case of black holes. With time, the list of similar bounds will grow longer and longer. Is there also a smallest entropy limit? So far, there does not seem to be a systemdependent minimum value for entropy; the approach gives no expression that is larger than k. The entropy limit is an important step in making the description of motion consistent. If space-time can move, as general relativity maintains, it also has an entropy. How could entropy be limited if space-time is continuous? Clearly, due to the minimum distance and a minimum time in nature, space-time is not continuous, but has a finite number of degrees of freedom. The number of degrees of freedom and thus the entropy of spacetime is thus finite. In addition, the Bekenstein limit also allows some interesting speculations. Let us speculate that the universe itself, being surrounded by a horizon, saturates the Bekenstein bound. The entropy bound gives a bound to all degrees of freedom inside a system; it tells us that the number N d.o.f . of degrees of freedom of the universe is roughly N d.o.f .  10132 .

(790)

This compares with the number N Pl. vol. of Planck volumes in the universe N Pl. vol.  10183

(791)

and with the number N part. of particles in the universe (792)

In other words, particles are only a tiny fraction of what moves around. Most motion must be that of space-time. At the same time, space-time moves much less than naively expected. Finding out how all this happens is the challenge of the unified description of motion. Temperature A lower limit for the temperature of a thermal system can be found using the idea that the number of degrees of freedom of a system is limited by its surface, more precisely, by

Copyright © Christoph Schiller November 1997–May 2006

N part.  1091 .

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Challenge 1493 ny

k ηV ħ

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upper and lower limits to observables

1091

the ratio between the surface and the Planck surface. One gets the limit TE

Page 310

Challenge 1496 ny

(793)

Alternatively, using the method given above, one can use the limit on the thermal energy kT~2 E ħc~(2πL) (the thermal wavelength must be smaller than the size of the system) together with the limit on mass c 2 ~4G E M~L and deduce the same result. We know the limit already: when the system is a black hole, it gives the temperature of the emitted radiation. In other words, the temperature of black holes is the lower limit for all physical systems for which a temperature can be defined, provided they share the same boundary gravity. As a criterion, boundary gravity makes sense: boundary gravity is accessible from the outside and describes the full physical system, since it makes use both of its boundary and its content. So far, no exception to this claim is known. All systems from everyday life comply with it, as do all stars. Even the coldest known systems in the universe, namely Bose–Einstein condensates and other cold matter in various laboratories, are much hotter than the limit, and thus much hotter than black holes of the same surface gravity. (Since a consistent Lorentz transformation for temperature is not possible, as we saw earlier, the limit of minimum temperature is only valid for an observer at the same gravitational potential and at zero relative speed to the system under consideration.) There seems to be no consistent way to define an upper limit for a system-dependent temperature. However, limits for other thermodynamic quantities can be found, but are not discussed here. Electromagnetic observables When electromagnetism plays a role, the involved system also needs to be characterized by a charge Q. The method used so far then gives the following lower limit for the electric field E: M2 (794) E E 4Ge 2 2 Q L

BE

Challenge 1498 ny

4Ge M 2 c Q 2 L2

(795)

Again, this limit is satisfied all known systems in nature. Similar limits can be found for the other electromagnetic observables. In fact, several of the earlier limits are modified when electrical charge is included. Can you show how the size limit changes when electric charge is taken into account? In fact, a dedicated research field is concerned only with the deduction of the most general limits valid in

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1497 ny

We write the limitºusing the elementary charge e, though writing it using the fine structure constant via e = 4πε 0 αħc would be more appropriate. Experimentally, this limit is not exceeded in any system in nature. Can you show whether it is achieved by maximally charged black holes? For the magnetic field we get

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Challenge 1495 n

4Għ M πkc L 2

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1092

xi general rel ativity versus quantum mechanics

nature. Curiosities and fun challenges about limits to observables The limits yield a plethora of interesting paradoxes that can be discussed in lectures and student exercises. All paradoxes can be solved by carefully taking into account the combination of effects from general relativity and quantum theory. All apparent violations only appear when one of the two aspects is somehow forgotten. We study a few examples.

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**

**

Challenge 1499 r

The content of a system is not only characterized by its mass and charge, but also by its strangeness, isospin, colour charge, charge and parity. Can you deduce the limits for these quantities? **

Challenge 1500 n

In our discussion of black hole limits, we silently assumed that they interact, like any thermal system, in an incoherent way with the environment. What changes in the results of this section when this condition is dropped? Which limits can be overcome? **

Challenge 1501 e

Can you find a general method to deduce all limits? **

Copyright © Christoph Schiller November 1997–May 2006

Page 1068

Brushing some important details aside, we can take the following summary of our study of nature. Galilean physics is that description for which the difference between composed and elementary systems does not exist. Quantum theory is the description of nature with no (really large) composed systems; general relativity is the description of nature with no elementary systems. This distinction leads to the following interesting conclusion. A unified theory of nature has to unify quantum theory and general relativity. Since the first theory affirms that no (really large) composed systems exist, while the other that no elementary systems exist, a unified theory should state that no systems exist. This strange result indeed seems to be one way to look at the issue. The conclusion is corroborated by the result that in the unified description of nature, the observables time, space and mass cannot be distinguished clearly from each other, which implies that systems cannot be clearly distinguished from their surroundings. To be precise, systems thus do not really exist at unification energy.

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Several black hole limits are of importance to the universe itself. For example, the observed average mass density of the universe is not far from the corresponding limit. Also the lifetime limit is obviously valid for the universe and provides an upper limit for its age. However, the age of the universe is far from that limit by a large factor. In fact, since the universe’s size and age still increase, the age limit is pushed further into the future with every second that passes. The universe evolves in a way to escape its own decay.

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limits to measurement precision and their challenge to thought

1093

L imits to measurement precision and their challenge to thought

Challenge 1502 r

Ref. 1102

Measurement precision and the existence of sets

Copyright Š Christoph Schiller November 1997â&#x20AC;&#x201C;May 2006

Ref. 1102, Ref. 1105

The impossibility of completely eliminating measurement errors has an additional and important consequence. In physics, it is assumed that nature is a set of components. These components are assumed to be separable from each other. This tacit assumption is introduced in three main situations: it is assumed that matter consists of separable particles, that space-time consists of separable events or points, and that the set of states consists of separable initial conditions. So far, all of physics has thus built its complete description of nature on the concept of set. A fundamentally limited measurement precision implies that nature is not a set of such separable elements. A limited measurement precision implies that distinguishing physical entities is possible only approximately. The approximate distinction is only possible at energies much lower than the Planck energy. As humans we do live at such smaller energies; thus we can safely make the approximation. Indeed, the approximation is excel-

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Ref. 1102

We now know that in nature, every physical measurement has a lower and an upper bound. One of the bounds is size-dependent, the other is absolute. So far, no violation of these claims is known. The smallest relative measurement error possible in nature is thus the ratio of the two bounds. In short, a smallest length, a highest force and a smallest action, when taken together, imply that all measurements are limited in precision. A fundamental limit to measurement precision is not a new result any more. But it raises many issues. If the mass and the size of any system are themselves imprecise, can you imagine or deduce what happens then to the limit formulae given above? Due to the fundamental limits to measurement precision, the measured values of physical observables do not require the full set of real numbers. In fact, limited precision implies that no observable can be described by real numbers. We thus recover again a result that appears whenever quantum theory and gravity are brought together. In addition, we found that measurement errors increase when the characteristic measurement energy approaches the Planck energy. In that domain, the measurement errors of any observable are comparable with the measurement values themselves. Limited measurement precision thus implies that at the Planck energy it is impossible to speak about points, instants, events or dimensionality. Limited measurement precision also implies that at the Planck length it is impossible to distinguish positive and negative time values: particle and anti-particles are thus not clearly distinguished at Planck scales. A smallest length in nature thus implies that there is no way to define the exact boundaries of objects or elementary particles. However, a boundary is what separates matter from vacuum. In short, a minimum measurement error means that, at Planck scales, it is impossible to distinguish objects from vacuum with complete precision. To put it bluntly, at Planck scales, time and space do not exist. The mentioned conclusions are the same as those that are drawn by modern research on unified theories. The force limit, together with the other limits, makes it possible to reach the same conceptual results found by string theory and the various quantum gravity approaches. To show the power of the maximum force limit, we now explore a few conclusions which go beyond present approaches.

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1094

Ref. 1102

Ref. 1105

lent; we do not notice any error when performing it. But the discussion of the situation at Planck energies shows that a perfect separation is impossible in principle. In particular, at the cosmic horizon, at the big bang, and at Planck scales any precise distinction between two events or two particles becomes impossible. Another way to reach this result is the following. Separation of two entities requires different measurement results, such as different positions, different masses, different velocities, etc. Whatever observable is chosen, at the Planck energy the distinction becomes impossible, due to the large measurements errors. Only at everyday energies is a distinction approximately possible. Any distinction between two physical systems, such as between a toothpick and a mountain, is thus possible only approximately; at Planck scales, a boundary cannot be drawn. A third argument is the following. In order to count any entities in nature – a set of particles, a discrete set of points, or any other discrete set of physical observables – the entities have to be separable. The inevitable measurement errors, however, contradict separability. At the Planck energy, it is thus impossible to count physical objects with precision. Nature has no parts. In short, at Planck energies a perfect separation is impossible in principle. We cannot distinguish observations at Planck energies. In short, at Planck scale it is impossible to split nature into separate entities. There are no mathematical elements of any kind – or of any set – in nature. Elements of sets cannot be defined. As a result, in nature, neither discrete nor continuous sets can be constructed. Nature does not contain sets or elements. Since sets and elements are only approximations, the concept of ‘set’, which assumes separable elements, is already too specialized to describe nature. Nature cannot be described at Planck scales – i.e., with full precision – if any of the concepts used for its description presupposes sets. However, all concepts used in the past twenty-five centuries to describe nature – space, time, particles, phase space, Hilbert space, Fock space, particle space, loop space or moduli space – are based on sets. They all must be abandoned at Planck energy. No approach used so far in theoretical physics, not even string theory or the various quantum gravity approaches, satisfies the requirement to abandon sets. Nature is one and has no parts. Nature must be described by a mathematical concept that does not contain any set. This requirement must guide us in the future search for the unification of relativity and quantum theory.

Why are observers needed?

Certain papers on quantum theory give the impression that observers are indispensable for quantum physics. We have debunked this belief already, showing that the observer in quantum theory is mainly a bath with a definite interaction. Often, humans are observers. At Planck scales, observers also play a role. At Planck scales, quantum theory is mandatory. In these domains, observers must realize an additional requirement: they must * ‘It is almost impossible to carry the torch of truth through a crowd without scorching somebody’s beard.’

Copyright © Christoph Schiller November 1997–May 2006

Es ist fast unmöglich, die Fackel der Wahrheit durch ein Gedränge zu tragen, ohne jemandem den Bart zu sengen.* Georg Christoph Lichtenberg (1742–1799)

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Ref. 1102

xi general rel ativity versus quantum mechanics

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limits to measurement precision and their challenge to thought

1095

function at low energies. Only at low energy, an observer can introduce sets for the description of nature. Introducing observers is thus the same as introducing sets. To put it in another way, the limits of human observers is that they cannot avoid using sets. However, human observers share this limitation with video recorders, cameras, computers and pencil and paper. Nothing singles out humans in this aspect. In simple terms, observers are needed to describe nature at Planck scales only in so far as they are and use sets. We should not get too bloated about our own importance.

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A solution to Hilbert’s sixth problem Ref. 1121

Physics can be summarized in a few limit statements. They imply that in nature every physical observable is limited by a value near the Planck value. The speed limit is equivalent to special relativity, the force limit to general relativity, and the action limit to quantum theory. Even though this summary could have been made (or at least conjectured) by Planck, Einstein or the fathers of quantum theory, it is much more recent. The numerical factors for most limit values are new. The limits provoke interesting Gedanken experiments, none of which leads to violations of the limits. On the other hand, the force limit is not yet within direct experimental reach. The existence of limit values to all observables implies that the description of spacetime with a continuous manifold is not correct at Planck scales; it is only an approx-

Copyright © Christoph Schiller November 1997–May 2006

Outlook

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Ref. 1105

In the year 1900, David Hilbert gave a well-known lecture in which he listed twentythree of the great challenges facing mathematics in the twentieth century. Most problems provided challenges to many mathematicians for decades afterwards. Of the still unsolved ones, Hilbert’s sixth problem challenges mathematicians and physicists to find an axiomatic treatment of physics. The challenge has remained in the minds of many physicists since that time. Since nature does not contain sets, we can deduce that such an axiomatic description of nature does not exist! The reasoning is simple; we only have to look at the axiomatic systems found in mathematics. Axiomatic systems define mathematical structures. These structures are of three main types: algebraic systems, order systems or topological systems. Most mathematical structures – such as symmetry groups, vector spaces, manifolds or fields – are combinations of all three. But all mathematical structures contain sets. Mathematics does not provide axiomatic systems that do not contain sets. The underlying reason is that every mathematical concept contains at least one set. Furthermore, all physical concepts used so far in physics contain sets. For humans, it is difficult even simply to think without first defining a set of possibilities. However, nature is different; nature does not contain sets. Therefore, an axiomatic formulation of physics is impossible. Of course, this conclusion does not rule out unification in itself; however, it does rule out an axiomatic version of it. The result surprises, as separate axiomatic treatments of quantum theory or general relativity (see above) are possible. Indeed, only their unification, not the separate theories, must be approached without an axiomatic systems. Axiomatic systems in physics are always approximate. The requirement to abandon axiomatic systems is one of the reasons for the difficulties in reaching the unified description of nature.

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imation. For the same reason, is predicted that elementary particles are not point-like. Nature’s limits also imply the non-distinguishability of matter and vacuum. As a result, the structure of particles and of space-time remains to be clarified. So far, we can conclude that nature can be described by sets only approximately. The limit statements show that Hilbert’s sixth problem cannot be solved and that unification requires fresh approaches, taking unbeaten paths into unexplored territory. We saw that at Planck scales there is no time, no space, and there are no particles. Motion is a low energy phenomenon. Motion only appears for low-energy observers. These are observers who use sets. The (inaccurate) citation of Zeno at the beginning of our walk, stating that motion is an illusion, turns out to be correct! Therefore, we now need to find out how motion actually arises. The discussion so far hints that motion appears as soon as sets are introduced. To check this hypothesis, we need a description of nature without sets. The only way to avoid the use of sets seems a description of empty space-time, radiation and matter as being made of the same underlying entity. The inclusion of space-time dualities and of interaction dualities is most probably a necessary step. Indeed, both string theory and modern quantum gravity attempt this, though possibly not yet with the full radicalism necessary. Realizing this unification is our next task.

Copyright © Christoph Schiller November 1997–May 2006

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Biblio graphy 1093 Bohr explained the indivisibilty of the quantum of action in his famous Como lecture. N.

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B ohr, Atomtheorie und Naturbeschreibung, Springer, Berlin, 1931. More statements about the indivisibility of the quantum of action can be found in N. B ohr, Atomic Physics and Human Knowledge, Science Editions, New York, 1961. For summaries of Bohr’s ideas by others see Max Jammer, The Philosophy of Quantum Mechanics, Wiley, first edition, 1974, pp. 90–91, and John Honner, The Description of Nature – Niels Bohr and the Philosophy of Quantum Physics, Clarendon Press, 1987, p. 104. Cited on page 1069. For an overview of the quantum of action as basis of quantum theory, see the introduction to quantum theory in this textbook, available as by C. Schiller, An appetizer – quantum theory for poets and lawyers, http://www.motionmountain.net/C-5-QEDA.pdf. Cited on page 1070. The first published statements of the principle of maximum force were in the present textbook and independently in the paper by G.W. Gibbons, The maximum tension principle in general relativity, Foundations of Physics 32, pp. 1891–1901, 2002, or http://www.arxiv. org/abs/hep-th/0210109. Cited on pages 1071 and 1084. Maximal luminosity of often mentioned in studies around gravitational wave detection; nevertheless, the general maximum is never mentioned. See for example L. Ju, D.G. Bl air & C. Zhao, Detection of gravitational waves, Reports on Progress in Physics 63, pp. 1317–1427, 2000. See also C. Misner, K. Thorne & J.A. Wheeler, Gravitation, Freeman, 1973, page 980. Cited on page 1072. See for example Wolf gang R indler, Relativity – Special, General and Cosmological, Oxford University Press, 2001, p. 70 ff, or R ay d’Inverno Introducing Einstein’s Relativity, Clarendon Press, 1992, p. 36 ff. Cited on page 1073. T. Jacobson, Thermodynamics of spacetime: the Einstein equation of state, Physical Review Letters 75, pp. 1260–1263, 1995, or preprint http://www.arxiv.org/abs/gr-qc/9504004 Cited on page 1073. C. Schiller, Maximum force: a simple principle encompassing general relativity, part of this text and downloadable as http://www.motionmountain.net/C-2-MAXF.pdf. Cited on page 1075. Indeterminacy relations in general relativity are discussed in C.A. Mead, Possible connection between gravitation and fundamental length, Physical Review B 135, pp. 849–862, 1964. See also P.K. Townsend, Small-scale structure of space-time as the origin of the gravitational constant, Physical Review D 15, pp. 2795–2801, 1977, or the paper by M.T. Jaekel & S. R enaud, Gravitational quantum limit for length measurement, Physics Letters A 185, pp. 143–148, 1994. Cited on page 1075. Minimal length and minimal time intervals are discussed for example by G. Amelino– Camelia, Limits on the measurability of space-time distances in (the semiclassical approximation of) quantum gravity, Modern Physics Letters A 9, pp. 3415–3422, 1994, and by Y.J. Ng & H. Van Dam, Limit to space-time measurement, Modern Physics Letters A 9, pp. 335–340, 1994. Many other authors have explored the topic. Cited on page 1077. A pedagogical summary is C. Schiller, http://www.motionmountain.net/C10-QMGR. pdf or C. Schiller, Le vide diffère-t-il de la matière? in E. Gunzig & S. Diner, editeurs, Le vide – Univers du tout et du rien – Des physiciens et des philosophes s’interrogent, Les Editions de l’Université de Bruxelles, 1998. An older English version is also available as C. Schiller, Does matter differ from vacuum? http://www.arxiv.org/abs/gr-qc/9610066.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006

Cited on pages 1077, 1078, 1081, 1093, and 1094. 1103 Maximal curvature, as well as area and volume quantization are discussed by A. Ashtekar, Quantum geometry and gravity: recent advances, http://www.arxiv.org/ abs/gr-qc/0112038 and by A. Ashtekar, Quantum geometry in action: big bang and black holes, preprint downloadable at http://www.arxiv.org/abs/math-ph/0202008. Cited on page 1077. 1104 This was kindly pointed out by Achim Kempf. The story is told in A.D. Sakharov, General Relativity and Gravitation 32, p. 365, 2000, a reprint of a paper of 1969. Cited on page 1078. 1105 About set definition and Hilbert’s sixth problem see C. Schiller, Nature at large scales – is the universe something or nothing? http://www.motionmountain.net/C11-LGSM.pdf. Cited on pages 1078, 1081, 1093, 1094, and 1095. 1106 Several incorrect counterclaims to the entropy limit were made in R. B ousso, The holographic principle, Review of Modern Physics 74, pp. 825–874, 2002, also available as downloadable preprint at http://www.arxiv.org/abs/hep-th/0203101. Bousso has changed position in the meantime. Cited on pages 1078 and 1089. 1107 G. Amelino–Camelia, Doubly-special relativity: first results and key open problems, International Journal of Modern Physics 11, pp. 1643–1669, 2002, electronic preprint available as http://www.arxiv.org/abs/gr-qc/0210063. See also J. Magueijo & L. Smolin, Lorentz invariance with an invariant energy scale, Physical Review Letters 88, p. 190403, 2002, or http://www.arxiv.org/abs/hep-th/0112090. Cited on page 1078. 1108 Maximons, elementary particles of Planck mass, are discussed by A.D. Sakharov, Vacuum quantum fluctuations in curved space and the theory of gravitation, Soviet Physics – Doklady, 12, pp. 1040–1041, 1968. Cited on page 1079. 1109 Minimal entropy is discussed by L. Szil ard, Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen, Zeitschrift für Physik 53, pp. 840–856, 1929. This classic paper can also be found in English translation in his collected works. Cited on page 1079. 1110 See for example A.E. Shalyt-Margolin & A.Ya. Tregubovich, Generalized uncertainty relation in thermodynamics, http://www.arxiv.org/abs/gr-qc/0307018, or J. Uffink & J. van Lith-van Dis, Thermodynamic uncertainty relations, Foundations of Physics 29, pp. 655–692, 1999. Cited on page 1080. 1111 Gamma ray bursts are discussed by G. Preparata, R. Ruffini & S.-S. Xue, The dyadosphere of black holes and gamma-ray bursts, Astronomy and Astrophysics 338, pp. L87– L90, 1998, and C.L. Bianco, R. Ruffini & S.-S. Xue, The elementary spike produced by a pure e+ e− pair-electromagnetic pulse from a black hole: the PEM pulse, Astronomy and Astrophysics 368, pp. 377–390, 2001. Cited on page 1081. 1112 See for example the review in C.W.J. Beenakker & al., Quantum transport in semiconductor nanostructures, pp. 1–228, in H. Ehrenreich & D. Turnbull, eds., Solid State Physics, 44, Academic Press, Cambridge, 1991. The prediction of a future Nobel Prize was made to the authors by C. Schiller in 1992. Cited on page 1081. 1113 Brian Greene, The Elegant Universe - Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory, Vintage, 2000, Cited on page 1082. 1114 A discussion of a different electric uncertainty relation, namely between current and charge, can be found in Y-Q. Li & B. Chen, Quantum theory for mesoscopic electronic circuits and its applications, http://www.arxiv.org/abs/cond-mat/9907171. Cited on page 1082.

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1115 Wolf gang R indler, Relativity – Special, General and Cosmological, Oxford University 1116 1117

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1119 1120

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Press, 2001, p. 230. Cited on page 1083. Hans C. Ohanian & R emo Ruffini, Gravitation and Spacetime, W.W. Norton & Co., 1994. Cited on page 1085. A maximal acceleration for microscopic systems was stated by E.R. Caianiello, Lettere al Nuovo Cimento 41, p. 370, 1984. It is discussed by G. Papini, Shadows of a maximal acceleration, http://www.arxiv.org/abs/gr-qc/0211011. Cited on page 1087. The entropy limit for black holes is discussed by J.D. Bekenstein, Entropy bounds and black hole remnants, Physical Review D 49, pp. 1912–1921, 1994. See also J.D. Bekenstein, Universal upper bound on the entropy-to-energy ration for bounded systems, Physical Review D 23, pp. 287–298, 1981. Cited on page 1089. Private communications with Jos Uffink and Bernard Lavenda. Cited on page 1089. P. Kovtun, D.T. Son & A.O. Starinets, A viscosity bound conjecture, preprint found at http://www.arxiv.org/abs/hep-th/0405231. Cited on page 1090. H. Wussing & P.S. Alexandrov, (eds.) Die Hilbertschen Probleme, Akademische Verlagsgesellschaft Geest & Portig, 1983, or Ben H. Yandell, The Honours Class: Hilbert’s Problems and their Solvers, A.K. Peters, 2002. Cited on page 1095.

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the shape of points – extension in nature

37.

1101

the shape of p oints – extension in nature

Nil tam difficile est, quin quaerendo investigari possiet. Terence*

Copyright © Christoph Schiller November 1997–May 2006

* ‘Nothing is so difficult that it could not be investigated.’ Terence is Publius Terentius Afer (c. 190–159 bce ), important roman poet. He writes this in his play Heauton Timorumenos, verse 675. ** This section describes a research topic and as such is not a compendium of generally accepted results (yet). It was written between December 2001 and May 2002.

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The expressions for the Compton wavelength λ = h~mc and for the Schwarzschild radius r s = 2Gm~c 2 imply a number of arguments which lead to the conclusion that at Planck energies, space-time points and point particles must be described, in contrast to their name, by extended entities. These arguments point towards a connection between microscopic and macroscopic scales, confirming the present results of string theory and quantum gravity. At the same time, they provide a pedagogical summary of this aspect of present day theoretical physics.** 1 – It is shown that any experiment trying to measure the size or the shape of an elementary particle with high precision inevitably leads to the result that at least one dimension of the particle is of macroscopic size. 2 – There is no data showing that empty space-time is continuous, but enough data showing that it is not. It is then argued that in order to build up an entity such as the vacuum, that is extended in three dimensions, one necessarily needs extended building blocks 3 – The existence of minimum measurable distances and time intervals is shown to imply the existence of space-time duality, i.e. a symmetry between very large and very small distances. Space-time duality in turn implies that the fundamental entities that make up vacuum and matter are extended. 4 – It is argued that the constituents of the universe and thus of space-time, matter and radiation cannot form a set. As a consequence any precise description of nature must use extended entities. 5 – The Bekenstein–Hawking expression for the entropy of black holes, in particular its surface dependence, confirms that both space-time and particles are composed of extended entities. 6 – Trying to extend statistical properties to Planck scales shows that both particles and space-time points behave as braids at high energies, a fact which also requires extended entities. 7 – The Dirac construction for spin provides a model for fermions, without contradiction with experiments, that points to extended entities. An overview of other arguments in favour of extended entities provided by present research efforts is also given. To complete the discussion, experimental and theoretical checks for the idea of extended building blocks of nature are presented.

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1102 xi general rel ativity versus quantum mechanics • the shape of points

Introduction – vacuum and particles Our greatest pretenses are built up not to hide the evil and the ugly in us, but our emptyness. The hardest thing to hide is something that is not there. Eric Hoffer, The Passionate State of Mind.

“ Ref. 1122, Ref. 1123

Copyright © Christoph Schiller November 1997–May 2006

The existence of a minimal length and space interval in nature implies that points in space, time or space-time have no experimental backing and that we are forced to part from the traditional idea of continuity. Even though, properly speaking, points do not exist, and thus space points, events or point particles do not exist either, we can still ask what happens when we study these entities in detail. The results provide many fascinating surprises. Using a simple Gedanken experiment, we have found above that particles and spacetime cannot be distinguished from each other at Planck scales. The argument was the following. The largest mass that can be put in a box of size R is a black hole with a Schwarzschild radius of the same value. That is also the largest possible mass measurement error. But any piece of vacuum also has a smallest mass measurement error. The issue of a smallest mass measurement error is so important that it merits special attention. Mass measurement errors always prevent humans to state that a region of space has zero mass. In exactly the same way, also nature cannot ‘know’ that the mass of a region is zero, provided that this error is due to quantum indeterminacy. Otherwise nature would circumvent quantum theory itself. Energy and mass are always unsharp in quantum theory. We are not used to apply this fact to vacuum itself, but at Planck scales we have to. We remember from quantum field theory that the vacuum, like any other system, has a mass; of course, its value is zero for long time averages. For finite measuring times, the mass value will be uncertain and thus different from zero. Not only limitations in time, but also limitations in space lead to mass indeterminacy for the vacuum. These indeterminacies in turn lead to a minimum mass errors for vacuum regions of finite size. Quantum theory implies that nobody, not even nature, knows the exact mass value of a system or of a region of empty space. A box is empty if it does not contain anything. But emptiness is not well defined for photons with wavelength of the size R of the box or larger. Thus the mass measurement error for an ‘empty’ box – corresponding to what we call vacuum – is due to the indeterm-

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Page 1013

Only the separation of the description of nature into general relativity and quantum theory allows us to use continuous space-time and point particles as basic entities. When the two theories are united, what we use to call ‘point’ turns out to have quite counterintuitive properties. We explore a few of them in the following. Above, we have given a standard argument showing that points do not exist in nature. The Compton wavelength and the Schwarzschild radius together determine a minimal length and a minimal time interval in nature. They are given (within a factor of order one, usually omitted in the following) by the Planck length and the Planck time, with the values » l Pl = »ħG~c 3 = 1.6 ë 10−35 m (796) t Pl = ħG~c 5 = 5.4 ë 10−44 s .

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1103

inacy relation and is given by that mass whose Compton wavelength matches the size of the box. As shown earlier on, the same mass value is found by every other Gedanken experiment: trying to determine the gravitational mass by weighing the ‘piece’ of vacuum or by measuring its kinetic energy gives the same result. Another, but in the end equivalent way to show that a region of vacuum has a finite mass error is to study how the vacuum energy depends on the position indeterminacy of the border of the region. Any region is defined through its border. The position indeterminacy of the border will induce a mass error for the contents of the box, in the same way that a time limit does. Again, the resulting mass error value for a region of vacuum is the one for which the box size is the Compton wavelength. Summarizing, for a box of size R, nature allows only mass values and mass measurement error value m between two limits: (full box)

(797)

We see directly that for sizes R of the order of the Planck length, the two limits coincide; they both give the Planck mass ħ M Pl = = c l Pl

¾

ħc  10−8 kg  1019 GeV~c2 . G

(798)

In other words, for boxes of Planck size, we cannot distinguish a full box from an empty one. This means that there is no difference between vacuum and matter at Planck scales. Of course, a similar statement holds for the distinction between vacuum and radiation. At Planck scales, vacuum and particles cannot be distinguished.

where the differences are taken between the values before and after the collision. The error δM of such a measurement is simply given by δM δ∆v δm δ∆V = + + . M ∆v m ∆V

(800)

At Planck scales we have δ∆v~∆v  1, because the velocity error is always, like the velocities themselves, of the order of the speed of light. In other words, at Planck scales the mass measurement error is so large that we cannot determine whether a mass is different

Copyright © Christoph Schiller November 1997–May 2006

How else can we show that matter and vacuum cannot be distinguished? The impossibility to distinguish vacuum from particles is a strong statement. A strong statement needs additional proof. Mass can also be measured by probing its inertial aspect, i.e. by colliding the unknown mass M with known velocity V with a known probe particle of mass m and momentum p = mv. We then have ∆p , (799) M= ∆V

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ħ c2 R EmE (empty box) . G cR

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1104 xi general rel ativity versus quantum mechanics • the shape of points

from zero: vacuum is indistinguishable from matter. The same conclusion arises if we take light with a wavelength λ as the probe particle. In this case, expression (799) leads to a mass error δM δ∆λ δ∆V = + . M ∆λ ∆V

Challenge 1503 ny

In order that photon scattering can probe Planck dimensions, we need a wavelength of the order of the Planck value; but in this case the first term is approximately unity. Again we find that at Planck scales the energy indeterminacy is always of the same size as the energy value to be measured. Measurements cannot distinguish between vanishing and non-vanishing mass M at Planck scales. In fact, this result appears for all methods of mass measurement that can be imagined. At Planck scales, matter cannot be distinguished from vacuum. Incidentally, the same arguments are valid if instead of the mass of matter we measure the energy of radiation. In other words, it is also impossible to distinguish radiation from vacuum at high energies. In short, no type of particle differs from vacuum at high energies. The indistinguishability of particles and vacuum, together with the existence of minimum space-time intervals, suggest that space, time, radiation and matter are macroscopic approximations of an underlying, common and discrete structure. This structure is often called quantum geometry. How do the common constituents of the two aspects of nature look like? We will provide several arguments showing that these constituents are extended and fluctuating. Also, die Aufgabe ist nicht zu sehen, was noch nie jemand gesehen hat, sondern über dasjenige was jeder schon gesehen hat zu denken was noch nie jemand gedacht hat.* Erwin Schrödinger

Argument 1: The size and shape of elementary particles

* ‘Our task is not to see what nobody has ever seen, but to think what nobody has ever thought about that which everybody has seen already.’

Copyright © Christoph Schiller November 1997–May 2006

Size is the length of vacuum taken by an object. This definition comes natural in everyday life, quantum theory and relativity. However, approaching Planck energy, vacuum and matter cannot be distinguished: it is impossible to define the boundary between the two, and thus it is impossible to define the size of an object. As a consequence, every object becomes as extended as the vacuum. Care is therefore required. What happens if Planck energy is approached, advancing step by step to higher energy? Every measurement requires comparison with a standard. A standard is made of matter and comparison is performed using radiation (see Figure 387). Thus any measurement requires to distinguish between matter, radiation and space-time. However, the distinction between matter and radiation is possible only up to the (grand) unification energy, which is about an 800th of the Planck energy. Measurements do not allow us to prove that particles are point-like. Let us take a step back and check whether measurements allow us to say whether particles can at least be contained inside small spheres.

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Ref. 1124

(801)

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1105

object

comparison

standard F I G U R E 387 Measurement requires matter

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and radiation

The first and simplest way to determine the size of a compact particle such as a sphere or something akin to it, is to measure the size of a box it fits in. To be sure that the particle fits inside, we first of all must be sure that the box is tight. This is done by checking whether something, such as matter or radiation, can leave the box. However, nature does not provide a way to ensure that a box has no holes. Potential hills cannot get higher than the maximum energy, namely the Planck energy. The tunnelling effect cannot be ruled out. In short, there is no way to make fully tight boxes. In addition, already at the unification energy there is no way to distinguish between the box and the object enclosed in it, as all particles can be transformed from any one type into any other. Let us cross-check this result. In everyday life, we call particles ‘small’ because they can be enclosed. Enclosure is possible because in daily life walls are impenetrable. However, walls are impenetrable for matter particles only up to roughly 10 MeV and for photons only up to 10 keV. In fact, boxes do not even exist at medium energies. We thus cannot extend the idea of ‘box’ to high energies at all. In summary, we cannot state that particles are compact or of finite size using boxes. We need to try other methods. Can the Greeks help? – The limits of knives

Copyright © Christoph Schiller November 1997–May 2006

The Greeks deduced the existence of atoms by noting that division of matter must end. In contrast, whenever we think of space (or space-time) as made of points, we assume that it can be subdivided without end. Zeno noted this already long time ago and strongly criticized this assumption. He was right: at Planck energy, infinite subdivision is impossible. Any attempt to divide space stops at Planck dimensions at the latest. The process of cutting is the insertion of a wall. Knifes are limited in the same ways that walls are. The limits of walls imply limits to size determination. In particular, the limits to walls and knives imply that at Planck energies, a cut does not necessarily lead to two separate parts. One cannot state that the two parts have been really separated; a thin connection between can never be excluded. In short, cutting objects at Planck scales does not prove compactness.

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Do boxes exist?

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1106 xi general rel ativity versus quantum mechanics • the shape of points

Are cross-sections finite?

Ref. 1125

Can one take a photograph of a point?

Ref. 1126 Ref. 1122, Ref. 1127

check the greek

Pittacus.

Humans or any other types of observers can only observe a part of the world with finite resolution in time and in space. In this, humans resemble a film camera. The highest possible resolution has (almost) been discovered in 1899: the Planck time and the Planck length. No human, no film camera and no measurement apparatus can measure space or time intervals smaller than the Planck values. But what would happen if we took photographs with shutter times approaching the Planck time? Imagine that you have the world’s best shutter and that you are taking photographs at increasingly shorter times. Table 79 gives a rough overview of the possibilities. For * ‘Recognize the right moment.’ also rendered as: ‘Recognize thine opportunity.’ Pittacus (Πιττακος) of Mitylene, (c. 650–570 BCE ) was the Lesbian tyrant that was also one of the ancient seven sages.

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Καιρὸν γνῶθι.*

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Particles are not point like. Particles are not compact. Are particles are at least of finite size? To determine particle size, we can take try to determine their departure from pointlikeness. At high energy, detecting this departure requires scattering. For example, we can suspend the particle in some trap and then shoot some probe at it. What happens in a scattering experiment at high energies? The question has been studied already by Leonard Susskind and his coworkers. When shooting at the particle with a high energy probe, the scattering process is characterized by an interaction time. Extremely short interaction times imply sensitivity to the size and shape fluctuations due to the quantum of action. An extremely short interaction time provides a cut-off for high energy shape and size fluctuations and thus determines the measured size. As a result, the size measured for any microscopic, but extended object increases when the probe energy is increased towards the Planck value. In summary, even though at experimentally achievable energies the size is always smaller than measurable, when approaching the Planck energy, size increases above all bounds. As a result, at high energies we cannot give a limit to sizes of particles! In other words, since particles are not point-like at everyday energy, at Planck energies they are enormous: particles are extended. That is quite a deduction. Right at the start of our mountain ascent, we distinguished objects from their environment. Objects are by definition localized, bounded and compact. All objects have a boundary, i.e. a surface which itself does not have a boundary. Objects are also bounded in abstract ways; boundedness is also a property of the symmetries of any object, such as its gauge group. In contrast, the environment is not localized, but extended and unbounded. All these basic assumptions disappear at Planck scales. At Planck energy, it is impossible to determine whether something is bounded or compact. Compactness and localisation are only approximate properties; they are not correct at high energies. The idea of a point particle is a low energy, approximated concept. Particles at Planck scales are as extended as the vacuum. Let us perform another check of this conclusion.

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argument 1: the size and shape of elementary particles

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TA B L E 79 Effects of various camera shutter times on photographs

O b s e r va t i o n p o s s i b i l i t i e s

1h 1s 20 ms 10 ms

high high lower lower

0.25 ms

lower

1 µs c. 10 ps

very low lowest

10 fs 100 zs

higher high

shorter times

very high

10−41 s

highest

Ability to see faint quasars at night if motion is compensated Everyday motion is completely blurred Interruption by eyelids; impossibility to see small changes Effective eye/brain shutter time; impossibility to see tennis ball when hitting it Shortest commercial photographic camera shutter time; allows to photograph fast cars Ability to photograph flying bullets; requires strong flashlight Study of molecular processes; ability to photograph flying light pulses; requires laser light to get sufficient illumination Light photography becomes impossible due to wave effects X-ray photography becomes impossible; only γ-ray imaging is left over Photographs get darker as illumination gets dimmer; gravitational effects start playing a role imaging makes no sense

Copyright © Christoph Schiller November 1997–May 2006

shorter and shorter shutter times, photographs get darker and darker. Once the shutter time reaches the oscillation time of light, strange things happen: light has no chance to pass undisturbed; signal and noise become impossible to distinguish; in addition, the moving shutter will produce colour shifts. In contrast to our intuition, the picture would get blurred at extremely short shutter times. Photography is not only impossible at long but also at short shutter times. The difficulty of taking photographs is independent of the used wavelength. The limits move, but do not disappear. A short shutter time τ does not allow photons of energy lower than ħ~τ to pass undisturbed. The blur is small when shutter times are those of everyday life, but increases when shutter times are shortened towards Planck times. As a result, there is no way to detect or confirm the existence of point objects by taking pictures. Points in space, as well as instants of time, are imagined concepts; they do not allow a precise description of nature. At Planck shutter times, only signals with Planck energy can pass through the shutter. Since at these energies matter cannot be distinguished from radiation or from empty space, all objects, light and vacuum look the same. As a result, it becomes impossible to say how nature looks at shortest times. But the situation is much worse: a Planck shutter does not exist at all, as it would need to be as small as a Planck length. A camera using it could not be built, as lenses do not work at this energy. Not even a camera obscura – without any lens – would work, as diffraction effects would make image production impossible. In other words, the idea that at short shutter times, a photograph of nature shows a frozen version of everyday life, like a stopped film, is completely wrong! Zeno criticized this image already in ancient Greece, in his discussions about motion, though not so clearly as we can do now. Indeed,

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D u r at i o n B lu r

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1108 xi general rel ativity versus quantum mechanics • the shape of points

at a single instant of time nature is not frozen at all.* At short times, nature is blurred and fuzzy. This is also the case for point particles. In summary, whatever the intrinsic shape of what we call a ‘point’ might be, we know that being always blurred, it is first of all a cloud. Whatever method to photograph of a point particle is used, it always shows an extended entity. Let us study its shape in more detail. What is the shape of an electron?

Copyright © Christoph Schiller November 1997–May 2006

* In fact, a shutter does not exist even at medium energy, as shutters, like walls, stop existing at around 10 MeV.

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Ref. 1122

Given that particles are not point-like, they have a shape. How can we determine it? Everyday object shape determination is performed by touching the object from all sides. This works with plants, people or machines. It works with molecules, such as water molecules. We can put them (almost) at rest, e.g. in ice, and then scatter small particles off them. Scattering is just a higher energy version of touching. However, scattering cannot determine shapes smaller than the wavelength of the used probes. To determine a size as small as that of an electron, we need the highest energies available. But we already know what happens when approaching Planck scales: the shape of a particle becomes the shape of all the space surrounding it. Shape cannot be determined in this way. Another method to determine the shape is to build a tight box filled of wax around the system under investigation. We let the wax cool and and observe the hollow part. However, near Planck energies boxes do not exist. We are unable to determine the shape in this way. A third way to measure shapes is cutting something into pieces and then study the pieces. But cutting is just a low-energy version of a scattering process. It does not work at high energies. Since the term ‘atom’ means ‘uncuttable’ or ‘indivisible’, we have just found out that neither atoms nor indivisible particles can exist. Indeed, there is no way to prove this property. Our everyday intuition leads us completely astray at Planck energies. A fourth way to measure shapes could appear by distinguishing transverse and longitudinal shape, with respect to the direction of motion. However, for transverse shape we get the same issues as for scattering; transverse shape diverges for high energy. To determine longitudinal shape, we need at least two infinitely high potential walls. Again, we already know that this is impossible. A further, indirect way of measuring shapes is the measurement of the moment of inertia. A finite moment of inertia means a compact, finite shape. However, when the measurement energy is increased, rotation, linear motion and exchange become mixed up. We do not get meaningful results. Still another way to determine shapes is to measure the entropy of a collection of particles we want to study. This allows to determine the dimensionality and the number of internal degrees of freedom. At high energies, a collection of electrons would become a black hole. We study the issue separately below, but again we find no new information. Are these arguments water-tight? We assumed three dimensions at all scales, and assumed that the shape of the particle itself is fixed. Maybe these assumptions are not valid at Planck scales. Let us check the alternatives. We have already shown above that due

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argument 1: the size and shape of elementary particles

Ref. 1122

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to the fundamental measurement limits, the dimensionality of space-time cannot be determined at Planck scales. Even if we could build perfect three-dimensional boxes, holes could remain in other dimensions. It does not take long to see that all the arguments against compactness work even if space-time has additional dimensions. Is the shape of an electron fixed?

Ref. 1129

* Examples are the neutron, positronium, or the atoms. Note that the argument does not change when the elementary particle itself is unstable, such as the muon. Note also that the possibility that all components be heavier than the composite, which would avoid this argument, does not seem to lead to satisfying physical properties; e.g. it leads to intrinsically unstable composites. ** Thus at Planck scales there is no quantum Zeno effect any more.

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Ref. 1128

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Page 761

Only an object composed of localized entities, such as a house or a molecule, can have a fixed shape. The smaller a system gets, the more quantum fluctuations play a role. Any entity with a finite size, thus also an elementary particle, cannot have a fixed shape. Every Gedanken experiment leading to finite shape also implies that the shape itself fluctuates. But we can say more. The distinction between particles and environment resides in the idea that particles have intrinsic properties. In fact, all intrinsic properties, such as spin, mass, charge and parity, are localized. But we saw that no intrinsic property is measurable or definable at Planck scales. It is impossible to distinguish particles from the environment, as we know already. In addition, at Planck energy particles have all properties the environment also has. In particular, particles are extended. In short, we cannot prove by experiments that at Planck energy elementary particles are finite in size in all directions. In fact, all experiments one can think of are compatible with extended particles, with ‘infinite’ size. We can also say that particles have tails. More precisely, a particle always reaches the borders of the region of space-time under exploration. Not only are particles extended; in addition, their shape cannot be determined by the methods just explored. The only possibility left over is also suggested by quantum theory: The shape of particles is fluctuating. We note that for radiation particles we reach the same conclusions. The box argument shows that also radiation particles are extended and fluctuating. In our enthusiasm we have also settled an important detail about elementary particles. We saw above that any particle which is smaller than its own Compton wavelength must be elementary. If it were composite, there would be a lighter component inside it; this lighter particle would have a larger Compton wavelength than the composite particle. This is impossible, since the size of a composite particle must be larger than the Compton wavelength of its components.* However, an elementary particle can have constituents, provided that they are not compact. The difficulties of compact constituents were already described by Sakharov in the 1960s. But if the constituents are extended, they do not fall under the argument, as extended entities have no localized mass. As a result, a flying arrow, Zeno’s famous example, cannot be said to be at a given position at a given time, if it is made of extended entities. Shortening the observation time towards the Planck time makes an arrow disappear in the same cloud that also makes up space-time.**

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1110 xi general rel ativity versus quantum mechanics • the shape of points

Ref. 1122

Challenge 1504 e

In summary, only the idea of points leads to problems at Planck scales. If space-time and matter are imagined to be made, at Planck scales, of extended and fluctuating entities, all problems disappear. We note directly that for extended entities the requirement of a non-local description is realized. Similarly, the entities being fluctuating, the requirement of a statistical description of vacuum is realized. Finally, the argument forbidding composition of elementary particles is circumvented, as extended entities have no clearly defined mass. Thus the concept of Compton wavelength cannot be defined or applied. Elementary particles can thus have constituents if they are extended. But if the components are extended, how can compact ‘point’ particles be formed with them? A few options will be studied shortly.

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Argument 2: The shape of points in vacuum

Ref. 1130

Ref. 1131

* Imagining the vacuum as a collection of entities with Planck size in all directions, such as spheres, would

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1122

We are used to think that empty space is made of spatial points. Let us check whether this is true at high energy. At Planck scales no measurement can give zero length, zero mass, zero area or zero volume. There is no way to state that something in nature is a point without contradicting experimental results. In addition, the idea of a point is an extrapolation of what is found in small empty boxes getting smaller and smaller. However, we just saw that at high energies small boxes cannot be said to be empty. In fact, boxes do not exist at all, as they are never tight and do not have impenetrable walls at high energies. Also the idea of a point as a continuous subdivision of empty space is untenable. At small distances, space cannot be subdivided, as division requires some sort of dividing wall, which does not exist. Even the idea of repeatedly putting a point between two others cannot be applied. At high energy, it is impossible to say whether a point is exactly on the line connecting the outer two points; and near Planck energy, there is no way to find a point between them at all. In fact, the term ‘in between’ makes no sense at Planck scales. We thus find that space points do not exist, in the same way that point particles do not exist. But there is more; space cannot be made of points for additional reasons. Common sense tells us that points need to be kept apart somehow, in order to form space. Indeed, mathematicians have a strong argument stating why physical space cannot be made of mathematical points: the properties of mathematical spaces described by the Banach– Tarski paradox are quite different from that of the physical vacuum. The Banach–Tarski paradox states states that a sphere made of mathematical points can be cut into 5 pieces which can be reassembled into two spheres each of the same volume as the original sphere. Mathematically, volume makes no sense. Physically speaking, we can say that the concept of volume does not exist for continuous space; it is only definable if an intrinsic length exists. This is the case for matter; it must also be the case for vacuum. But any concept with an intrinsic length, also the vacuum, must be described by one or several extended components.* In summary, we need extended entities to build up space-time!

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Thus, since there is an impossibility that [finite] quantities are built from contacts and points, it is necessary that there be indivisible material elements and [finite] quantities. Aristotle, Of Generation and Corruption.

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argument 2: the shape of points in vacuum

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Measuring the void

What is the maximum number of particles that fits inside a piece of vacuum? The maximum mass that fits into a piece of vacuum is a black hole. But also in this case, the maximum mass depends only on the surface of the given vacuum piece. The maximum

Challenge 1505 n

avoid the Banach–Tarski paradox, but would not allow to deduce the numbers of dimensions of space and time. It would also contradict all other results of this section. We therefore do not explore it further.

Copyright © Christoph Schiller November 1997–May 2006

To check whether space-time constituents are extended, let us perform a few additional Gedanken experiments. First, let us measure the size of a point of space-time. The clearest definition of size is through the cross-section. How can we determine the cross-section of a point? We can determine the cross section of a piece of vacuum and determine the number of points inside it. From the two determinations we can deduce the cross-section of a single point. At Planck energies however, we get a simple result: the cross-section of a volume of empty space is depth independent. At Planck energies, vacuum has a surface, but no depth. In other words, at Planck energy we can only state that a Planck layer covers the surface of a volume. We cannot say anything about its interior. One way to picture the result is to say that space points are long tubes. Another way to determine the size of a point is to count the points found in a given volume of space-time. One approach is to count the possible positions of a point particle in a volume. However, point particles are extended at Planck energies and indistinguishable from vacuum. At Planck energy, the number of points is given by surface area of the volume divided by the Planck area. Again, the surface dependence suggests that particles are long tubes. Another approach to count the number of points in a volume is to fill a piece of vacuum with point particles.

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Not only is it impossible to generate a volume with mathematical points; it is also impossible to generate exactly three physical dimensions with mathematical points. Mathematics shows that any compact one-dimensional set has as many points as any compact three-dimensional set. And the same is true for any other pair of dimension values. To build up the physical three-dimensional vacuum we need entities which organize their neighbourhood. This cannot be done with purely mathematical points. The fundamental entities must possess some sort of bond forming ability. Bonds are needed to construct or fill three dimensions instead of any other number. Bonds require extended entities. But also a collection of tangled entities extending to the maximum scale of the region under consideration would work perfectly. Of course the precise shape of the fundamental entities is not known at this point in time. In any case we again find that any constituents of physical three-dimensional space must be extended. In summary, we need extension to define dimensionality and to define volume. We are not surprised. Above we deduced that the constituents of particles are extended. Since vacuum is not distinguishable from matter, we expect the constituents of vacuum to be extended as well. Stated simply, if elementary particles are not point-like, then space-time points cannot be either.

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Page 223 Ref. 1132

mass increases less rapidly than the volume. In other words, the number of points in a volume is only proportional to the surface area of that volume. There is only one solution: vacuum must be made of extended entities crossing the whole volume, independently of the shape of the volume. Two thousand years ago, the Greek argued that matter must be made of particles because salt can be dissolved in water and because fish can swim through water. Now that we know more about Planck scales, we have to reconsider the argument. Like fish through water, particles can move through vacuum; but since vacuum has no bounds and since it cannot be distinguished from matter, vacuum cannot be made of particles. However, there is another possibility that allows for motion of particles through vacuum: both vacuum and particles can be made of a web of extended entities. Let us study this option in more detail.

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Argument 3: The l arge, the small and their connection

* There is also an S-duality, which connects large and small coupling constants, and a U-duality, which is the combination of S- and T-duality.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1135

If two observables cannot be distinguished, there is a symmetry transformation connecting them. For example, when switching observation frame, an electric field may change into a magnetic one. A symmetry transformation means that we can change the viewpoint (i.e. the frame of observation) with the consequence that the same observation is described by one quantity from one viewpoint and by the other quantity from the other viewpoint. When measuring a length at Planck scales it is impossible to say whether we are measuring the length of a piece of vacuum, the Compton wavelength of a body, or the Schwarzschild diameter of a body. For example, the maximum size for an elementary object is its Compton wavelength. The minimum size for an elementary object is its Schwarzschild radius. The actual size of an elementary object is somewhere in between. If we want to measure the size precisely, we have to go to Planck energy: but then all these quantities are the same. In other words, at Planck scales, there is a symmetry transformation between Compton wavelength and Schwarzschild radius. In short, at Planck scales there is a symmetry between mass and inverse mass. As a further consequence, at Planck scales there is a symmetry between size and inverse size. Matter–vacuum indistinguishability means that there is a symmetry between length and inverse length at Planck energies. This symmetry is called space-time duality or T-duality in the literature of superstrings.* Space-time duality is a symmetry between situations at scale n l Pl and at scale f l Pl ~n, or, in other words, between R and ( f l Pl )2 ~R, where the experimental number f has a value somewhere between 1 and 1000. Duality is a genuine non-perturbative effect. It does not exist at low energy, since dual2 ity automatically also relates energy E and energy E Pl ~E = ħc 3 ~GE, i.e. it relates energies below and above Planck scale. Duality is a quantum symmetry. It does not exist in everyday life, as Planck’s constant appears in its definition. In addition, it is a general relativistic

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I could be bounded in a nutshell and count myself a king of infinite space, were it not that I have bad dreams. William Shakespeare, Hamlet.

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effect, as it includes the gravitational constant and the speed of light. Let us study duality in more detail. Small is large?

“ Ref. 1133

[Zeno of Elea maintained:] If the existing are many, it is necessary that they are at the same time small and large, so small to have no size, and so large to be without limits. Simplicius, Commentary on the Physics of Aristotle, 140, 34.

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Unification and total symmetry

* A symmetry between size and Schwarzschild radius, i.e. a symmetry between length and mass, will lead to general relativity. Additionally, at Planck energy there is a symmetry between size and Compton wavelength. In other words, there is a symmetry between length and 1/mass. It means that there is a symmetry between coordinates and wave functions. Note that this is a symmetry between states and observables. It leads to quantum theory.

Copyright © Christoph Schiller November 1997–May 2006

So far, we have shown that at Planck energy, time and length cannot be distinguished. Duality has shown that mass and inverse mass cannot be distinguished. As a consequence, length, time and mass cannot be distinguished from each other. Since every observable is a combination of length, mass and time, space-time duality means that there is a symmetry between all observables. We call it the total symmetry.*

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To explore the consequences of duality, we can compare it to the 2π rotation symmetry in everyday life. Every object in daily life is symmetrical under a full rotation. For the rotation of an observer, angles make sense only as long as they are smaller than 2π. If a rotating observer would insist on distinguishing angles of 0, 2π, 4π etc., he would get a new copy of the universe at each full turn. 2 Similarly, in nature, scales R and l Pl ~R cannot be distinguished. Lengths make no sense when they are smaller than l Pl . If however, we insist on using even smaller values and insist on distinguishing them from large ones, we get a new copy of the universe at those small scales. Such an insistence is part of the standard continuum description of motion, where it is assumed that space and time are described by the real numbers, which are defined for arbitrary small intervals. Whenever the (approximate) continuum description 2 with infinite extension is used, the R  l Pl ~R symmetry pops up. Duality implies that diffeomorphism invariance is only valid at medium scales, not at extremal ones. At extremal scales, quantum theory has to be taken into account in the proper manner. We do not know yet how to do this. Space-time duality means that introducing lengths smaller than the Planck length (like when one defines space points, which have size zero) means at the same time introducing things with very large (‘infinite’) value. Space-time duality means that for every small enough sphere the inside equals outside. Duality means that if a system has a small dimension, it also has a large one. And vice versa. There are thus no small objects in nature. As a result, space-time duality is consistent with the idea that the basic entities are extended.

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Page 556 Ref. 1134

Ref. 1135

* Renormalization energy does connect different energies, but not in the correct way; in particular, it does not include duality.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1507 e

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Challenge 1506 d

Total symmetry implies that there are many types of specific dualities, one for each pair of quantities under investigation. Indeed, in string theory, the number of duality types discovered is increasing every year. It includes, among others, the famous electric–magnetic duality we first encountered in the chapter on electrodynamics, coupling constant duality, surface–volume duality, space-time duality and many more. All this confirms that there is an enormous symmetry at Planck scales. Similar statements are also well-known right from the beginning of string theory. Most importantly, total symmetry implies that gravity can be seen as equivalent to all other forces. Space-time duality shows that unification is possible. Physicist have always dreamt about unification. Duality tells us that this dream can indeed be realized. It may seem that total symmetry is in complete contrast with what was said in the previous section, where we argued that all symmetries are lost at Planck scales. Which result is correct? Obviously, both of them are. At Planck scales, all low energy symmetries are indeed lost. In fact, all symmetries that imply a fixed energy are lost. Duality and its generalizations however, combine both small and large dimensions, or large and small energies. Most symmetries of usual physics, such as gauge, permutation and space-time symmetries, are valid at each fixed energy separately. But nature is not made this way. The precise description of nature requires to take into consideration large and small energies at the same time. In everyday life, we do not do that. Everyday life is a low and fixed energy approximation of nature. For most of the twentieth century, physicists aimed to reach higher and higher energies. We believed that precision increases with increasing energy. But when we combine quantum theory and gravity we are forced to change this approach; to achieve high precision, we must take both high and low energy into account at the same time.* The large differences in phenomena at low and high energies are the main reason why unification is so difficult. So far, we were used to divide nature along the energy scale. We thought about high energy physics, atomic physics, chemistry, biology, etc. The differences between these sciences is the energy of the processes involved. But now we are not allowed to think in this way any more. We have to take all energies into account at the same time. That is not easy, but we do not have to despair. Important conceptual progress has been achieved in the last decade of the twentieth century. In particular, we now know that we need only one single concept for all things which can be measured. Since there is only one concept, there are many ways to study it. We can start from any (low-energy) concept in physics and explore how it looks and behaves when we approach Planck scales. In the present section, we are looking at the concept of point. Obviously, the conclusions must be the same, independently of the concept we start with, be it electric field, spin, or any other. Such studies thus provide a check for the results in this section. Unification thus implies to think using duality and using concepts which follow from it. In particular, we need to understand what exactly happens to duality when we restrict ourselves to low energy only, as we do in everyday life. This question is left for the next section.

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Argument 4: D oes nature have parts?

Ref. 1136

Another argument, independent of the ones given above, underlines the correctness of a model of nature made of extended entities. Let us take a little broader view. Any concept for which we can distinguish parts is described by a set. We usually describe nature as a set of objects, positions, instants, etc. The most famous set description of nature is the oldest known, given by Democritus: ‘The world is made of indivisible particles and void.’ This description was extremely successful in the past; there were no discrepancies with observations yet. However, after 2500 years, the conceptual difficulties of this approach are obvious. We now know that Democritus was wrong, first of all, because vacuum and matter cannot be distinguished at Planck scales. Thus the word ‘and’ in his sentence is already mistaken. Secondly, due to the existence of minimal scales, the void cannot be made of ‘points,’ as we usually assume nowadays. Thirdly, the description fails because particles are not compact objects. Fourth, the total symmetry implies that we cannot distinguish parts in nature; nothing can really be distinguished from anything else with complete precision, and thus the particles or points in space making up the naive model of void cannot exist. In summary, quantum theory and general relativity together show that in nature, all differences are only approximate. Nothing can really be distinguished from anything else with complete precision. In other words, there is no way to define a ‘part’ of nature, neither for matter, nor for space, nor for time, nor for radiation. Nature cannot be a set. The conclusion that nature is not a set does not come as a surprise. We have already encountered another reason to doubt that nature is a set. Whatever definition we use for the term ‘particle’, Democritus cannot be correct for a purely logical reason. The description he provided is not complete. Every description of nature defining nature as a set of parts necessarily misses certain aspects. Most importantly, it misses the number of these parts. In particular, the number of particles and the number of dimensions of space-time must be specified if we describe nature as made from particles and vacuum. Above we saw that it is rather dangerous to make fun of the famous statement by Arthur Eddington

In fact, practically all physicists share this belief; usually they either pretend to favour some other number, or worse, they keep the number unspecified. We have seen during * ‘Multitude should not be introduced without necessity.’ This famous principle is commonly called Occam’s razor. William of Ockham (b. 1285/1295 Ockham, d. 1349/50 München), or Occam in the common Latin spelling, was one of the great thinkers of his time. In his famous statement he expresses that only those concepts which are strictly necessary should be introduced to explain observations. It can be seen as the requirement to abandon beliefs when talking about nature. In addition, at this stage of our mountain ascent it has an even more direct interpretation.

Copyright © Christoph Schiller November 1997–May 2006

I believe there are 15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,914,527, 116,709,366,231,425,076,185,631,031,296 protons in the universe and the same number of electrons.

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Page 681

Pluralitas non est ponenda sine necessitate.* William of Occam

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1116 xi general rel ativity versus quantum mechanics • the shape of points

* As a curiosity, practically the same discussion can already be found, in Plato’s Parmenides, written in the fourth century bce. There, Plato musically ponders different arguments on whether nature is or can be a unity or a multiplicity, i.e. a set. It seems that the text is based on the real visit by Parmenides and Zeno in Athens, where they had arrived from their home city Elea, which lies near Naples. Plato does not reach a conclusion. Modern physics however, does.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1137

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Ref. 1138

our walk that in modern physics many specialized sets are used to describe nature. We have used vector spaces, linear spaces, topological spaces and Hilbert spaces. But very consistently we refrained, like all physicists, from asking about the origin of their sizes (mathematically speaking of their dimensionality or their cardinality). In fact, it is equally unsatisfying to say that the universe contains some specific number of atoms as it is to say that space-time is made of point-like events arranged in 3+1 dimensions. Both statements are about set sizes in the widest sense. In a complete, i.e. in a unified description of nature the number of smallest particles and the number of space-time points must not be added to the description, but must result from the description. Only in this case is unification achieved. Requiring a complete explanation of nature leads to a simple consequence. Any part of nature is by definition smaller than the whole of nature and different from other parts. As a result, any description of nature by a set cannot possibly yield the number of particles nor space-time dimensionality. As long as we insist in using space-time or Hilbert spaces for the description of nature, we cannot understand the number of dimensions or the number of particles. Well, that is not too bad, as we know already that nature is not made of parts. We know that parts are only approximate concepts. In short, if nature were made of parts, it could not be a unity, or a ‘one.’ If however, nature is a unity, a one, it cannot have parts.* Nature cannot be separable exactly. It cannot be made of particles. To sum up, nature cannot be a set. Sets are lists of distinguishable elements. When general relativity and quantum theory are unified, nature shows no elements: nature stops being a set at Planck scales. The result confirms and clarifies a discussion we have started in classical physics. There we had discovered that matter objects were defined using space and time, and that space and time were defined using objects. Including the results of quantum theory, this implies that in modern physics particles are defined with the help of the vacuum and the vacuum with particles. That is not a good idea. We have just seen that since the two concepts are not distinguishable from each other, we cannot define them with each other. Everything is the same; in fact, there is no ‘every’ and no ‘thing.’ Since nature is not a set, the circular reasoning is dissolved. Space-time duality also implies that space is not a set. Duality implies that events cannot be distinguished from each other. They thus do not form elements of some space. Phil Gibbs has given the name event symmetry to this property of nature. This thoughtprovoking term, even though still containing the term ‘event’, underlines that it is impossible to use a set to describe space-time. In summary, nature cannot be made of vacuum and particles. That is bizarre. People propagating this idea have been persecuted for 2000 years. This happened to the atomists from Democritus to Galileo. Were their battles it all in vain? Let us continue to clarify our thoughts.

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argument 4: d oes nature have parts?

1117

Does the universe contain anything? Stating that the universe contains something implies that we are able to distinguish the universe from its contents. However, we now know that precise distinctions are impossible. If nature is not made of parts, it is wrong to say that the universe contains something. Let us go further. As nothing can be distinguished, we need a description of nature which allows to state that at Planck energies nothing can be distinguished from anything else. For example, it must be impossible to distinguish particles from each other or from the vacuum. There is only one solution: everything, or at least what we call ‘everything’ in everyday life, must be made of the same single entity. All particles are made of one same ‘piece.’ Every point in space, every event, every particle and every instant of time must be made of the same single entity.

Dvipsbugw

An amoeba

“ ”

Page 651

Copyright © Christoph Schiller November 1997–May 2006

We found that parts are approximate concepts. The parts of nature are not strictly smaller than nature itself. As a result, any ‘part’ must be extended. Let us try to extract more information about the constituents of nature. The search for a unified theory is the search for a description in which all concepts appearing are only approximately parts of the whole. Thus we need an entity Ω, describing nature, which is not a set but which can be approximated by one. This is unusual. We all are convinced very early in our life that we are a part of nature. Our senses provide us with this information. We are not used to think otherwise. But now we have to. Let us eliminate straight away a few options for Ω. One concept without parts is the empty set. Perhaps we need to construct a description of nature from the empty set? We could be inspired by the usual construction of the natural numbers from the empty set. However, the empty set makes only sense as the opposite of some full set. That is not the case here. The empty set is not a candidate for Ω. Another possibility to define approximate parts is to construct them from multiple copies of Ω. But in this way we would introduce a new set through the back door. In addition, new concepts defined in this way would not be approximate. We need to be more imaginative. How can we describe a whole which has no parts, but which has parts approximately? Let us recapitulate. The world must be described by a single entity, sharing all properties of the world, but which can be approximated into a set of parts. For example, the approximation should yield a set of space points and a set of particles. We also saw that whenever we look at any ‘part’ of nature without any approximation, we should not be able to distinguish it from the whole world. In other words, composed entities are not always larger than constituents. On the other hand, composed entities must usually appear to be larger than their constituents. For example, space ‘points’ or ‘point’ particles are tiny, even though they are only approximations. Which concept without boundaries can be at their origin? Using usual concepts the world is everywhere at the same time; if nature is to be described by a single constituent, this entity must be

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A theory of nothing describing everything is better than a theory of everything describing nothing.

Dvipsbugw


1118 xi general rel ativity versus quantum mechanics • the shape of points

Argument 5: The entropy of bl ack holes

* Is this the only method to describe nature? Is it possible to find another description, in particular if space and time are not used as background? The answers are unclear at present.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1508 ny

We are still collecting arguments to determine particle shape. For a completely different way to explore the shape of particles it is useful to study situations where they appear in large numbers. Collections of high numbers of constituents behave differently if they are point-like or extended. In particular, their entropy is different. Studying large-number entropy thus allows to determine component shape. The best approach is to study situations in which large numbers of particles are crammed in a small volume. This leads to study the entropy of black holes. A black hole is a body whose gravity is so strong that even light cannot escape. Black holes tell us a lot about the fundamental entities of nature. It is easily deduced from general relativity that any body whose mass m fits inside the so-called Schwarzschild radius r S = 2Gm~c 2 (802)

Dvipsbugw

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extended. The entity has to be a single one, but it must seem to be multiple, i.e. it has to be multiple approximately, as nature shows multiple aspects. The entity must be something folded. It must be possible to count the folds, but only approximately. (An analogy is the question of how many tracks are found on an LP or a CD; depending on the point of view, local or global, one gets different answers.) Counting folds would correspond to a length measurement. The simplest model would be the use of a single entity which is extended, fluctuating, going to infinity and allowing approximate localization, thus allowing approximate definition of parts and points.* In more vivid imagery, nature could be described by some deformable, folded and tangled up entity: a giant, knotted amoeba. An amoeba slides between the fingers whenever one tries to grab a part of it. A perfect amoeba flows around any knife trying to cut it. The only way to hold it would be to grab it in its entirety. However, for an actor himself made of amoeba strands this is impossible. He can only grab it approximately, by catching part of it and approximately blocking it, e.g. using a small hole so that the escape takes a long time. To sum up, nature is modelled by an entity which is a single unity (to eliminate distinguishability), extended (to eliminate localizability) and fluctuating (to ensure approximate continuity). A far-reaching, fluctuating fold, like an amoeba. The tangled branches of the amoeba allow a definition of length via counting of the folds. In this way, discreteness of space, time and particles could also be realized; the quantization of space-time, matter and radiation thus follows. Any flexible and deformable entity is also a perfect candidate for the realization of diffeomorphism invariance, as required by general relativity. A simple candidate for the extended fold is the image of a fluctuating, flexible tube. Counting tubes implies to determine distances or areas. The minimum possible count of one gives the minimum distance, and thus allows us to deduce quantum theory. In fact, we can use as model any object which has flexibility and a small dimension, such as a tube, a thin sheet, a ball chain or a woven collection of rings. These options give the different but probably equivalent models presently explored in simplicial quantum gravity, in Ashtekar’s variables and in superstrings.

Dvipsbugw


argument 5: the entropy of bl ack holes

Ref. 1139, Ref. 1140

is a black hole. A black hole can be formed when a whole star collapses under its own weight. A black hole is thus a macroscopic body with a large number of constituents. For black holes, like for every macroscopic body, an entropy can be defined. The entropy S of a macroscopic black hole was determined by Bekenstein and Hawking and is given by S=

Ref. 1141

Ref. 1143

Ref. 1145 Ref. 1146

k A 2 4 l Pl

or

S=k

4πGm 2 ħc

(803)

where k is the Boltzmann constant and A = 4πr S2 is the surface of the black hole horizon. This important result has been derived in many different ways. The various derivations also confirm that space-time and matter are equivalent, by showing that the entropy value can be seen both as an entropy of matter and as one of space-time. In the present context, the two main points of interest are that the entropy is finite, and that it is proportional to the area of the black hole horizon. In view of the existence of minimum lengths and times, the finiteness of entropy is not surprising any more. A finite black hole entropy confirms the idea that matter is made of a finite number of discrete entities per volume. The existence of an entropy also shows that these entities behave statistically; they fluctuate. In fact, quantum gravity leads to a finite entropy for any object, not only for black holes; Bekenstein has shown that the entropy of any object is always smaller than the entropy of a (certain type of) black hole of the same mass. The entropy of a black hole is also proportional to its horizon area. Why is this the case? This question is the topic of a stream of publications up to this day.* A simple way to understand the entropy–surface proportionality is to look for other systems in nature with the property that entropy is proportional to system surface instead of system volume. In general, the entropy of any collection of one-dimensional flexible objects, such as polymer chains, shows this property. Indeed, the expression for the entropy of a polymer chain made of N monomers, each of length a, whose ends are kept a distance r apart, is given by º 3r 2 S(r) = k for Na Q Na Q r . (804) 2Na 2

Ref. 1144

* The result can be derived from quantum statistics alone. However, this derivation does not yield the proportionality coefficient.

Copyright © Christoph Schiller November 1997–May 2006

The formula is derived in a few lines from the properties of a random walk on a lattice, using only two assumptions: the chains are extended, and they have a characteristic internal length a given by the smallest straight segment. Expression (804) is only valid if the polymers are effectively infinite, i.e. if the length Na of the chain and their effective º average size, the elongation a N , are much larger than the radius r of the region of interest; if the chain length is comparable or smaller than the region of interest, one gets the usual extensive entropy, fulfilling S  r 3 . Thus only flexible extended entities yield a S  r 2 dependence. However, there is º a difficulty. From º the entropy expression of a black hole we deduce that the elongation a N is given by a N  l Pl ; thus it is much smaller than the radius of a general, macroscopic black hole which can have diameters of several kilometres. On

Dvipsbugw

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Ref. 1142

1119

Dvipsbugw


1120 xi general rel ativity versus quantum mechanics • the shape of points

Ref. 1125

Argument 6: Exchanging space points or particles at Pl anck scales We are still collecting arguments for the extension of fundamental entities in nature. Let us focus on their exchange behaviour. We saw above that points in space have to be eliminated in favour of continuous, fluctuating entities common to space, time and matter. Is such a space ‘point’ or space entity a boson or a fermion? If we exchange two points of empty space in everyday life, nothing happens. Indeed, quantum field theory is based – among others – on the relation [x, y] = x y − yx = 0

(805)

Ref. 1122

Ref. 1147 Ref. 1147

This means that ‘points’ are neither bosons nor fermions.* They have more complex exchange properties. In fact, the term on the right hand side will be energy dependent, with an effect increasing towards Planck scales. In particular, we saw that gravity implies that double exchange does not lead back to the original situation at Planck scales. Entities following this or similar relations have been studied in mathematics for many decades: braids. In summary, at Planck scales space-time is not made of points, but of braids or some of their generalizations. Thus quantum theory and general relativity taken together again show that vacuum must be made of extended entities. * The same reasoning destroys the fermionic or Grassmann coordinates used in supersymmetry.

Copyright © Christoph Schiller November 1997–May 2006

between any two points with coordinates x and y, making them bosons. But at Planck scale, due to the existence of minimal distances and areas, this relation is at least changed to 2 [x, y] = l Pl + ... . (806)

Dvipsbugw

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the other hand, the formula for long entities is only valid when the chains are longer than the distance r between the end points. This difficulty disappears once we remember that space near a black hole is strongly curved. All lengths have to be measured in the same coordinate system. It is well known that for an outside observer, any object of finite size falling into a black hole seems to cover the complete horizon for long times (whereas it falls into the hole in its original size for an observer attached to the object). In short, an extended entity can have a proper length of Planck size but still, when seen by an outside observer, be as long as the horizon of the black hole in question. We thus find that black holes are made of extended entities. Another viewpoint can confirm the result. Entropy is (proportional to) the number of yes/no questions needed to know the exact state of the system. This view of black holes has been introduced by Gerard ’t Hooft. But if a system is defined by its surface, like a black hole is, its components must be extended. Finally, imagining black holes as made of extended entities is also consistent with the so-called no-hair theorem: black holes’ properties do not depend on what material falls into them, as all matter and radiation particles are made of the same extended components. The final state only depends on the number of entities and on nothing else. In short, the entropy of a black hole is consistent with the idea that it is made of a big tangle of extended entities, fluctuating in shape.

Dvipsbugw


argument 7: the meaning of spin

1121

Ref. 1122

Ref. 1147

At Planck energies this cannot be correct. At those energies, quantum gravity effects appear and modify the right hand side; they add an energy dependent term that is negligible at experimentally accessible energies, but which becomes important at Planck energy. We know from our experience with Planck scales that exchanging particles twice cannot lead back to the original situation, in contrast to everyday life. It is impossible that a double exchange at Planck energy has no effect, because at planck energy such statements are impossible. The simplest extension of the commutation relation (807) satisfying the requirement that the right side does not vanish is again braid symmetry. Thus Planck scales suggest that particles are also made of extended entities.

Argument 7: The meaning of spin

* With a flat (or other) background, it is possible to define a local energy–momentum tensor. Thus particles can be defined. Without background, this is not possible, and only global quantities can be defined. Without background, even particles cannot be defined! Therefore, we assume that we have a slowly varying spacetime background in this section.

Copyright © Christoph Schiller November 1997–May 2006

In the last argument, we will show that even at everyday energy, the extension of particles makes sense. Any particle is a part of the universe. A part is something which is different from anything else. Being ‘different’ means that exchange has some effect. Distinction means possibility of exchange. In other words, any part of the universe is described by its exchange behaviour. Everyday life tells us that exchange can be seen as composed of rotation. In short, distinguishing parts are described by their rotation behaviour. For this reason, for microscopic particles, exchange behaviour is specified by spin. Spin distinguishes particles from vacuum.* We note that volume does not distinguish vacuum from particles; neither does rest mass or charge: there are particles without rest mass or without charge, such as photons. The only candidate observables to distinguish particles from vacuum are spin and momentum. In fact, linear momentum is only a limiting case of angular momentum. We again find that rotation behaviour is the basic aspect distinguishing particles from vacuum. If spin is the central property distinguishing particles from vacuum, finding a model for spin is of central importance. But we do not have to search for long. An well-known model for spin 1/2 is part of physics folklore. Any belt provides an example, as we dis-

Dvipsbugw

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Particles behave in a similar way. We know that at low, everyday energies, particles of the same type are identical. Experiments sensitive to quantum effects show that there is no way to distinguish them: any system of several identical particles obeys permutation symmetry. On the other hand we know that at Planck energy all low-energy symmetries disappear. We also know that, at Planck energy, permutation cannot be carried out, as it implies exchanging positions of two particles. At Planck energy, nothing can be distinguished from vacuum; thus no two entities can be shown to have identical properties. Indeed, no two particles can be shown to be indistinguishable, as they cannot even be shown to be separate. What happens when we slowly approach Planck energy? At everyday energies, permutation symmetry is defined by commutation or anticommutation relations of any two particle creation operators a† b†  b† a† = 0 . (807)

Dvipsbugw


1122 xi general rel ativity versus quantum mechanics • the shape of points

flexible bands in unspecified number reaching the border of space

particle position

Dvipsbugw J=1/2 F I G U R E 388 A possible model for a spin 1/2

particle

Ref. 1148

cussed in detail in chapter VI on permutation symmetry. Any localized structure with any number of long tails attached to it – and reaching the border of the region of space under consideration – has the same properties as a spin 1/2 particle. It is a famous exercise to show that such a model, shown in Figure 388, is indeed invariant under 4π rotation but not under 2π rotations, that two such particles get entangled when exchanged, but get untangled when exchanged twice. The model has all properties of spin 1/2 particles, independently of the precise structure of the central region, which remains unknown at this point. The tail model even has the same problems with highly curved space as real spin 1~2 particles have. We explore the idea in detail shortly. The tail model thus confirms that rotation is partial exchange. More interestingly, it shows that rotation implies connection with the border of space-time. Extended particles can be rotating. Particles can have spin 1~2 provided that they have tails going to the border of space-time. If the tails do not reach the border, the model does not work. Spin 1~2 thus even seems to require extension. We again reach the conclusion that extended entities are a good description for particles.

Present research

The Greek deduced the existence of atoms because fish can swim through water. They argued that only if water is made of atoms, can a fish find its way through it by pushing the atoms aside. We can ask a similar question when a particle flies through vacuum: why are particles able to move through vacuum at all? Vacuum cannot be a fluid or a solid of small entities, as this would not fix its dimensionality. Only one possibility remains: both vacuum and particles are made of a web of extended entities. Describing matter as composed of extended entities is an idea from the 1960s. Describing nature as composed of ‘infinitely’ extended entities is an idea from the 1980s. Indeed, in addition to the arguments presented so far, present research provides several other approaches that arrive at the same conclusion.

Copyright © Christoph Schiller November 1997–May 2006

“ Ref. 1132

To understand is to perceive patterns. Isaiah Berlin

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Page 771

Dvipsbugw


present research

1123

** Ref. 1149

Bosonization, the construction of fermions using an infinite number of bosons, is a central property of modern unification attempts. It also implies coupling duality, and thus the extension of fundamental constituents. **

Ref. 1150

String theory and in particular its generalization to membranes are explicitly based on extended entities, as the name already states. The fundamental entities are indeed assumed to reach the limits of space-time.

Dvipsbugw

** Ref. 1151

Research into quantum gravity, in particular the study of spin networks and spin foams, has shown that the vacuum must be thought as a collection of extended entities.

Ref. 1152

In the 1990s, Dirk Kreimer has shown that high-order QED diagrams are related to knot theory. He thus proved that extension appears through the back door even when electromagnetism is described using point particles. **

Ref. 1153

A recent discovery in particle physics, ‘holography’, connects the surface and volume of physical systems at high energy. Even if it were not part of string theory, it would still imply extended entities. **

Page 805

Other fundamental nonlocalities in the description of nature, such as wave function collapse, can be seen as the result of extended entities. ** The start of the twenty-first century has brought forwards a number of new approaches, such as string net condensation or knotted particle models. All these make use of extended entities. Conceptual checks of extension

— As Ed Witten likes to say, any unified model of nature must be supersymmetric and dual. The idea of extended entities would be dead as soon as it is shown not to be compatible with these requirements. — Any model of nature must be easily extendible to a model for black holes. If not, it cannot be correct. — Showing that the results on quantum gravity, such as the results on the area and volume quantization, are in contradiction with extended entities would directly invalidate the

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1509 ny

Is nature really described by extended entities? The idea is taken for granted by all present approaches in theoretical physics. How can we be sure about this result? The arguments presented above provide several possible checks.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

**

Dvipsbugw


1124 xi general rel ativity versus quantum mechanics • the shape of points

model. — The same conclusion of extended entities must appear if one starts from any physical (low-energy) concept – not only from length measurements – and continues to study how it behaves at Planck scales. If the conclusion were not reached, the idea of extension would not be consistent and thus incorrect. — Showing that any conclusion of the idea of extension is in contrast with string theory or with M-theory would lead to strong doubts. — Showing that the measurement of length cannot be related to the counting of folds would invalidate the model. — Finding a single Gedanken experiment invalidating the extended entity idea would prove it wrong.

Dvipsbugw

Experimental falsification of models based on extended entities

Ref. 1154

Ref. 1155

— Observing a single particle in cosmic rays with energy above the Planck energy would bring this approach to a sudden stop. (The present record is a million times lower.) — Finding any property of nature not consistent with extended entities would spell the end of the idea. — Finding an elementary particle of spin 0 would invalidate the idea. In particular, finding the Higgs boson and showing that it is elementary, i.e. that its size is smaller than its own Compton wavelength, would invalidate the model. — Most proposed experimental checks of string theory can also yield information on the ideas presented. For example, Paul Mende has proposed a number of checks on the motion of extended objects in space-time. He argues that an extended object moves differently from a mass point; the differences could be noticeable in scattering or dispersion of light near masses. — In July 2002, the Italian physicist Andrea Gregori has made a surprising prediction valid for any model using extended entities that reach the border of the universe: if particles are extended in this way, their mass should depend on the size of the universe. Particle masses should thus change with time, especially around the big bang. This completely new point is still a topic of research.

Possibilities for confirmation of extension models — The best confirmation would be to find a concrete model for the electron, muon, tau and for their neutrinos. In addition, a concrete model for photons and gravitons is needed. With these models, finding a knot-based definition for the electrical charge and the lepton number would be a big step ahead. All quantum numbers should be topological quantities deduced from these models and should behave as observed. — Estimating the coupling constants and comparing them with the experimental values is of course the main dream of modern theoretical physics.

Copyright © Christoph Schiller November 1997–May 2006

Incidentally, most of these scenarios would spell the death penalty for almost all present unification attempts.

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Physics is an experimental science. What kind of data could show that extension is incorrect?

Dvipsbugw


present research

1125 Nature's energy scale

Eeveryday

EPl

F I G U R E 389 Planck effects make the energy

axis an approximation

Dvipsbugw — Proving in full detail that extended entities imply exactly three plus one space-time dimensions is still necessary. — Estimating the total number of particles in the visible universe would provide the final check of any extended entity model. Generally speaking, the only possible confirmations are those from the one-page table of unexplained properties of nature given in Chapter X. No other confirmations are possible. The ones mentioned here are the main ones. Curiosities and fun challenges about extension

No problem is so formidable that you can’t walk away from it. Charles Schultz

Even though this section already provided sufficient food for thought, here is some more. ** Challenge 1510 ny

If measurements become impossible near Planck energy, we cannot even draw a diagram with an energy axis reaching that value. (See Figure 389) Is this conclusion valid in all cases? **

Challenge 1511 n

Quantum theory implies that even if tight walls would exist, the lid of such a box can never be tightly shut. Can you provide the argument? ** Is it correct that a detector able to detect Planck mass particles would be of infinite size? What about a detector to detect a particle moving with Planck energy? **

Challenge 1513 e

Can you provide an argument against the idea of extended entities?* **

Challenge 1514 ny

Does duality imply that the cosmic background fluctuations (at the origin of galaxies and clusters) are the same as vacuum fluctuations?

* If so, please email it to the author

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1512 n

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Page 959

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1126 xi general rel ativity versus quantum mechanics • the shape of points

** Challenge 1515 ny

Does duality imply that a system with two small masses colliding is the same as one with two large masses gravitating? **

Challenge 1516 d

It seems that in all arguments so far we assumed and used continuous time, even though we know it is not. Does this change the conclusions so far?

Dvipsbugw

** Duality also implies that large and small masses are equivalent in some sense. A mass 2 2 m in a radius r is equivalent to a mass m Pl ~m in a radius l Pl ~r. In other words, duality 2 transforms mass density from ρ to ρ Pl ~ρ. Vacuum and maximum density are equivalent. Vacuum is thus dual to black holes. Duality implies that initial conditions for the big bang make no sense. Duality again shows the uselessness of the idea, as minimal distance did before. As duality implies a symmetry between large and small energies, the big bang itself becomes an unclearly defined concept. ** Challenge 1517 n

The total symmetry, as well as space-time duality, imply that there is a symmetry between all values an observable can take. Do you agree? **

Challenge 1518 n

Can supersymmetry be an aspect or special case of total symmetry or is it something else? **

Challenge 1519 n

Any description is a mapping from nature to mathematics, i.e. from observed differences (and relations) to thought differences (and relations). How can we do this accurately, if differences are only approximate? Is this the end of physics? ** What do extended entities imply for the big-bang? **

Challenge 1521 d

Can you show that going to high energies or selecting a Planck size region of space-time is equivalent to visiting the big-bang? **

Ref. 1155, Ref. 1156 Challenge 1522 ny

Additionally, one needs a description for the expansion of the universe in terms of extended entities. First approaches are being explored; no final conclusions can be drawn yet. Can you speculate about the solution?

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1520 d

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**

Dvipsbugw


present research

1127

An intermediate status report

Many efforts for unification advance by digging deeper and deeper into details of quantum field theory and general relativity. Here we took the opposite approach: we took a step back and looked at the general picture. Guided by this idea we found several arguments, all leading to the same conclusion: space-time points and point particles are made of extended entities. Somehow it seems that the universe is best described by a fluctuating, multi-branched entity, a crossing between a giant amoeba and a heap of worms. Another analogy is a big pot of boiling and branched spaghetti. Such an extended model of quantum geometry is beautiful and simple, and these two criteria are often taken as indication, before any experimental tests, of the correctness of a description. We toured topics such as the existence of Planck limits, 3-dimensionality, curvature, renormalization, spin, bosonization, the cosmological constant problem, as well as the search for a background free description of nature. We will study and test specific details of the model in the next section. All these tests concern one of only three possible topics: the construction of space-time, the construction of particles and the construction of the universe. These are the only issues remaining on our mountain ascent of Motion Mountain. We will discuss them in the next section. Before we do so, we enjoy two small thoughts. Sexual preferences in physics

Page 223

Ref. 1158

A physical aphorism Ref. 1159, Ref. 1160

To ‘show’ that we are not far from the top of Motion Mountain, we give a less serious argument as final curiosity. Salecker and Wigner and then Zimmerman formulated the fundamental» limit for the measurement precision τ attainable by a clock of mass M. It is given by τ = ħT~Mc 2 , where T is the time to be measured. We can then ask what time * ‘We must know, we will know.’ This was Hilbert’s famous personal credo.

Copyright © Christoph Schiller November 1997–May 2006

Fluctuating entities can be seen to answer an old and not so serious question. When nature was defined as made of tiny balls moving in vacuum, we described this as a typically male idea. Suggesting that it is male implies that the female part is missing. Which part would that be? From the present point of view, the female part of physics might be the quantum description of the vacuum. The unravelling of the structure of the vacuum, as extended container of localized balls, could be seen as the female half of physics. If women had developed physics, the order of the discoveries would surely have been different. Instead of studying matter, as men did, women might have studied the vacuum first. In any case, the female and the male approaches, taken together, lead us to the description of nature by extended entities. Extended entities, which show that particles are not balls and that the vacuum is not a container, transcend the sexist approaches and lead to the unified description. In a sense, extended entities are thus the politically correct approach to nature.

Dvipsbugw

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Ref. 1157

Wir müssen wissen, wir werden wissen.* David Hilbert, 1930.

Dvipsbugw


1128

xi general rel ativity versus quantum mechanics • string theory

T can be measured with a precision of a Planck time t Pl , given a clock of the mass of the whole universe. We get a maximum time of T=

Challenge 1523 e

2 2 t Pl c M. ħ

(808)

Inserting numbers, we find rather precisely that the time T is the present age of the universe. With the right dose of humour we can see the result as an omen for the belief that time is now ripe, after so much waiting, to understand the universe down to the Planck scale. We are getting nearer to the top of Motion Mountain. Be prepared for even more fun.

38.

string theory – a web of dualities

By incorporating these four ideas, string theory acquires a number of appealing properties. First of all, the theory is unique. It has no adjustable parameters. Furthermore, as expected from any theory with extended entities,string theory contains gravity. In addition, string theory describes interactions. It does describe gauge fields such as the electromagnetic field. The theory is thus an extension of quantum field theory. All essential points of quantum field theory are retained.

Copyright © Christoph Schiller November 1997–May 2006

— String theory describes particles as extended entities. In a further generalization, Mtheory describes particles as higher-dimensional membranes. — String theory uses supersymmetry. Supersymmetry is the symmetry that relates matter to radiation, or equivalently, fermions to bosons. Supersymmetry is the most general local interaction symmetry that is known. — String theory uses higher dimensions to introduce extended entities and to unify interactions. A number of dimensions higher than 3 + 1 seems to be necessary for a consistent quantum field theory. At present however, the topology and size of the dimensions above 4 is still unclear. (In fact, to be honest, the topology of the first 4 dimensions is also unknown.) — String theory makes heavy use of dualities. Dualities, in the context of high energy physics, are symmetries that map large to small values of physical observables. Important examples are space-time duality and coupling constant duality. Dualities are global interaction and space-time symmetries. They are essential to include gravitation in the description of nature. With dualities, string theory integrates the equivalence between space-time and matter-radiation.

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String theory, superstring theory or membrane theory – all are names for related approaches – can be characterized as the most investigated theory ever. String theory explores the idea of particles as extended entities. String theory contains a maximum speed, a minimum action and a maximum force. It thus incorporates special relativity, quantum theory and general relativity. In its attempt to achieve a unified description, string theory uses four ideas that go beyond usual general relativity and quantum theory: extension, supersymmetry, higher dimensions and dualities.

Ref. 1161

Dvipsbugw

Dvipsbugw


strings and membranes – why string theory is so difficult

1129

String theory has many large symmetries, a consequence of its many dualities. These symmetries connect many wildly different situations; that makes the theory fascinating but also difficult to picture. Finally, string theory shows special cancellations of anomalies and inconsistencies. Historically, the first example was the Green-Schwarz anomaly cancellation; however, many other mathematical problems of quantum field theory are solved. In short, string theory research has shown that general relativity and quantum field theory can be unified using extended entities.

Dvipsbugw

Strings and membranes – why string theory is so difficult

– CS – more to be added in the future – CS –

Matrix models and M-theory

– CS – the rest of the chapter will be made available in the future – CS – In other words, M-theory complies with all conceptual requirements that a unified theory must fulfil. Let us now turn to experiment. * The history of string theory was characterized by short periods of excitement followed by long periods of disappointment. The main reason for this ineffective evolution was that most researchers only studied topics that everybody else was also studying. Due to the fear of unemployment of young researchers and out of the fear of missing out something, people were afraid to research topics that nobody else was looking at. Thus string theory had an extremely difficult birth.

Copyright © Christoph Schiller November 1997–May 2006

One of the most beautiful results of string theory research is the fact that string theory can be deduced from an extremely small number of fundamental ideas. The approach is to take a few basic ideas and to deduce M-theory form them. M-theory is the most recent formulation of string theory. M-theory assumes that the basic entities of nature are membranes of different dimensions.M-theory incorporates dualities and holography. The existence of interactions can been deduced. Supersymmetry and, in particular, super Yang–Mills theory can be deduced. Also the entropy of black holes follows from M-theory. Finally, supergravity has been deduced from M-theory. This means that the existence of space-time follows from M-theory. M-theory is thus a promising candidate for a theory of motion.

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What is string theory? There are two answers. The first answer is to follow its historical development. Unfortunately, that is almost the hardest of all possible ways to learn and understand string theory.* Thus we only give a short overview of this approach here. The full description of string theory along historical lines starts from classical strings, then proceeds to string field theory, incorporating dualities, higher dimensions and supersymmetry on the way. Each step can be described by a Lagrangian. Let us have a look at them. The first idea of string theory, as the name says, is the use of strings to describe particles. Strings replace the idea of point particles. The simplest form of string theory can be given if the existence of a background space-time is already taken for granted.

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1130

xi general rel ativity versus quantum mechanics • string theory

Masses and couplings Ref. 1166

One of the main results of QCD, the theory of strong interactions, is the explanation of mass relations such as m proton  e−k~α unif m Pl

k = 11~2π , α unif = 1~25 .

(809)

Here, the value of the coupling constant α unif is taken at the unifying energy. This energy is known to be a factor of about 800 below the Planck energy. In other words, a general understanding of masses of bound states of the strong interaction, such as the proton, requires almost only a knowledge of the unification energy and the coupling constant at that energy. The approximate value α unif = 1~25 is an extrapolation from the low energy value, using experimental data. Any unified theory must allow one to calculate the coupling constants as function of energy, including the value α unif at the unification energy itself. At present, researchers state that the main difficulty of string theory and of M-theory is the search for the vacuum state. Without the vacuum state, no calculations of the masses and coupling constants are possible. The vacuum state of string theory or M-theory is expected by researchers to be one of an extremely involved set of topologically distinct manifolds. At present, is it estimated that there are around 10750250 candidate vacuum states. One notes that the error alone is as large as the number of Planck 4-volumes in the history of the universe. This poorness of the estimate reflects the poor understanding of the theory. On can describe the situation also in the following way. String and M-theory predict states with Planck mass and with zero mass. The zero mass particles are then thought to get a tiny mass (compared to the Planck mass) due to the Higgs mechanism. However, the Higgs mechanism, with all its coupling constants, has not yet been deduced from string theory. In other words, so far, string theory predicts no masses for elementary particles. This disappointing situation is the reason that several scientists are dismissing string theory altogether. Future will show. Outlook

Copyright © Christoph Schiller November 1997–May 2006

It is estimated that 10 000 man-years have been invested in the search for string theory; compare this with about 5 for Maxwell’s electrodynamics, 10 for general relativity and a similar number for the foundation of quantum theory. Historically, the community of string theorists took over 10 years to understand that strings were not the basic entities of string theory. The fundamental entities are membranes. Then string theorists took another 10 years and more to understand that membranes are not the most practical fundamental entities for calculation. However, the search for the most practical entities is still ongoing. It is probable that knotted membranes must be taken into consideration. Why is M-theory so hard? Doing calculations on knotted membranes is difficult. In fact, it is extremely difficult. Mathematicians and physicists all over the world are still struggling to find simple ways for such calculations. In short, a new approach for calculations with extended entities is needed. We then can ask the following question. M-theory is based on several basic ideas: extension, higher di-

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Ref. 1167

and

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1131

mensions, supersymmetry and duality. Which of these basic assumptions is confusing? Dualities seem to be related to extension, for which experimental and theoretical evidence exists. However, supersymmetry and higher dimensions are completely speculative. We thus leave them aside, and continue our adventure.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006

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1132

Biblio graphy 1122 C. Schiller, Does matter differ from vacuum? http://www.arxiv.org/abs/gr-qc/9610066.

1123

1124

1126

1127 1128 1129 1130

1132 1133 1134 1135

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A French language version has been published as C. Schiller, Le vide diffère-t-il de la matière? in E. Gunzig & S. Diner, editeurs, Le vide – Univers du tout et du rien – Des physiciens et des philosophes s’interrogent, Les Éditions de l’Université de Bruxelles, 1998. Cited on pages 1102, 1103, 1106, 1108, 1109, 1110, 1120, and 1121. Most arguments for minimal temporal and spatial intervals are due to M. Schön, Operative time definition and principal uncertainty, preprint http://www.arxiv.org/abs/gr-qc/ 9304024, and to T. Padmanabhan, Limitations on the operational definition of spacetime events and quantum gravity, Classical and Quantum Gravity 4, pp. L107–L113, 1987; see also his earlier papers referenced there. Cited on page 1102. A similar point of view, often called monism, was proposed by Baruch Spinoza, Ethics Demonstrated in Geometrical Order, 1677, originally in Latin; an affordable French edition is B. Spinoza, L’Ethique, Folio-Gallimard, 1954. For a discussion of his ideas, especially his monism, see Don Garret, ed., The Cambridge Companion to Spinoza, Cambridge University Press, 1996, or any general text on the history of philosophy. Cited on page 1104. See L. Susskind & J. Uglum, Black holes, interactions, and strings, http://www.arxiv. org/abs/hep-th/9410074, or L. Susskind, String theory and the principle of black hole complementarity, Physical Review Letters 71, pp. 2367–2368, 1993, and M. Karliner, I. Klebanov & L. Susskind, Size and shape of strings, International Journal of Modern Physics A 3, pp. 1981–1996, 1988, as well as L. Susskind, Structure of hadrons implied by duality, Physical Review D 1, pp. 1182–1186, 1970. Cited on pages 1106 and 1120. M. Pl anck, Über irreversible Strahlungsvorgänge, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften zu Berlin, pp. 440–480, 1899. Today it is commonplace to use ħ = h~2π instead of Planck’s h, which he originally called b in his paper. Cited on page 1106. C.A. Mead, Possible connection between gravitation and fundamental length, Physical Review B 135, pp. 849–862, 1964. Cited on page 1106. A.D. Sakharov, Vacuum quantum fluctuations in curved space and the theory of gravitation, Soviet Physics - Doklady 12, pp. 1040–1041, 1968. Cited on page 1109. P. Facchi & S. Pascazio, Quantum Zeno and inverse quantum Zeno effects, pp. 147– 217, in E. Wolf, editor, Progress in Optics, 42, 2001. Cited on page 1109. Aristotle, Of Generation and Corruption, I, II, 316 a 13. See Jean-Paul Dumont, Les écoles présocratiques, Folio Essais, Gallimard, p. 427, 1991. Cited on page 1110. See for example the speculative model of vacuum as composed of Planck-size spheres proposed by F. Winterberg, Zeitschrift für Naturforschung 52a, p. 183, 1997. Cited on page 1110. The Greek salt and fish arguments are given by Lucrece, or Titus Lucretius Carus, De natura rerum, around 50 bce. Cited on pages 1112 and 1122. Cited in Jean-Paul Dumont, Les écoles présocratiques, Folio Essais, Gallimard, p. 379, 1991. Cited on page 1113. D. Olive & C. Montonen, Magnetic monopoles as gauge particles, Physics Letters 72B, pp. 117–120, 1977. Cited on page 1114. J.H. Schwarz, The second superstring revolution, http://www.arxiv.org/abs/hep-th/ 9607067. More on dualities in ... Cited on pages 1112 and 1114.

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bibliography

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1136 The famous quote is found at the beginning of chapter XI, ‘The Physical Universe’, in Ar-

thur Eddington, The Philosophy of Physical Science, Cambridge, 1939. Cited on page 1115. 1137 Pl ato, Parmenides, c. 370 bce. It has been translated into most languages. Reading it aloud, like a song, is a beautiful experience. A pale reflection of these ideas is Bohm’s concept of ‘unbroken wholeness’. Cited on page 1116. 1138 P. Gibbs, Event-symmetric physics, http://www.arxiv.org/abs/hep-th/9505089; see also 1139 1140

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1143 1144 1145 1146 1147

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his http://www.weburbia.com/pg/contents.htm website. Cited on page 1116. J.D. Bekenstein, Black holes and entropy, Physical Review D 7, pp. 2333–2346, 1973. Cited on page 1119. S.W. Hawking, Particle creation by black holes, Communications in Mathematical Physics 43, pp. 199–220, 1975; see also S.W. Hawking, Black hole thermodynamics, Physical Review D13, pp. 191–197, 1976. Cited on page 1119. J.B. Hartle & S.W. Hawking, Path integral derivation of black hole radiance, Physical Review D 13, pp. 2188–2203, 1976. See also A. Strominger & C. Vafa, Microscopic origin of Bekenstein–Hawking entropy, Physics Letters B379, pp. 99–104, 1996, or the downloadable version found at http://www.arxiv.org/abs/hep-th/9601029. For another derivation of black hole entropy, see Gary T. Horowitz & J. Polchinski, A correspondence principle for black holes and strings, Physical Review D 55, pp. 6189–6197, 1997, or http://www.arxiv.org/abs/hep-th/9612146. Cited on page 1119. J.D. Bekenstein, Entropy bounds and black hole remnants, Physical Review D 49, pp. 1912–1921, 1994. Cited on page 1119. J. Madd ox, When entropy does not seem extensive, Nature 365, p. 103, 1993. Other references on the proportionality of entropy and surface are ... Cited on page 1119. L. B ombelli, R.K. Koul, J. Lee & R.D. Sorkin, Quantum source of entropy of black holes, Physical Review D 34, pp. 373–383, 1986. Cited on page 1119. The analogy between polymers and black holes is by G. Weber, Thermodynamics at boundaries, Nature 365, p. 792, 1993. Cited on page 1119. P.-G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, 1979. Cited on page 1119. See for example S. Majid, Introduction to braided geometry and q-Minkowski space, preprint found at http://www.arxiv.org/abs/hep-th/9410241 or S. Majid, Duality principle and braided geometry, preprint http://www.arxiv.org/abs/gr-qc/940957. Cited on pages 1120 and 1121. The relation between spin and statistics has been studied recently by M.V. Berry & J.M. Robbins, Quantum indistinguishability: spin–statistics without relativity or field theory?, in R.C. Hilborn & G.M. Tino, editors, Spin–Statistics Connection and Commutation Relations, American Institute of Physics CP545 2000. Cited on page 1122. An example is given by A.A. Sl avnov, Fermi–Bose duality via extra dimension, preprint http://www.arxiv.org/abs/hep-th/9512101. See also the standard work by Michael Stone, editor, Bosonization, World Scientific, 1994. Cited on page 1123. A popular account making the point is the bestseller by the US-american string theorist Brian Greene, The Elegant Universe – Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory, Jonathan Cape 1999. Cited on page 1123. The weave model of space-time appears in certain approaches of quantum gravity, as told by A. Ashtekar, Quantum geometry and gravity: recent advances, December 2001,

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1157 1158

1159 1160 1161

1163 1164

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preprint found at http://www.arxiv.org/abs/gr-qc/0112038. See also A. Ashtekar, Quantum geometry in action: big bang and black holes, http://www.arxiv.org/abs/ math-ph/0202008. On a slightly different topic, see also S.A. Major, A spin network primer, http://www.arxiv.org/abs/gr-qc/9905020. Cited on page 1123. A good introduction into his work is the paper D. Kreimer, New mathematical structures in renormalisable quantum field theories, http://www.arxiv.org/abs/hep-th/0211136 to appear in Annals of Physics Cited on page 1123. Introductions to holography are E. Alvarez, J. Conde & L. Hernandez, Rudiments of Holography, http://www.arxiv.org/abs/hep-th/0205075 or R. B ousso, The holographic principle, Review of Modern Physics 74, pp. 825–874, 2002, also at http://www.arxiv. org/abs/hep-th/0203101. (However, this otherwise good review has some argumentation errors, as explained on page 871.) The importance of holography in theoretical high energy physics was underlined by the discovery of J. Maldacena, The large N limit of superconformal field theories and supergravity, http://www.arxiv.org/abs/hep-th/9711200. Cited on page 1123. P.F. Mende, String theory at short distance and the principle of equivalence, preprint available at http://www.arxiv.org/abs/hep-th/9210001. Cited on page 1124. A. Gregori, Entropy, string theory and our world, http://www.arxiv.org/abs/hep-th/ 0207195 preprint. Cited on pages 1124 and 1126. String cosmology is a passtime for many. Examples are N.E. Mavromatos, String cosmology, http://www.arxiv.org/abs/hep-th/0111275 and N.G. Sanchez, New developments in string gravity and string cosmology - a summary report, http://www.arxiv.org/ abs/hep-th/0209016. Cited on page 1126. Searches for background free approaches are described by E. Witten, .... Cited on page 1127. Not even the most combative feminists have ever said how women-centered physics should look like. This lack of alternatives pervades also the agressive text by Margaret Wertheim, Pythagoras’ Trousers – God, Physics and the Gender Wars, Fourth Estate, 1997, and thus makes its author look rather silly. Cited on page 1127. H. Salecker & E.P. Wigner, Quantum limitations of the measurement of space-time distances, Physical Review 109, pp. 571–577, 1958. Cited on page 1127. E.J. Zimmerman, The macroscopic nature of space-time, American Journal of Physics 30, pp. 97–105, 1962. Cited on page 1127. See A. Sen, An introduction to duality symmetries in string theory, in Les Houches Summer School: Unity from Duality: Gravity, Gauge Theory and Strings, (Les Houches, France, 2001), Springer Verlag, 76, pp. 241–322, 2002. Cited on page 1128. See for example C. Rovelli, Notes for a brief history of quantum gravity, electronic preprint available at http://www.arxiv.org/abs/gr-qc/0006061. No citations. See the http://www.nuclecu.unam.mx/~alberto/physics/stringrev.html website for an upto date list of string theory and M theory reviews. No citations. W.G. Unruh, Notes on black hole evaporation, Physical Review D 14, pp. 870–875, 1976. W.G. Unruh & R.M. Wald, What happens when an accelerating observer detects a Rindler particle, Physical Review D 29, pp. 1047–1056, 1984. No citations. The most prominent proposer of the idea that particles might be knots was William Thomson. See W. Thomson, On vortex motion, Transactions of the Royal Society in Edinburgh pp. 217–260, 1868. This paper stimulated work on knot theory, as it proposed the idea that different particles might be differently knotted vortices in the aether. No citations.

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1166 F. Wilczek, Getting its from bits, Nature 397, pp. 303–306, 1999. Cited on page 1130. 1167 See the well-known blog by Peter Woit, Not even wrong, at http://www.math.columbia.

edu/~woit/blog. Cited on page 1130.

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C h a p t e r XII

UNI FIC ATION (NOT YET AVAI L ABLE)

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– CS – this chapter will appear in the near future – CS –

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C h a p t e r XIII

THE TOP OF THE MOUNTAI N (NOT YET AVAI L ABLE)

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– CS – this chapter will appear in the near future – CS –

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Fourth Part

Appendices

Where necessary reference information for mountain ascents is provided, preparing the reader for this and any other future adventure.


Appendix A

NOTATION AND C ONV ENTIONS

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N

The L atin alphabet What is written without effort is in general read without pleasure. Samuel Johnson

Copyright © Christoph Schiller November 1997–May 2006

Books are collections of symbols. Writing was probably invented between 3400 and 3300 bce by the Sumerians in Mesopotamia (though other possibilities are also discussed). It then took over a thousand years before people started using symbols to represent sounds instead of concepts: this is the way in which the first alphabet was created. This happened between 2000 and 1600 bce (possibly in Egypt) and led to the Semitic alphabet. The use of an alphabet had so many advantages that it was quickly adopted in all neighbouring cultures, though in different forms. As a result, the Semitic alphabet is the forefather of all alphabets used in the world. This text is written using the Latin alphabet. At first sight, this seems to imply that its pronunciation cannot be explained in print, in contrast to the pronunciation of other alphabets or of the International Phonetic Alphabet (IPA). (They can be explained using the alphabet of the main text.) However, it is in principle possible to write a text that describes exactly how to move lips, mouth and tongue for each letter, using physical concepts where necessary. The descriptions of pronunciations found in dictionaries make indirect use of this method: they refer to the memory of pronounced words or sounds found in nature. Historically, the Latin alphabet was derived from the Etruscan, which itself was a derivation of the Greek alphabet. There are two main forms.

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Ref. 1168

ewly introduced and defined concepts in this text are indicated by italic typeface. ew definitions are also referred to in the index by italic page numbers. We naturally use SI units throughout the text; these are defined in Appendix B. Experimental results are cited with limited precision, usually only two digits, as this is almost always sufficient for our purposes. High-precision reference values can be found in Appendices B and C. In relativity we use the time convention, where the metric has the signature (+ − − − ). This is used in about 70 % of the literature worldwide. We use indices i, j, k for 3-vectors and indices a, b, c, etc. for 4-vectors. Other conventions specific to general relativity are explained as they arise in the text.

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The ancient Latin alphabet, used from the sixth century bce onwards: A B C D E F Z

H I K L M N O P Q R S T V X

The classical Latin alphabet, used from the second century bce until the eleventh century: A B C D E F G H I K L M N O P Q R S T V X Y Z

The Latin alphabet was derived from the Etruscan; the Etruscan from the Greek. The Greek alphabet was itself derived from the Phoenician or a similar northern Semitic al-

Ref. 1170

* To meet Latin speakers and writers, go to http://www.alcuinus.net/. ** The Runic script, also called Futhark or futhorc, a type of alphabet used in the Middle Ages in Germanic, Anglo–Saxon and Nordic countries, probably also derives from the Etruscan alphabet. The name derives from the first six letters: f, u, th, a (or o), r, k (or c). The third letter is the letter thorn mentioned above; it is often written ‘Y’ in Old English, as in ‘Ye Olde Shoppe.’ From the runic alphabet Old English also took the letter wyn to represent the ‘w’ sound, and the already mentioned eth. (The other letters used in Old English – not from futhorc – were the yogh, an ancient variant of g, and the ligatures æ or Æ, called ash, and œ or Œ, called ethel.)

Copyright © Christoph Schiller November 1997–May 2006

The Greek alphabet

Outside a dog, a book is a man’s best friend. Inside a dog, it’s too dark to read. Groucho Marx

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Ref. 1169

The letter G was added in the third century bce by the first Roman to run a fee-paying school, Spurius Carvilius Ruga. He added a horizontal bar to the letter C and substituted the letter Z, which was not used in Latin any more, for this new letter. In the second century bce, after the conquest of Greece, the Romans included the letters Y and Z from the Greek alphabet at the end of their own (therefore effectively reintroducing the Z) in order to be able to write Greek words. This classical Latin alphabet was stable for the next thousand years.* The classical Latin alphabet was spread around Europe, Africa and Asia by the Romans during their conquests; due to its simplicity it began to be used for writing in numerous other languages. Most modern ‘Latin’ alphabets include a few other letters. The letter W was introduced in the eleventh century in French and was then adopted in most European languages. The letters J and U were introduced in the sixteenth century in Italy, to distinguish certain sounds which had previously been represented by I and V. In other languages, these letters are used for other sounds. The contractions æ and œ date from the Middle Ages. Other Latin alphabets include more letters, such as the German sharp s, written ß, a contraction of ‘ss’ or ‘sz’, the and the Nordic letters thorn, written Þ or þ, and eth, written Ð or ð, taken from the futhorc,** and other signs. Lower-case letters were not used in classical Latin; they date only from the Middle Ages, from the time of Charlemagne. Like most accents, such as ê, ç or ä, which were also first used in the Middle Ages, lower-case letters were introduced to save the then expensive paper surface by shortening written words.

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TA B L E 80 The ancient and classical Greek alphabets, and the correspondence with Latin and Indian

digits

Α Β Γ ∆ Ε F ϝ, Ϛ

Α Β Γ ∆ Ε

Ζ Η Θ Ι Κ Λ Μ

Ζ Η Θ Ι Κ Λ Μ

α β γ δ ε

ζ η θ ι κ λ µ

alpha beta gamma delta epsilon digamma, stigma2 zeta eta theta iota kappa lambda mu

Corresp.

Anc.

Cl ass. Name

a b g, n1 d e w

1 2 3 4 5 6

Ν ν Ξ ξ Ο ο Π π

z e th i, j k l m

7 8 9 10 20 30 40

Ν Ξ Ο Π P Ϟ, Ρ Σ Τ

Λ Ϡ v

Cl ass. Name

Ρ Σ Τ Υ Φ Χ Ψ Ω

ρ σ, ς τ υ φ χ ψ ω

nu xi omicron pi qoppa3 rho sigma4 tau upsilon phi chi psi omega sampi6

Corresp. n x o p q r, rh s t y, u5 ph, f ch ps o s

50 60 70 80 90 100 200 300 400 500 600 700 800 900

The regional archaic letters yot, sha and san are not included in the table. The letter san was the ancestor of sampi. 1. Only if before velars, i.e. before kappa, gamma, xi and chi. 2. ‘Digamma’ is the name used for the F-shaped form. It was mainly used as a letter (but also sometimes, in its lower-case form, as a number), whereas the shape and name ‘stigma’ is used only for the number. Both names were derived from the respective shapes; in fact, the stigma is a medieval, uncial version of the digamma. The name ‘stigma’ is derived from the fact that the letter looks like a sigma with a tau attached under it – though unfortunately not in all modern fonts. The original letter name, also giving its pronunciation, was ‘waw’. 3. The version of qoppa that looks like a reversed and rotated z is still in occasional use in modern Greek. Unicode calls this version ‘koppa’. 4. The second variant of sigma is used only at the end of words. 5. Uspilon corresponds to ‘u’ only as the second letter in diphthongs. 6. In older times, the letter sampi was positioned between pi and qoppa.

phabet in the tenth century bce. The Greek alphabet, for the first time, included letters also for vowels, which the Semitic alphabets lacked (and often still lack). In the Phoenician alphabet and in many of its derivatives, such as the Greek alphabet, each letter has a proper name. This is in contrast to the Etruscan and Latin alphabets. The first two Greek letter names are, of course, the origin of the term alphabet itself. In the tenth century bce, the Ionian or ancient (eastern) Greek alphabet consisted of the upper-case letters only. In the sixth century bce several letters were dropped, while a few new ones and the lower-case versions were added, giving the classical Greek alphabet. Still later, accents, subscripts and breathings were introduced. Table 80 also gives the values signified by the letters took when they were used as numbers. For this special use, the obsolete ancient letters were kept during the classical period; thus they also acquired

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Ref. 1171

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Anc.

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TA B L E 81 The beginning of the Hebrew abjad

Letter

Name

Correspondence

ℵ ° ‫ג‬ ² etc.

aleph beth gimel daleth

a b g d

1 2 3 4

Copyright © Christoph Schiller November 1997–May 2006

* The Greek alphabet is also the origin of the Gothic alphabet, which was defined in the fourth century by Wulfila for the Gothic language, using also a few signs from the Latin and futhorc scripts. The Gothic alphabet is not to be confused with the so-called Gothic letters, a style of the Latin alphabet used all over Europe from the eleventh century onwards. In Latin countries, Gothic letters were replaced in the sixteenth century by the Antiqua, the ancestor of the type in which this text is set. In other countries, Gothic letters remained in use for much longer. They were used in type and handwriting in Germany until 1941, when the National Socialist government suddenly abolished them, in order to comply with popular demand. They remain in sporadic use across Europe. In many physics and mathematics books, Gothic letters are used to denote vector quantities.

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lower-case forms. The Latin correspondence in the table is the standard classical one, used for writing Greek words. The question of the correct pronunciation of Greek has been hotly debated in specialist circles; the traditional Erasmian pronunciation does not correspond either to the results of linguistic research, or to modern Greek. In classical Greek, the sound that sheep make was βη–βη. (Erasmian pronunciation wrongly insists on a narrow η; modern Greek pronunciation is different for β, which is now pronounced ‘v’, and for η, which is now pronounced as ‘i’ – a long ‘i’.) Obviously, the pronunciation of Greek varied from region to region and over time. For Attic Greek, the main dialect spoken in the classical period, the question is now settled. Linguistic research has shown that chi, phi and theta were less aspirated than usually pronounced in English and sounded more like the initial sounds of ‘cat’, ‘perfect’ and ‘tin’; moreover, the zeta seems to have been pronounced more like ‘zd’ as in ‘buzzed’. As for the vowels, contrary to tradition, epsilon is closed and short whereas eta is open and long; omicron is closed and short whereas omega is wide and long, and upsilon is really a ‘u’ sound as in ‘boot’, not like a French ‘u’ or German ‘ü.’ The Greek vowels can have rough or smooth breathings, subscripts, and acute, grave, circumflex or diaeresis accents. Breathings – used also on ρ – determine whether the letter is aspirated. Accents, which were interpreted as stresses in the Erasmian pronunciation, actually represented pitches. Classical Greek could have up to three of these added signs per letter; modern Greek never has more than one. Another descendant of the Greek alphabet* is the Cyrillic alphabet, which is used with slight variations, in many Slavic languages, such as Russian and Bulgarian. However, there is no standard transcription from Cyrillic to Latin, so that often the same Russian name is spelled differently in different countries or even in the same country on different occasions.

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1146

The Hebrew alphabet and other scripts Ref. 1172 Page 649

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Novem figure indorum he sunt 9 8 7 6 5 4 3 2 1. Cum his itaque novem fig-

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* A well-designed website on the topic is http://www.omniglot.com. The main present and past writing systems are encoded in the Unicode standard, which at present contains 52 writing systems. See http:// www.unicode.org. ** The story of the development of the numbers is told most interestingly by G. Ifrah, Histoire universelle des chiffres, Seghers, 1981, which has been translated into several languages. He sums up the genealogy in ten beautiful tables, one for each digit, at the end of the book. However, the book contains many factual errors, as explained in the http://www.ams.org/notices/200201/rev-dauben.pdf and http://www.ams.org/notices/ 200202/rev-dauben.pdf review. It is not correct to call the digits 0 to 9 Arabic. Both the actual Arabic digits and the digits used in Latin texts such as this one derive from the Indian digits. Only the digits 0, 2, 3 and 7 resemble those used in Arabic writing, and then only if they are turned clockwise by 90°. *** Leonardo di Pisa, called Fibonacci (b. c. 1175 Pisa, d. 1250 Pisa), Italian mathematician, and the most important mathematician of his time.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1173

The phoenician alphabet is also the origin of the Hebrew consonant alphabet or abjad. Its first letters are given in Table 81. Only the letter aleph is commonly used in mathematics, though others have been proposed. Around one hundred writing systems are in use throughout the world. Experts classify them into five groups. Phonemic alphabets, such as Latin or Greek, have a sign for each consonant and vowel. Abjads or consonant alphabets, such as Hebrew or Arabic, have a sign for each consonant (sometimes including some vowels, such as aleph), and do not write (most) vowels; most abjads are written from right to left. Abugidas, also called syllabic alphabets or alphasyllabaries, such as Balinese, Burmese, Devanagari, Tagalog, Thai, Tibetan or Lao, write consonants and vowels; each consonant has an inherent vowel which can be changed into the others by diacritics. Syllabaries, such as Hiragana or Ethiopic, have a sign for each syllable of the language. Finally, complex scripts, such as Chinese, Mayan or the Egyptian hieroglyphs, use signs which have both sound and meaning. Writing systems can have text flowing from right to left, from bottom to top, and can count book pages in the opposite sense to this book. Even though there are about 7000 languages on Earth, there are only about one hundred writing systems used today. About fifty other writing systems have fallen out of use. * For physical and mathematical formulae, though, the sign system used in this text, based on Latin and Greek letters, written from left to right and from top to bottom, is a standard the world over. It is used independently of the writing system of the text containing it. Digits and numbers Both the digits and the method used in this text to write numbers originated in India. They were brought to the Mediterranean by Arabic mathematicians in the Middle Ages. The number system used in this text is thus much younger than the alphabet.** The Indian numbers were made popular in Europe by Leonardo of Pisa, called Fibonacci,*** in his book Liber Abaci or ‘Book of Calculation’, which he published in 1202. That book revolutionized mathematics. Anybody with paper and a pen (the pencil had not yet been invented) was now able to calculate and write down numbers as large as reason allowed, or even larger, and to perform calculations with them. Fibonacci’s book started:

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uris, et cum hoc signo 0, quod arabice zephirum appellatur, scribitur quilibet numerus, ut inferius demonstratur.*

The symbols used in the text To avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or Gemowe lines of one lengthe, thus: = , bicause noe .2. thynges, can be moare equalle. Robert Recorde***

Ref. 1172

Ref. 1174

* ‘The nine figures of the Indians are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated below.’ ** Currently, the shortest time for finding the thirteenth (integer) root of a hundred-digit number, a result with 8 digits, is 11.8 seconds. For more about the stories and the methods of calculating prodigies, see the fascinating book by Steven B. Smith, The Great Mental Calculators – The Psychology, Methods and Lives of the Calculating Prodigies, Columbia University Press, 1983. The book also presents the techniques that they use, and that anybody else can use to emulate them. *** Robert Recorde (c. 1510–1558), English mathematician and physician; he died in prison, though not for his false pretence to be the inventor of the ‘equal’ sign, which he took over from his Italian colleagues, but for a smaller crime, namely debt. The quotation is from his The Whetstone of Witte, 1557. An image showing the quote in manuscript can be found at the http://members.aol.com/jeff94100/witte.jpg website.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1174

Besides text and numbers, physics books contain other symbols. Most symbols have been developed over hundreds of years, so that only the clearest and simplest are now in use. In this mountain ascent, the symbols used as abbreviations for physical quantities are all taken from the Latin or Greek alphabets and are always defined in the context where they are used. The symbols designating units, constants and particles are defined in Appendices B and C. There is an international standard for them (ISO 31), but it is virtually inaccessible; the symbols used in this text are those in common use. The mathematical symbols used in this text, in particular those for operations and relations, are given in the following list, together with their origins. The details of their history have been extensively studied in the literature.

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The Indian method of writing numbers consists of a large innovation, the positional system, and a small one, the digit zero. The positional system, as described by Fibonacci, was so much more efficient that it completely replaced the previous Roman number system, which writes 1996 as IVMM or MCMIVC or MCMXCVI, as well as the Greek number system, in which the Greek letters were used for numbers in the way shown in Table 80, thus writing 1996 as ͵αϠϞϚʹ. Compared to these systems, the Indian numbers are a much better technology. Indeed, the Indian system proved so practical that calculations done on paper completely eliminated the need for the abacus, which therefore fell into disuse. The abacus is still in use only in those countries which do not use a positional system to write numbers. (The Indian system also eliminated the need fir systems to represent numbers with fingers. Such systems, which could show numbers up to 10 000 and more, have left only one trace: the term ‘digit’ itself, which derives from the Latin word for finger.) Similarly, only the positional number system allows mental calculations and made – and still makes – calculating prodigies possible.**

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Symbol

Meaning

+, −

plus, minus

Origin

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J. Regiomontanus 1456; the plus sign is derived from Latin ‘et’ º read as ‘square root’ used by C. Rudolff in 1525; the sign stems from a deformation of the letter ‘r’, initial of the Latin radix = equal to Italian mathematicians, early sixteenth century, then brought to England by R. Recorde { }, [ ], ( ) grouping symbols use starts in the sixteenth century A, < larger than, smaller than T. Harriot 1631  multiplied with, times W. Oughtred 1631  divided by G. Leibniz 1684 ë multiplied with, times G. Leibniz 1698 an power R. Descartes 1637 x, y, z coordinates, unknowns R. Descartes 1637 ax +by + c = 0 constants and equations for unknowns R. Descartes 1637 d~dx, d2 x, derivative, differential, integral G. Leibniz 1675 ∫ y dx φx function of x J. Bernoulli 1718 f x, f (x) function of x L. Euler 1734 ∆x, P difference, sum L. Euler 1755 x is different from L. Euler eighteenth century ∂~∂x partial derivative, read like ‘d~dx’ it was derived from a cursive form of ‘d’ or of the letter ‘dey’ of the Cyrillic alphabet by A. Legendre in 1786 ∆ Laplace operator R. Murphy 1833 SxS absolute value K. Weierstrass 1841 ∇ read as ‘nabla’ (or ‘del’) introduced by William Hamilton in 1853 and P.G. Tait in 1867, named after the shape of an old Egyptian musical instrument [x] the measurement unit of a quantity x twentieth century ª infinity J. Wallis 1655 π 4 arctan 1 H. Jones 1706 ª 1 n L. Euler 1736 e Pn=0 n! = lim n ª (1 + 1~n) º L. Euler 1777 i + −1 8, 9 set union and intersection G. Peano 1888 > element of G. Peano 1888 g empty set André Weil as member of the N. Bourbaki group in the early twentieth century `ψS, Sψe bra and ket state vectors Paul Dirac 1930 a dyadic product or tensor product or unknown outer product

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Ref. 1175

Ref. 1176 Ref. 1177

Ref. 1179

Other signs used here have more complicated origins. The & sign is a contraction of Latin et meaning ‘and’, as is often more clearly visible in its variations, such as &, the common italic form. Each of the punctuation signs used in sentences with modern Latin alphabets, such as , . ; : ! ? ‘ ’ » « – ( ) ... has its own history. Many are from ancient Greece, but the question mark is from the court of Charlemagne, and exclamation marks appear first in the sixteenth century.* The @ or at-sign probably stems from a medieval abbreviation of Latin ad, meaning ‘at’, similarly to how the & sign evolved from Latin et. In recent years, the smiley :-) and its variations have become popular. The smiley is in fact a new version of the ‘point of irony’ which had been formerly proposed, without success, by A. de Brahm (1868–1942). The section sign § dates from the thirteenth century in northern Italy, as was shown by the German palaeographer Paul Lehmann. It was derived from ornamental versions of the capital letter C for capitulum, i.e. ‘little head’ or ‘chapter.’ The sign appeared first in legal texts, where it is still used today, and then spread into other domains. The paragraph sign ¶ was derived from a simpler ancient form looking like the Greek letter Γ, a sign which was used in manuscripts from ancient Greece until well into the Middle Ages to mark the start of a new text paragraph. In the Middle Ages it took the modern form, probably because a letter c for caput was added in front of it. One of the most important signs of all, the white space separating words, was due to Celtic and Germanic influences when these people started using the Latin alphabet. It became commonplace between the ninth and the thirteenth century, depending on the language in question. Calendars

* On the parenthesis see the beautiful book by J. L ennard, But I Digress, Oxford University Press, 1991.

Copyright © Christoph Schiller November 1997–May 2006

The many ways to keep track of time differ greatly from civilization to civilization. The most common calendar, and the one used in this text, is also one of the most absurd, as it is a compromise between various political forces who tried to shape it. In ancient times, independent localized entities, such as tribes or cities, preferred lunar calendars, because lunar timekeeping is easily organized locally. This led to the use of the month as a calendar unit. Centralized states imposed solar calendars, based on the year. Solar calendars require astronomers, and thus a central authority to finance them. For various reasons, farmers, politicians, tax collectors, astronomers, and some, but not all, religious groups wanted the calendar to follow the solar year as precisely as possible. The compromises necessary between days and years are the origin of leap days. The compromises necessary between months and year led to the varying lengths are different in different calendars. The most commonly used year–month structure was organized over 2000 years ago by Gaius Julius Ceasar, and is thus called the Julian calendar. The system was destroyed only a few years later: August was lengthened to 31 days when it was named after Augustus. Originally, the month was only 30 days long; but in order to show that Augustus was as important as Caesar, after whom July is named, all

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Ref. 1178

1149

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Copyright © Christoph Schiller November 1997–May 2006

* Remembering the intermediate result for the current year can simplify things even more, especially since the dates 4.4, 6.6, 8.8, 10.10, 12.12, 9.5, 5.9, 7.11, 11.7 and the last day of February all fall on the same day of the week, namely on the year’s intermediate result plus 4. ** The present counting of years was defined in the Middle Ages by setting the date for the foundation of Rome to the year 753 bce, or 753 before the Common Era, and then counting backwards, so that the bce years behave almost like negative numbers. However, the year 1 follows directly after the year 1 bce: there was no year 0. Some other standards set by the Roman Empire explain several abbreviations used in the text: - c. is a Latin abbreviation for circa and means ‘roughly’; - i.e. is a Latin abbreviation for id est and means ‘that is’; - e.g. is a Latin abbreviation for exempli gratia and means ‘for the sake of example’; - ibid. is a Latin abbreviation for ibidem and means ‘at that same place’; - inf. is a Latin abbreviation for infra and means ‘(see) below’; - op. cit. is a Latin abbreviation for opus citatum and means ‘the cited work’; - et al. is a Latin abbreviation for et alii and means ‘and others’. By the way, idem means ‘the same’ and passim means ‘here and there’ or ‘throughout’. Many terms used in physics, like frequency, acceleration, velocity, mass, force, momentum, inertia, gravitation and temperature, are derived from Latin. In fact, it is arguable that the language of science has been Latin for over two thousand years. In Roman times it was Latin vocabulary with Latin grammar, in modern times it switched to Latin vocabulary with French grammar, then for a short time to Latin vocabulary with German grammar, after

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month lengths in the second half of the year were changed, and February was shortened by an additional day. The week is an invention of Babylonia, from where it was spread through the world by various religious groups. (The way astrology and astronomy cooperated to determine the order of the weekdays is explained in the section on gravitation.) Although it is about three thousand years old, the week was fully included into the Julian calendar only around the year 300, towards the end of the Western Roman Empire. The final change in the Julian calendar took place between 1582 and 1917 (depending on the country), when more precise measurements of the solar year were used to set a new method to determine leap days, a method still in use today. Together with a reset of the date and the fixation of the week rhythm, this standard is called the Gregorian calendar or simply the modern calendar. It is used by a majority of the world’s population. Despite its complexity, the modern calendar does allow you to determine the day of the week of a given date in your head. Just execute the following six steps: 1. take the last two digits of the year, and divide by 4, discarding any fraction; 2. add the last two digits of the year; 3. subtract 1 for January or February of a leap year; 4. add 6 for 2000s or 1600s, 4 for 1700s or 2100s, 2 for 1800s and 2200s, and 0 for 1900s or 1500s; 5. add the day of the month; 6. add the month key value, namely 144 025 036 146 for JFM AMJ JAS OND. The remainder after division by 7 gives the day of the week, with the correspondence 1-23-4-5-6-0 meaning Sunday-Monday-Tuesday-Wednesday-Thursday-Friday-Saturday.* When to start counting the years is a matter of choice. The oldest method not attached to political power structures was that used in ancient Greece, when years were counted from the first Olympic games. People used to say, for example, that they were born in the first year of the twenty-third Olympiad. Later, political powers always imposed the counting of years from some important event onwards.** Maybe reintroducing the Olympic

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counting is worth considering? Abbreviations and eponyms or concepts? Sentences like the following are the scourge of modern physics: The EPR paradox in the Bohm formulation can perhaps be resolved using the GRW approach, using the WKB approximation of the Schrödinger equation.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1180

which it changed to Latin vocabulary with British/American grammar. Many units of measurement also date from Roman times, as explained in the next appendix. Even the infatuation with Greek technical terms, as shown in coinages such as ‘gyroscope’, ‘entropy’ or ‘proton’, dates from Roman times.

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Ref. 1181

Using such vocabulary is the best way to make language unintelligible to outsiders. First of all, it uses abbreviations, which is a shame. On top of this, the sentence uses people’s names to characterize concepts, i.e. it uses eponyms. Originally, eponyms were intended as tributes to outstanding achievements. Today, when formulating radical new laws or variables has become nearly impossible, the spread of eponyms intelligible to a steadily decreasing number of people simply reflects an increasingly ineffective drive to fame. Eponyms are a proof of scientist’s lack of imagination. We avoid them as much as possible in our walk and give common names to mathematical equations or entities wherever possible. People’s names are then used as appositions to these names. For example, ‘Newton’s equation of motion’ is never called ‘Newton’s equation’; ‘Einstein’s field equations’ is used instead of ‘Einstein’s equations’; and ‘Heisenberg’s equation of motion’ is used instead of ‘Heisenberg’s equation’. However, some exceptions are inevitable: certain terms used in modern physics have no real alternatives. The Boltzmann constant, the Planck scale, the Compton wavelength, the Casimir effect, Lie groups and the Virasoro algebra are examples. In compensation, the text makes sure that you can look up the definitions of these concepts using the index. In addition, it tries to provide pleasurable reading.

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Biblio graphy 1168 For a clear overview of the various sign conventions in general relativity, see the front cover

of Charles W. Misner, Kip S. Thorne & John A. Wheeler, Gravitation, Freeman, 1973. We use the gravitational sign conventions of Hans C. Ohanian & R emo Ruffini, Gravitazione e spazio-tempo, Zanichelli, 1997. Cited on page 1142. 1169 For more information about the letters thorn and eth, have a look at the extensive report

to be found on the website http://www.everytype.com/standards/wynnyogh/thorn.html. Cited on page 1143.

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1170 For a modern history of the English language, see David Crystal, The Stories of English,

Allen Lane, 2004. Cited on page 1143. 1171 Hans Jensen, Die Schrift, Berlin, 1969, translated into English as Sign, Symbol and Script:

an Account of Man’s Efforts to Write, Putnam’s Sons, 1970. Cited on page 1144. 1172 David R. Lide, editor, CRC Handbook of Chemistry and Physics, 78th edition, CRC Press,

1173 See the mighty text by Peter T. Daniels & William Bright, The World’s Writing

Systems, Oxford University Press, 1996. Cited on page 1146. 1174 See for example the article ‘Mathematical notation’ in the Encyclopedia of Mathematics,

10 volumes, Kluwer Academic Publishers, 1988–1993. But first all, have a look at the informative and beautiful http://members.aol.com/jeff570/mathsym.html website. The main source for all these results is the classic and extensive research by Florian Cajori, A History of Mathematical Notations, 2 volumes, The Open Court Publishing Co., 1928–1929. The square root sign is used in Christoff Rud olff, Die Coss, Vuolfius Cephaleus Joanni Jung: Argentorati, 1525. (The full title was Behend vnnd Hubsch Rechnung durch die kunstreichen regeln Algebre so gemeinlicklich die Coss genent werden. Darinnen alles so treülich an tag gegeben, das auch allein auss vleissigem lesen on allen mündtliche vnterricht mag begriffen werden, etc.) Cited on page 1147. 1175 J. Tschichold, Formenwamdlungen der et-Zeichen, Stempel AG, 1953. Cited on page 1176 Malcolm B. Parkes, Pause and Effect: An Introduction to the History of Punctuation

in the West, University of California Press, 1993. Cited on page 1149. 1177 This is explained by Berthold Louis Ullman, Ancient Writing and its Influence, 1932.

Cited on page 1149. 1178 Paul Lehmann, Erforschung des Mittelalters – Ausgewählte Abhandlungen und Aufsätze,

Anton Hiersemann, 1961, pp. 4–21. Cited on page 1149. 1179 Bernard Bischoff, Paläographie des römischen Altertums und des abendländischen

Mittelalters, Erich Schmidt Verlag, 1979, pp. 215–219. Cited on page 1149. 1180 The connections between Greek roots and many French words – and thus many English

ones – can be used to rapidly build up a vocabulary of ancient Greek without much study, as shown by the practical collection by J. Chaineux, Quelques racines grecques, Wetteren

Copyright © Christoph Schiller November 1997–May 2006

1149.

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1997. This classic reference work appears in a new edition every year. The full Hebrew alphabet is given on page 2-90. The list of abbreviations of physical quantities for use in formulae approved by ISO, IUPAP and IUPAC can also be found there. However, the ISO 31 standard, which defines these abbreviations, costs around a thousand euro, is not available on the internet, and therefore can safely be ignored, like any standard that is supposed to be used in teaching but is kept inaccessible to teachers. Cited on pages 1146 and 1147.

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– De Meester, 1929. See also Donald M. Ayers, English Words from Latin and Greek Elements, University of Arizona Press, 1986. Cited on page 1151. 1181 To write well, read William Strunk & E.B. White, The Elements of Style, Macmillan, 1935, 1979, or Wolf Schneider, Deutsch für Kenner – Die neue Stilkunde, Gruner und Jahr, 1987. Cited on page 1151.

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Appendix B

UNI T S, MEA SUR EMENT S AND C ONSTANT S

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M

Copyright © Christoph Schiller November 1997–May 2006

* The respective symbols are s, m, kg, A, K, mol and cd. The international prototype of the kilogram is a

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Ref. 1182

easurements are comparisons with standards. Standards are based on a unit. any different systems of units have been used throughout the world. ost standards confer power to the organization in charge of them. Such power can be misused; this is the case today, for example in the computer industry, and was so in the distant past. The solution is the same in both cases: organize an independent and global standard. For units, this happened in the eighteenth century: to avoid misuse by authoritarian institutions, to eliminate problems with differing, changing and irreproducible standards, and – this is not a joke – to simplify tax collection, a group of scientists, politicians and economists agreed on a set of units. It is called the Système International d’Unités, abbreviated SI, and is defined by an international treaty, the ‘Convention du Mètre’. The units are maintained by an international organization, the ‘Conférence Générale des Poids et Mesures’, and its daughter organizations, the ‘Commission Internationale des Poids et Mesures’ and the ‘Bureau International des Poids et Mesures’ (BIPM), which all originated in the times just before the French revolution. All SI units are built from seven base units, whose official definitions, translated from French into English, are given below, together with the dates of their formulation: ‘The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.’ (1967)* ‘The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.’ (1983) ‘The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram.’ (1901)* ‘The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 ë 10−7 newton per metre of length.’ (1948) ‘The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.’ (1967)* ‘The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12.’ (1971)* ‘The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 ë 1012 hertz and has a radiant intensity in that direction of (1/683) watt per steradian.’ (1979)*

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platinum–iridium cylinder kept at the BIPM in Sèvres, in France. For more details on the levels of the caesium atom, consult a book on atomic physics. The Celsius scale of temperature θ is defined as: θ~°C = T~K−273.15; note the small difference with the number appearing in the definition of the kelvin. SI also states: ‘When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.’ In the definition of the mole, it is understood that the carbon 12 atoms are unbound, at rest and in their ground state. In the definition of the candela, the frequency of the light corresponds to 555.5 nm, i.e. green colour, around the wavelength to which the eye is most sensitive. * Jacques Babinet (1794–1874), French physicist who published important work in optics.

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Ref. 1183

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Note that both time and length units are defined as certain properties of a standard example of motion, namely light. This is an additional example making the point that the observation of motion as the fundamental type of change is a prerequisite for the definition and construction of time and space. By the way, the use of light in the definitions had been proposed already in 1827 by Jacques Babinet.* From these basic units, all other units are defined by multiplication and division. Thus, all SI units have the following properties: SI units form a system with state-of-the-art precision: all units are defined with a precision that is higher than the precision of commonly used measurements. Moreover, the precision of the definitions is regularly being improved. The present relative uncertainty of the definition of the second is around 10−14 , for the metre about 10−10 , for the kilogram about 10−9 , for the ampere 10−7 , for the mole less than 10−6 , for the kelvin 10−6 and for the candela 10−3 . SI units form an absolute system: all units are defined in such a way that they can be reproduced in every suitably equipped laboratory, independently, and with high precision. This avoids as much as possible any misuse by the standard-setting organization. (The kilogram, still defined with help of an artefact, is the last exception to this requirement; extensive research is under way to eliminate this artefact from the definition – an international race that will take a few more years. There are two approaches: counting particles, or fixing ħ. The former can be achieved in crystals, the latter using any formula where ħ appears, such as the formula for the de Broglie wavelength or that of the Josephson effect.) SI units form a practical system: the base units are quantities of everyday magnitude. Frequently used units have standard names and abbreviations. The complete list includes the seven base units, the supplementary units, the derived units and the admitted units. The supplementary SI units are two: the unit for (plane) angle, defined as the ratio of arc length to radius, is the radian (rad). For solid angle, defined as the ratio of the subtended area to the square of the radius, the unit is the steradian (sr). The derived units with special names, in their official English spelling, i.e. without capital letters and accents, are:

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Name hertz pascal watt volt ohm weber henry lumen becquerel sievert

Hz = 1~s Pa = N~m2 = kg~m s2 W = kg m2 ~s3 V = kg m2 ~As3 Ω = V~A = kg m2 ~A2 s3 Wb = Vs = kg m2 ~As2 H = Vs~A = kg m2 ~A2 s2 lm = cd sr Bq = 1~s Sv = J~kg = m2 ~s2

Name

A b b r e v i at i o n N = kg m~s2 J = Nm = kg m2 ~s2 C = As F = As~V = A2 s4 ~kg m2 S = 1~Ω T = Wb~m2 = kg~As2 = kg~Cs °C (see definition of kelvin) lx = lm~m2 = cd sr~m2 Gy = J~kg = m2 ~s2 kat = mol~s

newton joule coulomb farad siemens tesla degree Celsius lux gray katal

We note that in all definitions of units, the kilogram only appears to the powers of 1, 0 and -1. The final explanation for this fact appeared only recently. Can you try to formulate the reason? The admitted non-SI units are minute, hour, day (for time), degree 1X = π~180 rad, minute 1′ = π~10 800 rad, second 1′′ = π~648 000 rad (for angles), litre and tonne. All other units are to be avoided. All SI units are made more practical by the introduction of standard names and abbreviations for the powers of ten, the so-called prefixes:* Power Name Power Name

Power Name

101 102 103 106 109 1012 1015

1018 Exa 1021 Zetta 1024 Yotta unofficial: 1027 Xenta 1030 Wekta 1033 Vendekta 36 10 Udekta

deca da hecto h kilo k Mega M Giga G Tera T Peta P

10−1 10−2 10−3 10−6 10−9 10−12 10−15

deci d centi c milli m micro µ nano n pico p femto f

Power Name E Z Y

10−18 10−21 10−24

atto zepto yocto

a z y

xenno weko vendeko udeko

x w v u

Ref. 1184

X W V U

10−27 10−30 10−33 10−36

Challenge 1525 e

* Some of these names are invented (yocto to sound similar to Latin octo ‘eight’, zepto to sound similar to Latin septem, yotta and zetta to resemble them, exa and peta to sound like the Greek words for six and five, the unofficial ones to sound similar to the Greek words for nine, ten, eleven and twelve); some are from Danish/Norwegian (atto from atten ‘eighteen’, femto from femten ‘fifteen’); some are from Latin (from mille ‘thousand’, from centum ‘hundred’, from decem ‘ten’, from nanus ‘dwarf ’); some are from Italian (from piccolo ‘small’); some are Greek (micro is from µικρός ‘small’, deca/deka from δέκα ‘ten’, hecto from ἑκατόν ‘hundred’, kilo from χίλιοι ‘thousand’, mega from µέγας ‘large’, giga from γίγας ‘giant’, tera from τέρας ‘monster’). Translate: I was caught in such a traffic jam that I needed a microcentury for a picoparsec and that my car’s fuel consumption was two tenths of a square millimetre.

Copyright © Christoph Schiller November 1997–May 2006

SI units form a complete system: they cover in a systematic way the complete set of observables of physics. Moreover, they fix the units of measurement for all other sciences as well.

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Challenge 1524 ny

A b b r e v i at i o n

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b units, measurements and constants

Page 1104

1157

SI units form a universal system: they can be used in trade, in industry, in commerce, at home, in education and in research. They could even be used by extraterrestrial civilizations, if they existed. SI units form a coherent system: the product or quotient of two SI units is also an SI unit. This means that in principle, the same abbreviation, e.g. ‘SI’, could be used for every unit. The SI units are not the only possible set that could fulfil all these requirements, but they are the only existing system that does so.* Remember that since every measurement is a comparison with a standard, any measurement requires matter to realize the standard (yes, even for the speed standard), and radiation to achieve the comparison. The concept of measurement thus assumes that matter and radiation exist and can be clearly separated from each other.

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Planck’s natural units

TA B L E 83 Planck’s (uncorrected) natural units

Name

Definition

Basic units the Planck length

l Pl

the Planck time

t Pl

the Planck mass

m Pl

the Planck current

I Pl

the Planck temperature

TPl

=

=

=

=

¼

¼

¼

¼

ħG~c 3 ħG~c 5 ħc~G

4πε 0 c ~G 6

ħc 5 ~Gk 2

= = = = =

1.616 0(12) ë 10−35 m 5.390 6(40) ë 10−44 s 21.767(16) µg 3.479 3(22) ë 1025 A 1.417 1(91) ë 1032 K

* Most non-SI units still in use in the world are of Roman origin. The mile comes from milia passum, which used to be one thousand (double) strides of about 1480 mm each; today a nautical mile, once defined as minute of arc on the Earth’s surface, is exactly 1852 m). The inch comes from uncia/onzia (a twelfth – now of a foot). The pound (from pondere ‘to weigh’) is used as a translation of libra – balance – which is the origin of its abbreviation lb. Even the habit of counting in dozens instead of tens is Roman in origin. These and all other similarly funny units – like the system in which all units start with ‘f ’, and which uses furlong/fortnight as its unit of velocity – are now officially defined as multiples of SI units. ** The natural units x Pl given here are those commonly used today, i.e. those defined using the constant ħ, and not, as Planck originally did, by using the constant h = 2πħ. The electromagnetic units can also be defined with other factors than 4πε 0 in the expressions: for example, using 4πε 0 α, with the fine structure constant α, gives q Pl = e. For the explanation of the numbers between brackets, the standard deviations, see page 1164.

Copyright © Christoph Schiller November 1997–May 2006

=

¼

Va l u e

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Challenge 1526 e

Since the exact form of many equations depends on the system of units used, theoretical physicists often use unit systems optimized for producing simple equations. The chosen units and the values of the constants of nature are related. In microscopic physics, the system of Planck’s natural units is frequently used. They are defined by setting c = 1, ħ = 1, G = 1, k = 1, ε 0 = 1~4π and µ 0 = 4π. Planck units are thus defined from combinations of fundamental constants; those corresponding to the fundamental SI units are given in Table 83.** The table is also useful for converting equations written in natural units back to SI units: just substitute every quantity X by X~X Pl .

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b units, measurements and constants

1158

Name

Definition

Trivial units the Planck velocity the Planck angular momentum the Planck action the Planck entropy

v Pl L Pl S aPl S ePl

Composed units the Planck energy

E Pl

the Planck momentum

p Pl

the Planck force the Planck power

FPl PPl

the Planck acceleration

a Pl

the Planck frequency

f Pl

the Planck electric charge

q Pl

the Planck voltage

U Pl

the Planck resistance

R Pl

the Planck capacitance

C Pl

the Planck inductance

L Pl

the Planck electric field

E Pl

the Planck magnetic flux density

B Pl

=

=

=

= =

=

=

=

=

=

=

=

=

=

c 5 ~G 2 ħ ¼ ħc 5 ~G ¼ ħc 3 ~G

c 4 ~G c 5 ~G ¼ ¼

¼

¼

c 7 ~ħG c 5 ~ħG

4πε 0 cħ

c 4 ~4πε 0 G

1~4πε¼ 0c 4πε 0 ħG~c 3 ¼ (1~4πε 0 ) ħG~c 7 ¼ c 7 ~4πε 0 ħG 2 ¼ c 5 ~4πε 0 ħG 2

= = = =

=

= =

= =

= = = =

=

= = = =

0.3 Gm~s 1.1 ë 10−34 Js 1.1 ë 10−34 Js 13.8 yJ~K

Dvipsbugw 5.2 ë 1096 kg~m3

2.0 GJ = 1.2 ë 1028 eV 6.5 Ns

1.2 ë 1044 N 3.6 ë 1052 W

5.6 ë 1051 m~s2 1.9 ë 1043 Hz

1.9 aC = 11.7 e 1.0 ë 1027 V 30.0 Ω

1.8 ë 10−45 F 1.6 ë 10−42 H 6.5 ë 1061 V~m 2.2 ë 1053 T

The natural units are important for another reason: whenever a quantity is sloppily called ‘infinitely small (or large)’, the correct expression is ‘as small (or as large) as the corresponding corrected Planck unit’. As explained throughout the text, and especially in the third part, this substitution is possible because almost all Planck units provide, within a correction factor of order 1, the extremal value for the corresponding observable – some an upper and some a lower limit. Unfortunately, these correction factors are not yet widely known. The exact extremal value for each observable in nature is obtained when G is substituted by 4G, ħ by ħ~2, k by k~2 and 4πε 0 by 8πε 0 α in all Planck quantities. These extremal values, or corrected Planck units, are the true natural units. To exceeding the extremal values is possible only for some extensive quantities. (Can you find out which ones?) Other unit systems A central aim of research in high-energy physics is the calculation of the strengths of all interactions; therefore it is not practical to set the gravitational constant G to unity, as in the Planck system of units. For this reason, high-energy physicists often only set

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1527 n

ρ Pl

c ħ ħ k

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Page 1076

the Planck mass density

= = = =

Va l u e

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b units, measurements and constants

1159

c = ħ = k = 1 and µ 0 = 1~ε 0 = 4π,* leaving only the gravitational constant G in the equations. In this system, only one fundamental unit exists, but its choice is free. Often a standard length is chosen as the fundamental unit, length being the archetype of a measured quantity. The most important physical observables are then related by

1 eV 

Challenge 1529 e

Ref. 1187

Ref. 1186 Challenge 1528 n

aJ

(811)

which is easily remembered. The simplification c = ħ = 1 yields G = 6.9 ë 10−57 eV−2 and allows one to use the unit eV also for mass, momentum, temperature, frequency, time and length, with the respective correspondences 1 eV  1.8 ë 10−36 kg  5.4 ë 10−28 Ns  242 THz  11.6 kK and 1 eV−1  4.1 fs  1.2 µm. To get some feeling for the unit eV, the following relations are useful. Room temperature, usually taken as 20°C or 293 K, corresponds to a kinetic energy per particle of 0.025 eV or 4.0 zJ. The highest particle energy measured so far belongs to a cosmic ray with an energy of 3 ë 1020 eV or 48 J. Down here on the Earth, an accelerator able to produce an energy of about 105 GeV or 17 nJ for electrons and antielectrons has been built, * Other definitions for the proportionality constants in electrodynamics lead to the Gaussian unit system often used in theoretical calculations, the Heaviside–Lorentz unit system, the electrostatic unit system, and the electromagnetic unit system, among others. ** In the list, l is length, E energy, F force, E electric the electric and B the magnetic field, m mass, p momentum, a acceleration, f frequency, I electric current, U voltage, T temperature, v speed, q charge, R resistance, P power, G the gravitational constant. The web page http://www.chemie.fu-berlin.de/chemistry/general/units_en.html provides a tool to convert various units into each other. Researchers in general relativity often use another system, in which the Schwarzschild radius rs = 2Gm~c 2 is used to measure masses, by setting c = G = 1. In this case, mass and length have the same dimension, and ħ has the dimension of an area. Already in the nineteenth century, George Stoney had proposed to use as length, time and mass units » 2 4 » 2 6 the Ge ~(c 4πε 0 ) = 1.4 ë 10−36 m, t S = Ge ~(c 4πε 0 ) = 4.6 ë 10−45 s and m S = » 2quantities l S = e ~(G4πε 0 ) = 1.9 µg. How are these units related to the Planck units?

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1185

1 6

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1~[l 2 ] = [E]2 = [F] = [B] = [E electric ] , 1~[l] = [E] = [m] = [p] = [a] = [ f ] = [I] = [U] = [T] , 1 = [v] = [q] = [e] = [R] = [S action ] = [S entropy ] = ħ = c = k = [α] , [l] = 1~[E] = [t] = [C] = [L] and [l]2 =1~[E]2 = [G] = [P] (810) where we write [x] for the unit of quantity x. Using the same unit for time, capacitance and inductance is not to everybody’s taste, however, and therefore electricians do not use this system.** Often, in order to get an impression of the energies needed to observe an effect under study, a standard energy is chosen as fundamental unit. In particle physics the most common energy unit is the electronvolt (eV), defined as the kinetic energy acquired by an electron when accelerated by an electrical potential difference of 1 volt (‘protonvolt’ would be a better name). Therefore one has 1 eV = 1.6 ë 10−19 J, or roughly

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1160

Ref. 1188

b units, measurements and constants

and one able to produce an energy of 10 TeV or 1.6 µJ for protons will be finished soon. Both are owned by CERN in Geneva and have a circumference of 27 km. The lowest temperature measured up to now is 280 pK, in a system of rhodium nuclei held inside a special cooling system. The interior of that cryostat may even be the coolest point in the whole universe. The kinetic energy per particle corresponding to that temperature is also the smallest ever measured: it corresponds to 24 feV or 3.8 vJ = 3.8 ë 10−33 J. For isolated particles, the record seems to be for neutrons: kinetic energies as low as 10−7 eV have been achieved, corresponding to de Broglie wavelengths of 60 nm.

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Curiosities and fun challenges about units Here are a few facts making the concept of physical unit more vivid. **

**

Ref. 1189

A gray is the amount of radioactivity that deposits 1 J on 1 kg of matter. A sievert is the unit of radioactivity adjusted to humans by weighting each type of human tissue with a factor representing the impact of radiation deposition on it. Four to five sievert are a lethal dose to humans. In comparison, the natural radioactivity present inside human bodies leads to a dose of 0.2 mSv per year. An average X-ray image implies an irradiation of 1 mSv; a CAT scan 8 mSv. **

* This story revived an old (and false) urban legend that states that only three countries in the world do not use SI units: Liberia, the USA and Myanmar.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1530 e

Are you confused by the candela? The definition simply says that 683 cd = 683 lm~sr corresponds to 1 W~sr. The candela is thus a unit for light power per (solid) angle, except that it is corrected for the eye’s sensitivity: the candela measures only visible power per angle. Similarly, 683 lm = 683 cd sr corresponds to 1 W. So both the lumen and the watt measure power, or energy flux, but the lumen measures only the visible part of the power. This difference is expressed by inserting ‘radiant’ or ‘luminous’: thus, the Watt measures radiant flux, whereas the lumen measures luminous flux. The factor 683 is historical. An ordinary candle emits a luminous intensity of about a candela. Therefore, at night, a candle can be seen up to a distance of 10 or 20 kilometres. A 100 W incandescent light bulb produces 1700 lm, and the brightest light emitting diodes about 5 lm. Cinema projectors produce around 2 Mlm, and the brightest flashes, like lightning, 100 Mlm. The irradiance of sunlight is about 1300 W~m2 on a sunny day; on the other hand, the illuminance is only 120 klm~m2 = 120 klux or 170 W~m2 . (A cloud-covered summer day or a clear winter day produces about 10 klux.) These numbers show that most of the energy from the Sun that reaches the Earth is outside the visible spectrum.

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Not using SI units can be expensive. In 1999, NASA lost a satellite on Mars because some software programmers had used imperial units instead of SI units in part of the code. As a result, the Mars Climate Orbiter crashed into the planet, instead of orbiting it; the loss was around 100 million euro.*

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b units, measurements and constants

Ref. 1190

1161

On a glacier, near the sea shore, on the top of a mountain, or in particular weather condition the brightness can reach 150 klux. The brightest lamps, those used during surgical operations, produce 120 klux. Humans need about 30 lux for comfortable reading. Museums are often kept dark because water-based paintings are degraded by light above 100 lux, and oil paintings by light above 200 lux. The full moon produces 0.1 lux, and the sky on a dark moonless night about 1 mlux. The eyes lose their ability to distinguish colours somewhere between 0.1 lux and 0.01 lux; the eye stops to work below 1 nlux. Technical devices to produce images in the dark, such as night goggles, start to work at 1 µlux. By the way, the human body itself shines with about 1 plux, a value too small to be detected with the eye, but easily measured with specialized apparatus. The origin of this emission is still a topic of research.

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**

** The luminous density is a quantity often used by light technicians. Its unit is 1 cd~m2 , unofficially called 1 Nit and abbreviated 1 nt. Eyes see purely with rods from 0.1 µcd~m2 to 1 mcd~m2 ; they see purely with cones above 5 cd~m2 ; they see best between 100 and 50 000 cd~m2 ; and they get completely overloaded above 10 Mcd~m2 : a total range of 15 orders of magnitude. **

Ref. 1192

The Planck length is roughly the de Broglie wavelength λ B = h~mv of a man walking comfortably (m = 80 kg, v = 0.5 m~s); this motion is therefore aptly called the ‘Planck stroll.’ ** The Planck mass is equal to the mass of about 1019 protons. This is roughly the mass of a human embryo at about ten days of age.

The second does not correspond to 1/86 400th of the day any more, though it did in the year 1900; the Earth now takes about 86 400.002 s for a rotation, so that the International Earth Rotation Service must regularly introduce a leap second to ensure that the Sun is at the highest point in the sky at 12 o’clock sharp.* The time so defined is called Universal Time Coordinate. The speed of rotation of the Earth also changes irregularly from day to day due to the weather; the average rotation speed even changes from winter to summer because of the changes in the polar ice caps; and in addition that average decreases over * Their website at http://hpiers.obspm.fr gives more information on the details of these insertions, as does http://maia.usno.navy.mil, one of the few useful military websites. See also http://www.bipm.fr, the site of the BIPM.

Copyright © Christoph Schiller November 1997–May 2006

**

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Ref. 1191

The highest achieved light intensities are in excess of 1018 W~m2 , more than 15 orders of magnitude higher than the intensity of sunlight. They are produced by tight focusing of pulsed lasers. The electric field in such light pulses is of the same order as the field inside atoms; such a beam therefore ionizes all matter it encounters.

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1162

b units, measurements and constants

time, because of the friction produced by the tides. The rate of insertion of leap seconds is therefore higher than once every 500 days, and not constant in time. ** The most precisely measured quantities in nature are the frequencies of certain millisecond pulsars,* the frequency of certain narrow atomic transitions, and the Rydberg constant of atomic hydrogen, which can all be measured as precisely as the second is defined. The caesium transition that defines the second has a finite linewidth that limits the achievable precision: the limit is about 14 digits.

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** Ref. 1193 Ref. 1194

** Ref. 1195 Challenge 1531 n

The shortest times measured are the lifetimes of certain ‘elementary’ particles. In particular, the lifetime of certain D mesons have been measured at less than 10−23 s. Such times are measured using a bubble chamber, where the track is photographed. Can you estimate how long the track is? (This is a trick question – if your length cannot be observed with an optical microscope, you have made a mistake in your calculation.) **

Ref. 1196

The longest times encountered in nature are the lifetimes of certain radioisotopes, over 1015 years, and the lower limit of certain proton decays, over 1032 years. These times are thus much larger than the age of the universe, estimated to be fourteen thousand million years. **

**

Challenge 1532 n

The precision of mass measurements of solids is limited by such simple effects as the adsorption of water. Can you estimate the mass of a monolayer of water – a layer with thickness of one molecule – on a metal weight of 1 kg?

* An overview of this fascinating work is given by J.H. Taylor, Pulsar timing and relativistic gravity, Philosophical Transactions of the Royal Society, London A 341, pp. 117–134, 1992.

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Page 1167

The least precisely measured of the fundamental constants of physics are the gravitational constant G and the strong coupling constant α s . Even less precisely known are the age of the universe and its density (see Table 86).

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The most precise clock ever built, using microwaves, had a stability of 10−16 during a running time of 500 s. For longer time periods, the record in 1997 was about 10−15 ; but values around 10−17 seem within technological reach. The precision of clocks is limited for short measuring times by noise, and for long measuring times by drifts, i.e. by systematic effects. The region of highest stability depends on the clock type; it usually lies between 1 ms for optical clocks and 5000 s for masers. Pulsars are the only type of clock for which this region is not known yet; it certainly lies at more than 20 years, the time elapsed at the time of writing since their discovery.

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b units, measurements and constants

1163

**

Ref. 1197

Ref. 1198

Variations of quantities are often much easier to measure than their values. For example, in gravitational wave detectors, the sensitivity achieved in 1992 was ∆l~l = 3 ë 10−19 for lengths of the order of 1 m. In other words, for a block of about a cubic metre of metal it is possible to measure length changes about 3000 times smaller than a proton radius. These set-ups are now being superseded by ring interferometers. Ring interferometers measuring frequency differences of 10−21 have already been built; and they are still being improved.

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**

Ref. 1199

** In the previous millennium, thermal energy used to be measured using the unit calorie, written as cal. 1 cal is the energy needed to heat 1 g of water by 1 K. To confuse matters, 1 kcal was often written 1 Cal. (One also spoke of a large and a small calorie.) The value of 1 kcal is 4.1868 kJ. ** SI units are adapted to humans: the values of heartbeat, human size, human weight, human temperature and human substance are no more than a couple of orders of magnitude near the unit value. SI units thus (roughly) confirm what Protagoras said 25 centuries ago: ‘Man is the measure of all things.’ **

Challenge 1534 n

The table of SI prefixes covers 72 orders of magnitude. How many additional prefixes will be needed? Even an extended list will include only a small part of the infinite range of possibilities. Will the Conférence Générale des Poids et Mesures have to go on forever, defining an infinite number of SI prefixes? Why?

The French philosopher Voltaire, after meeting Newton, publicized the now famous story that the connection between the fall of objects and the motion of the Moon was discovered by Newton when he saw an apple falling from a tree. More than a century later, just before the French Revolution, a committee of scientists decided to take as the unit of force precisely the force exerted by gravity on a standard apple, and to name it after the English scientist. After extensive study, it was found that the mass of the standard apple was 101.9716 g; its weight was called 1 newton. Since then, visitors to the museum in Sèvres near Paris have been able to admire the standard metre, the standard kilogram and the standard apple.* * To be clear, this is a joke; no standard apple exists. It is not a joke however, that owners of several apple

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**

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Challenge 1533 n

The Swedish astronomer Anders Celsius (1701–1744) originally set the freezing point of water at 100 degrees and the boiling point at 0 degrees. Later the scale was reversed. However, this is not the whole story. With the official definition of the kelvin and the degree Celsius, at the standard pressure of 1013.25 Pa, water boils at 99.974°C. Can you explain why it is not 100°C any more?

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b units, measurements and constants

1164 N number of measurements

standard deviation full width at half maximum (FWHM)

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limit curve for a large number of measurements

x average value

x measured values

Precision and accuracy of measurements As explained on page 264, precision means how well a result is reproduced when the measurement is repeated; accuracy is the degree to which a measurement corresponds to the actual value. Lack of precision is due to accidental or random errors; they are best measured by the standard deviation, usually abbreviated σ; it is defined through σ2 =

Challenge 1535 n

1 n 2 Q(x i − x¯ ) , n − 1 i=1

where x¯ is the average of the measurements x i . (Can you imagine why n − 1 is used in the formula instead of n?) For most experiments, the distribution of measurement values tends towards a normal distribution, also called Gaussian distribution, whenever the number of measurements is increased. The distribution, shown in Figure 390, is described by the expression

Ref. 1201

Ref. 1200

¯ 2 (x− x) 2σ 2

.

(813)

The square σ 2 of the standard deviation is also called the variance. For a Gaussian distribution of measurement values, 2.35σ is the full width at half maximum. Lack of accuracy is due to systematic errors; usually these can only be estimated. This estimate is often added to the random errors to produce a total experimental error, sometimes also called total uncertainty. The following tables give the values of the most important physical constants and particle properties in SI units and in a few other common units, as published in the standtrees in Britain and in the US claim descent, by rerooting, from the original tree under which Newton had his insight. DNA tests have even been performed to decide if all these derive from the same tree. The result was, unsurprisingly, that the tree at MIT, in contrast to the British ones, is a fake.

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N(x)  e−

Challenge 1536 e

(812)

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F I G U R E 390 A precision experiment and its measurement distribution

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b units, measurements and constants

Ref. 1202

Challenge 1537 ny Page 1035

1165

Basic physical constants Ref. 1202

In principle, all quantitative properties of matter can be calculated with quantum theory. For example, colour, density and elastic properties, can be predicted using the values of the following constants using the equations of the standard model of high-energy physics. TA B L E 84 Basic physical constants

U n c e r t. a

number of space-time dimensions

3+1

0b

vacuum speed of light c vacuum permeabilityc

299 792 458 m~s 4π ë 10−7 H~m = 1.256 637 061 435 ... µH~m 8.854 187 817 620 ... pF~m 6.626 068 76(52) ë 10−34 Js 1.054 571 596(82) ë 10−34 Js 0.160 217 646 2(63) aC 1.380 650 3(24) ë 10−23 J~K 6.673(10) ë 10−11 Nm2 ~kg2 2.076(3) ë 10−43 s2 ~kg m 1~137.035 999 76(50)

0 0 0 0 7.8 ë 10−8 7.8 ë 10−8 3.9 ë 10−8 1.7 ë 10−6 1.5 ë 10−3 1.5 ë 10−3 3.7 ë 10−9

vacuum permittivityc original Planck constant reduced Planck constant positron charge Boltzmann constant gravitational constant gravitational coupling constant fine structure constant,d e.m. coupling constant

Symbol

c µ0 ε 0 = 1~µ 0 c 2 h ħ e k G κ = 8πG~c 4 2 α = 4πεe 0 ħc

= дem (m e2 c 2 )

= 0.007 297 352 533(27)

3.7 ë 10−9

* Some of the stories can be found in the text by N.W. Wise, The Values of Precision, Princeton University Press, 1994. The field of high-precision measurements, from which the results on these pages stem, is a world on its own. A beautiful introduction to it is J.D. Fairbanks, B.S. Deaver, C.W. Everitt & P.F. Michaelson, eds., Near Zero: Frontiers of Physics, Freeman, 1988.

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Va l u e i n S I u n i t s

Q ua n t i t y

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ard references. The values are the world averages of the best measurements made up to the present. As usual, experimental errors, including both random and estimated systematic errors, are expressed by giving the standard deviation in the last digits; e.g. 0.31(6) means – roughly speaking – 0.31 0.06. In fact, behind each of the numbers in the following tables there is a long story which is worth telling, but for which there is not enough room here.* What are the limits to accuracy and precision? There is no way, even in principle, to measure a length x to a precision higher than about 61 digits, because ∆x~x A l Pl ~d horizon = 10−61 . (Is this valid also for force or for volume?) In the third part of our text, studies of clocks and metre bars strengthen this theoretical limit. But it is not difficult to deduce more stringent practical limits. No imaginable machine can measure quantities with a higher precision than measuring the diameter of the Earth within the smallest length ever measured, about 10−19 m; that is about 26 digits of precision. Using a more realistic limit of a 1000 m sized machine implies a limit of 22 digits. If, as predicted above, time measurements really achieve 17 digits of precision, then they are nearing the practical limit, because apart from size, there is an additional practical restriction: cost. Indeed, an additional digit in measurement precision often means an additional digit in equipment cost.

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b units, measurements and constants

1166

Q ua n t i t y Fermi coupling constant,d weak coupling constant weak mixing angle weak mixing angle strong coupling constantd

Page 860

Symbol G F ~(ħc)3 α w (M Z ) = дw2 ~4π sin2 θ W (MS) sin2 θ W (on shell) = 1 − (m W ~m Z )2 α s (M Z ) = дs2 ~4π

Va l u e i n S I u n i t s

U n c e r t. a

1.166 39(1) ë 10−5 GeV−2 1~30.1(3) 0.231 24(24) 0.2224(19)

8.6 ë 10−6 1 ë 10−2 1.0 ë 10−3 8.7 ë 10−3

0.118(3)

25 ë 10−3

Why do all these constants have the values they have? The answer is different in each case. For any constant having a dimension, such as the quantum of action ħ, the numerical value has only historical meaning. It is 1.054 ë 10−34 Js because of the SI definition of the joule and the second. The question why the value of a dimensional constant is not larger or smaller always requires one to understand the origin of some dimensionless number giving the ratio between the constant and the corresponding natural unit. Understanding the sizes of atoms, people, trees and stars, the duration of molecular and atomic processes, or the mass of nuclei and mountains, implies understanding the ratios between these values and the corresponding natural units. The key to understanding nature is thus the understanding of all ratios, and thus of all dimensionless constants. The story of the most important ratios is told in the third part of the adventure. The basic constants yield the following useful high-precision observations. TA B L E 85 Derived physical constants

Q ua n t i t y

Va l u e i n S I u n i t s

U n c e r t.

376.730 313 461 77... Ω 6.022 141 99(47) ë 1023 10 973 731.568 549(83) m−1 77.480 916 96(28) µS 2.067 833 636(81) pWb 483.597 898(19) THz~V 25 812.807 572(95) Ω 9.274 008 99(37) yJ~T 27.992 4925(11) GHz~T

0 7.9 ë 10−8 7.6 ë 10−12 3.7 ë 10−9 3.9 ë 10−8 3.9 ë 10−8 3.7 ë 10−9 4.0 ë 10−8 4.0 ë 10−8

r e = e 2 ~4πε 0 m e c 2 2.817 940 285(31) fm λ c = h~m e c 2.426 310 215(18) pm

1.1 ë 10−8 7.3 ë 10−9

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Vacuum wave resistance Avogadro’s number Rydberg constant a conductance quantum magnetic flux quantum Josephson frequency ratio von Klitzing constant Bohr magneton cyclotron frequency of the electron classical electron radius Compton wavelength

Symbol » Z 0 = µ 0 ~ε 0 NA Rª = m e cα 2 ~2h G 0 = 2e 2 ~h φ 0 = h~2e 2e~h h~e 2 = µ 0 c~2α µ B = eħ~2m e f c ~B = e~2πm e

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a. Uncertainty: standard deviation of measurement errors. b. Only down to 10−19 m and up to 1026 m. c. Defining constant. d. All coupling constants depend on the 4-momentum transfer, as explained in the section on renormalization. Fine structure constant is the traditional name for the electromagnetic coupling constant дem in the case of a 4-momentum transfer of Q 2 = m e2 c 2 , which is the smallest one possible. At higher 2 2 momentum transfers it has larger values, e.g. дem (Q 2 = M W c )  1~128. The strong coupling constant has higher values at lower momentum transfers; e.g., α s (34 GeV) = 0.14(2).

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b units, measurements and constants

Q ua n t i t y of the electron Bohr radius a nuclear magneton proton–electron mass ratio Stefan–Boltzmann constant Wien displacement law constant bits to entropy conversion const. TNT energy content

Symbol λ c = ħ~m e c = r e ~α aª = r e ~α 2 µ N = eħ~2m p m p ~m e σ = π 2 k 4 ~60ħ 3 c 2 b = λ max T

1167

Va l u e i n S I u n i t s

U n c e r t.

0.386 159 264 2(28) pm 52.917 720 83(19) pm 5.050 783 17(20) ë 10−27 J~T 1 836.152 667 5(39) 56.704 00(40) nW~m2 K4 2.897 768 6(51) mmK 1023 bit = 0.956 994 5(17) J~K 3.7 to 4.0 MJ~kg

7.3 ë 10−9 3.7 ë 10−9 4.0 ë 10−8 2.1 ë 10−9 7.0 ë 10−6 1.7 ë 10−6 1.7 ë 10−6 4 ë 10−2

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a. For infinite mass of the nucleus.

Some properties of nature at large are listed in the following table. (If you want a challenge, can you determine whether any property of the universe itself is listed?) TA B L E 86 Astrophysical constants

Q ua n t i t y

Symbol

Va l u e

Copyright © Christoph Schiller November 1997–May 2006

gravitational constant G 6.672 59(85) ë 10−11 m3 ~kg s2 cosmological constant Λ c. 1 ë 10−52 m−2 tropical year 1900 a a 31 556 925.974 7 s tropical year 1994 a 31 556 925.2 s mean sidereal day d 23 h 56′ 4.090 53′′ light year al 9.460 528 173 ... Pm b astronomical unit AU 149 597 870.691(30) km parsec pc 30.856 775 806 Pm = 3.261 634 al c age of the universe t0 4.333(53) ë 1017 s = 13.73(0.17) ë 109 a (determined from space-time, via expansion, using general relativity) age of the universe c t0 over 3.5(4) ë 1017 s = 11.5(1.5) ë 109 a (determined from matter, via galaxies and stars, using quantum theory) Hubble parameter c H0 2.3(2) ë 10−18 s−1 = 0.73(4) ë 10−10 a−1 H 0 = h 0 ë 100 km~sMpc = h 0 ë 1.0227 ë 10−10 a−1 reduced Hubble parameter c h0 0.71(4) ¨ 0 ~H 02 −0.66(10) deceleration parameter q 0 = −( a~a) universe’s horizon distance c d 0 = 3ct 0 40.0(6) ë 1026 m = 13.0(2) Gpc universe’s topology unknown number of space dimensions 3, for distances up to 1026 m 2 h 02 ë 1.878 82(24) ë 10−26 kg~m3 critical density ρ c = 3H 0 ~8πG of the universe = 0.95(12) ë 10−26 kg~m3 c (total) density parameter Ω 0 = ρ 0 ~ρ c 1.02(2) c baryon density parameter Ω B0 = ρ B0 ~ρ c 0.044(4) cold dark matter density parameter c Ω CDM0 = ρ CDM0 ~ρ c 0.23(4) neutrino density parameter c Ω ν0 = ρ ν0 ~ρ c 0.001 to 0.05

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Challenge 1538 n

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b units, measurements and constants

1168

Q ua n t i t y

Symbol Ω X0 = ρ X0 ~ρ c w = p X ~ρ X

dark energy density parameter c dark energy state parameter baryon mass baryon number density luminous matter density stars in the universe baryons in the universe

mb

ns nb

Planck length Planck time Planck mass instants in history c space-time points inside the horizon c mass inside horizon

º S º T σ8 n dn~d ln k ¼ l Pl = ħG~c 3 ¼ t Pl = ħG~c 5 » m Pl = ħc~G t 0 ~t Pl N 0 = (R 0 ~l Pl )3 ë (t 0 ~t Pl ) M

1.67 ë 10−27 kg 0.25(1) ~m3 3.8(2) ë 10−28 kg~m3 10221 10811 2.725(1) K 1089 4.6 ë 10−31 kg~m3 410.89 ~cm3 or 400 ~cm3 (T0 ~2.7 K)3 5.6(1.5) ë 10−6 º < 0.71 S 0.84(4) 0.93(3) -0.03(2) 1.62 ë 10−35 m 5.39 ë 10−44 s 21.8 µg 8.7(2.8) ë 1060 102441 10541 kg

a. Defining constant, from vernal equinox to vernal equinox; it was once used to define the second. (Remember: π seconds is about a nanocentury.) The value for 1990 is about 0.7 s less, corresponding to a slowdown of roughly 0.2 ms~a. (Watch out: why?) There is even an empirical formula for the change of the length of the year over time. b. Average distance Earth–Sun. The truly amazing precision of 30 m results from time averages of signals sent from Viking orbiters and Mars landers taken over a period of over twenty years. c. The index 0 indicates present-day values. d. The radiation originated when the universe was 380 000 years old and had a temperature of about 3000 K; the fluctuations ∆T0 which led to galaxy formation are today about 16  4 µK = 6(2) ë 10−6 T0 .

Warning: in the third part of this text it is shown that many of the constants in Table 86 are not physically sensible quantities. They have to be taken with many grains of salt. The more specific constants given in the following table are all sensible, though. TA B L E 87 Astronomical constants

Q ua n t i t y

Symbol

Va l u e

Earth’s mass

M

5.972 23(8) ë 1024 kg

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Copyright © Christoph Schiller November 1997–May 2006

Page 451

T0 nγ ρ γ = π 2 k 4 ~15T04

0.73(4) −1.0(2)

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microwave background temperature d photons in the universe photon energy density photon number density density perturbation amplitude gravity wave amplitude mass fluctuations on 8 Mpc scalar index running of scalar index

Challenge 1539 n Ref. 1203

Va l u e

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b units, measurements and constants

Q ua n t i t y Earth’s gravitational length Earth radius, equatorial a Earth’s radius, polar a Equator–pole distance a Earth’s flattening a Earth’s av. density Earth’s age Moon’s radius Moon’s radius Moon’s mass Moon’s mean distance b Moon’s distance at perigee b

Moon’s angular size c Moon’s average density Sun’s mass Sun’s gravitational length Sun’s luminosity solar radius, equatorial Sun’s angular size

Va l u e

l = 2GM~c 2 R eq Rp

8.870(1) mm 6378.1367(1) km 6356.7517(1) km 10 001.966 km (average) e 1~298.25231(1) ρ 5.5 Mg~m3 4.55 Ga = 143 Ps R mv 1738 km in direction of Earth R mh 1737.4 km in other two directions Mm 7.35 ë 1022 kg dm 384 401 km typically 363 Mm, hist. minimum 359 861 km typically 404 Mm, hist. maximum 406 720 km average 0.5181X = 31.08′ , minimum 0.49X , maximum - shortens line 0.55X ρ 3.3 Mg~m3 Mb 1.988 43(3) ë 1030 kg 2 lb = 2GMb ~c 2.953 250 08 km Lb 384.6 YW Rb 695.98(7) Mm 0.53X average; minimum on fourth of July (aphelion) 1888′′ , maximum on fourth of January (perihelion) 1952′′ ρ 1.4 Mg~m3 AU 149 597 870.691(30) km 4.6 Ga v bg 220(20) km~s 370.6(5) km~s 8.0(5) kpc = 26.1(1.6) kal 13.6 Ga c. 1021 m or 100 kal 1012 solar masses, c. 2 ë 1042 kg 1.90 ë 1027 kg 71.398 Mm 67.1(1) Mm 778 412 020 km 13.2 ë 109 al = 1.25 ë 1026 m, red-shift 10

Copyright © Christoph Schiller November 1997–May 2006

Sun’s average density Sun’s average distance Sun’s age solar velocity around centre of galaxy solar velocity v bb against cosmic background distance to galaxy centre Milky Way’s age Milky Way’s size Milky Way’s mass Jupiter’s mass M Jupiter’s radius, equatorial R Jupiter’s radius, polar R Jupiter’s average distance from Sun D most distant galaxy known 1835 IR1916

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Moon’s distance at apogee b

Symbol

1169

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b units, measurements and constants

1170

a. The shape of the Earth is described most precisely with the World Geodetic System. The last edition dates from 1984. For an extensive presentation of its background and its details, see the http://www.wgs84.com website. The International Geodesic Union refined the data in 2000. The radii and the flattening given here are those for the ‘mean tide system’. They differ from those of the ‘zero tide system’ and other systems by about 0.7 m. The details constitute a science in itself. b. Measured centre to centre. To find the precise position of the Moon at a given date, see the http://www. fourmilab.ch/earthview/moon_ap_per.html page. For the planets, see the page http://www.fourmilab.ch/ solar/solar.html and the other pages on the same site. c. Angles are defined as follows: 1 degree = 1X = π~180 rad, 1 (first) minute = 1′ = 1X ~60, 1 second (minute) = 1′′ = 1′ ~60. The ancient units ‘third minute’ and ‘fourth minute’, each 1~60th of the preceding, are not in use any more. (‘Minute’ originally means ‘very small’, as it still does in modern English.)

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Useful numbers

Challenge 1540 n

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 375105 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 699959 0.57721 56649 01532 86060 65120 90082 40243 10421 59335 939923 0.69314 71805 59945 30941 72321 21458 17656 80755 00134 360255 2.30258 50929 94045 68401 79914 54684 36420 76011 01488 628772 3.16227 76601 68379 33199 88935 44432 71853 37195 55139 325216

If the number π is normal, i.e. if all digits and digit combinations in its decimal expansion appear with the same limiting frequency, then every text ever written or yet to be written, as well as every word ever spoken or yet to be spoken, can be found coded in its sequence. The property of normality has not yet been proven, although it is suspected to hold. Does this mean that all wisdom is encoded in the simple circle? No. The property is nothing special: it also applies to the number 0.123456789101112131415161718192021... and many others. Can you specify a few examples? By the way, in the graph of the exponential function ex , the point (0, 1) is the only point with two rational coordinates. If you imagine painting in blue all points on the plane with two rational coordinates, the plane would look quite bluish. Nevertheless, the graph goes through only one of these points and manages to avoid all the others.

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Ref. 1204

π e γ ln 2 ln 10 º 10

Copyright © Christoph Schiller November 1997–May 2006

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bibliography

1171

Biblio graphy 1182 Le Système International d’Unités, Bureau International des Poids et Mesures, Pavillon de

Breteuil, Parc de Saint Cloud, 92310 Sèvres, France. All new developments concerning SI units are published in the journal Metrologia, edited by the same body. Showing the slow pace of an old institution, the BIPM launched a website only in 1998; it is now reachable at http://www.bipm.fr. See also the http://www.utc.fr/~tthomass/Themes/Unites/index.html website; this includes the biographies of people who gave their names to various units. The site of its British equivalent, http://www.npl.co.uk/npl/reference, is much better; it provides many details as well as the English-language version of the SI unit definitions. Cited on page 1154.

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1183 The bible in the field of time measurement is the two-volume work by J. Vanier & C.

1184 The unofficial prefixes were first proposed in the 1990s by Jeff K. Aronson of University of

Oxford, and might come into general usage in the future. Cited on page 1156. 1185 For more details on electromagnetic unit systems, see the standard text by John David

Jackson, Classical Electrodynamics, 3rd edition, Wiley, 1998. Cited on page 1159. 1186 G.J. Stoney, On the physical units of nature, Philosophical Magazine 11, pp. 381–391,

1881. Cited on page 1159. 1187 D.J. Bird & al., Evidence for correlated changes in the spectrum and composition of

cosmic rays at extremely high energies, Physical Review Letters 71, pp. 3401–3404, 1993. Cited on page 1159. 1188 P.J. Hakonen, R.T. Vuorinen & J.E. Martikainen, Nuclear antiferromagnetism

in rhodium metal at positive and negative nanokelvin temperatures, Physical Review Letters 70, pp. 2818–2821, 1993. See also his article in Scientific American, January 1994. Cited on page 1160. 1189 G. Charpak & R.L. Garwin, The DARI, Europhysics News 33, pp. 14–17, Janu-

ary/February 2002. Cited on page 1160. 1190 Measured values and ranges for physical quantities are collected in Horst Völz &

1191 See e.g. K. Codling & L.J. Frasinski, Coulomb explosion of simple molecules in in-

tense laser fields, Contemporary Physics 35, pp. 243–255, 1994. Cited on page 1161. 1192 A. Zeilinger, The Planck stroll, American Journal of Physics 58, p. 103, 1990. Can you Challenge 1541 e

find another similar example? Cited on page 1161. 1193 The most precise clock built in 2004, a caesium fountain clock, had a precision of one

part in 1015 . Higher precision has been predicted to be possible soon, among others by M. Takamoto, F.-L. Hong, R. Higashi & H. Katori, An optical lattice clock, Nature 435, pp. 321–324, 2005. Cited on page 1162. 1194 J. Bergquist, ed., Proceedings of the Fifth Symposium on Frequency Standards and Met-

rology, World Scientific, 1997. Cited on page 1162.

Copyright © Christoph Schiller November 1997–May 2006

Peter Ackermann, Die Welt in Zahlen, Spektrum Akademischer Verlag, 1996. Cited on page 1161.

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Aud oin, The Quantum Physics of Atomic Frequency Standards, Adam Hilge, 1989. A popular account is Tony Jones, Splitting the Second, Institute of Physics Publishing, 2000. The site http://opdaf1.obspm.fr/www/lexique.html gives a glossary of terms used in the field. On length measurements, see ... On electric current measurements, see ... On mass and atomic mass measurements, see page 314. On high-precision temperature measurements, see page 270. Cited on page 1155.

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b units, measurements and constants

1172

1195 See the information on Ds mesons from the particle data group at http://pdg.web.cern.ch/

pdg. Cited on page 1162.

1196 About the long life of tantalum 180, see D. Belic & al., Photoactivation of 180 Tam and its

implications for the nucleosynthesis of nature’s rarest naturally occurring isotope, Physical Review Letters 83, pp. 5242–5245, 20 December 1999. Cited on page 1162. 1197 See the review by L. Ju, D.G. Bl air & C. Zhao, The detection of gravitational waves,

Reports on Progress in Physics 63, pp. 1317–1427, 2000. Cited on page 1163. 1198 See the clear and extensive paper by G.E. Stedman, Ring laser tests of fundamental

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physics and geophysics, Reports on Progress in Physics 60, pp. 615–688, 1997. Cited on page 1163. 1199 Following a private communication by Richard Rusby, this is the value of 1997, whereas it

1200 J. Short, Newton’s apples fall from grace, New Scientist, 2098, p. 5, 6 September 1997.

More details can be found in R.G. Keesing, The history of Newton’s apple tree, Contemporary Physics 39, pp. 377–391, 1998. Cited on page 1164. 1201 The various concepts are even the topic of a separate international standard, ISO 5725, with

the title Accuracy and precision of measurement methods and results. A good introduction is John R. Taylor, An Introduction to Error Analysis: the Study of Uncertainties in Physical Measurements, 2nd edition, University Science Books, Sausalito, 1997. Cited on page 1164. 1202 P.J. Mohr & B.N. Taylor, CODATA recommended values of the fundamental physical

constants: 1998, Reviews of Modern Physics 59, p. 351, 2000. This is the set of constants resulting from an international adjustment and recommended for international use by the Committee on Data for Science and Technology (CODATA), a body in the International Council of Scientific Unions, which brings together the International Union of Pure and Applied Physics (IUPAP), the International Union of Pure and Applied Chemistry (IUPAC) and other organizations. The website of IUPAC is http://www.iupac.org. Cited on page 1165. 1203 The details are given in the well-known astronomical reference, P. Kenneth Seidel-

mann, Explanatory Supplement to the Astronomical Almanac, 1992. Cited on page 1168. 1204 For information about the number π, as well as about other constants, the website http://

or the beautiful formula discovered in 1996 by Bailey, Borwein and Plouffe 1 4 2 1 1 ( − − − ). n 8n + 1 16 8n + 4 8n + 5 8n +6 n=0

π=Q ª

(815)

The site also explains the newly discovered methods for calculating specific binary digits of π without having to calculate all the preceding ones. By the way, the number of (consecutive) digits known in 1999 was over 1.2 million million, as told in Science News 162, p. 255, 14 December 2002. They pass all tests of randomness, as the http://mathworld.wolfram.com/ PiDigits.html website explains. However, this property, called normality, has never been

Copyright © Christoph Schiller November 1997–May 2006

oldweb.cecm.sfu.ca/pi/pi.html provides lots of data and references. It also has a link to the overview at http://mathworld.wolfram.com/Pi.html and to many other sites on the topic. Simple formulae for π are ª n 2n (814) π + 3 = Q 2n n=1 ‰ n Ž

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was estimated as 99.975°C in 1989, as reported by Gareth Jones & R ichard Rusby, Official: water boils at 99.975°C, Physics World 2, pp. 23–24, September 1989, and R.L. Rusby, Ironing out the standard scale, Nature 338, p. 1169, March 1989. For more details on temperature measurements, see page 270. Cited on page 1163.

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bibliography

Challenge 1542 r Challenge 1543 n

1173

proven; it is the biggest open question about π. It is possible that the theory of chaotic dynamics will lead to a solution of this puzzle in the coming years. Another method to calculate π and other constants was discovered and published by D.V. Chudnovsky & G.V. Chudnovsky, The computation of classical constants, Proceedings of the National Academy of Sciences (USA) 86, pp. 8178–8182, 1989. The Chudnowsky brothers have built a supercomputer in Gregory’s apartment for about 70 000 euros, and for many years held the record for calculating the largest number of digits of π. They have battled for decades with Kanada Yasumasa, who held the record in 2000, calculated on an industrial supercomputer. New formulae to calculate π are still occasionally discovered. For the calculation of Euler’s constant γ see also D.W. DeTemple, A quicker convergence to Euler’s constant, The Mathematical Intelligencer, pp. 468–470, May 1993. Note that little is known about the basic properties of some numbers; for example, it is still not known whether π + e is a rational number or not! (It is believed that it is not.) Do you want to become a mathematician? Cited on page 1170.

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Appendix C

PARTIC LE PROPERTI ES

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T

he following table lists the known and predicted elementary particles. he list has not changed since the mid-1970s, mainly because of the inefficient use hat was made of the relevant budgets across the world.

Radiation

electromagnetic interaction

weak interaction

strong interaction

γ

W+ , W−

д1 ... д8

Z0 photon

intermediate vector bosons

gluons

Radiation particles are bosons with spin 1. W − is the antiparticle of W + ; all others are their own antiparticles. Matter

generation 1

generation 2

generation 3

Leptons

e

µ

τ

νe

νµ

ντ

Quarks

d

s

t

(each in three colours)

u

c

b

Hypothetical matter and radiation Higgs boson Supersymmetric partners

H se ...

predicted to have spin 0 and to be elementary

h

one partner for each of the above

Copyright © Christoph Schiller November 1997–May 2006

Matter particles are fermions with spin 1/2; all have a corresponding antiparticle.

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TA B L E 88 The elementary particles

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c particle properties

1175

The following table lists all properties for the elementary particles (for reasons of space, the colour quantity is not given explicitely).* The table and its header thus allow us, in principle, to deduce a complete characterization of the intrinsic properties of any composed moving entity, be it an object or an image. TA B L E 89 Elementary particle properties

Particle

Mass m

a

photon γ

0 (<10−52 kg)

W

80.425(38) GeV~c 2

Z

91.1876(21) GeV~c 2

gluon

0

Isospin I, spin J, c parity P, charge parity C

I(J PC ) = 0, 1(1−− ) 2.124(41) GeV J = 1 67.96(35)% hadrons, 32.04(36)% l + ν J=1 2.4952(43) GeV~c 2 69.91(6)% hadrons 10.0974(69)% l+l− stable I(J P ) = 0(1− ) stable

Charge, isospin, strangeness, c charm, beau ty, topness: QISCBT

Lepton & baryon e numbers L B, Rparity

000000

0, 0, 1

100000

0, 0, 1

000000

0, 0, 1

000000

0, 0, 1

Elementary matter (fermions): leptons electron e

−100000 2.197 03(4) µs J = 21 99 % e − ν¯e ν µ = 105.658 3568(52) MeV~c 2 = 0.113 428 9168(34) u gyromagnetic ratio µ µ ~(eħ~2m µ ) = −1.001 165 916 02(64)

0.188 353 109(16) yg

1, 0, 1

1, 0, 1

* The official reference for all these data, worth a look for every physicist, is the massive collection of information compiled by the Particle Data Group, with the website http://pdg.web.cern.ch/pdg containing the most recent information. A printed review is published about every two years, with updated data, in one of the major journals on elementary particle physics. See for example S. Eidelman & al., Review of Particle Physics, Physics Letters B 592, p. 1, 2004. For many measured properties of these particles, the official reference is the set of CODATA values. The most recent list was published by P.J. Mohr & B.N. Taylor, Reviews of Modern Physics 59, p. 351, 2000.

Copyright © Christoph Schiller November 1997–May 2006

muon µ

−100 000 9.109 381 88(72) ë A 13 ë 1030 s J = 21 −31 2 10 kg = 81.871 0414(64) pJ~c = 0.510 998 902(21) MeV~c 2 = 0.000 548 579 9110(12) u gyromagnetic ratio µ e ~µ B = −1.001 159 652 1883(42) electric dipole moment d = (−0.3  0.8) ë 10−29 e m f

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Elementary radiation (bosons)

Lifetime τ or energy width, b main decay modes

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c particle properties

1176

Particle

tau τ

Mass m

a

Lifetime τ or energy width, b main decay modes

electric dipole moment d = (3.7  3.4) ë 10−22 e m 1.777 05(29) GeV~c 2

290.0(1.2) fs

J=

J=

Elementary matter (fermions): quarks д 1.5 to 5 MeV~c 2 3 to 9 MeV~c 2 60 to 170 MeV~c 2 1.25(15) GeV~c 2 4.25(15) GeV~c 2 173.8(5.2) GeV~c 2

J=

see proton see proton τ = 1.33(11) ps

Hypothetical, maybe elementary (boson)

−100000

1, 0, 1

1 2

1, 0, 1

1 2

1, 0, 1

I(J P ) = 21 ( 21 ) 23 21 0000 + I(J P ) = 21 ( 21 ) − 31 − 21 0000 + I(J P ) = 0( 21 ) − 31 0−1000 + I(J P ) = 0( 21 ) 23 00+100 + I(J P ) = 0( 21 ) − 31 000−10 + I(J P ) = 0( 21 ) 23 0000+1 +

J J J J J

J J J J J

=0 =0 =0 =0 =0 = = = = =

1 2 1 2 1 2 1 2 1 2

0 31 1 0 31 1 0 31 1 0 31 1 0 31 1 0 31 1

R = −1 R = −1 R = −1 R = −1 R = −1 R = −1 R = −1 R = −1 R = −1 R = −1

Notes: a. See also the table of SI prefixes on page 1155. About the eV~c 2 mass unit, see page 1159. b. The energy width Γ of a particle is related to its lifetime τ by the indeterminacy relation Γτ = ħ. There

Copyright © Christoph Schiller November 1997–May 2006

Selectron h Smuon h Stauon h Sneutrinos h Squark h Hypothetical elementary matter (fermions)

1, 0, 1

1 2

Hypothetical elementary radiation (bosons)

Higgsino(s) h Wino h (a chargino) Zino h (a neutralino) Photino h Gluino h

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1 2

J=0

Higgs h H

A 79.3 GeV~c 2

J=

Lepton & baryon e numbers L B, Rparity

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el. neutrino < 3 eV~c 2 νe muon < 0.19 MeV~c 2 neutrino ν µ tau neutrino < 18.2 MeV~c 2 ντ up u down d strange s charm c bottom b top t

Charge, isospin, strangeness, c charm, beau ty, topness: QISCBT

Isospin I, spin J, c parity P, charge parity C

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c particle properties

Challenge 1544 n

is a difference between the half-life t 1~2 and the lifetime τ of a particle: they are related by t 1~2 = τ ln 2, where ln 2  0.693 147 18; the half-life is thus shorter than the lifetime. The unified atomic mass unit u is defined as 1/12 of the mass of a carbon 12 atom at rest and in its ground state. One has 1 u = 121 m(12 C) = 1.660 5402(10) yg. c. To keep the table short, the header does not explicitly mention colour, the charge of the strong interactions. This has to be added to the list of basic object properties. Quantum numbers containing the word ‘parity’ are multiplicative; all others are additive. Time parity T (not to be confused with topness T), better called motion inversion parity, is equal to CP. The isospin I (or I Z ) is defined only for up and down quarks and their composites, such as the proton and the neutron. In the literature one also sees references to the so-called G-parity, defined as G = (−1)IC . d. ‘Beauty’ is now commonly called bottomness; similarly, ‘truth’ is now commonly called topness. The signs of the quantum numbers S, I, C, B, T can be defined in different ways. In the standard assignment shown here, the sign of each of the non-vanishing quantum numbers is given by the sign of the charge of the corresponding quark. e. R-parity is a quantum number important in supersymmetric theories; it is related to the lepton number L, the baryon number B and the spin J through the definition R = (−1)3B+L+2J . All particles from the standard model are R-even, whereas their superpartners are odd. f . The electron radius is less than 10−22 m. It is possible to store single electrons in traps for many months. д. See page 933 for the precise definition and meaning of the quark masses. h. Currently a hypothetical particle.

Using the table of elementary particle properties, together with the standard model and the fundamental constants, in principle all properties of composite matter and radiation can be deduced, including all those encountered in everyday life. (Can you explain how the size of an object follows from them?) In a sense, this table contains all our knowledge of matter and radiation, including materials science, chemistry and biology. The most important examples of composites are grouped in the following table. TA B L E 90 Properties of selected composites

C o m p o si t e

M a ss m, q ua n t u m L i f e t i m e τ , m a i n numbers d e c ay m o d e s

mesons (hadrons, bosons) (selected from over 130 known types) º ¯ pion π 0 (u u¯ − d d)~ 2 134.976 4(6) MeV~c 2 84(6) as, 2γ 98.798(32)% ¯ pion π+ (u d)

kaon K S0 kaon K L0 ¯ kaon K (u¯s , us) 

PC

−+

m K S0 + 3.491(9) µeV~c 2

493.677(16) MeV~c

2

51.7(4) ns

12.386(24) ns, µ + ν µ 63.51(18)% π+ π 0 21.16(14)% n.a.

kaon K 0 (d¯s) (50 % K S , 50 % 497.672(31) MeV~c 2 KL ) all kaons K  , K 0 , K S0 , K L0 I(J P ) = 21 (0− ), S = 1, B = C = 0

baryons (hadrons, fermions) (selected from over 100 known types)

 1 fm

 1 fm  1 fm

 1 fm

 1 fm

 1 fm

Copyright © Christoph Schiller November 1997–May 2006

I (J ) = 1 (0 ), S = C = B = 0 139.569 95(35) MeV~c 2 26.030(5) ns, µ + ν µ 99.987 7(4)% G P − − I (J ) = 1 (0 ), S = C = B = 0 m K S0 89.27(9) ps G

Size (diam.)

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Ref. 1205, Ref. 1206

1177

Dvipsbugw


c particle properties

1178

C o m p o si t e proton p or N + (uud)

neutron n or N 0 (udd)

1.672 621 58(13) yg τ total A 1.6 ë 1025 a, = 1.007 276 466 88(13) u τ(p e + π 0 ) A5.5 ë 1032 a = 938.271 998(38) MeV~c 2 + I(J P ) = 21 ( 21 ), S = 0 gyromagnetic ratio µ p ~µ N = 2.792 847 337(29) electric dipole moment d = (−4  6) ë 10−26 e m electric polarizability α e = 12.1(0.9) ë 10−4 fm3 magnetic polarizability α m = 2.1(0.9) ë 10−4 fm3

1.674 927 16(13) yg 887.0(2.0) s, pe − ν¯e 100 % = 1.008 664 915 78(55) u = 939.565 330(38) MeV~c 2 + I(J P ) = 21 ( 21 ), S = 0 gyromagnetic ratio µ n ~µ N = −1.913 042 72(45) electric dipole moment d n = (−3.3  4.3) ë 10−28 e m electric polarizability α = 0.98(23) ë 10−3 fm3 1672.43(32) MeV~c 2 82.2(1.2) ps, ΛK − 67.8(7)%, Ξ 0 π− 23.6(7)% gyromagnetic ratio µ Ω ~µ N = −1.94(22)

composite radiation: glueballs glueball f 0 (1500)

1503(11) MeV I G (J PC ) = 0+ (0++ )

full width 120(19) MeV

atoms (selected from 114 known elements with over 2000 known nuclides) 1

1.007 825 032(1) u = 1.6735 yg 1.007 u = 1.67 yg 4.002 603250(1) u = 6.6465 yg 12 u = 19.926 482(12) yg 209 u 0.1 ps 180 u

 1 fm

 1 fm

 1 fm

Ref. 1208

Ref. 1209

223 u 289 u

22 min 0.6 ms

A 1025 a A 1025 a

Dvipsbugw

Ref. 1210

1.9(2)1019 a

 2u  18 u

Ref. 1207

2 ë 30 pm 2 ë 30 pm 2 ë 32 pm 2 ë 77 pm

209 u

molecules (selected from over 107 known types) hydrogen (H2 ) water (H2 O)

A 1015 a

0.89(1) fm

Ref. 1210

2 ë 0.28 nm

Copyright © Christoph Schiller November 1997–May 2006

hydrogen ( H) [lightest] antihydrogen helium (4 He) [smallest] carbon (12 C) bismuth (209 Bi‡ ) [shortest living and rarest] tantalum (180 Ta) [second longest living radioactive] bismuth (209 Bi) [longest living radioactive] francium (223 Fr) [largest] atom 116 (289 Uuh) [heaviest]

Size (diam.)

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omega Ω− (sss)

M a ss m, q ua n t u m L i f e t i m e τ , m a i n numbers d e c ay m o d e s

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c particle properties

1179

C o m p o si t e

M a ss m, q ua n t u m L i f e t i m e τ , m a i n numbers d e c ay m o d e s

ATP

507 u

(adenosinetriphosphate) human Y chromosome

70 ë 106 base pairs

other composites

 50 a 7 plus 120 days not fecundated:  5 d

cell (ovule)

1 µg

cell (E. coli) adult human

1 pg 35 kg < m < 350 kg

fecundated: over 4000 million years 4000 million years body: 2 µm 9 τ  2.5 ë 10 s Ref. 1211  1.7 m  600 million breaths  2 500 million heartbeats < 122 a, 60 % H2 O and 40 % dust A 130 a A 4 km

Notes (see also those of the previous table): G-parity is defined only for mesons and given by G = (−1) L+S+I = (−1) I C. In 2003, experiments provided candidates for tetraquarks, namely the X(3872), Ds(2317), f(980) and Ds(2460), and for pentaquarks, namely the Θ+ (1500) particle. Time will tell whether these interpretations are correct. Neutrons bound in nuclei have a lifetime of at least 1020 years. The f 0 (1500) resonance is a candidate for the glueball ground state and thus for a radiation composite. The Y (3940) resonance is a candidate for a hybrid meson, a composite of a gluon and a quark– antiquark pair. This prediction of 1980 seems to have been confirmed in 2005. In 2002, the first evidence for the existence of tetra-neutrons was published by a French group. However, more recent investigations seem to have refuted the claim. The number of existing molecules is several orders of magnitude larger than the number of molecules that have been analysed and named. Some nuclei have not yet been observed; in 2006 the known nuclei ranged from 1 to 116, but 113 and 115 were still missing. The first anti-atoms, made of antielectrons and antiprotons, were made in January 1996 at CERN in Geneva. All properties of antimatter checked so far are consistent with theoretical predictions. The charge parity C is defined only for certain neutral particles, namely those that are different from their antiparticles. For neutral mesons, the charge parity is given by C = (−1) L+S , where L

Dvipsbugw

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1214

See the table on page 185.

20 m  10 µm length 60 µm, head 3 µm5 µm  120 µm

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Ref. 1213

c. 50 mm (uncoiled)

 1 kg 0.1 ng 10 pg

larger composites

Ref. 1212

A 106 a

c. 3 nm

blue whale nerve cell cell (red blood) cell (sperm)

heaviest living thing: colony 6.6 ë 106 kg of aspen trees

Page 1176

A 1010 a

Size (diam.)

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1180

c particle properties

is the orbital angular momentum. P is the parity under space inversion r −r. For mesons, it is related to the orbital angular momentum L through P = (−1) L+1 . The electric polarizability, defined on page 594, is predicted to vanish for all elementary particles.

Copyright © Christoph Schiller November 1997–May 2006

* The ‘average formula’ of life is approximately C5 H40 O18 N.

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Ref. 1217

The most important matter composites are the atoms. Their size, structure and interactions determine the properties and colour of everyday objects. Atom types, also called elements in chemistry, are most usefully set out in the so-called periodic table, which groups together atoms with similar properties in rows and columns. It is given in Table 91 and results from the various ways in which protons, neutrons and electrons can combine to form aggregates. Comparable to the periodic table of the atoms, there are tables for the mesons (made of two quarks) and the baryons (made of three quarks). Neither the meson nor the baryon table is included here; they can both be found in the cited Review of Particle Physics at http://pdg.web.cern.ch/pdg. In fact, the baryon table still has a number of vacant spots. However, the missing particles are extremely heavy and short-lived (which means expensive to make and detect), and their discovery is not expected to yield deep new insights. The atomic number gives the number of protons (and electrons) found in an atom of a given element. This number determines the chemical behaviour of an element. Most – but not all – elements up to 92 are found on Earth; the others can be produced in laboratories. The highest element discovered is element 116. (In a famous case of research fraud, a scientist in the 1990s tricked two whole research groups into claiming to have made and observed elements 116 and 118. Element 116 was independently made and observed by another group later on.) Nowadays, extensive physical and chemical data are available for every element. Elements in the same group behave similarly in chemical reactions. The periods define the repetition of these similarities. More elaborate periodic tables can be found on the http://chemlab.pc.maricopa.edu/periodic website. The most beautiful of them all can be found on page 838 of this text. Group 1 are the alkali metals (though hydrogen is a gas), group 2 the Earth-alkali metals. Actinoids, lanthanoids and groups 3 to 13 are metals; in particular, groups 3 to 12 are transition or heavy metals. The elements of group 16 are called chalkogens, i.e. oreformers; group 17 are the halogens, i.e. the salt-formers, and group 18 are the inert noble gases, which form (almost) no chemical compounds. The groups 13, 14 and 15 contain metals, semimetals, a liquid and gases; they have no special name. Groups 1 and 13 to 17 are central for the chemistry of life; in fact, 96 % of living matter is made of C, O, N, H;* almost 4 % of P, S, Ca, K, Na, Cl; trace elements such as Mg, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Cd, Pb, Sn, Li, Mo, Se, Si, I, F, As, B form the rest. Over 30 elements are known to be essential for animal life. The full list is not yet known; candidate elements to extend this list are Al, Br, Ge and W. Many elements exist in versions with different numbers of neutrons in their nucleus, and thus with different mass; these various isotopes – so called because they are found at the same place in the periodic table – behave identically in chemical reactions. There are

Dvipsbugw


c particle properties

1181

TA B L E 91 The periodic table of the elements known in 2006, with their atomic numbers

Group 1 I

2 II

3 4 5 6 7 IIIa IVa Va VIa VIIa

8

9 10 VIIIa

11 Ia

12 IIa

13 III

14 IV

15 V

16 VI

17 18 VII VIII

Period 1

2

4

5

6

7

2

H

He

3

4

5

6

7

8

9

10

Li

Be

B

C

N

O

F

Ne

11

12

13

14

15

16

17

18

Al

Si

P

S

Cl

Ar

31

32

33

Na Mg 19

20

21

22

23

24

K

Ca

Sc

Ti

V

Cr Mn Fe

37

38

39

40

41

42

Rb

Sr

Y

Zr

55

56

72

73

74

75

76

77

78

Cs

Ba

‡

Hf

Ta

W

Re

Os

Ir

Pt

104

105

106

107

108

109

110

Rf

Db Sg Bh Hs Mt Ds Uuu Uub Uut Uuq Uup Uuh

87

88

Fr

Ra

‡‡

Lanthanoids ‡ Actinoids

43

Nb Mo Tc

57

58

59

La

Ce

Pr

60

89

90

91

92

Ac

Th

Pa

U

26

44

27

28

Co

Ni

45

46

29

30

Cu Zn Ga Ge As 47

48

62

63

94

95

36

Kr

50

51

52

53

54

In

Sn

Sb

Te

I

Xe

81

82

83

84

85

86

Au Hg

Tl

Pb

Bi

Po

At

Rn

111

113

114

115

116

117

118

79

64

80

112

65

66

67

Nd Pm Sm Eu Gd Tb Dy Ho 93

35

Br

49

Ru Rh Pd Ag Cd

61

34

Se

96

97

Np Pu Am Cm Bk

68

Er 100

69

70

Tm Yb 101

71

Lu

98

99

102

103

Cf

Es Fm Md No

Lr

over 2000 of them. TA B L E 92 The elements, with their atomic number, average mass, atomic radius and main properties

Name

Sym- At. bol n.

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Actiniumb

Ac

(227.0277(1)) 21.77(2) a

(188)

89

highly radioactive metallic rare Earth (Greek aktis ray) 1899, used as alphaemitting source

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1208, Ref. 1215

‡‡

25

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3

1

Dvipsbugw


c particle properties

1182

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Aluminium

Al

13

26.981 538 (8) stable

118c, 143m

Americiumb

Am

95

(184)

Antimony

Sb

51

(243.0614(1)) 7.37(2) ka 121.760(1) f stable

Argon

Ar

18

39.948(1) f stable

(71n)

Arsenic

As

33

74.921 60(2) stable

120c, 185v

Astatineb

At

85

(140)

Barium

Ba

56

(209.9871(1)) 8.1(4) h 137.327(7) stable

Berkeliumb

Bk

97

Beryllium

Be

4

Bismuth

Bi

Bohriumb

137c, 159m, 205v

224m

(247.0703(1)) 1.4(3) ka 9.012 182(3) stable

n.a. 106c, 113m

83

208.980 40(1) stable

170m, 215v

Bh

107

(264.12(1)) 0.44 s д

n.a.

Boron

B

5

10.811(7) f stable

83c

Bromine

Br

35

79.904(1) stable

120c, 185v

light metal (Latin alumen alum) 1827, used in machine construction and living beings radioactive metal (Italian America from Amerigo) 1945, used in smoke detectors toxic semimetal (via Arabic from Latin stibium, itself from Greek, Egyptian for one of its minerals) antiquity, colours rubber, used in medicines, constituent of enzymes noble gas (Greek argos inactive, from anergos without energy) 1894, third component of air, used for welding and in lasers poisonous semimetal (Greek arsenikon tamer of males) antiquity, for poisoning pigeons and doping semiconductors radioactive halogen (Greek astatos unstable) 1940, no use Earth-alkali metal (Greek bary heavy) 1808, used in vacuum tubes, paint, oil industry, pyrotechnics and X-ray diagnosis made in lab, probably metallic (Berkeley, US town) 1949, no use because rare toxic Earth-alkali metal (Greek beryllos, a mineral) 1797, used in light alloys, in nuclear industry as moderator diamagnetic metal (Latin via German weisse Masse white mass) 1753, used in magnets, alloys, fire safety, cosmetics, as catalyst, nuclear industry made in lab, probably metallic (after Niels Bohr) 1981, found in nuclear reactions, no use semimetal, semiconductor (Latin borax, from Arabic and Persian for brilliant) 1808, used in glass, bleach, pyrotechnics, rocket fuel, medicine red-brown liquid (Greek bromos strong odour) 1826, fumigants, photography, water purification, dyes, medicines

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Sym- At. bol n.

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Name

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c particle properties

1183

Name

Sym- At. bol n.

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Cadmium

Cd

112.411(8) f stable

157m

48

55

132.905 4519(2) 273m stable

Calcium

Ca

20

40.078(4) f stable

197m

Californiumb

Cf

98

(251.0796(1)) 0.90(5) ka

n.a.

Carbon

C

6

12.0107(8) f stable

77c

Cerium

Ce

58

140.116(1) f stable

183m

Chlorine

Cl

17

35.453(2) f stable

102c, 175v

Chromium

Cr

24

51.9961(6) stable

128m

Cobalt

Co

27

58.933 195(5) stable

125m

Copper

Cu

29

63.546(3) f stable

128m

Curiumb

Cm

96

(247.0704(1)) 15.6(5) Ma

n.a.

Copyright © Christoph Schiller November 1997–May 2006

Cs

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Caesium

heavy metal, cuttable and screaming (Greek kadmeia, a zinc carbonate mineral where it was discovered) 1817, electroplating, solder, batteries, TV phosphors, dyes alkali metal (Latin caesius sky blue) 1860, getter in vacuum tubes, photoelectric cells, ion propulsion, atomic clocks Earth-alkali metal (Latin calcis chalk) antiquity, pure in 1880, found in stones and bones, reducing agent, alloying made in lab, probably metallic, strong neutron emitter (Latin calor heat and fornicare have sex, the land of hot sex :-) 1950, used as neutron source, for well logging makes up coal and diamond (Latin carbo coal) antiquity, used to build most life forms rare Earth metal (after asteroid Ceres, Roman goddess) 1803, cigarette lighters, incandescent gas mantles, glass manufacturing, self-cleaning ovens, carbon-arc lighting in the motion picture industry, catalyst, metallurgy green gas (Greek chloros yellow-green) 1774, drinking water, polymers, paper, dyes, textiles, medicines, insecticides, solvents, paints, rubber transition metal (Greek chromos colour) 1797, hardens steel, makes steel stainless, alloys, electroplating, green glass dye, catalyst ferromagnetic transition metal (German Kobold goblin) 1694, part of vitamin B12 , magnetic alloys, heavy-duty alloys, enamel dyes, ink, animal nutrition red metal (Latin cuprum from Cyprus island) antiquity, part of many enzymes, electrical conductors, bronze, brass and other alloys, algicides, etc. highly radioactive, silver-coloured (after Pierre and Marie Curie) 1944, used as radioactivity source

Dvipsbugw


c particle properties

1184

Name

Sym- At. bol n.

Darmstadtiumb Ds Dubniumb Db

110 105

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

(271) 1.6 min д n.a. (262.1141(1)) n.a. 34(5) s

66

162.500(1) f stable

177m

Einsteiniumb

Es

99

n.a.

Erbium

Er

68

Europium

Eu

63

(252.0830(1)) 472(2) d 167.259(3) f stable 151.964(1) f stable

Fermiumb

Fm

100

Fluorine

F

9

Franciumb

Fr

87

Gadolinium

Gd

64

Gallium

Ga

31

Germanium

Ge

Gold

176m 204m

(257.0901(1)) n.a. 100.5(2) d 18.998 4032(5) 62c, stable 147v (223.0197(1)) 22.0(1) min 157.25(3) f stable 69.723(1) stable

125c, 141m

32

72.64(1) stable

122c, 195v

Au

79

196.966 569(4) 144m stable

Hafnium

Hf

72

Hassiumb

Hs

108

178.49(2) c 158m stable (277) 16.5 min д n.a.

Helium

He

2

4.002 602(2) f stable

(278) 180m

(31n)

gaseous halogen (from fluorine, a mineral, from Greek fluo flow) 1886, used in polymers and toothpaste radioactive metal (from France) 1939, no use (after Johan Gadolin) 1880, used in lasers and phosphors almost liquid metal (Latin for both the discoverer’s name and his nation, France) 1875, used in optoelectronics semiconductor (from Germania, as opposed to gallium) 1886, used in electronics heavy noble metal (Sanskrit jval to shine, Latin aurum) antiquity, electronics, jewels metal (Latin for Copenhagen) 1923, alloys, incandescent wire radioactive element (Latin form of German state Hessen) 1984, no use noble gas (Greek helios Sun) where it was discovered 1895, used in balloons, stars, diver’s gas and cryogenics

Copyright © Christoph Schiller November 1997–May 2006

Dy

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Dysprosium

(after the German city) 1994, no use made in lab in small quantities, radioactive (Dubna, Russian city) 1967, no use (once known as Hahnium) rare Earth metal (Greek dysprositos difficult to obtain) 1886, used in laser materials, as infrared source material, and in nuclear industry made in lab, radioactive (after Albert Einstein) 1952, no use rare Earth metal (Ytterby, Swedish town) 1843, used in metallurgy and optical fibres rare Earth metal (named after the continent) 1901, used in red screen phosphor for TV tubes (after Enrico Fermi) 1952, no use

Dvipsbugw


c particle properties

1185

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Holmium

Ho

67

177m

Hydrogen

H

1

Indium

In

49

Iodine

I

53

Iridium

Ir

77

Iron

Fe

26

Krypton

Kr

36

Lanthanum

La

57

164.930 32(2) stable 1.007 94(7) f stable 114.818(3) stable 126.904 47(3) stable 192.217(3) stable 55.845(2) stable 83.798(2) f stable 138.905 47(7) c, f

Lawrenciumb

Lr

103

Lead

Pb

82

Lithium

Li

3

6.941(2) f stable

156m

Lutetium

Lu

71

173m

Magnesium

Mg

12

174.967(1) f stable 24.3050(6) stable

Manganese

Mn

25

54.938 045(5) stable

126m

Meitneriumb

Mt

109

n.a.

Mendeleviumb Md

101

(268.1388(1)) 0.070 s д (258.0984(1)) 51.5(3) d

30c 141c, 166m 140c, 198v 136m 127m (88n) 188m

stable (262.110 97(1)) n.a. 3.6(3) h 207.2(1) c, f 175m stable

160m

n.a.

metal (Stockholm, Swedish capital) 1878, alloys reactive gas (Greek for water-former) 1766, used in building stars and universe soft metal (Greek indikon indigo) 1863, used in solders and photocells blue-black solid (Greek iodes violet) 1811, used in photography precious metal (Greek iris rainbow) 1804, electrical contact layers metal (Indo-European ayos metal, Latin ferrum) antiquity, used in metallurgy noble gas (Greek kryptos hidden) 1898, used in lasers reactive rare Earth metal (Greek lanthanein to be hidden) 1839, used in lamps and in special glasses appears in reactions (after Ernest Lawrence) 1961, no use poisonous, malleable heavy metal (Latin plumbum) antiquity, used in car batteries, radioactivity shields, paints light alkali metal with high specific heat (Greek lithos stone) 1817, used in batteries, anti-depressants, alloys and many chemicals rare Earth metal (Latin Lutetia for Paris) 1907, used as catalyst light common alkaline Earth metal (from Magnesia, a Greek district in Thessalia) 1755, used in alloys, pyrotechnics, chemical synthesis and medicine, found in chlorophyll brittle metal (Italian manganese, a mineral) 1774, used in alloys, colours amethyst and permanganate appears in nuclear reactions (after Lise Meitner) 1982, no use appears in nuclear reactions (after Dimitri Ivanovitch Mendeleiev) 1955, no use

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Copyright © Christoph Schiller November 1997–May 2006

Sym- At. bol n.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Name

Dvipsbugw


c particle properties

1186

Name

Sym- At. bol n.

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Mercury

Hg

200.59(2) stable

157m

80

42

95.94(2) f stable

140m

Neodymium

Nd

60

182m

Neon

Ne

10

Neptuniumb

Np

93

144.242(3) c, f stable 20.1797(6) f stable (237.0482(1)) 2.14(1) Ma

Nickel

Ni

28

58.6934(2) stable

125m

Niobium

Nb

41

92.906 38(2) stable

147m

Nitrogen

N

7

14.0067(2) f stable

70c, 155v

Nobeliumb

No

102

n.a.

Osmium

Os

76

(259.1010(1)) 58(5) min 190.23(3) f stable

Oxygen

O

8

15.9994(3) f stable

66c, 152v

Palladium

Pd

46

106.42(1) f stable

138m

Phosphorus

P

15

30.973 762(2) stable

109c, 180v

(36n) n.a.

135m

noble gas (Greek neos new) 1898, used in lamps, lasers and cryogenics radioactive metal (planet Neptune, after Uranus in the solar system) 1940, appears in nuclear reactors, used in neutron detection and by the military metal (German Nickel goblin) 1751, used in coins, stainless steels, batteries, as catalyst ductile metal (Greek Niobe, mythical daughter of Tantalos) 1801, used in arc welding, alloys, jewellery, superconductors diatomic gas (Greek for nitre-former) 1772, found in air, in living organisms, Viagra, fertilizers, explosives (after Alfred Nobel) 1958, no use heavy metal (from Greek osme odour) 1804, used for fingerprint detection and in very hard alloys transparent, diatomic gas (formed from Greek to mean ‘acid former’) 1774, used for combustion, blood regeneration, to make most rocks and stones, in countless compounds, colours auroras red heavy metal (from asteroid Pallas, after the Greek goddess) 1802, used in alloys, white gold, catalysts, for hydride storage poisonous, waxy, white solid (Greek phosphoros light bearer) 1669, fertilizers, glasses, porcelain, steels and alloys, living organisms, bones

Copyright © Christoph Schiller November 1997–May 2006

Mo

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Molybdenum

liquid heavy metal (Latin god Mercurius, Greek hydrargyrum liquid silver) antiquity, used in switches, batteries, lamps, amalgam alloys metal (Greek molybdos lead) 1788, used in alloys, as catalyst, in enzymes and lubricants (Greek neos and didymos new twin) 1885

Dvipsbugw


c particle properties

1187

Name

Sym- At. bol n.

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Platinum

Pt

195.084(9) stable

139m

78

94

(244.0642(1)) 80.0(9) Ma

n.a.

Polonium

Po

84

(208.9824(1)) 102(5) a

(140)

Potassium

K

19

39.0983(1) stable

238m

Praeseodymium Pr

59

140.907 65(2) stable

183m

Promethiumb

Pm

61

(144.9127(1)) 17.7(4) a

181m

Protactinium

Pa

91

(231.035 88(2)) n.a. 32.5(1) ka

Copyright © Christoph Schiller November 1997–May 2006

Pu

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Plutonium

silvery-white, ductile, noble heavy metal (Spanish platina little silver) pre-Columbian, again in 1735, used in corrosion-resistant alloys, magnets, furnaces, catalysts, fuel cells, cathodic protection systems for large ships and pipelines; being a catalyst, a fine platinum wire glows red hot when placed in vapour of methyl alcohol, an effect used in hand warmers extremely toxic alpha-emitting metal (after the planet) synthesized 1940, found in nature 1971, used as nuclear explosive, and to power space equipment, such as satellites and the measurement equipment brought to the Moon by the Apollo missions alpha-emitting, volatile metal (from Poland) 1898, used as thermoelectric power source in space satellites, as neutron source when mixed with beryllium; used in the past to eliminate static charges in factories, and on brushes for removing dust from photographic films reactive, cuttable light metal (German Pottasche, Latin kalium from Arabic quilyi, a plant used to produce potash) 1807, part of many salts and rocks, essential for life, used in fertilizers, essential to chemical industry white, malleable rare Earth metal (Greek praesos didymos green twin) 1885, used in cigarette lighters, material for carbon arcs used by the motion picture industry for studio lighting and projection, glass and enamel dye, darkens welder’s goggles radioactive rare Earth metal (from the Greek mythical figure of Prometheus) 1945, used as beta source and to excite phosphors radioactive metal (Greek protos first, as it decays into actinium) 1917, found in nature, no use

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c particle properties

1188

Name

Sym- At. bol n.

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Radium

Ra

(226.0254(1)) 1599(4) a

(223)

88

Rn

86

(222.0176(1)) 3.823(4) d

Rhenium

Re

75

186.207(1) c stable

Rhodium

Rh

45

102.905 50(2) stable

Roentgeniumb Rg

111

Rubidium

Rb

37

(272.1535(1)) 1.5 ms д 85.4678(3) f stable

Ruthenium

Ru

44

101.107(2) f stable

134m

Rutherfordiumb Rf

104

n.a.

Samarium

Sm

62

(261.1088(1)) 1.3 min д 150.36(2) c, f stable

Scandium

Sc

21

44.955 912(6) stable

164m

Seaborgiumb

Sg

106

n.a.

Selenium

Se

34

266.1219(1) 21 s д 78.96(3) f stable

255m

180m

120c, 190v

silvery-white, reactive alkali metal (Latin rubidus red) 1861, used in photocells, optical glasses, solid electrolytes white metal (Latin Rhuthenia for Russia) 1844, used in platinum and palladium alloys, superconductors, as catalyst; the tetroxide is toxic and explosive radioactive transactinide (after Ernest Rutherford) 1964, no use silver-white rare Earth metal (from the mineral samarskite, after Wassily Samarski) 1879, used in magnets, optical glasses, as laser dopant, in phosphors, in high-power light sources silver-white metal (from Latin Scansia Sweden) 1879, the oxide is used in highintensity mercury vapour lamps, a radioactive isotope is used as tracer radioactive transurane (after Glenn Seaborg) 1974, no use red or black or grey semiconductor (Greek selene Moon) 1818, used in xerography, glass production, photographic toners, as enamel dye

Copyright © Christoph Schiller November 1997–May 2006

Radon

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highly radioactive metal (Latin radius ray) 1898, no use any more; once used in luminous paints and as radioactive source and in medicine (130n) radioactive noble gas (from its old name ‘radium emanation’) 1900, no use (any more), found in soil, produces lung cancer 138m (Latin rhenus for Rhine river) 1925, used in filaments for mass spectrographs and ion gauges, superconductors, thermocouples, flash lamps, and as catalyst 135m white metal (Greek rhodon rose) 1803, used to harden platinum and palladium alloys, for electroplating, and as catalyst n.a. 1994, no use

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c particle properties

1189

Name

Sym- At. bol n.

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Silicon

Si

28.0855(3) f stable

105c, 210v

14

Silver

Ag

47

107.8682(2) f stable

Sodium

Na

11

22.989 769 28(2)191m

145m

Sr

38

87.62(1) f stable

215m

Sulphur

S

16

32.065(5) f stable

105c, 180v

Tantalum

Ta

73

180.947 88(2) stable

147m

Technetiumb

Tc

43

(97.9072(1)) 6.6(10) Ma

136m

Tellurium

Te

52

127.60(3) f stable

139c, 206v

Terbium

Tb

65

158.925 35(2) stable

178m

Thallium

Tl

81

204.3833(2) stable

172m

Thorium

Th

90

232.038 06(2)d , f 180m 14.0(1) Ga

Copyright © Christoph Schiller November 1997–May 2006

Strontium

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stable

grey, shiny semiconductor (Latin silex pebble) 1823, Earth’s crust, electronics, sand, concrete, bricks, glass, polymers, solar cells, essential for life white metal with highest thermal and electrical conductivity (Latin argentum, Greek argyros) antiquity, used in photography, alloys, to make rain light, reactive metal (Arabic souwad soda, Egyptian and Arabic natrium) component of many salts, soap, paper, soda, salpeter, borax, and essential for life silvery, spontaneously igniting light metal (Strontian, Scottish town) 1790, used in TV tube glass, in magnets, and in optical materials yellow solid (Latin) antiquity, used in gunpowder, in sulphuric acid, rubber vulcanization, as fungicide in wine production, and is essential for life; some bacteria use sulphur instead of oxygen in their chemistry heavy metal (Greek Tantalos, a mythical figure) 1802, used for alloys, surgical instruments, capacitors, vacuum furnaces, glasses radioactive (Greek technetos artificial) 1939, used as radioactive tracer and in nuclear technology brittle, garlic-smelling semiconductor (Latin tellus Earth) 1783, used in alloys and as glass component malleable rare Earth metal (Ytterby, Swedish town) 1843, used as dopant in optical material soft, poisonous heavy metal (Greek thallos branch) 1861, used as poison and for infrared detection radioactive (Nordic god Thor, as in ‘Thursday’) 1828, found in nature, heats Earth, used as oxide, in alloys and as coating

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c particle properties

1190

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Thulium

Tm

69

175m

Tin

Sn

50

168.934 21(2) stable 118.710(7) f stable

Titanium

Ti

22

Tungsten

W

74

Ununbiumb Ununtrium Ununquadiumb Ununpentium Ununhexiumb Ununseptium Ununoctium Uranium

Uub Uut Uuq Uup Uuh Uus Uuo U

112 113 114 115 116 117 118 92

Vanadium

V

23

Xenon

Xe

54

Ytterbium

Yb

70

Yttrium

Y

39

Zinc

Zn

30

Zirconium

Zr

40

rare Earth metal (Thule, mythical name for Scandinavia) 1879, found in monazite 139c, grey metal that, when bent, allows one 210v, to hear the ‘tin cry’ (Latin stannum) an162m tiquity, used in paint, bronze and superconductors 47.867(1) 146m metal (Greek hero Titanos) 1791, alloys, stable fake diamonds 183.84(1) 141m heavy, highest-melting metal (Swedish stable tung sten heavy stone, German name Wolfram) 1783, lightbulbs (285) 15.4 min д n.a. 1996, no use n.a. 2004, no use (289) 30.4 s д n.a. 1999, no use n.a. 2004, no use (289) 0.6 ms д n.a. 2000 (earlier claim was false), no use n.a. not yet observed n.a. not yet observed, but false claim in 1999 238.028 91(3)d , f 156m radioactive and of high density (planet Ur4.468(3) ë 109 a anus, after the Greek sky god) 1789, found in pechblende and other minerals, used for nuclear energy 50.9415(1) 135m metal (Vanadis, scandinavian goddess of stable beauty) 1830, used in steel 131.293(6) f (103n) noble gas (Greek xenos foreign) 1898, stable 200v used in lamps and lasers f 173.04(3) 174m malleable heavy metal (Ytterby, Swedish stable town) 1878, used in superconductors 88.905 85(2) 180m malleable light metal (Ytterby, Swedish stable town) 1794, used in lasers 65.409(4) 139m heavy metal (German Zinke protuberstable ance) antiquity, iron rust protection 91.224(2) f 160m heavy metal (from the mineral zircon, stable after Arabic zargum golden colour) 1789, chemical and surgical instruments, nuclear industry

a. The atomic mass unit is defined as 1 u = 121 m(12 C), making 1 u = 1.660 5402(10) yg. For elements found on Earth, the average atomic mass for the naturally occurring isotope mixture is given, with the error in the last digit in brackets. For elements not found on Earth, the mass of the longest living isotope is given; as it is

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Sym- At. bol n.

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Ref. 1215

Name

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c particle properties

Ref. 1216 Ref. 1216

Challenge 1545 n

not an average, it is written in brackets, as is customary in this domain. b. The element is not found on Earth because of its short lifetime. c. The element has at least one radioactive isotope. d. The element has no stable isotopes. e. Strictly speaking, the atomic radius does not exist. Because atoms are clouds, they have no boundary. Several approximate definitions of the ‘size’ of atoms are possible. Usually, the radius is defined in such a way as to be useful for the estimation of distances between atoms. This distance is different for different bond types. In the table, radii for metallic bonds are labelled m, radii for (single) covalent bonds with carbon c, and Van der Waals radii v. Noble gas radii are labelled n. Note that values found in the literature vary by about 10 %; values in brackets lack literature references. The covalent radius can be up to 0.1 nm smaller than the metallic radius for elements on the (lower) left of the periodic table; on the (whole) right side it is essentially equal to the metallic radius. In between, the difference between the two decreases towards the right. Can you explain why? By the way, ionic radii differ considerably from atomic ones, and depend both on the ionic charge and the element itself. All these values are for atoms in their ground state. Excited atoms can be hundreds of times larger than atoms in the ground state; however, excited atoms do not form solids or chemical compounds. f . The isotopic composition, and thus the average atomic mass, of the element varies depending on the place where it was mined or on subsequent human treatment, and can lie outside the values given. For example, the atomic mass of commercial lithium ranges between 6.939 and 6.996 u. The masses of isotopes are known in atomic mass units to nine or more significant digits, and usually with one or two fewer digits in kilograms. The errors in the atomic mass are thus mainly due to the variations in isotopic composition. д. The lifetime errors are asymmetric or not well known. h. Extensive details on element names can be found on http://elements.vanderkrogt.net.

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Ref. 1208 Ref. 1215

1191

Copyright © Christoph Schiller November 1997–May 2006

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c particle properties

1192

Biblio graphy 1205 The electron radius limit is deduced from the д − 2 measurements, as explained in the

1206

1207

1209

1210

1211

1212

1214 1215

Copyright © Christoph Schiller November 1997–May 2006

1213

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1208

Nobel Prize talk by Hans Dehmelt, Experiments with an isolated subatomic particle at rest, Reviews of Modern Physics 62, pp. 525–530, 1990, or in Hans Dehmelt, Is the electron a composite particle?, Hyperfine Interactions 81, pp. 1–3, 1993. Cited on page 1177. G. Gabrielse, H. Dehmelt & W. Kells, Observation of a relativistic, bistable hysteresis in the cyclotron motion of a single electron, Physical Review Letters 54, pp. 537–540, 1985. Cited on page 1177. The proton charge radius was determined by measuring the frequency of light emitted by hydrogen atoms to high precision by T. Udem, A. Huber, B. Gross, J. R eichert, M. Prevedelli, M. Weitz & T.W. Hausch, Phase-coherent measurement of the hydrogen 1S–2S transition frequency with an optical frequency interval divider chain, Physical Review Letters 79, pp. 2646–2649, 1997. Cited on page 1178. For a full list of isotopes, see R.B. Firestone, Table of Isotopes, Eighth Edition, 1999 Update, with CD-ROM, John Wiley & Sons, 1999. For a list of isotopes on the web, see the Korean website by J. Chang, http://atom.kaeri.re.kr/. For a list of precise isotope masses, see the http://csnwww.in2p3.fr website. Cited on pages 1178, 1181, and 1191. For information on the long life of tantalum 180, see D. Belic & al., Photoactivation of 180 m Ta and its implications for the nucleosynthesis of nature’s rarest naturally occurring isotope, Physical Review Letters 83, pp. 5242–5245, 20 December 1999. Cited on page 1178. The ground state of bismuth 209 was thought to be stable until early 2003. It was then discovered that it was radioactive, though with a record lifetime, as reported by P. de Marcill ac, N. Coron, G. Dambier, J. Lebl anc & J.-P. Moalic, Experimental detection of α-particles from the radioactive decay of natural bismuth, Nature 422, pp. 876– 878, 2003. By coincidence, the excited state 83 MeV above the ground state of the same bismuth 209 nucleus is the shortest known radioactive nuclear state. Cited on page 1178. Stephen J. Gould, The Panda’s thumb, W.W. Norton & Co., 1980. This is one of several interesting and informative books on evolutionary biology by the best writer in the field. Cited on page 1179. A tetraquark is thought to be the best explanation for the f0 (980) resonance at 980 MeV. However, this assignment is still under investigation. Pentaquarks were first predicted by Maxim Polyakov, Dmitri Diakonov, and Victor Petrov in 1997. Two experimental groups in 2003 claimed to confirm their existence, with a mass of 1540 MeV; see K. Hicks, An experimental review of the Θ+ pentaquark, http://www.arxiv.org/abs/hep-ex/0412048. Results from 2005, however, seem to rule out that the 1540 MeV particle is a pentaquark. Cited on page 1179. F. Marques & al., Detection of neutron clusters, Physical Review C 65, p. 044006, 2002. Opposite results have been obtained by B.M. Sherrill & C.A. Bertul ani, Proton-tetraneutron elastic scattering, Physical Review C 69, p. 027601, 2004, and D.V. Aleksandrov & al., Search for resonances in the three- and four-neutron systems in the 7Li(7Li, 11C)3n and 7Li(7Li, 10C)4n reactions, JETP Letters 81, p. 43, 2005. Cited on page 1179. For a good review, see the article by P.T. Greenl and, Antimatter, Contemporary Physics 38, pp. 181–203, 1997. Cited on page 1179. The atomic masses, as given by IUPAC, can be found in Pure and Applied Chemistry 73, pp. 667–683, 2001, or on the http://www.iupac.org website. For an isotope mass list, see

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bibliography

1193

the http://csnwww.in2p3.fr website. Cited on pages 1181, 1190, and 1191. 1216 The metallic, covalent and Van der Waals radii are from Nathaniel W. Alcock, Bonding and Structure, Ellis Horwood, 1999. This text also explains in detail how the radii are defined and measured. Cited on page 1191. 1217 Almost everything known about each element and its chemistry can be found in the encyclopaedic Gmelin, Handbuch der anorganischen Chemie, published from 1817 onwards. There are over 500 volumes, now all published in English under the title Handbook of Inorganic and Organometallic Chemistry, with at least one volume dedicated to each chemical element. On the same topic, an incredibly expensive book with an equally bad layout is Per Enhag, Encyclopedia of the Elements, Wiley–VCH, 2004. Cited on page 1180.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006

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Appendix D

NUMBER S AND SPAC ES

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M

athematical concepts can all be expressed in terms of ‘sets’ and ‘relations.’ any fundamental concepts were presented in the first Intermezzo. Why does athematics, given this simple basis, grow into a passion for certain people? The following pages present a few more advanced concepts as simply* and vividly as possible, for all those who want to smell the passion for mathematics. In particular, in this appendix we shall expand the range of algebraic and of topological structures; the third basic type of mathematical structures, order structures, are not so important in physics. Mathematicians are concerned not only with the exploration of concepts, but also with their classification. Whenever a new mathematical concept is introduced, mathematicians try to classify all the possible cases and types. This has been achieved most spectacularly for the different types of numbers, for finite simple groups, and for many types of spaces and manifolds.

Numbers as mathematical structures Challenge 1546 ny

A person who can solve x 2 − 92y 2 = 1 in less than a year is a mathematician. Brahmagupta (b. 598 Sindh, d. 668) (implied: solve in integers)

— R is a commutative group with respect to addition, i.e. a + b > R, a + b = b + a, a + 0 = a, a + (−a) = a − a = 0 and a + (b + c) = (a + b) + c; — R is closed under multiplication, i.e. ab > R; * The opposite approach can have the same effect: it is taken in the delightful text by Carl E. L inderholm, Mathematics Made Difficult, Wolfe Publishing, 1971.

Copyright © Christoph Schiller November 1997–May 2006

We start with a short introduction to the vocabulary. Any mathematical system with the same basic properties as the natural numbers is called a semi-ring. Any mathematical system with the same basic properties as the integers is called a ring. (The term is due to David Hilbert. Both structures can also be finite rather than infinite.) More precisely, a ring (R, +, ë) is a set R of elements with two binary operations, called addition and multiplication, usually written + and ë (the latter may simply be understood without notation), for which the following properties hold for all elements a, b, c > R:

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Ref. 1218

A mathematician is a machine that transforms coffee into theorems. Paul Erdős (b. 1913 Budapest, d. 1996 Warsaw)

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numbers as mathematical structures

1195

— multiplication is associative, i.e. a(bc) = (ab)c; — distributivity holds, i.e. a(b + c) = ab + ac and (b + c)a = ba + ca.

Defining properties such as these are called axioms. Note that axioms are not basic beliefs, as is often stated; axioms are the basic properties used in the definition of a concept: in this case, of a ring. A semi-ring is a set satisfying all the axioms of a ring, except that the existence of neutral and negative elements for addition is replaced by the weaker requirement that if a + c = b + c then a = b. To incorporate division and define the rational numbers, we need another concept. A field K is a ring with

Ref. 1220

A ring or field is said to be commutative if the multiplication is commutative. A noncommutative field is also called a skew field. Fields can be finite or infinite. (A field or a ring is characterized by its characteristic p. This is the smallest number of times one has to add 1 to itself to give zero. If there is no such number the characteristic is set to 0. p is always a prime number or zero.) All finite fields are commutative. In a field, all equations of the type cx = b and xc = b (c x 0) have solutions for x; there is a unique solution if b x 0. To sum up sloppily by focusing on the most important property, a field is a set of elements for which, together with addition, subtraction and multiplication, a division (by non-zero elements) is also defined. The rational numbers are the simplest field that incorporates the integers. The system of the real numbers is the minimal extension of the rationals which is complete and totally ordered.* However, the concept of ‘number’ is not limited to these examples. It can be generalized in several ways. The simplest generalization is achieved by extending the real numbers to manifolds of more than one dimension.

— if a D b and b D c, then a D c; — if a D b and b D a, then a = b; — a D b or b D a holds. In summary, a set is totally ordered if there is a binary relation that allows to say about any two elements which one is the predecessor of the other in a consistent way.

Copyright © Christoph Schiller November 1997–May 2006

* A set is mathematically complete if physicists call it continuous. More precisely, a set of numbers is complete if every non-empty subset that is bounded above has a lest upper bound. A set is totally ordered if there exists a binary relation D between pairs of elements such that for all elements a and b

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— an identity 1, such that all elements a obey 1a = a; — at least one element different from zero; and most importantly — a (multiplicative) inverse a −1 for every element a =~ 0.

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d numbers and spaces

1196

Complex numbers A complex number is defined by z = a + ib, where a and b are real numbers, and i is a new symbol. Under multiplication, the generators of the complex numbers, 1 and i, obey ë 1 i

1 i 1 i i −1

(816)

(a 12 + a 22 )(b 12 + b 22 ) = (a 1 b 1 − a 2 b 2 )2 + (a 1 b 2 + a 2 b 1 )2

Challenge 1547 e

valid for all real numbers a i , b i . It was already known, in its version for integers, to Diophantus of Alexandria. Complex numbers can also be written as ordered pairs (a, A) of real numbers, with their addition defined as (a, A) + (b, B) = (a + b, A + ic B) and their multiplication defined as (a, A) ë (b, B) = (ab − AB, aB + bA). This notation allows us to identify the complex numbers with the points on a plane. Translating the definition of multiplication into geometrical language allows ih = − i ab c us to rapidly prove certain geometrical theorems, such as the one of Figure 391. Complex numbers a + ib can also be repres0 a b ented as 2  2 matrices

Page 1204 Challenge 1549 ny

witha, b > R .

F I G U R E 391 A property of triangles easily

(818)

provable with complex numbers

Matrix addition and multiplication then correspond to complex addition and multiplication. In this way, complex numbers can be represented by a special type of real matrix. What is SzS in matrix language? The set C of complex numbers with addition and multiplication as defined above forms both a commutative two-dimensional field and a vector space over R. In the field of complex numbers, quadratic equations az 2 + bz + c = 0 for an unknown z always have two solutions (for a x 0 and counting multiplicity). Complex numbers can be used to describe the points of a plane. A rotation around the origin can be described by multiplication by a complex number of unit length. Other twodimensional quantities can also be described with complex numbers. Electrical engineers

Copyright © Christoph Schiller November 1997–May 2006

a b ‘ Œ −b a

Challenge 1548 n

(817)

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º often summarized as i = + −1 . The complex conjugate z ‡ , also written z¯, of a complex number z = a + ib isºdefined as z ‡ = aº− ib. The absolute value SzS of a complex number is defined as SzS = zz ‡ = º z ‡ z = a 2 + b 2 . It defines a norm on the vector space of the complex numbers. From SwzS = SwS SzS follows the two-squares theorem

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numbers as mathematical structures

Challenge 1550 e

Challenge 1551 n

use complex numbers to describe quantities with phases, such as alternating currents or electrical fields in space. Writing complex numbers of unit length as cos θ + i sin θ is a useful method for remembering angle addition formulae. Since one has cos nθ + i sin nθ = (cos θ + i sin θ)n , one can easily deduce formulae cos 2θ = cos2 θ − sin2 θ and sin 2θ = 2 sin θ cos θ. By the way, there are exactly as many complex numbers as there are real numbers. Can you show this?

Quaternions

The positions of the points on a line can be described by real numbers. Complex numbers can be used to describe the positions of the points of a plane. It is natural to try to generalize the idea of a number to higher-dimensional spaces. However, it turns out that no useful number system can be defined for three-dimensional space. A new number system, the quaternions, can be constructed which corresponds the points of four-dimensional space, but only if the commutativity of multiplication is sacrificed. No useful number system can be defined for dimensions other than 1, 2 and 4. The quaternions were discovered by several mathematicians in the nineteenth century, among them Hamilton,* who studied them for much of his life. In fact, Maxwell’s theory of electrodynamics was formulated in terms of quaternions before three-dimensional vectors were used. Under multiplication, the quaternions H form a 4-dimensional algebra over the reals with a basis 1, i, j, k satisfying ë 1 i j k

1 i j k 1 i j k i −1 k − j . j −k −1 i k j −i −1

(819)

These relations are also often written i 2 = j2 = k 2 = −1, i j = − ji = k, jk = −k j = i, ki = −ik = j. The quaternions 1, i, j, k are also called basic units or generators. The lack of symmetry across the diagonal of the table shows the non-commutativity of quaternionic multiplication. With the quaternions, the idea of a non-commutative product appeared for the first time in mathematics. However, the multiplication of quaternions is associative. As a consequence of non-commutativity, polynomial equations in quaternions have many more solutions than in complex numbers: just search for all solutions of the equation X 2 + 1 = 0 to convince yourself of it. Every quaternion X can be written in the form X = x 0 + x 1 i + x 2 j + x 3 k = x 0 + v = (x 0 , x 1 , x 2 , x 3 ) = (x 0 , v) ,

(820)

* William Rowan Hamilton (b. 1805 Dublin, d. 1865 Dunsink), Irish child prodigy and famous mathematician, named the quaternions after an expression from the Vulgate (Acts. 12: 4).

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Copyright © Christoph Schiller November 1997–May 2006

Challenge 1552 ny

Love is complex: it has real and imaginary parts.

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Ref. 1221 Page 1207

1197

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d numbers and spaces

1198

where x 0 is called the scalar part and v the vector part. The multiplication is thus defined as (x, v)(y, w) = (x y−vëw, xw+yv+vw). The multiplication of two general quaternions can be written as (a 1 , b 1 , c 1 , d 1 )(a 2 , b 2 , c 2 , d 2 ) = (a 1 a 2 − b 1 b 2 − c 1 c 2 − d 1 d 2 , a 1 b 2 + b 1 a 2 + c 1 d 2 − d 1 c 2 , a1 c2 − b1 d2 + c1 a2 + d1 b2 , a1 d2 + b1 c2 − c1 b2 + d1 a2 ) . (821)

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The conjugate quaternion X is defined as X = x 0 − v, so that XY = Y X. The norm SXS of a quaternion X is defined as SXS2 = X X = X X = x 02 + x 12 + x 22 + x 32 = x 02 + v 2 . The norm is multiplicative, i.e. SXYS = SXS SYS. Unlike complex numbers, every quaternion is related to its complex conjugate by

(822)

No relation of this type exists for complex numbers. In the language of physics, a complex number and its conjugate are independent variables; for quaternions, this is not the case. As a result, functions of quaternions are less useful in physics than functions of complex variables. The relation SXYS = SXS SYS implies the four-squares theorem (a 12 + a 22 + a 32 + a 42 )(b 12 + b 22 + b 32 + b 42 )

= (a 1 b 1 − a 2 b 2 − a 3 b 3 − a 4 b 4 )2 + (a 1 b 2 + a 2 b 1 + a 3 b 4 − a 4 b 3 )2

+(a 1 b 3 + a 3 b 1 + a 4 b 2 − a 2 b 4 )2 + (a 1 b 4 + a 4 b 1 + a 2 b 3 − a 3 b 2 )2

valid for all real numbers a i and b i , and thus also for any set of eight integers. It was discovered in 1748 by Leonhard Euler (1707–1783) when trying to prove that each integer is the sum of four squares. (That fact was proved only in 1770, by Joseph Lagrange.) Hamilton thought that a quaternion with zero scalar part, which he simply called a vector (a term which he invented), could be identified with an ordinary three-dimensional translation vector; but this is wrong. Such a quaternion is now called a pure, or homogeneous, or imaginary quaternion. The product of two pure quaternions V = (0, v) and W = (0, w) is given by V W = (−v ë w, v  w), where ë denotes the scalar product and  denotes the vector product. Note that any quaternion can be written as the ratio of two pure quaternions. In reality, a pure quaternion (0, v) does not behave like a translation vector under coordinate transformations; in fact, a pure quaternion represents a rotation by the angle π or 180° around the axis defined by the direction v = (v x , v y , v z ). It turns out that in three-dimensional space, a general rotation about the origin can be described by a unit quaternion Q, also called a normed quaternion, for which SQS = 1. Such a quaternion can be written as (cos θ~2, n sin θ~2), where n = (n x , n y , n z ) is the normed vector describing the direction of the rotation axis and θ is the rotation angle. Such a unit quaternion Q = (cos θ~2, n sin θ~2) rotates a pure quaternion V = (0, v) into

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Challenge 1553 ny

(823)

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X = − 21 (X + iXi + jX j + kXk) .

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numbers as mathematical structures

1199

k

i j

1

k

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i

j

back of right hand

palm of right hand

another pure quaternion W = (0, w) given by

W = QV Q ‡ .

(824)

Thus, if we use pure quaternions such as V or W to describe positions, we can use unit quaternions to describe rotations and to calculate coordinate changes. The concatenation of two rotations is then given by the product of the corresponding unit quaternions. Indeed, a rotation by an angle α about the axis l followed by a rotation by an angle β about the axis m gives a rotation by an angle γ about the axis n, with the values determined by (cos γ~2, sin γ~2n) = (cos α~2, sin α~2l)(cos β~2, sin β~2m) .

Challenge 1554 e

Challenge 1555 n

Page 783

One way to show the result graphically is given in Figure 392. By drawing a triangle on a unit sphere, and taking care to remember the factor 1~2 in the angles, the combination of two rotations can be simply determined. The interpretation of quaternions as rotations is also illustrated, in a somewhat different way, in the motion of any hand. To see this, take a green marker and write the letters 1, i, j and k on your hand as shown in Figure 393. Defining the three possible 90° rotations axes as shown in the figure and taking concatenation as multiplication, the motion of the right hand follows the same ‘laws’ as those of pure unit quaternions. (One still needs to distinguish +i and −i, and the same for the other units, by the sense of the arm twist. And the result of a multiplication is that letter that can be read by a person facing you.) You can show that i 2 = j2 = k 2 = −1, that i 4 = 1, and all other quaternion relations.) The model also shows that the rotation angle of the arm is half the rotation angle of the corresponding quaternion. In other words, quaternions can be used to describe the belt trick, if the multiplication V W of two quaternions is taken to mean that rotation V is performed after rotation W. Quaternions, or the human hand, thus behaves like a spin

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1219

(825)

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F I G U R E 393 The hand and the quaternions

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1200

Page 1218

Ref. 1222

Œ

A B ‘ with A, B > C, ‡ −B A‡

c d “ ’ a b – −b a −d c — — with a, b, c, d > R, (826) or as – – −c d a −b — — – ” −d −c b a •

where A = a + ib, B = c + id and the quaternion X is X = A + B j = a + ib + jc + kd; matrix addition and multiplication then corresponds to quaternionic addition and multiplication.

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Copyright © Christoph Schiller November 1997–May 2006

1/2 particle. Quaternions and spinors are isomorphic. The reason for the half-angle behaviour of rotations can be specified more precisely using mathematical language. The rotations in three α~2 dimensions around a point form the ‘special orthogonal group’ in three dimensions, which l π − γ~2 is called SO(3). But the motions of a hand attached to a shoulder via an arm form a different group, isomorphic to the Lie group SU(2). n The difference is due to the appearance of half angles in the parametrization of rotations; inβ~2 deed, the above parametrizations imply that a m rotation by 2π corresponds to a multiplication by −1. Only in the twentieth century was it real- F I G U R E 392 Combinations of rotations ized that there exist fundamental physical observables that behaves like hands attached to arms: they are called spinors. More on spinors can be found in the section on permutation symmetry, where belts are used as an analogy as well as arms. In short, the group SU(2) of the quaternions is the double cover of the rotation group SO(3). The simple representation of rotations and positions with quaternions is used in by computer programmes in robotics, in astronomy and in flight simulation. In the software used to create three-dimensional images and animations, visualization software, quaternions are often used to calculate the path taken by repeatedly reflected light rays and thus give surfaces a realistic appearance. The algebra of the quaternions is the only associative, non-commutative. finite-dimensional normed algebra with an identity over the field of real numbers. Quaternions form a non-commutative field, i.e. a skew field, in which the inverse of a quaternion X is X~SXS. We can therefore define division of quaternions (while being careful to distinguish XY −1 and Y −1 X). Therefore quaternions are said to form a division algebra. In fact, the quaternions H, the complex numbers C and the reals R are the only three finite-dimensional associative division algebras. In other words, the skew-field of quaternions is the only finite-dimensional real associative non-commutative algebra without divisors of zero. The centre of the quaternions, i.e. the set of quaternions that commute with all other quaternions, is just the set of real numbers. Quaternions can be represented as matrices of the form

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Page 783

d numbers and spaces

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numbers as mathematical structures

1201

The generators of the quaternions can be realized as 1  σ0

,

i  −iσ1

,

j  −iσ2

,

k  −iσ3

(827)

where the σn are the Pauli spin matrices.* Real 4  4 representations are not unique, as the alternative representation

Challenge 1556 ny

Ref. 1221 Challenge 1557 n

Dvipsbugw (829)

shows; however, no representation by 3  3 matrices is possible. These matrices contain real and complex elements, which pose no special problems. In contrast, when matrices with quaternionic elements are constructed, care has to be taken, because quaternionic multiplication is not commutative, so that simple relations such as trAB = trBA are not generally valid. What can we learn from quaternions about the description of nature? First of all, we see that binary rotations are similar to positions, and thus to translations: all are represented by 3-vectors. Are rotations the basic operations of nature? Is it possible that translations are only ‘shadows’ of rotations? The connection between translations and rotations s is investigated in the third part of our mountain ascent. When Maxwell wrote down his equations of electrodynamics, he used quaternion notation. (The now usual 3-vector notation was introduced later by Hertz and Heaviside.) The equations can be written in variousº ways using quaternions. The simplest is achieved when one keeps a distinction between −1 and the units i, j, k of the quaternions. One then can write all of electrodynamics in a single equation: dF = −

Q ε0

(830)

where F is the generalized electromagnetic field and Q the generalized charge. These are

σ0 = 1 =

Page 1209

’ 1 ” 0

0 “ 1 •

,

σ1 =

’ 0 ” 1

1 “ 0 •

,

σ2 =

’ 0 ” i

−i “ 0 •

,

σ3 =

’ 1 ” 0

0 “ −1 •

(828)

all of whose eigenvalues are 1; they satisfy the relations [σ i , σ k ]+ = 2 δ i k and [σ i , σ k ] = 2i ε i k l σ l . The linear combinations σ = 21 (σ1  σ2 ) are also frequently used. By the way, another possible representation of the quaternions is i  iσ3 , j  iσ2 , k  iσ1 .

Copyright © Christoph Schiller November 1997–May 2006

* The Pauli spin matrices are the complex Hermitean matrices

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Page 545

b −d −c “ ’ a – −b a −c d — — – — – c a b — – d ” c −d −b a •

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d numbers and spaces

1202

defined by

º F = E + −1 cB E = iE x + jE y + kE z B = iB x + jB y + kB z º d = δ + −1 ∂ t ~c δ = i∂ x + j∂ y + k∂ z º Q = ρ + −1 J~c

(831)

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Octonions In the same way that quaternions are constructed from complex numbers, octonions can be constructed from quaternions. They were first investigated by Arthur Cayley (1821–1895). Under multiplication, octonions (or octaves) are the elements of an eightdimensional algebra over the reals with the generators 1, i n with n = 1 . . . 7 satisfying ë 1 i1 i2 i3 i4 i5 i6 i7

1 1

i1 i2 i3 i4 i5 i6 i7

i1 i2 i3 i4 i5 i6 i7 i1 i2 i3 i4 i5 i6 i7 −1 i 3 −i 2 i 5 −i 4 i 7 −i 6 −i 3 −1 i 1 −i 6 i7 i 4 −i 5 i 2 −i 1 −1 i7 i 6 −i 5 −i 4 −i 5 i 6 −i 7 −1 i 1 −i 2 i3 i 4 −i 7 −i 6 −i 1 −1 i3 i2 −i 7 −i 4 i5 i 2 −i 3 −1 i1 i6 i5 i 4 −i 3 −i 2 −i 1 −1

(832)

Copyright © Christoph Schiller November 1997–May 2006

Nineteen other, equivalent multiplication tables are also possible. This algebra is called the Cayley algebra; it has an identity and a unique division. The algebra is non-commutative, and also non-associative. It is, however, alternative, meaning that for all elements x and y, one has x(x y) = x 2 y and (x y)y = x y 2 : a property somewhat weaker than associativity. It is the only 8-dimensional real alternative algebra without zero divisors. Because it is not associative, the set Ω of all octonions does not form a field, nor even a ring, so that the old designation of ‘Cayley numbers’ has been abandoned. The octonions are the most general hypercomplex ‘numbers’ whose norm is multiplicative. Its generators obey since (i n i m )i l = i n (i m i l ), where the minus sign, which shows the non-associativity, is valid for combinations of indices, such as 1-2-4, which are not quaternionic.

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where the fields E and B and the charge distributions ρ and J have the usual meanings. The content of equation 830 for the electromagnetic field is exactly the same as the usual formulation. Despite their charm, quaternions do not seem to be ready for the reformulation of special relativity; the main reason for this is the sign in the expression for their norm. Therefore, relativity and space-time are usually described using real numbers.

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numbers as mathematical structures

1203

Octonions can be represented as matrices of the form Œ

A B ‘ where A, B > H , or as real 8  8 matrices. −B¯ A¯

(833)

Matrix multiplication then gives the same result as octonionic multiplication. The relation SwzS = SwS SzS allows one to deduce the impressive eight-squares theorem (a 12 + a 22 + a 32 + a 42 + a 52 + a 62 + a 72 + a 82 )(b 12 + b 22 + b 32 + b 42 + b 52 + b 62 + b 72 + b 82 )

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= (a 1 b 1 − a 2 b 2 − a 3 b 3 − a 4 b 4 − a 5 b 5 − a 6 b 6 − a 7 b 7 − a 8 b 8 )2

+ (a 1 b 2 + a 2 b 1 + a 3 b 4 − a 4 b 3 + a 5 b 6 − a 6 b 5 + a 7 b 8 − a 8 b 7 )2

+ (a 1 b 4 + a 2 b 3 − a 3 b 2 + a 4 b 1 + a 5 b 8 + a 6 b 7 − a 7 b 6 − a 8 b 5 )2

+ (a 1 b 5 − a 2 b 6 + a 3 b 7 − a 4 b 8 + a 5 b 1 + a 6 b 2 − a 7 b 3 + a 8 b 4 )2

+ (a 1 b 6 + a 2 b 5 − a 3 b 8 − a 4 b 7 − a 5 b 2 + a 6 b 1 + a 7 b 4 + a 8 b 3 )2

+ (a 1 b 7 − a 2 b 8 − a 3 b 5 + a 4 b 6 + a 5 b 3 − a 6 b 4 + a 7 b 1 + a 8 b 2 )2

+ (a 1 b 8 + a 2 b 7 + a 3 b 6 + a 4 b 5 − a 5 b 4 − a 6 b 3 − a 7 b 2 + a 8 b 1 )2

(834)

valid for all real numbers a i and b i and thus in particular also for all integers. (There are many variations of this expression, with different possible sign combinations.) The theorem was discovered in 1818 by Carl Ferdinand Degen (1766–1825), and then rediscovered in 1844 by John Graves and in 1845 by Arthur Cayley. There is no generalization to higher numbers of squares, a fact proved by Adolf Hurwitz (1859–1919) in 1898. The octonions can be used to show that a vector product can be defined in more than three dimensions. A vector product or cross product is an operation  satisfying

Using the definition

anticommutativity exchange rule.

1 X  Y = (XY − Y X) , 2

(835)

(836)

the -products of imaginary quaternions, i.e. of quaternions of the type (0, u), are again imaginary, and correspond to the usual vector product, thus fulfilling (835). Interestingly, it is possible to use definition (836) for octonions as well. In that case, the product of imaginary octonions is also imaginary, and (835) is again satisfied. In fact, this is the only other non-trivial example of a vector product. Thus a vector product exists only in three and in seven dimensions.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1220 Challenge 1558 e

u  v = −v  u (u  v)w = u(v  w)

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+ (a 1 b 3 − a 2 b 4 + a 3 b 1 + a 4 b 2 − a 5 b 7 + a 6 b 8 + a 7 b 5 − a 8 b 6 )2

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d numbers and spaces

1204

Other types of numbers

Ref. 1223 Challenge 1559 n

Challenge 1560 ny Ref. 1220

Grassmann numbers

Challenge 1561 ny

θ2 = 0

and

θi θ j + θ jθi = 0 .

(837)

You may want to look for a matrix representation of these numbers.

Vector spaces Vector spaces, also called linear spaces, are mathematical generalizations of certain aspects of the intuitive three-dimensional space. A set of elements any two of which can * Hermann Günther Grassmann (1809–1877), schoolteacher in Stettin, and one of the most profound mathematical thinkers of the nineteenth century.

Copyright © Christoph Schiller November 1997–May 2006

With the discovery of supersymmetry, another system of numbers became important, called the Grassmann numbers.* They are in fact a special type of hypercomplex numbers. In supersymmetric Lagrangians, fields depend on two types of coordinates: on the usual real space-time coordinates and additionally on the Grassmann coordinates. Grassmann numbers, also called fermionic coordinates, θ have the defining properties

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Ref. 1224 Page 653

The process of constructing new systems of hypercomplex ‘numbers’ or real algebras by ‘doubling’ a given one can be continued ad infinitum. However, octonions, sedenions and all the following doublings are neither rings nor fields, but only non-associative algebras with unity. Other finite-dimensional algebras with unit element over the reals, once called hypercomplex ‘numbers’, can also be defined: they include the so-called such as ‘dual numbers’, ‘double numbers’, ‘Clifford–Lifshitz numbers’ etc. They play no special role in physics. Mathematicians have also defined number fields which have ‘one and a bit’ dimensions, such as algebraic number fields. There is also a generalization of the concept of integers to the complex domain: the Gaussian integers, defined as n+im, where n and m are ordinary integers. Gauss even defined what are now known as Gaussian primes. (Can you find out how?) They are not used in the description of nature, but are important in number theory. Physicists used to call quantum-mechanical operators ‘q-numbers.’ But this term has now fallen out of fashion. Another way in which the natural numbers can be extended is to include numbers larger infinite numbers. The most important such classes of transfinite number are the ordinals, the cardinals and the surreals (which were defined in the first intermezzo). The ordinals are essentially an extension of the integers beyond infinity, whereas the surreals are a continuous extension of the reals, also beyond infinity. Loosely speaking, among the transfinites, the ordinals have a similar role as the integers have among the reals; the surreals fill in all the gaps between the ordinals, like the reals do for integers. Interestingly, many series that diverge in R converge n the surreals. Can you find one example? The surreals include infinitely small numbers, as do the numbers of nonstandard analysis, also called hyperreals. In both the number systems, in contrast to real numbers, the numbers 1 and 0.999 999 9 (with an infinite string of nines) do not coincide, and indeed are separated by infinitely many other numbers.

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vector spaces

1205

be added together and any one of which can be multiplied by a number is called a vector space, if the result is again in the set and the usual rules of calculation hold. More precisely, a vector space over a number field K is a set of elements, called vectors, for which a vector addition and a scalar multiplication is defined, such that for all vectors a, b, c and for all numbers s and r from K one has (a + b) + c = a + (b + c) = a + b + c n+a=a (−a) + a = n 1a = a (s + r)(a + b) = sa + sb + ra + rb

If the field K, whose elements are called scalars in this context, is taken to be the real (or complex, or quaternionic) numbers, one speaks of a real (or complex, or quaternionic) vector space. Vector spaces are also called linear vector spaces or simply linear spaces. The complex numbers, the set of all real functions defined on the real line, the set of all polynomials, the set of matrices with a given number of rows and columns, all form vector spaces. In mathematics, a vector is thus a more general concept than in physics. (What is the simplest possible mathematical vector space?) In physics, the term ‘vector’ is reserved for elements of a more specialized type of vector space, namely normed inner product spaces. To define these, we first need the concept of a metric space. A metric space is a set with a metric, i.e. a way to define distances between elements. A real function d(a, b) between elements is called a metric if d(a, b) E 0 d(a, b) + d(b, c) E d(a, c) d(a, b) = 0 if and only if a = b

Challenge 1563 n

(839)

A non-trivial example is the following. We define a special distance d between cities. If the two cities lie on a line going through Paris, we use the usual distance. In all other cases, we define the distance d by the shortest distance from one to the other travelling via Paris. This strange method defines a metric between all cities in France. A normed vector space is a linear space with a norm, or ‘length’, associated to each a vector. A norm is a non-negative number SSaSS defined for each vector a with the properties SSraSS = SrS SSaSS linearity of norm SSa + bSS D SSaSS + SSbSS triangle inequality SSaSS = 0 only if a = 0 regularity

(840)

Usually there are many ways to define a norm for a given space. Note that a norm can always be used to define a metric by setting d(a, b) = SSa − bSS

(841)

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1564 ny

positivity of metric triangle inequality regularity of metric

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Challenge 1562 ny

associativity of vector addition existence of null vector existence of negative vector (838) regularity of scalar multiplication complete distributivity of scalar multiplication

Dvipsbugw


d numbers and spaces

1206

so that all normed spaces are also metric spaces. This is the natural distance definition (in contrast to unnatural ones like that between French cities). The norm is often defined with the help of an inner product. Indeed, the most special class of linear spaces are the inner product spaces. These are vector spaces with an inner product, also called scalar product ë (not to be confused with the scalar multiplication!) which associates a number to each pair of vectors. An inner product space over R satisfies commutativity of scalar product bilinearity of scalar product left distributivity of scalar product right distributivity of scalar product positivity of scalar product regularity of scalar product

Dvipsbugw

(842)

for all vectors a, b, c and all scalars r, s. A real inner product space of finite dimension is also called a Euclidean vector space. The set of all velocities, the set of all positions, or the set of all possible momenta form such spaces. An inner product space over C satisfies* aëb =bëa =bëa (ra) ë (sb) = rs(a ë b) (a + b) ë c = a ë c + b ë d a ë (b + c) = a ë b + a ë c aëaE0 a ë a = 0 if and only if a = 0

Page 1195

Challenge 1566 ny

(843)

for all vectors a, b, c and all scalars r, s. A complex inner product space (of finite dimension) is also called a unitary or Hermitean vector space. If the inner product space is complete, it is called, especially in the infinite-dimensional complex case, a Hilbert space. The space of all possible states of a quantum system forms a Hilbert space. All inner product spaces are also metric spaces, and thus normed spaces, if the metric is defined by » d(a, b) = (a − b) ë (a − b) . (844) Only in the context of an inner product spaces we can speak about angles (or phase differences) between vectors, as we are used to in physics. Of course, like in normed spaces, inner product spaces also allows us to speak about the length of vectors and to define a basis, the mathematical concept necessary to define a coordinate system. The dimension of a vector space is the number of linearly independent basis vectors. Can you define these terms precisely? Which vector spaces are of importance in physics? * Two inequivalent forms of the sequilinearity axiom exist. The other is (ra) ë (sb) = rs(a ë b). The term sesquilinear is derived from Latin and means for ‘one-and-a-half-linear’.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1565 ny

Hermitean property sesquilinearity of scalar product left distributivity of scalar product right distributivity of scalar product positivity of scalar product regularity of scalar product

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aëb =bëa (ra) ë (sb) = rs(a ë b) (a + b) ë c = a ë c + b ë c a ë (b + c) = a ë b + a ë c aëaE0 a ë a = 0 if and only if a = 0

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algebras

1207

Algebras

Page 1194

The term algebra is used in mathematics with three different, but loosely related, meanings. First, it denotes a part of mathematics, as in ‘I hated algebra at school’. Secondly, it denotes a set of formal rules that are obeyed by abstract objects, as in the expression ‘tensor algebra’. Finally – and this is the only meaning used here – an algebra denotes a specific type of mathematical structure. Intuitively, an algebra is a set of vectors with a vector multiplication defined on it. More precisely, an algebra is a vector space (over a field K) that is also a ring. (The concept is due to Benjamin Peirce (1809–1880), father of Charles Sanders Peirce.) A ring is a set for which an addition and a multiplication is defined – like the integers. Thus, in an algebra, there are (often) three types of multiplications:

A precise definition of an algebra thus only needs to define properties of the (main) multiplication and to specify the number field K. An algebra is defined by the following axioms x(y + z) = x y + xz

, (x + y)z = xz + yz c(x y) = (cx)y = x(c y)

distributivity of multiplication bilinearity (845)

for all vectors x, y, z and all scalars c > K. To stress their properties, algebras are also called linear algebras. For example, the set of all linear transformations of an n-dimensional linear space (such as the translations on a plane, in space or in time) is a linear algebra, if the composition is taken as multiplication. So is the set of observables of a quantum mechanical system.* An associative algebra is an algebra whose multiplication has the additional property

Challenge 1567 n

T(e) = λe

Page 744

Page 805

(846)

where λ is a scalar, then the vector e is called an eigenvector of T, and λ is associated eigenvalue. The set of all eigenvalues of a transformation T is called the spectrum of T. Physicists did not pay much attention to these mathematical concepts until they discovered quantum theory. Quantum theory showed that observables are transformations in Hilbert space, because any measurement interacts with a system and thus transforms it. Quantum-mechanical experiments also showed that a measurement result for an observable must be an eigenvalue of the corresponding transformation. The state of the system after the measurement is given by the eigenvector corresponding to the measured eigenvalue. Therefore every expert on motion must know what an eigenvalue is.

Copyright © Christoph Schiller November 1997–May 2006

* Linear transformations are mappings from the vector space to itself, with the property that sums and scalar multiples of vectors are transformed into the corresponding sums and scalar multiples of the transformed vectors. Can you specify the set of all linear transformations of the plane? And of three-dimensional space? And of Minkowski space? All linear transformations transform some special vectors, called eigenvectors (from the German word eigen meaning ‘self ’) into multiples of themselves. In other words, if T is a transformation, e a vector, and

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— scalar multiplication: the c-fold multiple of a vector x is another vector y = cx; — the (main) algebraic multiplication: the product of two vectors x and y is another vector z = x y; — if the vector space is a inner product space, the scalar product: the scalar product of two algebra elements (vectors) x and y is a scalar c = x ë y;

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d numbers and spaces

1208

that

Challenge 1568 e

Challenge 1570 n

associativity .

(847)

Most algebras that arise in physics are associative.* Therefore, in mathematical physics, a linear associative algebra is often simply called an algebra. The set of multiples of the unit 1 of the algebra is called the field of scalars scal(A) of the algebra A. The field of scalars is also a subalgebra of A. The field of scalars and the scalars themselves behave in the same way. We explore a few examples. The set of all polynomials in one variable (or in several variables) forms an algebra. It is commutative and infinite-dimensional. The constant polynomials form the field of scalars. The set of n  n matrices, with the usual operations, also forms an algebra. It is n 2 dimensional. Those diagonal matrices (matrices with all off-diagonal elements equal to zero) whose diagonal elements all have the same value form the field of scalars. How is the scalar product of two matrices defined? The set of all real-valued functions over a set also forms an algebra. Can you specify the multiplication? The constant functions form the field of scalars. A star algebra, also written ‡-algebra, is an algebra over the complex numbers for which there is a mapping ‡  A A, x ( x ‡ , called an involution, with the properties (x ‡ )‡ = x (x + y)‡ = x ‡ + y‡ (cx)‡ = cx ‡ for all (x y)‡ = y‡ x ‡

c>C

(848)

valid for all elements x, y of the algebra A. The element x ‡ is called the adjoint of x. Star algebras are the main type of algebra used in quantum mechanics, since quantummechanical observables form a ‡-algebra. A C‡-algebra is a Banach algebra over the complex numbers with an involution ‡ (a function that is its own inverse) such that the norm YxY of an element x satisfies

Challenge 1571 n

(849)

(A Banach algebra is a complete normed algebra; an algebra is complete if all Cauchy sequences converge.) In short, C‡-algebra is a nicely behaved algebra whose elements form a continuous set and a complex vector space. The name C comes from ‘continuous functions’. Indeed, the bounded continuous functions form such an algebra, with a properly defined norm. Can you find it? Every C‡-algebra contains a space of Hermitean elements (which have a real spectrum), a set of normal elements, a multiplicative group of unitary elements and a set of positive elements (with non-negative spectrum). We should mention one important type of algebra used in mathematics. A division algebra is an algebra for which the equations ax = b and ya = b are uniquely solvable in * Note that a non-associative algebra does not possess a matrix representation.

Copyright © Christoph Schiller November 1997–May 2006

YxY2 = x ‡ x .

Dvipsbugw

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Challenge 1569 ny

x(yz) = (x y)z

Dvipsbugw


algebras

1209

Lie algebras A Lie algebra is special type of algebra (and thus of vector space). Lie algebras are the most important type of non-associative algebra. A vector space L over the field R (or C) with an additional binary operation [ , ], called Lie multiplication or the commutator, is called a real (or complex) Lie algebra if this operation satisfies

Challenge 1572 e

Challenge 1573 ny

antisymmetry (left-)linearity Jacobi identity

(850)

for all elements X, Y , Z > L and for all a, b > R (or C). (Lie algebras are named after Sophus Lie.) The first two conditions together imply bilinearity. A Lie algebra is called commutative if [X, Y] = 0 for all elements X and Y. The dimension of the Lie algebra is the dimension of the vector space. A subspace N of a Lie algebra L is called an ideal* if [L, N] ⊂ N; any ideal is also a subalgebra. A maximal ideal M which satisfies [L, M] = 0 is called the centre of L. A Lie algebra is called a linear Lie algebra if its elements are linear transformations of another vector space V (intuitively, if they are ‘matrices’). It turns out that every finitedimensional Lie algebra is isomorphic to a linear Lie algebra. Therefore, there is no loss * Can you explain the notation [L, N]? Can you define what a maximal ideal is and prove that there is only one?

Copyright © Christoph Schiller November 1997–May 2006

[X, Y] = −[Y , X] [aX + bY , Z] = a[X, Z] + b[Y , Z] [X, [Y ,Z]] + [Y , [Z, X]] + [Z, [X, Y]] = 0

Dvipsbugw

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x or y for all b and all a x 0. Obviously, all type of continuous numbers must be division algebras. Division algebras are thus one way to generalize the concept of a number. One of the important results of modern mathematics states that (finite-dimensional) division algebras can only have dimension 1, like the reals, dimension 2, like the complex numbers, dimension 4, like the quaternions, or dimension 8, like the octonions. There is thus no way to generalize the concept of (continuous) ‘number’ to other dimensions. And now for some fun. Imagine a ring A which contains a number field K as a subring (or ‘field of scalars’). If the ring multiplication is defined in such a way that a general ring element multiplied with an element of K is the same as the scalar multiplication, then A is a vector space, and thus an algebra – provided that every element of K commutes with every element of A. (In other words, the subring K must be central.) For example, the quaternions H are a four-dimensional real division algebra, but although H is a two-dimensional complex vector space, it is not a complex algebra, because i does not commute with j (one has i j = − ji = k). In fact, there are no finite-dimensional complex division algebras, and the only finite-dimensional real associative division algebras are R, C and H. Now, if you are not afraid of getting a headache, think about this remark: every Kalgebra is also an algebra over its field of scalars. For this reason, some mathematicians prefer to define an (associative) K-algebra simply as a ring which contains K as a central subfield. In physics, it is the algebras related to symmetries which play the most important role. We study them next.

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Page 1220

of generality in picturing the elements of finite-dimensional Lie algebras as matrices. The name ‘Lie algebra’ was chosen because the generators, i.e. the infinitesimal elements of every Lie group, form a Lie algebra. Since all important symmetries in nature form Lie groups, Lie algebras appear very frequently in physics. In mathematics, Lie algebras arise frequently because from any associative finite-dimensional algebra (in which the symbol ë stands for its multiplication) a Lie algebra appears when we define the commutator by [X, Y] = X ë Y − Y ë X . (851)

k

Page 1201

The numbers c ikj are called the structure constants of the Lie algebra. They depend on the choice of basis. The structure constants determine the Lie algebra completely. For example, the algebra of the Lie group SU(2), with the three generators defined by Ta = σ a ~2i, where the σ a are the Pauli spin matrices, has the structure constants C abc = ε abc .* Classification of Lie algebras

Finite-dimensional Lie algebras are classified as follows. Every finite-dimensional Lie algebra is the (semidirect) sum of a semisimple and a solvable Lie algebra. A Lie algebra is called solvable if, well, if it is not semisimple. Solvable Lie algebras have not yet been classified completely. They are not important in physics. A semisimple Lie algebra is a Lie algebra which has no non-zero solvable ideal. Other

c

The definition implies that ad(a i ) jk = c ikj , where c ikj are the structure constants of the Lie algebra. For a real Lie algebra, all elements of ad(a) are real for all a > L. Note that for any Lie algebra, a scalar product can be defined by setting X ë Y = Tr( adX ë adY ) .

(854)

This scalar product is symmetric and bilinear. (Can you show that it is independent of the representation?) The corresponding bilinear form is also called the Killing form, after the German mathematician Wilhelm Killing (1847–1923), the discoverer of the ‘exceptional’ Lie groups. The Killing form is invariant under the action of any automorphism of the Lie algebra L. In a given basis, one has X ë Y = Tr( (adX) ë (adY)) = c li k c ski x l y s = дl s x l y s

where дl s = c li k c ski is called the Cartan metric tensor of L.

(855)

Copyright © Christoph Schiller November 1997–May 2006

* Like groups, Lie algebras can be represented by matrices, i.e. by linear operators. Representations of Lie algebras are important in physics because many continuous symmetry groups are Lie groups. The adjoint representation of a Lie algebra with basis a 1 ...a n is the set of matrices ad(a) defined for each element a by (853) [a, a j ] = Q ad(a)c j a c .

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(This fact gave the commutator its name.) Lie algebras are non-associative in general; but the above definition of the commutator shows how to build one from an associative algebra. Since Lie algebras are vector spaces, the elements Ti of a basis of the Lie algebra always obey a relation of the form: [Ti , T j ] = Q c ikj Tk . (852)

Dvipsbugw

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algebras

1211

equivalent definitions are possible, depending on your taste: — a semisimple Lie algebra does not contain non-zero abelian ideals; — its Killing form is non-singular, i.e. non-degenerate; — it splits into the direct sum of non-abelian simple ideals (this decomposition is unique); — every finite-dimensional linear representation is completely reducible; — the one-dimensional cohomology of д with values in an arbitrary finite-dimensional д-module is trivial.

Lie superalgebras

[x, y] = −(−1)SxSSyS [y, x]

0 = (−1)SzSSxS [x, [y, z]] + (−1)SxSSyS [y, [z, x]] + (−1)SySSzS [z, [x, y]]

(856)

Copyright © Christoph Schiller November 1997–May 2006

Lie superalgebras arise when the concept of Lie algebra is extended to the case of supersymmetry. A Lie superalgebra contains even and odd elements; the even elements correspond to bosons and the odd elements to fermions. Supersymmetry applies to systems with anticommuting coordinates. Lie superalgebras are a natural generalization of Lie algebras to supersymmetry, and simply add a Z2 -grading. In detail, a Lie superalgebra is a Z2 -graded algebra over a field of characteristic 0 – usually R or C – with a product [., .], called the (Lie) superbracket or supercommutator, that has the properties

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Ref. 1225

Finite-dimensional semisimple Lie algebras have been completely classified. They decompose uniquely into a direct sum of simple Lie algebras. Simple Lie algebras can be complex or real. The simple finite-dimensional complex Lie algebras all belong to four infinite classes and to five exceptional cases. The infinite classes are also called classical, and are: A n for n E 1, corresponding to the Lie groups SL(n + 1) and their compact ‘cousins’ SU(n + 1); B n for n E 1, corresponding to the Lie groups SO(2n + 1); C n for n E 1, corresponding to the Lie groups Sp(2n); and D n for n E 4, corresponding to the Lie groups SO(2n). Thus A n is the algebra of all skew-Hermitean matrices; B n and D n are the algebras of the symmetric matrices; and C n is the algebra of the traceless matrices. The exceptional Lie algebras are G 2 , F4 , E 6 , E 7 , E 8 . In all cases, the index gives the number of roots. The dimensions of these algebras are A n  n(n+2); B n and C n  n(2n+1); D n  n(2n − 1); G 2  14; F4  32; E 6  78; E 7  133; E 8  248. The simple and finite-dimensional real Lie algebras are more numerous; their classification follows from that of the complex Lie algebras. Moreover, corresponding to each complex Lie group, there is always one compact real one. Real Lie algebras are not so important in fundamental physics. The so-called superalgebras (see below) play a role in systems with supersymmetry. Of the large number of infinite-dimensional Lie algebras, only few are important in physics: among them are the Poincaré algebra, the Cartan algebra, the Virasoro algebra and a few other Kac–Moody algebras.

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where x, y and z are algebra elements that are ‘pure’ in the Z2 -grading. The expression SxS denotes the degree of the algebra element x and is either 0 or 1. The second condition is sometimes called the ‘super Jacobi identity’. Obviously, the even subalgebra of a Lie superalgebra forms a Lie algebra; in that case the superbracket becomes the usual Lie bracket. As in the case of Lie algebras, the simple Lie superalgebras have been completely classified. They fall into five infinite classes and some special cases, namely

The Virasoro algebra The Virasoro algebra is the infinite algebra of operators L n satisfying [L m , L n ] = (m − n)L m+n +

Challenge 1574 ny

c (m 3 − m)δ m,−n 12

(857)

where the number c, which may be zero, is called the central charge, and the factor 1~12 is introduced by historical convention. This rather specific algebra is important in physics because it is the algebra of conformal symmetries in two dimensions.* Can you find a representation in terms of infinite square matrices? Mathematically speaking, the Virasoro algebra is a special case of a Kac–Moody algebra. Kac–Moody algebras

Page 328

* Note that, in contrast, the conformal symmetry group in four dimensions has 15 parameters, and thus its Lie algebra is finite (15) dimensional.

Copyright © Christoph Schiller November 1997–May 2006

In physics, more general symmetries than Lie groups also appear. This happens in particular when general relativity is taken into account. Because of the symmetries of space-time, the number of generators becomes infinite. The concepts of Lie algebra and of superalgebra have to be extended to take space-time symmetry into account. The corresponding algebras are the Kac–Moody algebras. (‘Kac’ is pronounced like ‘Katz’.) Kac–Moody algebras are a particular class of infinite-parameter Lie algebras; thus they have an infinite number of generators. They used to be called associated affine algebras or affine Lie algebras; sometimes they are also called Z-graded Lie algebras. They were introduced independently in 1968 by Victor Kac and Robert Moody. We present some specific examples. Of basic physical importance are those Kac–Moody algebras associated to a symmetry (n) group G a C[t], where t is some continuous parameter. The generators M a of the corresponding algebra have two indices, one, a, that indexes the generators of the group G and another, n, that indexes the generators of the group of Laurent polynomials C[t]. The

Dvipsbugw

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— A(n, m) corresponding to the Lie supergroups SL(n + 1Sm + 1) and their compact ‘cousins’ SU(NSM); — B(n, m) corresponding to the Lie supergroups OSp(2n + 1S2m); — D(n, m) corresponding to the Lie supergroups OSp(2nS2m); — P(n); — Q(n); — the exceptional cases D α (2, 1), G(3), F(4); — and finally the Cartan superalgebras.

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1213

generators form a Kac–Moody algebra if they obey the relations: [M (n) a , Mb

(m)

] = C abc M c(n+m)

n, m = 0, 1, 2, . . . , ª.

for

(858)

For example, if we take G as SU(2) with its generators T n = σ a ~2i, σ a being the Pauli spin matrices, then the objects M (n) = Ta a t n , a

e.g.

(n)

M3

=

1 tn 0 ‘ Œ 2i 0 −t n

Dvipsbugw

(859)

form a Kac–Moody algebra with the structure constants C abc = ε abc of SU(2).

Topolo gy – what shapes exist?

Challenge 1575 n

In a simplified view of topology that is sufficient for physicists, only one type of entity can possess shape: manifolds. Manifolds are generalized examples of pullovers: they are locally flat, can have holes, boundaries and can often be turned inside out. Pullovers are subtle entities. For example, can you turn your pullover inside out while your hands are tied together? (A friend may help you.) By the way, the same feat is also possible with your trousers, while your feet are tied together. Certain professors like to demonstrate this during topology lectures – of course with a carefully selected pair of underpants. Topological spaces

Ref. 1227

— x is in N; — if N and M are neighbourhoods, so is N 9 M; — anything containing a neighbourhood of x is itself a neighbourhood of x. The choice of the subsets is free. All the subsets Yx , for all points x, that were chosen in the definition are then called open sets. (A neighbourhood and an open set are not necessarily the same; but all open sets are also neighbourhoods.) One also calls a topological space a ‘set with a topology’. In effect, a topology specifies the systems of ‘neighbourhoods’ of every point of the set. ‘Topology’ is also the name of the branch of mathematics that studies topological spaces.

Copyright © Christoph Schiller November 1997–May 2006

The study of shapes requires a good definition of a set made of ‘points’. To be able to talk about shape, these sets must be structured in such a way as to admit a useful concept of ‘neighbourhood’ or ‘closeness’ between the elements of the set. The search for the most general type of set which allows a useful definition of neighbourhood has led to the concept of topological space. A topological space is a finite or infinite set X of elements, called points, together with a neighbourhood for each point. A neighbourhood N of a point x is a collection of subsets Yx of X with the properties that

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Topology is group theory. The Erlangen program

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Challenge 1576 e

d numbers and spaces

For example, the real numbers together with the open intervals form the usual topology of R. If one takes all subsets of Ras open sets, one speaks of the discrete topology. If one takes only the full set X and the empty set as open sets, one speaks of the trivial or indiscrete topology. The concept of topological space allows us to define continuity. A mapping from one topological space X to another topological space Y is continuous if the inverse image of every open set in Y is an open set in X. You may verify that this definition is not satisfied by a real function that makes a jump. You may also check that the term ‘inverse’ is necessary in the definition; otherwise a function with a jump would be continuous, as such a function may still map open sets to open sets.* We thus need the concept of topological space, or of neighbourhood, if we want to express the idea that there are no jumps in nature. We also need the concept of topological space in order to be able to define limits.

Of the many special kinds of topological spaces that have been studied, one type is particularly important. A Hausdorff space is a topological space in which for any two points x and y there are disjoint open sets U and V such that x is in U and y is in V . A Hausdorff space is thus a space where, no matter how ‘close’ two points are, they can always be separated by open sets. This seems like a desirable property; indeed, non-Hausdorff spaces are rather tricky mathematical objects. (At Planck energy, it seems that vacuum appears to behave like a non-Hausdorff space; however, at Planck energy, vacuum is not really a space at all. So non-Hausdorff spaces play no role in physics.) A special case of Hausdorff space is well-known: the manifold. Manifolds

Challenge 1577 ny

* The Cauchy-Weierstass definition of continuity says that a real function f (x) is continuous at a point a if (1) f is defined on a open interval containing a, (2) f (x) tends to a limit as x tends to a, and (3) the limit is f (a). In this definition, the continuity of f is defined using the intuitive idea that the real numbers form the basic model of a set that has no gaps. Can you see the connection with the general definition given above?

Copyright © Christoph Schiller November 1997–May 2006

In physics, the most important topological spaces are differential manifolds. Loosely speaking, a differential manifold is a set of points that looks like Rn under the microscope – at small distances. For example, a sphere and a torus are both two-dimensional differential manifolds, since they look locally like a plane. Not all differential manifolds are that simple, as the examples of Figure 394 show. A differential manifold is called connected if any two points can be joined by a path lying in the manifold. (The term has a more general meaning in topological spaces. But the notions of connectedness and pathwise connectedness coincide for differential manifolds.) We focus on connected manifolds in the following discussion. A manifold is called simply connected if every loop lying in the manifold can be contracted to a point. For example, a sphere is simply connected. A connected manifold which is not simply connected is called multiply connected. A torus is multiply connected. Manifolds can be non-orientable, as the well-known Möbius strip illustrates. Nonorientable manifolds have only one surface: they do not admit a distinction between front

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– CS – more on topological spaces to be added – CS –

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topology – what shapes exist?

1215

Dvipsbugw

F I G U R E 395 Compact (left) and noncompact (right) manifolds of various dimensions

and back. If you want to have fun, cut a paper Möbius strip into two along a centre line. You can also try this with paper strips with different twist values, and investigate the regularities. In two dimensions, closed manifolds (or surfaces), i.e. surfaces that are compact and without boundary, are always of one of three types: — The simplest type are spheres with n attached handles; they are called n-tori or surfaces of genus n. They are orientable surfaces with Euler characteristic 2 − 2n. — The projective planes with n handles attached are non-orientable surfaces with Euler characteristic 1 − 2n.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1578 e

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F I G U R E 394 Examples of orientable and non-orientable manifolds of two dimensions: a disc, a Möbius strip, a sphere and a Klein bottle

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Dvipsbugw F I G U R E 396 Simply connected (left), multiply connected (centre) and

disconnected (right) manifolds of one (above) and two (below) dimensions

Therefore Euler characteristic and orientability describe compact surfaces up to homeomorphism (and if surfaces are smooth, then up to diffeomorphism). Homeomorphisms are defined below. The two-dimensional compact manifolds or surfaces with boundary are found by removing one or more discs from a surface in this list. A compact surface can be embedded in R3 if it is orientable or if it has non-empty boundary. In physics, the most important manifolds are space-time and Lie groups of observables. We study Lie groups below. Strangely enough, the topology of space-time is not known. For example, it is unclear whether or not it is simply connected. Obviously, the reason is that it is difficult to observe what happens at large distances form the Earth. However, a similar difficulty appears near Planck scales. If a manifold is imagined to consist of rubber, connectedness and similar global properties are not changed when the manifold is deformed. This fact is formalized by saying that two manifolds are homeomorphic (from the Greek words for ‘same’ and ‘shape’) if between them there is a continuous, one-to-one and onto mapping with a continuous inverse. The concept of homeomorphism is somewhat more general than that of rubber deformation, as can be seen from Figure 397. Holes, homotopy and homology

– CS – more on topology to be added – CS –

Copyright © Christoph Schiller November 1997–May 2006

Only ‘well-behaved’ manifolds play a role in physics: namely those which are orientable and connected. In addition, the manifolds associated with observables, are always compact. The main non-trivial characteristic of connected compact orientable manifolds is that they contain ‘holes’ (see Figure 398). It turns out that a proper description of the holes of manifolds allows us to distinguish between all different, i.e. non-homeomorphic, types of manifold. There are three main tools to describe holes of manifolds and the relations among them: homotopy, homology and cohomology. These tools play an important role in the study of gauge groups, because any gauge group defines a manifold.

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— The Klein bottles with n attached handles are non-orientable surfaces with Euler characteristic −2n.

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Dvipsbugw F I G U R E 397 Examples of homeomorphic pairs

of manifolds

Challenge 1579 d

In other words, through homotopy and homology theory, mathematicians can classify manifolds. Given two manifolds, the properties of the holes in them thus determine whether they can be deformed into each other. Physicists are now extending these results of standard topology. Deformation is a classical idea which assumes continuous space and time, as well as arbitrarily small action. In nature, however, quantum effects cannot be neglected. It is speculated that quantum effects can transform a physical manifold into one with a different topology: for example, a torus into a sphere. Can you find out how this can be achieved? Topological changes of physical manifolds happen via objects that are generalizations of manifolds. An orbifold is a space that is locally modelled by Rn modulo a finite group. Examples are the tear-drop or the half-plane. Orbifolds were introduced by Satake Ichiro in 1956; the name was coined by William Thurston. Orbifolds are heavily studied in string theory.

Page 192

We introduced groups early on because groups play an important role in many parts of physics, from the description of solids, molecules, atoms, nuclei, elementary particles and forces up to the study of shapes, cycles and patterns in growth processes. Group theory is also one of the most important branches of modern mathematics, and is still an active area of research. One of the aims of group theory is the classification of all groups. This has been achieved only for a few special types. In general, one distinguishes between finite and infinite groups. Finite groups are better understood. Every finite group is isomorphic to a subgroup of the symmetric group S N , for some number N. Examples of finite groups are the crystalline groups, used to classify crystal structures, or the groups used to classify wallpaper patterns in terms of their symmetries.

Copyright © Christoph Schiller November 1997–May 2006

Types and cl assification of groups

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F I G U R E 398 The first four two-dimensional compact connected orientable manifolds: 0-, 1-, 2- and 3-tori

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Ref. 1226

d numbers and spaces

Lie groups

Copyright © Christoph Schiller November 1997–May 2006

In nature, the Lagrangians of the fundamental forces are invariant under gauge transformations and under continuous space-time transformations. These symmetry groups are examples of Lie groups, which are a special type of infinite continuous group. They are named after the great Norwegian mathematician Sophus Lie (1849–1899). His name is pronounced like ‘Lee’. A (real) Lie group is an infinite symmetry group, i.e. a group with infinitely many elements, which is also an analytic manifold. Roughly speaking, this means that the elements of the group can be seen as points on a smooth (hyper-) surface whose shape can be described by an analytic function, i.e. by a function so smooth that it can be expressed as a power series in the neighbourhood of every point where it is defined. The points of the Lie group can be multiplied according to the group multiplication. Furthermore, the

Dvipsbugw

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The symmetry groups of Platonic and many other regular solids are also finite groups. Finite groups are a complex family. Roughly speaking, a general (finite) group can be seen as built from some fundamental bricks, which are groups themselves. These fundamental bricks are called simple (finite) groups. One of the high points of twentiethcentury mathematics was the classification of the finite simple groups. It was a collaborative effort that took around 30 years, roughly from 1950 to 1980. The complete list of finite simple groups consists of 1) the cyclic groups Z p of prime group order; 2) the alternating groups An of degree n at least five; 3) the classical linear groups, PSL(n; q), PSU(n; q), PSp(2n; q) and PΩ ε (n; q); 4) the exceptional or twisted groups of Lie type 3 D4 (q), E6 (q), 2 E6 (q), E7 (q), E8 (q), F4 (q), 2 F4 (2n ), G2 (q), 2 G2 (3n ) and 2 B2 (2n ); 5) the 26 sporadic groups, namely M11 , M12 , M22 , M23 , M24 (the Mathieu groups), J1 , J2 , J3 , J4 (the Janko groups), Co1 , Co2 , Co3 (the Conway groups), HS, Mc, Suz (the Co1 ‘babies’), Fi22 , Fi23 , Fi′24 (the Fischer groups), F1 = M (the Monster), F2 , F3 , F5 , He (= F7 ) (the Monster ‘babies’), Ru, Ly, and ON. The classification was finished in the 1980s after over 10 000 pages of publications. The proof is so vast that a special series of books has been started to summarize and explain it. The first three families are infinite. The last family, that of the sporadic groups, is the most peculiar; it consists of those finite simple groups which do not fit into the other families. Some of these sporadic groups might have a role in particle physics: possibly even the largest of them all, the so-called Monster group. This is still a topic of research. (The Monster group has about 8.1 ë 1053 elements; more precisely, its order is 808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000 or 246 ë 320 ë 59 ë 76 ë 112 ë 133 ë 17 ë 19 ë 23 ë 29 ë 31 ë 41 ë 47 ë 59 ë 71.) Of the infinite groups, only those with some finiteness condition have been studied. It is only such groups that are of interest in the description of nature. Infinite groups are divided into discrete groups and continuous groups. Discrete groups are an active area of mathematical research, having connections with number theory and topology. Continuous groups are divided into finitely generated and infinitely generated groups. Finitely generated groups can be finite-dimensional or infinite-dimensional. The most important class of finitely generated continuous groups are the Lie groups.

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Challenge 1580 n

A = UeH .

(860)

(H is given by H = 21 ln A† A, and U is given by U = Ae−H .) – CS – more on Lie groups to be added – CS –

Connectedness It is not hard to see that the Lie groups SU(N) are simply connected for all N = 2, 3 . . . ; they have the topology of a 2N-dimensional sphere. The Lie group U(1), having the topology of the 1-dimensional sphere, or circle, is multiply connected. The Lie groups SO(N) are not simply connected for any N = 2, 3 . . . . In general, SO(N,K) is connected, and GL(N,C) is connected. All the Lie groups SL(N,K) are connected; and SL(N,C) is simply connected. The Lie groups Sp(N,K) are connected; Sp(2N,C) is simply connected. Generally, all semi-simple Lie groups are connected.

Copyright © Christoph Schiller November 1997–May 2006

The simple Lie groups U(1) and SO(2,R) and the Lie groups based on the real and complex numbers are abelian (see Table 93); all others are non-abelian. Lie groups are manifolds. Therefore, in a Lie group one can define the distance between two points, the tangent plane (or tangent space) at a point, and the notions of integration and differentiations. Because Lie groups are manifolds, Lie groups have the same kind of structure as the objects of Figures 394, 395 and 396. Lie groups can have any number of dimensions. Like for any manifold, their global structure contains important information; let us explore it.

Dvipsbugw

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coordinates of the product have to be analytic functions of the coordinates of the factors, and the coordinates of the inverse of an element have to be analytic functions of the coordinates of the element. In fact, this definition is unnecessarily strict: it can be proved that a Lie group is just a topological group whose underlying space is a finite-dimensional, locally Euclidean manifold. A complex Lie group is a group whose manifold is complex and whose group operations are holomorphic (instead of analytical) functions in the coordinates. In short, a Lie group is a well-behaved manifold in which points can be multiplied (and technicalities). For example, the circle T = ˜z > C  SzS = 1, with the usual complex 1 multiplication, is a real Lie group. It is abelian. This group is also called S , as it is the onedimensional sphere, or U(1), which means ‘unitary group of one dimension’. The other one-dimensional Lie groups are the multiplicative group of non-zero real numbers and its subgroup, the multiplicative group of positive real numbers. So far, in physics, only linear Lie groups have played a role – that is, Lie groups which act as linear transformations on some vector space. (The cover of SL(2,R) or the complex compact torus are examples of non-linear Lie groups.) The important linear Lie groups for physics are the Lie subgroups of the general linear group GL(N,K), where K is a number field. This is defined as the set of all non-singular, i.e. invertible, NN real, complex or quaternionic matrices. All the Lie groups discussed below are of this type. Every complex invertible matrix A can be written in a unique way in terms of a unitary matrix U and a Hermitean matrix H:

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d numbers and spaces

1220

The Lie groups O(N,K), SO(N,M,K) and GL(N,R) are not connected; they contain two connected components. Note that the Lorentz group is not connected: it consists of four separate pieces. Like the Poincaré group, it is not compact, and neither is any of its four pieces. Broadly speaking, the non-compactness of the group of space-time symmetries is a consequence of the non-compactness of space-time. Compactness

TA B L E 93 Properties of the most important real and complex Lie groups

Description

Properties a

Lie algebra

Description of Lie algebra

real

1. Real groups Rn

R

Euclidean space with addition

abelian, simply connected, not compact; π0 = π1 = 0 non-zero real abelian, not numbers with connected, not compact; π 0 = Z2 , multiplicano π 1 tion

Dimension

Rn

abelian, thus Lie bracket is zero; not simple

n

R

abelian, thus Lie bracket is zero

1

Copyright © Christoph Schiller November 1997–May 2006

Lie group

Dvipsbugw

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Page 1209

A Lie group is compact if it is closed and bounded when seen as a manifold. For a given parametrization of the group elements, the Lie group is compact if all parameter ranges are closed and finite intervals. Otherwise, the group is called non-compact. Both compact and non-compact groups play a role in physics. The distinction between the two cases is important, because representations of compact groups can be constructed in the same simple way as for finite groups, whereas for non-compact groups other methods have to be used. As a result, physical observables, which always belong to a representation of a symmetry group, have different properties in the two cases: if the symmetry group is compact, observables have discrete spectra; otherwise they do not. All groups of internal gauge transformations, such as U(1) and SU(n), form compact groups. In fact, field theory requires compact Lie groups for gauge transformations. The only compact Lie groups are Tn , O(n), U(n), SO(n) and SU(n), their double cover Spin(n) and the Sp(n). In contrast, SL(n,R), GL(n,R), GL(n,C) and all others are not compact. Besides being manifolds, Lie groups are obviously also groups. It turns out that most of their group properties are revealed by the behaviour of the elements which are very close (as points on the manifold) to the identity. Every element of a compact and connected Lie group has the form exp(A) for some A. The elements A arising in this way form an algebra, called the corresponding Lie algebra. For any linear Lie group, every element of the connected subgroup can be expressed as a finite product of exponentials of elements of the corresponding Lie algebra. In short, Lie algebras express the local properties of Lie groups. That is the reason for their importance in physics.

Dvipsbugw


types and cl assification of groups

Lie group

Description

RA0

GL(n, R) general linear group: invertible n-by-n real matrices n-by-n real GL+ (n, R) matrices with positive determinant

Lie algebra

Description of Lie algebra

Dimension

abelian, thus Lie bracket is zero

1

abelian, thus Lie bracket is zero

1

H

quaternions, with Lie bracket the commutator

4

simply connected, compact; isomorphic to SU(2), Spin(3) and to double cover of SO(3); π 0 = π 1 = 0

Im(H)

not connected, not compact; π 0 = Z2 , no π 1

M(n, R)

quaternions with zero 3 real part, with Lie bracket the commutator; simple and semi-simple; isomorphic to real 3-vectors, with Lie bracket the cross product; also isomorphic to su(2) and to so(3) n-by-n matrices, with n 2 Lie bracket the commutator

R abelian, simply connected, not compact; π0 = π1 = 0 abelian, connected, R not simply connected, compact; π 0 = 0, π 1 = Z

simply connected, not compact; π0 = π1 = 0

n-by-n matrices, with n 2 Lie bracket the commutator

n-by-n matrices with trace 0, with Lie bracket the commutator

n2 − 1

Copyright © Christoph Schiller November 1997–May 2006

M(n, R) simply connected, not compact; π 0 = 0, for n = 2: π 1 = Z, for n C 2: π 1 = Z2 ; GL+ (1, R) isomorphic to RA0 sl(n, R) SL(n, R) special linear simply connected, not compact if n A 1; = An−1 group: real matrices with π 0 = 0, for n = 2: determinant 1 π 1 = Z, for n C 2: π 1 = Z2 ; SL(1, R) is a single point, SL(2, R) is isomorphic to SU(1, 1) and Sp(2, R)

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positive real numbers with multiplication S1 = R~Z complex = U(1) = numbers of absolute value T = SO(2) 1, with multi= Spin(2) plication H non-zero quaternions with multiplication 3 S quaternions of absolute value 1, with multiplication, also known as Sp(1); topologically a 3-sphere

Properties a

1221

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d numbers and spaces

1222

Lie group

Description

Properties a

Lie algebra

Description of Lie algebra

Dimension

n(n − 1)~2 skew-symmetric n-by-n real matrices, with Lie bracket the commutator; so(3, R) is isomorphic to su(2) and to R3 with the cross product connected, compact; so(n, R) skew-symmetric SO(n, R) special n(n − 1)~2 for n E 2 not simply = B n−1 = SO(n) orthogonal or n-by-n real matrices, 2 connected; π 0 = 0, group: real D n2 with Lie bracket the for n = 2: π 1 = Z, for orthogonal commutator; for n = 3 matrices with n C 2: π 1 = Z2 and n E 5 simple and determinant 1 semisimple; SO(4) is semisimple but not simple n(n − 1)~2 so(n, R) skew-symmetric simply connected Spin(n) spin group; n-by-n real matrices, double cover for n E 3, compact; with Lie bracket the for n = 3 and n E 5 of SO(n); commutator simple and Spin(1) is isomorphic to semisimple; for Q2 , Spin(2) to n A 1: π 0 = 0, for n A 2: π 1 = 0 S1 n(2n + 1) not compact; π 0 = 0, sp(2n, R) real matrices A that symplectic π1 = Z = Cn satisfy JA + AT J = 0 Sp(2n, R) group: real where J is the standard symplectic skew-symmetric matrices matrix;b simple and semisimple sp(n) n-by-n quaternionic n(2n + 1) compact, simply Sp(n) for compact connected; matrices A satisfying symplectic nE3 π0 = π1 = 0 A = −A‡ , with Lie group: bracket the quaternionic commutator; simple n  n unitary and semisimple matrices n2 u(n) n-by-n complex U(n) unitary group: not simply matrices A satisfying connected, compact; complex A = −A‡ , with Lie n  n unitary it is not a complex bracket the Lie group/algebra; matrices commutator π 0 = 0, π 1 = Z; isomorphic to S 1 for n=1 O(n, R) = O(n)

orthogonal group: real orthogonal matrices; symmetry of hypersphere

not connected, compact; π 0 = Z2 , no π 1

so(n, R)

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006

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types and cl assification of groups

1223

Lie group

Description

Properties a

Lie algebra

Description of Lie algebra

SU(n)

special unitary group: complex n  n unitary matrices with determinant 1

simply connected, compact; it is not a complex Lie group/algebra; π0 = π1 = 0

su(n)

n2 − 1 n-by-n complex matrices A with trace 0 satisfying A = −A‡ , with Lie bracket the commutator; for n E 2 simple and semisimple

2. Complex groups c n

group operation is addition

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complex n

n

C

abelian, thus Lie bracket is zero

1

M(n, C)

n-by-n matrices, with n 2 Lie bracket the commutator

sl(n, C)

n-by-n matrices with n 2 − 1 trace 0, with Lie bracket the commutator; simple, semisimple; sl(2, C) is isomorphic to su(2, C) a C

sl(2, C)

2-by-2 matrices with trace 0, with Lie bracket the commutator; sl(2, C) is isomorphic to su(2, C) a C

3

Copyright © Christoph Schiller November 1997–May 2006

abelian, thus Lie bracket is zero

C

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abelian, simply connected, not compact; π0 = π1 = 0  abelian, not simply C nonzero connected, not complex numbers with compact; π 0 = 0, π1 = Z multiplication GL(n, C) general linear simply connected, not compact; π 0 = 0, group: π 1 = Z; for n = 1 invertible isomorphic to C n-by-n complex matrices SL(n, C) special linear simply connected; for n E 2 not group: compact; complex matrices with π 0 = π 1 = 0; determinant 1 SL(2, C) is isomorphic to Spin(3, C) and Sp(2, C) projective not compact; π 0 = 0, PSL(2, C) special linear π 1 = Z2 group; isomorphic to the Möbius group, to the restricted Lorentz group SO+ (3, 1, R) and to SO(3, C) C

Dimension

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d numbers and spaces

1224

Lie group

Description

O(n, C)

orthogonal group: complex orthogonal matrices SO(n, C) special orthogonal group: complex orthogonal matrices with determinant 1

Lie algebra

Description of Lie algebra

not connected; for n E 2 not compact; π 0 = Z2 , no π 1

so(n, C)

n(n − 1)~2 skew-symmetric n-by-n complex matrices, with Lie bracket the commutator n(n − 1)~2 skew-symmetric n-by-n complex matrices, with Lie bracket the commutator; for n = 3 and n E 5 simple and semisimple

Dimension

so(n, C) for n E 2 not compact; not simply connected; π 0 = 0, for n = 2: π 1 = Z, for n C 2: π 1 = Z2 ; nonabelian for n A 2, SO(2, C) is abelian and isomorphic to C not compact; sp(2n, C) complex matrices that n(2n + 1) π0 = π1 = 0 satisfy JA + AT J = 0 where J is the standard skew-symmetric matrix;b simple and semi-simple

a. The group of components π 0 of a Lie group is given; the order of π 0 is the number of components of the Lie group. If the group is trivial (0), the Lie group is connected. The fundamental group π 1 of a connected Lie group is given. If the group π 1 is trivial (0), the Lie group is simply connected. This table is based on that in the Wikipedia, at http://en.wikipedia.org/wiki/Table_of_Lie_groups. b. The standard skew-symmetric matrix J of rank 2n is J k l = δ k,n+l − δ k+n, l . c. Complex Lie groups and Lie algebras can be viewed as real Lie groups and real Lie algebras of twice the dimension.

Mathematical curiosities and fun challenges Mathematics is a passion in itself.

Mathematics provides many counter-intuitive results. Reading a book on the topic, such as Bernard R. Gelbaum & John M.H. Olmsted, Theorems and Counter-examples in Mathematics, Springer, 1993, can help you sharpen your mind. ** The distinction between one, two and three dimensions is blurred in mathematics. This is well demonstrated in the text Hans Sagan, Space Filling Curves, Springer Verlag, 1994. ** There are at least seven ways to win a million dollars with mathematical research. The Clay Mathematics Institute at http://www.claymath.org offers them for major advances

Copyright © Christoph Schiller November 1997–May 2006

**

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symplectic Sp(2n, C) group: complex symplectic matrices

Properties a

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mathematical curiosities and fun challenges

1225

in seven topics: — proving the Birch and Swinnerton–Dyer conjecture about algebraic equations; — proving the Poincaré conjecture about topological manifolds; — solving the Navier–Stokes equations for fluids; — finding criteria distinguishing P and NP numerical problems; — proving the Riemann hypothesis stating that the nontrivial zeros of the zeta function lie on a line; — proving the Hodge conjectures; — proving the connection between Yang–Mills theories and a mass gap in quantum field theory.

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On each of these topics, substantial progress can buy you a house.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

No man but a blockhead ever wrote except for money. Samuel Johnson

Copyright © Christoph Schiller November 1997–May 2006

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d numbers and spaces

1226

Biblio graphy 1218 A good reference is the Encyclopedia of Mathematics, in 10 volumes, Kluwer Academic

Publishers, 1988−1993. It explains most concepts used in mathematics. Spending an hour with it looking up related keywords is an efficient way to get an introduction into any part of mathematics, especially into the vocabulary and the main connections. Cited on page 1194. 1219 S.L. Altman, Rotations, Quaternions and Double Groups, Clarendon Press, 1986, and

1220

1222

1223

1224 1225

1227

Copyright © Christoph Schiller November 1997–May 2006

1226

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

1221

also S.L. Altman, Hamilton, Rodriguez and the quaternion scandal, Mathematical Magazine 62, pp. 291–308, 1988. See also J.C. Hart, G.K. Francis & L.H. Kauffman, Visualzing quaternion rotation, ACM Transactions on Graphics 13, pp. 256–276, 1994. The latter can be downloaded in several places via the internet. Cited on page 1199. An excellent introduction into number systems in mathematics, including nonstandard numbers, quaternions, octonions, p-adic numbers and surreal numbers, is the book H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel & R. R emmert, Zahlen, 3rd edition, Springer Verlag, 1993. It is also available in English, under the title Numbers, Springer Verlag, 1990. Cited on pages 1195, 1203, and 1204. A. Waser, Quaternions in Electrodynamics, 2001. The text can be downloaded from various websites. Cited on pages 1197 and 1201. See the fine book by Louis H. Kauffman, Knots and Physics, World Scientific, 2nd edition, 1994, which gives a clear and visual introduction to the mathematics of knots and their main applications to physics. Cited on page 1200. Gaussian integers are explored by G.H. Hardy & E.M. Wright, An Introduction to the Theory of Numbers, 5th edition, Clarendon Press, Oxford, 1979, in the sections 12.2 ‘The Rational Integers, the Gaussian Integers, and the Integers’, pp. 178–180, and 12.6 ‘Properties of the Gaussian Integers’ pp. 182–183. For challenges relating to Gaussian integers, look at http://www.mathpuzzle.com/Gaussians.html. Cited on page 1204. About transfinite numbers, see the delightful paperback by Rudy Rucker, Infinity and the Mind – the Science and Philosophy of the Infinite, Bantam, 1983. Cited on page 1204. M. Fl ato, P. Sally & G. Zuckerman (editors), Applications of Group Theory in Physics and Mathematical Physics, Lectures in applied mathematics, volume 21, American Mathematical Society, 1985. This interesting book was written before the superstring revolution, so that the topic is missing from the otherwise excellent presentation. Cited on page 1211. An introduction to the classification theorem is R. Solomon, On finite simple groups and their classification, Notices of the AMS 42, pp. 231–239, 1995, also available on the web as http://www.ams.org/notices/199502/solomon.ps Cited on page 1218. For an introduction to topology, see for example Mikio Nakahara, Geometry, Topology and Physics, IOP Publishing, 1990. Cited on page 1213.

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Appendix E

SOURC ES OF I NFOR MATION ON MOTION

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In a consumer society there are inevitably two kinds of slaves: the prisoners of addiction and the prisoners of envy. Ivan Illich*

I

” ”

* Ivan Illich (b. 1926 Vienna, d. 2002 Bremen), Austrian theologian and social and political thinker.

Copyright © Christoph Schiller November 1997–May 2006

n the text, outstanding books introducing neighbouring domains are presented n footnotes. The bibliographies at the end of each chapter collect general material n order to satisfy further curiosity about what is encountered in this mountain ascent. All citations can also be found by looking up the author’s name in the index. To find additional information, either libraries or the internet can help. In a library, review articles of recent research appear in journals such as Reviews of Modern Physics, Reports on Progress in Physics, Contemporary Physics and Advances in Physics. Good pedagogical introductions are found in the American Journal of Physics, the European Journal of Physics and Physik in unserer Zeit. Another useful resource is Living Reviews in Relativity, found at http://www.livingreviews.org. Overviews on research trends occasionally appear in magazines such as Physics World, Physics Today, Europhysics Journal, Physik Journal and Nederlands tijdschrift voor natuurkunde. For coverage of all the sciences together, the best sources are the magazines Nature, New Scientist, Naturwissenschaften, La Recherche and the cheap but excellent Science News. Research papers appear mainly in Physics Letters B, Nuclear Physics B, Physical Review D, Physical Review Letters, Classical and Quantum Gravity, General Relativity and Gravitation, International Journal of Modern Physics and Modern Physics Letters. The newest results and speculative ideas are found in conference proceedings, such as the Nuclear Physics B Supplements. Research articles also appear in Fortschritte der Physik, Zeitschrift für Physik C, La Rivista del Nuovo Cimento, Europhysics Letters, Communications in Mathematical Physics, Journal of Mathematical Physics, Foundations of Physics, International Journal of Theoretical Physics and Journal of Physics G. There is also the purely electronic New Journal of Physics, which can be found at the http://www.njp.org website. Papers on the description of motion without time and space which appear after this text is published can be found via the Scientific Citation Index. It is published in printed form and as compact disc, and allows, one to search for all publications which cite a given

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“ “

No place affords a more striking conviction of the vanity of human hopes than a public library. Samuel Johnson

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e sources of information on motion

1228

TA B L E 94 The structure of the Arxiv preprint archive for physics and

related topics at http://www.arxiv.org

A b b r e v i at i o n

general relativity and quantum cosmology astrophysics experimental nuclear physics theoretical nuclear physics theoretical high-energy physics computational high-energy physics phenomenological high-energy physics experimental high-energy physics quantum physics general physics condensed matter physics nonlinear sciences mathematical physics mathematics computer science quantitative biology

gr-qc astro-ph nucl-ex nucl-th hep-th hep-lat hep-ph hep-ex quant-ph physics cond-mat nlin math-ph math CoRR q-bio

To receive preprints by email, send an email to an address of the form gr-qc@arxiv.org (or the corresponding abbreviation in place of ‘grqc’), with a subject line consisting simply of the word ‘help’, without the quotes.

* Several decades ago, the provocative book by Ivan Illich, Deschooling Society, Harper & Row, 1971, listed four basic ingredients for any educational system:

Copyright © Christoph Schiller November 1997–May 2006

paper. Then, using the bimonthly Physics Abstracts, which also exists in both paper and electronic form, you can look up the abstract of the paper and check whether it is of interest. But by far the simplest and most efficient way to keep in touch with ongoing research on motion is to use the internet, the international computer network. To anybody with a personal computer connected to a telephone, most theoretical physics papers are available free of charge, as preprints, i.e. before official publication and checking by referees, at the http://www.arxiv.org website. Details are given in Table 94. A service for finding subsequent preprints that cite a given one is also available. In the last decade of the twentieth century, the internet expanded into a combination of library, media store, discussion platform, order desk, brochure collection and time waster. Today, commerce, advertising and – unfortunately – crime of all kind are also an integral part of the web. With a personal computer, a modem and free browser software, one can look for information in millions of pages of documents. The various parts of the documents are located in various computers around the world, but the user does not need to be aware of this.*

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

To p i c

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e sources of information on motion

1229

To start using the web, ask a friend who knows.* Searching the web for authors, organizations, books, publications, companies or simple keywords using search engines can be a rewarding or a time-wasting experience. A selection of interesting servers are given below. TA B L E 95 Some interesting servers on the world-wide web

To p i c

W e b s i t e a d d r e s s o r ‘ URL ’

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General topics Wikipedia Information search engines

Physics Research preprints

http://www.arxiv.org – see page 1228

Copyright © Christoph Schiller November 1997–May 2006

1. access to resources for learning, e.g. books, equipment, games, etc. at an affordable price, for everybody, at any time in their life; 2. for all who want to learn, access to peers in the same learning situation, for discussion, comparison, cooperation and competition; 3. access to elders, e.g. teachers, for their care and criticism towards those who are learning; 4. exchanges between students and performers in the field of interest, so that the latter can be models for the former. For example, there should be the possibility to listen to professional musicians and reading the works of specialist writers. This also gives performers the possibility to share, advertise and use their skills. Illich develops the idea that if such a system were informal – he then calls it a ‘learning web’ or ‘opportunity web’ – it would be superior to formal, state-financed institutions, such as conventional schools, for the development of mature human beings. These ideas are deepened in his following works, Deschooling Our Lives, Penguin, 1976, and Tools for Conviviality, Penguin, 1973. Today, any networked computer offers one or more of the following: email (electronic mail), ftp (file transfer to and from another computer), access to Usenet (the discussion groups on specific topics, such as particle physics), and the powerful world-wide web. (Roughly speaking, each of those includes the ones before.) In a rather unexpected way, all these facilities of the internet have transformed it into the backbone of the ‘opportunity web’ discussed by Illich. However, as in any school, it strongly depends on the user’s discipline whether the internet actually does provide a learning web. * It is also possible to use both the internet and to download files through FTP with the help of email only. But the tools change too often to give a stable guide here. Ask your friend.

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http://www.wikipedia.org http://www.altavista.com http://www.metager.de http://www.google.com http://www.yahoo.com Search old usenet articles http://groups.google.com Frequently asked questions on http://www.faqs.org physics and other topics Libraries http://www.konbib.nl http://portico.bl.uk http://www.theeuropeanlibrary.org http://www.hero.ac.uk/uk/niss/niss_library4008.cfm http://www.bnf.fr http://www.grass-gis.de/bibliotheken http://www.loc.gov

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To p i c

W e b s i t e a d d r e s s o r ‘ URL ’ http://www.slac.stanford.edu/spires http://pdg.web.cern.ch/pdg http://www.aip.org/physnews/update http://www.innovations-report.de/berichte/physik.php http://star.tau.ac.il/QUIZ/ http://www.phy.duke.edu/~hsg/physics-challenges/challenges. html http://www.physics.umd.edu/lecdem/outreach/QOTW/active http://www.nyteknik.se/miniproblemet http://www.bipm.fr http://www.chemie.fu-berlin.de/chemistry/general/units.html http://www.sciam.com/askexpert_directory.cfm http://www.osti.gov

Copyright © Christoph Schiller November 1997–May 2006

http://www.sciencenews.org http://www.nobel.se/physics/laureates http://www.if.ufrj.br/famous/physlist.html http://www.phys.lsu.edu/mog.html http://www.livingreviews.org http://math.ucr.edu/home/baez/relativity.html http://www.tat.physik.uni-tuebingen.de/~weiskopf http://www.cern.ch/ http://www.hep.net http://www.nikhef.nl http://www.het.brown.edu/physics/review/index.html Physics textbooks on the web http://www.plasma.uu.se/CED/Book http://www.biophysics.org/education/resources.htm http://www.lightandmatter.com http://www.motionmountain.net Three beautiful French sets of http://feynman.phy.ulaval.ca/marleau/notesdecours.htm notes on classical mechanics and particle theory The excellent Radical Freshman http://www.physics.nmt.edu/~raymond/teaching.html Physics by David Raymond Physics course scripts from http://ocw.mit.edu/OcwWeb/Physics/index.html MIT Physics lecture scripts in http://www.akleon.de German and English ‘World lecture hall’ http://www.utexas.edu/world/lecture Engineering data and formulae http://www.efunda.com/

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Particle data Physics news, weekly Physics news, daily Physics problems by Yakov Kantor Physics problems by Henry Greenside Physics ‘question of the week’ Physics ‘miniproblem’ Official SI unit site Unit conversion ‘Ask the experts’ Abstracts of papers in physics journals Science News Nobel Prize winners Pictures of physicists Gravitation news Living Reviews in Relativity Information on relativity Relativistic imaging and films Physics organizations

e sources of information on motion

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e sources of information on motion

To p i c

1231

W e b s i t e a d d r e s s o r ‘ URL ’

Mathematics

Curiosities

Copyright © Christoph Schiller November 1997–May 2006

http://webmineral.com http://www.mindat.org ESA http://sci.esa.int NASA http://www.nasa.gov Hubble space telescope http://hubble.nasa.gov Sloan Digital Sky Survey http://skyserver.sdss.org The ‘cosmic mirror’ http://www.astro.uni-bonn.de/~dfischer/mirror Solar system simulator http://space.jpl.nasa.gov Observable satellites http://liftoff.msfc.nasa.gov/RealTime/JPass/20/ Astronomy picture of the day http://antwrp.gsfc.nasa.gov/apod/astropix.html The Earth from space http://www.visibleearth.nasa.gov Current solar data http://www.n3kl.org/sun Optical illusions http://www.sandlotscience.com Petit’s science comics http://www.jp-petit.com Physical toys http://www.e20.physik.tu-muenchen.de/~cucke/toylinke.htm Physics humour http://www.dctech.com/physics/humor/biglist.php Literature on magic http://www.faqs.org/faqs/magic-faq/part2/ Algebraic surfaces http://www.mathematik.uni-kl.de/~hunt/drawings.html Making paper aeroplanes http://www.pchelp.net/paper_ac.htm http://www.ivic.qc.ca/~aleexpert/aluniversite/klinevogelmann. html Small flying helicopters http://pixelito.reference.be Ten thousand year clock http://www.longnow.org Gesellschaft Deutscher http://www.gdnae.de/ Naturforscher und Ärzte Pseudoscience http://suhep.phy.syr.edu/courses/modules/PSEUDO/pseudo_ main.html Minerals

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

‘Math forum’ internet resource http://mathforum.org/library/ collection Biographies of mathematicians http://www-history.mcs.st-andrews.ac.uk/BiogIndex.html Purdue math problem of the http://www.math.purdue.edu/academics/pow/ week Macalester College maths http://mathforum.org/wagon/ problem of the week Mathematical formulae http://dlmf.nist.gov Functions http://functions.wolfram.com Symbolic integration http://www.integrals.com Weisstein’s World of http://mathworld.wolfram.com Mathematics

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e sources of information on motion

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To p i c

W e b s i t e a d d r e s s o r ‘ URL ’

Crackpots Mathematical quotations The ‘World Question Center’ Plagiarism Hoaxes

http://www.crank.net http://math.furman.edu/mwoodard/~mquot.html http://www.edge.org/questioncenter.html http://www.plagiarized.com http://www.museumofhoaxes.com

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Do you want to study physics without actually going to university? Nowadays it is possible to do so via email and internet, in German, at the University of Kaiserslautern.* In the near future, a nationwide project in Britain should allow the same for English-speaking students. As an introduction, use the latest update of this physics text!

Si tacuisses, philosophus mansisses.*** After Boethius.

” ”

Copyright © Christoph Schiller November 1997–May 2006

* See the http://www.fernstudium-physik.de website. ** ‘The internet is the most open form of a closed institution.’ *** ‘If you had kept quiet, you would have remained a philosopher.’ After the story Boethius tells in De consolatione philosophiae, 2.7, 67 ff.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

“ “

Das Internet ist die offenste Form der geschlossenen Anstalt.** Matthias Deutschmann

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Appendix F

C HALLENGE HI NT S AND SOLUTIONS

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Never make a calculation before you know the answer. John Wheeler’s motto

Challenge 2, page 23: These topics are all addressed later in the text.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 3, page 30: There are many ways to distinguish real motion from an illusion of motion: for example, only real motion can be used to set something else into motion. In addition, the motion illusions of the figures show an important failure; nothing moves if the head and the paper remain fixed with respect to each other. In other words, the illusion only amplifies existing motion, it does not create motion from nothing. Challenge 4, page 30: Without detailed and precise experiments, both sides can find examples to prove their point. Creation is supported by the appearance of mould or bacteria in a glass of water; creation is also supported by its opposite, namely traceless disappearance, such as the disappearance of motion. However, conservation is supported and creation falsified by all those investigations that explore assumed cases of appearance or disappearance in full detail. Challenge 6, page 32: Political parties, sects, helping organizations and therapists of all kinds are typical for this behaviour. Challenge 7, page 36: The issue is not yet completely settled for the motion of empty space, such as in the case of gravitational waves. In any case, empty space is not made of small particles of finite size, as this would contradict the transversality of gravity waves. Challenge 8, page 38: The circular definition is: objects are defined as what moves with respect to the background, and the background is defined as what stays when objects change. We shall return to this important issue several times in our adventure. It will require a certain amount of patience to solve it, though. Challenge 9, page 39: Holes are not physical systems, because in general they cannot be tracked. Challenge 10, page 39: See page 815. Challenge 11, page 40: A ghost can be a moving image; it cannot be a moving object, as objects cannot interpenetrate. See page 785. Challenge 12, page 40: Hint: yes, there is such a point. Challenge 13, page 40: Can one show at all that something has stopped moving? Challenge 14, page 40: How would you measure this?

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

John Wheeler wanted people to estimate, to try and to guess; but not saying it out loud. A correct guess reinforces the physics instinct, whereas a wrong one leads to the pleasure of surprise. This text contains 1580 challenges. Let me know the challenge for which you want a hint or a solution to be added next.

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1234

f challenge hints and solu tions

Challenge 15, page 40: The number of reliable digits of a measurement result is a simple quantification of precision. Challenge 16, page 40: No; memory is needed for observation and measurements. Challenge 17, page 40: Note that you never have observed zero speed.

Challenge 18, page 41: (264 − 1) = 18 446 744 073 700 551 615 grains of rice, given a world har-

vest of 500 million tons, are about 4000 years of rice harvests.

Challenge 19, page 41: Some books state that the flame leans inwards. But experiments are not

easy, and sometimes the flame leans outwards. Just try it. Can you explain your observations?

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Challenge 20, page 41: Accelerometers are the simplest motion detectors. They exist in form

Challenge 21, page 41: The ball rolls towards the centre of the table, as the centre is somewhat

lower than the border, shoots over, and then performs an oscillation around that centre. The period is 84 min, as shown in challenge 299. Challenge 22, page 41: Accelerations can be felt. Many devices measure accelerations and then

deduce the position. They are used in aeroplanes when flying over the atlantic. Challenge 23, page 41: The necessary rope length is nh, where n is the number of wheels/pulleys. Challenge 24, page 41: The block moves twice as fast as the cylinders, independently of their

radius. Challenge 25, page 41: This methods is known to work with other fears as well. Challenge 26, page 42: Three couples require 11 passages. Two couples require 5. For four or more couples there is no solution. What is the solution if there are n couples and n − 1 places on the boat? Challenge 27, page 42: In everyday life, this is correct; what happens when quantum effects are taken into account? Challenge 28, page 43: There is only one way: compare the velocity to be measured with the

Challenge 29, page 44: Take the average distance change of two neighbouring atoms in a piece

of quartz over the last million years. Do you know something still slower? Challenge 30, page 45: Equivalently: do points in space exist? The third part studies this issue

in detail; see page 1001. Challenge 31, page 46: All electricity sources must use the same phase when they feed electric power into the net. Clocks of computers on the internet must be synchronized. Challenge 32, page 46: Note that the shift increases quadratically with time, not linearly. Challenge 33, page 47: Natural time is measured with natural motion. Natural motion is the motion of light. Natural time is this defined with the motion of light.

Copyright © Christoph Schiller November 1997–May 2006

speed of light. In fact, almost all physics textbooks, both for schools and for university, start with the definition of space and time. Otherwise excellent relativity textbooks have difficulties avoiding this habit, even those that introduce the now standard k-calculus (which is in fact the approach mentioned here). Starting with speed is the logically cleanest approach.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

of piezo devices that produce a signal whenever the box is accelerated and can cost as little as one euro. Another accelerometer that might have a future is an interference accelerometer that makes use of the motion of an interference grating; this device might be integrated in silicon. Other, more precise accelerometers use gyroscopes or laser beams running in circles. Velocimeters and position detectors can also detect motion; they need a wheel or at least an optical way to look out of the box. Tachographs in cars are examples of velocimeters, computer mice are examples of position detectors.

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Challenge 34, page 48: Galileo measured time with a scale (and with other methods). His stopwatch was a water tube that he kept closed with his thumb, pointing into a bucket. To start the stopwatch, he removed his thumb, to stop it, he put it back on. The volume of water in the bucket then gave him a measure of the time interval. This is told in his famous book Galileo Galilei, Discorsi e dimostrazioni matematiche intorno a due nuove scienze attenenti alla mecanica e i movimenti locali, usually simply called the ‘Discorsi’, which he published in 1638 with Louis Elsevier in Leiden, in the Netherlands. Challenge 35, page 48: There is no way to define a local time at the poles that is consistent with

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all neighbouring points. Challenge 37, page 50: The forest is full of light and thus of light rays. Challenge 38, page 50: One pair of muscles moves the lens along the third axis by deforming the eye from prolate to spherical to oblate. Challenge 39, page 50: This you can solve trying to think in four dimensions. Try to imagine

how to switch the sequence when two pieces cross. Challenge 43, page 54: It is easier to work with the unit torus. Take the unit interval [0, 1] and

Ref. 35

equate the end points. Define a set B in which the elements are a given real number b from the interval plus all those numbers who differ from that real by a rational number. The unit circle can be thought as the union of all the sets B. (In fact, every set B is a shifted copy of the rational numbers Q.) Now build a set A by taking one element from each set B. Then build the set family consisting of the set A and its copies A q shifted by a rational q. The union of all these sets is the unit torus. The set family is countably infinite. Then divide it into two countably infinite set families. It is easy to see that each of the two families can be renumbered and its elements shifted in such a way that each of the two families forms a unit torus. Mathematicians say that there is no countably infinitely additive measure of Rn or that sets such as A are non-measurable. As a result of their existence, the ‘multiplication’ of lengths is possible. Later on we shall explore whether bread or gold can be multiplied in this way. Challenge 44, page 54: Hint: start with triangles. Challenge 45, page 54: An example is the region between the x-axis and the function which assigns 1 to every transcendental and 0 to every non-transcendental number. Challenge 46, page 55: We use the definition of the function of the text. The dihedral angle of

a regular tetrahedron is an irrational multiple of π, so the tetrahedron has a non-vanishing Dehn invariant. The cube has a dihedral angle of π~2, so the Dehn invariant of the cube is 0. Therefore, the cube is not equidecomposable with the regular tetrahedron.

Challenge 48, page 56: Obviously, we use light to check that the plumb line is straight, so the

two definitions must be the same. This is the case because the field lines of gravity are also possible paths for the motion of light. However, this is not always the case; can you spot the exceptions? Another way to check straightness is along the surface of calm water. Challenge 49, page 56: The hollow Earth theory is correct if the distance formula is used consistently. In particular, one has to make the assumption that objects get smaller as they approach the centre of the hollow sphere. Good explanations of all events are found on http://www.geocities. com/inversedearth/. Quite some material can be found on the internet, also under the names of

Copyright © Christoph Schiller November 1997–May 2006

Challenge 47, page 55: If you think you can show that empty space is continuous, you are wrong. Check your arguments. If you think you can prove the opposite, you might be right – but only if you already know what is explained in the third part of the text. If that is not the case, check your arguments.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Challenge 40, page 51: Measure distances using light.

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paper discs

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F I G U R E 399 A simple way to measure bullet

speeds

d

b w

F I G U R E 400 Leaving a parking space – the outer turning radius

as deduced from Figure 400. Challenge 55, page 58: A smallest gap does not exist: any value will do! Can you show this? Challenge 56, page 58: The first solution sent in will go here. Challenge 57, page 58: Clocks with two hands: 22 times. Clocks with three hands: 2 times. Challenge 58, page 59: For two hands, the answer is 143 times. Challenge 59, page 59: The Earth rotates with 15 minutes per minute. Challenge 60, page 59: You might be astonished, but no reliable data exist on this question. The highest speed of a throw measured so far seems to be a 45 m~s cricket bowl. By the way, much more data are available for speeds achieved with the help of rackets. The c. 70 m~s of fast badminton smashes seem to be a good candidate for record racket speed; similar speeds are achieved by golf balls.

Copyright © Christoph Schiller November 1997–May 2006

celestrocentric system, inner world theory or concave Earth theory. There is no way to prefer one description over the other, except possibly for reasons of simplicity or intellectual laziness. Challenge 51, page 57: A hint is given in Figure 399. For the measurement of the speed of light with almost the same method, see page 278. Challenge 52, page 57: Color is a property that applies only to objects, not to boundaries. The question shows that it is easy to ask questions that make no sense also in physics. Challenge 53, page 57: You can do this easily yourself. You can even find websites on the topic. Challenge 54, page 57: The required gap is ¼ » d = (L − b)2 − w 2 + 2w R 2 − (L − b)2 − L + b ,

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

L R

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f challenge hints and solu tions

F I G U R E 401 A simple drawing proving Pythagoras’ theorem

1237

F I G U R E 402 The trajectory of the middle point between the two ends of the hands of a clock

Challenge 62, page 59: 1.8 km~h or 0.5 m~s. Challenge 63, page 59: Nothing, neither a proof nor a disproof. Challenge 64, page 60: The different usage reflects the idea that we are able to determine our Page 154

position by ourselves, but not the time in which we are. The section on determinism will show how wrong this distinction is. Challenge 65, page 60: Yes, there is. However, this is not obvious, as it implies that space and

time are not continuous, in contrast to what we learn in primary school. The answer will be found in the third part of this text. Challenge 66, page 60: For a curve, use, at each point, the curvature radius of the circle approx-

imating the curve in that point; for a surface, define two directions in each point and use two such circles along these directions. Challenge 67, page 60: It moves about 1 cm in 50 ms. Challenge 68, page 60: The surface area of the lung. Challenge 69, page 60: The final shape is a full cube without any hole. Challenge 70, page 60: See page 279. Challenge 71, page 60: A hint for the solution is given by Figure 401. Challenge 72, page 60: Because they are or were fluid. Challenge 73, page 60: The shape is shown in Figure 402; it has eleven lobes.

cos φ~2).

Challenge 76, page 61: See Figure 403.

Challenge 78, page 62: Hint: draw all objects involved. Challenge 79, page 62: Hint: there is an infinite number of such shapes. Challenge 80, page 62: The curve is obviously called a catenary, from Latin ‘catena’ for chain. The formula for a catenary is y = a cosh(x~a). If you approximate the chain by short straight segments, you can make wooden blocks that can form an arch without any need for glue. The St. Louis arch is in shape of a catenary. A suspension bridge has the shape of a catenary before it is loaded, i.e. before the track is attached to it. When the bridge is finished, the shape is in between a catenary and a parabola.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 74, page 61: The cone angle φ is related to the solid angle Ω through Ω = 2π(1 −

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Challenge 61, page 59: Yes, it can. In fact, many cats can slip through as well.

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1238 1°

4° 3° 3°

10°

Copyright © Christoph Schiller November 1997–May 2006

Challenge 81, page 63: A limit does not exist in classical physics; however, there is one in nature which appears as soon as quantum effects are taken into account. Challenge 82, page 63: The inverse radii, or curvatures, obey a 2 + b 2 + c 2 + d 2 = (1~2)(a + b + c + d)2 . This formula was discovered by René Descartes. If one continues putting circles in the remaining spaces, one gets so-called circle packings, a pretty domain of recreational mathematics. They have many strange properties, such as intriguing relations between the coordinates of the circle centres and their curvatures. Challenge 83, page 63: One option: use the three-dimensional analogue of Pythagoras’s theorem. The answer is 9. Challenge 84, page 63: Draw a logarithmic scale, i.e., put every number at a distance corresponding to its natural logarithm. Challenge 85, page 63: Two more. Challenge 86, page 63: The Sun is exactly behind the back of the observer; it is setting, and the rays are coming from behind and reach deep into the sky in the direction opposite to that of the Sun. A slightly different situation – equally useful for getting used to perspective drawing – appears when you have a lighthouse in your back. Can you draw it? Challenge 87, page 63: Problems appear when quantum effects are added. A two-dimensional universe would have no matter, since matter is made of spin 1/2 particles. But spin 1/2 particles do not exist in two dimensions. Can you find other reasons? Challenge 88, page 65: From x = дt 2 ~2 you get the following rule: square the number of seconds, multiply by five and you get the depth in metres. Challenge 89, page 65: Just experiment. Challenge 90, page 65: The Academicians suspended one cannon ball with a thin wire just in front of the mouth of the cannon. When the shot was released, the second, flying cannon ball flew through the wire, thus ensuring that both balls started at the same time. An observer from far away then tried to determine whether both balls touched the Earth at the same time. The experiment is not easy, as small errors in the angle and air resistance confuse the results. Challenge 91, page 66: A parabola has a so-called focus or focal point. All light emitted from that point and reflected exits in the same direction: all light ray are emitted in parallel. The name

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F I G U R E 403 The angles defined by the hands against the sky, when the arms are extended

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ab = ab cos ∢(a, b) .

(861)

Copyright © Christoph Schiller November 1997–May 2006

How does this differ form the vector product? Challenge 104, page 70: Professor to student: What is the derivative of velocity? Acceleration! What is the derivative of acceleration? I don’t know. Jerk! The fourth, fifth and sixth derivatives of position are sometimes called snap, crackle and pop. Challenge 105, page 70: A candidate for low acceleration of a physical system might be the accelerations measured by gravitational wave detectors. They are below 10−13 m~s2 . Challenge 107, page 71: One can argue that any source of light must have finite size. Challenge 109, page 72: What the unaided human eye perceives as a tiny black point is usually about 50 µm in diameter. Challenge 110, page 72: See page 570. Challenge 111, page 72: One has to check carefully whether the conceptual steps that lead us to extract the concept of point from observations are correct. It will be shown in the third part of the adventure that this is not the case.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

‘focus’ – Latin for fireplace – expresses that it is the hottest spot when a parabolic mirror is illuminated. Where is the focus of the parabola y = 2x? (Ellipses have two foci, with a slightly different definition. Can you find it?) Challenge 92, page 66: Neglecting air resistance and approximating the angle by 45°, we get v = » d д , or about 3.8 m~s. This speed is created by a stead pressure build-up, using blood pressure, which is suddenly released with a mechanical system at the end of the digestive canal. The cited reference tells more about the details. Challenge 93, page 67: On horizontal ground, for a speed v and an angle from the horizontal α, neglecting air resistance and the height of the thrower, the distance d is d = v 2 sin 2α~д. Challenge 94, page 67: Walk or run in the rain, measure your own speed v and the angle from the vertical α with which the rain appears to fall. Then the speed of the rain is v rain = v~ tan α. Challenge 95, page 67: Check your calculation with the information that the 1998 world record is juggling with 9 balls. Challenge 96, page 67: The long jump record could surely be increased by getting rid of the sand stripe and by measuring the true jumping distance with a photographic camera; that would allow jumpers to run more closely to their top speed. The record could also be increased by a small inclined step or by a spring-suspended board at the take-off location, to increase the takeoff angle. Challenge 97, page 67: It is said so, as rain drops would then be ice spheres and fall with high speed. Challenge 98, page 67: It seems not too much. But the lead in them can poison the environment. Challenge 99, page 67: Stones never follow parabolas: when studied in detail, i.e. when the change of д with height is taken into account, their precise path turns out to be an ellipse. This shape appears most clearly for long throws, such as throws around the part of the Earth, or for orbiting objects. In short, stones follow parabolas only if the Earth is assumed to be flat. If its curvature is taken into account, they follow ellipses. Challenge 102, page 69: The set of all rotations around a point in a plane is indeed a vector space. What about the set of all rotations around all points in a plane? And what about the threedimensional cases? Challenge 103, page 70: The scalar product between two vectors a and b is given by

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Challenge 112, page 73: One can rotate the hand in a way that the arm makes the motion de-

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006

scribed. See also page 783. Challenge 113, page 73: Any number, without limit. Challenge 114, page 73: The blood and nerve supply is not possible if the wheel has an axle. The method shown to avoid tangling up connections only works when the rotating part has no axle: the ‘wheel’ must float or be kept in place by other means. It thus becomes impossible to make a wheel axle using a single piece of skin. And if a wheel without an axle could be built (which might be possible), then the wheel would periodically run over the connection. Could such a axle-free connection realize a propeller? By the way, it is still thinkable that animals have wheels on axles, if the wheel is a ‘dead’ object. Even if blood supply technologies like continuous flow reactors were used, animals could not make such a detached wheel grow in a way tuned to the rest of the body and they would have difficulties repairing a damaged wheel. Detached wheels cannot be grown on animals; they must be dead. Challenge 115, page 74: The brain in the skull, the blood factories inside bones or the growth of the eye are examples. Challenge 116, page 74: One can also add the Sun, the sky and the landscape to the list. Challenge 117, page 75: Ghosts, hallucinations, Elvis sightings, or extraterrestrials must all be one or the other. There is no third option. Even shadows are only special types of images. Challenge 118, page 75: The issue was hotly discussed in the seventeenth century; even Galileo argued for them being images. However, they are objects, as they can collide with other objects, as the spectacular collision between Jupiter and the comet Shoemaker-Levy 9 in 1994 showed. In the meantime, satellites have been made to collide with comets and even to shoot at them (and hitting). Challenge 119, page 77: The minimum speed is roughly the one at which it is possible to ride without hands. If you do so, and then gently push on the steering wheel, you can make the experience described above. Watch out: too strong a push will make you fall badly. Challenge 120, page 79: If the ball is not rotating, after the collision the two balls will depart with a right angle between them. Challenge 121, page 79: Part of the energy is converted into heat; the rest is transferred as kinetic energy of the concrete block. As the block is heavy, its speed is small and easily stopped by the human body. This effect works also with anvils, it seems. In another common variation the person does not lie on nails, but on air: he just keeps himself horizontal, with head and shoulders on one chair, and the feet on a second one. Challenge 122, page 80: Yes, mass works also for magnetism, because the precise condition is not that the interaction be central, but that it realizes a more general condition, which includes accelerations such as those produced by magnetism. Can you deduce the condition from the definition of mass? Challenge 123, page 80: The weight decreased due to the evaporated water lost by sweating and, to a minor degree, due to the exhaled carbon bound in carbon dioxide. Challenge 124, page 80: Rather than using the inertial effects of the Earth, it is easier to deduce its mass from its gravitational effects. See challenge 234. Challenge 128, page 82: At first sight, relativity implies that tachyons have imaginary mass; however, the imaginary factor can be extracted from the mass–energy and mass–momentum relation, so that one can define a real mass value for tachyons; as a result, faster tachyons have smaller energy and smaller momentum. Both momentum and energy can be a negative number of any size.

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Challenge 129, page 82: Legs are never perfectly vertical; they would immediately glide away.

Once the cat or the person is on the floor, it is almost impossible to stand up again. Challenge 130, page 82: Momentum (or centre of mass) conservation would imply that the environment would be accelerated into the opposite direction. Energy conservation would imply that a huge amount of energy would be transferred between the two locations, melting everything in between. Teleportation would thus contradict energy and momentum conservation. Challenge 131, page 83: The part of the tides due to the Sun, the solar wind, and the interactions between both magnetic fields are examples of friction mechanisms between the Earth and the Sun.

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Challenge 132, page 84: With the factor 1~2, increase of (physical) kinetic energy is equal to

the (physical) work performed on a system: total energy is thus conserved only if the factor 1/2 is added. Challenge 134, page 85: It is a smart application of momentum conservation. Challenge 135, page 85: Neither. With brakes on, the damage is higher, but still equal for both Challenge 136, page 86: Heating systems, transport engines, engines in factories, steel plants, electricity generators covering the losses in the power grid, etc. By the way, the richest countries in the world, such as Sweden or Switzerland, consume only half the energy per inhabitant as the USA. This waste is one of the reasons for the lower average standard of living in the USA. Challenge 138, page 88: If the Earth changed its rotation speed ever so slightly we would walk

inclined, the water of the oceans would flow north, the atmosphere would be filled with storms and earthquakes would appear due to the change in Earth’s shape. Challenge 140, page 89: Just throw it into the air and compare the dexterity needed to make it turn around various axes. Challenge 141, page 90: Use the definition of the moment of inertia and Pythagoras’ theorem

for every mass element of the body. Challenge 142, page 90: Hang up the body, attaching the rope in two different points. The crossing point of the prolonged rope lines is the centre of mass. Challenge 143, page 90: Spheres have an orientation, because we can always add a tiny spot on

their surface. This possibility is not given for microscopic objects, and we shall study this situation in the part on quantum theory. Challenge 146, page 91: See Tables 14 and 15. Challenge 147, page 91: Self-propelled linear motion contradicts the conservation of mo-

Challenge 148, page 91: Yes, the ape can reach the banana. The ape just has to turn around its

own axis. For every turn, the plate will rotate a bit towards the banana. Of course, other methods, like blowing at a right angle to the axis, peeing, etc., are also possible. Challenge 150, page 92: The points that move exactly along the radial direction of the wheel form a circle below the axis and above the rim. They are the points that are sharp in Figure 38 of page 92. Challenge 151, page 92: Use the conservation of angular momentum around the point of con-

tact. If all the wheel’s mass is assumed in the rim, the final rotation speed is half the initial one; it is independent of the friction coefficient.

Copyright © Christoph Schiller November 1997–May 2006

mentum; self-propelled change of orientation (as long as the motion stops again) does not contradict any conservation law. But the deep, final reason for the difference will be unveiled in the third part of our adventure.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

cars.

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Challenge 155, page 96: A short pendulum of length L that swings in two dimensions (with amplitude ρ and orientation φ) shows two additional terms in the Lagrangian L:

1 ρ2 l2 1 ρ2 L = T − V = m ρ˙2 (1 + 2 ) + z 2 − mω 20 ρ 2 (1 + ) 2 L 2mρ 2 4 L2

(862)

˙ The where as usual the basic frequency is ω 20 = д~L and the angular momentum is l z = mρ 2 φ. two additional terms disappear when L ª; in that case, if the system oscillates in an ellipse with semiaxes a and b, the ellipse is fixed in space, and the frequency is ω 0 . For finite pendulum length L, the frequency changes to ω = ω 0 (1 −

most of all, the ellipse turns with a frequency

a2 + b2 ) 16 L 2 3 ab . 8 L2

(863)

(864)

v = 2ωr sin θ,

(865)

as long as friction is negligible. Here ω is the angular velocity of the Earth, θ the latitude and r the (larger) radius of the torus. For a tube with 1 m diameter in continental Europe, this gives a speed of about 6.3 ë 10−5 m~s.

Copyright © Christoph Schiller November 1997–May 2006

(These formulae can be derived using the least action principle, as shown by C.G. Gray, G. Karl & V.A. Novikov, Progress in classical and quantum variational principles, http://www. arxiv.org/abs/physics/0312071.) In other words, a short pendulum in elliptical motion shows a precession even without the Coriolis effect. Since this precession frequency diminishes with 1~L 2 , the effect is small for long pendulums, where only the Coriolis effect is left over. To see the Coriolis effect in a short pendulum, one thus has to avoid that it starts swinging in an elliptical orbit by adding a suppression method of elliptical motion. Challenge 156, page 96: The Coriolis acceleration is the reason for the deviation from the straight line. The Coriolis acceleration is due to the change of speed with distance from the rotation axis. Now think about a pendulum, located in Paris, swinging in the North-South direction with amplitude A. At the Southern end of the swing, the pendulum is further from the axis by A sin φ, where φ is the latitude. At that end of the swing, the central support point overtakes the pendulum bob with a relative horizontal speed given by v = 2πA sin φ~24 h. The period of precession is given by TF = v~2πA, where 2πA is the circumference 2πA of the envelope of the pendulum’s path (relative to the Earth). This yields TF = 24 h~ sin φ. Challenge 157, page 97: The axis stays fixed with respect to distant stars, not with respect to absolute space (which is an entity that cannot be observed at all). Challenge 158, page 97: Rotation leads to a small frequency and thus colour changes of the circulating light. Challenge 159, page 97: The weight changes when going east or when moving west due to the Coriolis acceleration. If the rotation speed is tuned to the oscillation frequency of the balance, the effect is increased by resonance. This trick was also used by Eötvös. Challenge 160, page 97: The Coriolis acceleration makes the bar turn, as every moving body is deflected to the side, and the two deflections add up in this case. The direction of the deflection depends on whether the experiments is performed on the northern or the southern hemisphere. Challenge 161, page 98: When rotated by π around an east–west axis, the Coriolis force produces a drift velocity of the liquid around the tube. It has the value

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Ω=ω

;

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light source

Ωt

t=0

F I G U R E 404 Deducing the expression for the Sagnac effect

t = 2πR~c

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Challenge 162, page 98: Imagine a circular light path (for example, inside a circular glass fibre) and two beams moving in opposite directions along it, as shown in Figure 404. If the fibre path rotates with rotation frequency Ω, we can deduce that, after one turn, the difference ∆L in path length is 4πR 2 Ω ∆L = 2RΩt = . (866) c

The phase difference is thus

∆φ =

8π 2 R 2 Ω cλ

(867)

if the refractive index is 1. This is the required formula for the main case of the Sagnac effect. Challenge 163, page 101: The original result by Bessel was 0.3136 ′′ , or 657.7 thousand orbital

radii, which he thought to be 10.3 light years or 97.5 Pm. Challenge 165, page 104: The galaxy forms a stripe in the sky. The galaxy is thus a flattened structure. This is even clearer in the infrared, as shown more clearly in Figure 197 on page 439. From the flattening (and its circular symmetry) we can deduce that the galaxy must be rotating. Thus other matter must exist in the universe. Challenge 166, page 104: Probably the ‘rest of the universe’ was meant by the writer. Indeed, a

moving a part never shifts the centre of gravity of a closed system. But is the universe closed? Or a system? The third part of the adventure centres on these issues. Challenge 170, page 105: The scale reacts to your heartbeat. The weight is almost constant over

time, except when the heart beats: for a short duration of time, the weight is somewhat lowered at each beat. Apparently it is due to the blood hitting the aortic arch when the heart pumps it upwards. The speed of the blood is about 0.3 m~s at the maximum contraction of the left ventricle. The distance to the aortic arch is a few centimetres. The time between the contraction and the reversal of direction is about 15 ms. Challenge 171, page 105: The conservation of angular momentum saves the glass. Try it. Challenge 172, page 105: The mass decrease could also be due to expelled air. The issue is still

open. Challenge 173, page 105: Assuming a square mountain, the height h above the surrounding

Copyright © Christoph Schiller November 1997–May 2006

Challenge 167, page 105: Hint: an energy per distance is a force.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

The measurement can be made easier if the tube is restricted in diameter at one spot, so that the velocity is increased there. A restriction by an area factor of 100 increases the speed by the same factor. When the experiment is performed, one has to carefully avoid any other effects that lead to moving water, such as temperature gradients across the system.

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1244 crust and the depth d below are related by

h ρm − ρc = d ρc

(868)

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006

where ρ c is the density of the crust and ρ m is the density of the mantle. For the density values given, the ratio is 6.7, leading to an additional depth of 6.7 km below the mountain. Challenge 177, page 107: The behaviour of the spheres can only be explained by noting that elastic waves propagate through the chain of balls. Only the propagation of these elastic waves, in particular their reflection at the end of the chain, explains that the same number of balls that hit on one side are lifted up on the other. For long times, friction makes all spheres oscillate in phase. Can you confirm this? Challenge 178, page 107: When the short cylinder hits the long one, two compression waves start to run from the point of contact through the two cylinders. When each compression wave arrives at the end, it is reflected as an expansion wave. If the geometry is well chosen, the expansion wave coming back from the short cylinder can continue into the long one (which is still in his compression phase). For sufficiently long contact times, waves from the short cylinder can thus depose much of their energy into the long cylinder. Momentum is conserved, as is energy; the long cylinder is oscillating in length when it detaches, so that not all its energy is translational energy. This oscillation is then used to drive nails or drills into stone walls. In commercial hammer drills, length ratios of 1:10 are typically used. Challenge 179, page 108: The momentum transfer to the wall is double when the ball rebounds perfectly. Challenge 180, page 108: If the cork is in its intended position: take the plastic cover off the cork, put the cloth around the bottle (this is for protection reasons only) and repeatedly hit the bottle on the floor or a fall in an inclined way, as shown in Figure 33 on page 83. With each hit, the cork will come out a bit. If the cork has fallen inside the bottle: put half the cloth inside the bottle; shake until the cork falls unto the cloth. Pull the cloth out: first slowly, until the cloth almost surround the cork, and then strongly. Challenge 182, page 108: The atomic force microscope. Challenge 183, page 108: Use Figure 42 on page 95 for the second half of the trajectory, and think carefully about the first half. Challenge 184, page 108: Hint: starting rockets at the Equator saves a lot of energy, thus of fuel and of weight. Challenge 185, page 109: Running man: E  0.5 ë 80 kg ë (5 m~s)2 = 1 kJ; rifle bullet: E  0.5 ë 0.04 kg ë (500 m~s)2 = 5 kJ. Challenge 186, page 109: The flame leans towards the inside. Challenge 187, page 109: The ball leans in the direction it is accelerated to. As a result, one could imagine that the ball in a glass at rest pulls upwards because the floor is accelerated upwards. We will come back to this issue in the section of general relativity. Challenge 188, page 109: It almost doubles in size. Challenge 189, page 109: For your exam it is better to say that centrifugal force does not exist. But since in each stationary system there is a force balance, the discussion is somewhat a red herring. Challenge 191, page 110: Place the tea in cups on a board and attach the board to four long ropes that you keep in your hand.

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Challenge 192, page 110: The friction if the tides on Earth are the main cause.

to repair; in addition, legs do not require flat surfaces (so-called ‘streets’) to work. Challenge 217, page 115: The staircase formula is an empirical result found by experiment, used by engineers world-wide. Its origin and explanation seems to be lost in history. Challenge 218, page 115: Classical or everyday nature is right-left symmetric and thus requires an even number of legs. Walking on two-dimensional surfaces naturally leads to a minimum of four legs. Challenge 220, page 116: The length of the day changes with latitude. So does the length of a shadow or the elevation of stars at night, facts that are easily checked by telephoning a friend.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 216, page 114: As said before, legs are simpler than wheels to grow, to maintain and

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Challenge 195, page 111: An earthquake with Richter magnitude of 12 is 1000 times the energy of the 1960 Chile quake with magnitude 10; the latter was due to a crack throughout the full 40 km of the Earth’s crust along a length of 1000 km in which both sides slipped by 10 m with respect to each other. Only the impact of a meteorite could lead to larger values than 12. Challenge 196, page 111: This is not easy; a combination of friction and torques play a role. See for example the article J. Sauer, E. Schörner & C. Lennerz, Real-time rigid body simulation of some classical mechanical toys, 10th European Simulation and Symposium and Exhibition (ESS ’98) 1998, pp. 93–98, or http//www.lennerz.de/paper_ess98.pdf. Challenge 198, page 111: If a wedding ring rotates on an axis that is not a principal one, angular momentum and velocity are not parallel. Challenge 199, page 111: Yes; it happens twice a year. To minimize the damage, dishes should be dark in colour. Challenge 200, page 111: A rocket fired from the back would be a perfect defence against planes attacking from behind. However, when released, the rocket is effectively flying backwards with respect to the air, thus turns around and then becomes a danger to the plane that launched it. Engineers who did not think about this effect almost killed a pilot during the first such tests. Challenge 202, page 111: Whatever the ape does, whether it climbs up or down or even lets himself fall, it remains at the same height as the mass. Now, what happens if there is friction at the wheel? Challenge 204, page 111: Weigh the bullet and shoot it against a mass hanging from the ceiling. From the mass and the angle it is deflected to, the momentum of the bullet can be determined. Challenge 206, page 112: Yes, if he moves at a large enough angle to the direction of the boat’s motion. Challenge 208, page 112: The moment of inertia is Θ = 25 mr 2 . Challenge 209, page 112: The moments of inertia are equal also for the cube, but the values are Θ = 16 ml 2 . The efforts required to put a sphere and a cube into rotation are thus different. Challenge 210, page 112: See the article by C. Ucke & H.-J. Schlichting, Faszinierendes Dynabee, Physik in unserer Zeit 33, pp. 230–231, 2002. Challenge 211, page 112: See the article by C. Ucke & H.-J. Schlichting, Die kreisende Büroklammer, Physik in unserer Zeit 36, pp. 33–35, 2005. Challenge 212, page 113: Yes. Can you imagine what happens for an observer on the Equator? Challenge 213, page 113: A straight line at the zenith, and circles getting smaller at both sides. See an example on the website http://antwrp.gsfc.nasa.gov/apod/ap021115.html. Challenge 215, page 114: The plane is described in the websites cited; for a standing human the plane is the vertical plane containing the two eyes.

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Page 669

Ships appear at the horizon first be showing only their masts. These arguments, together with the round shadow of the earth during a lunar eclipse and the observation that everything falls downwards everywhere, were all given already by Aristotle, in his text On the Heavens. It is now known that everybody in the last 2500 years knew that the Earth is s sphere. The myth that many people used to believe in a flat Earth was put into the world – as rhetorical polemic – by Copernicus. The story then continued to be exaggerated more and more during the following centuries, because a new device for spreading lies had just been invented: book printing. Fact is that since 2500 years the vast majority of people knew that the Earth is a sphere.

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Challenge 221, page 116: Robert Peary had forgotten that on the date he claimed to be at the

North Pole, 6th of April 1909, the Sun is very low on the horizon, casting very long shadows, about ten times the height of objects. But on his photograph the shadows are much shorter. (In fact, the picture is taken in such a way to hide all shadows as carefully as possible.) Interestingly, he had even convinced the US congress to officially declare him the first man on the North Pole in 1911. (A rival had claimed to have reached it earlier on, but his photograph has the same mistake.) Challenge 222, page 116: Yes, the effect has been measured for skyscrapers. Can you estimate Challenge 223, page 117: The tip of the velocity arrow, when drawn over time, produces a circle

around the centre of motion.

Challenge 226, page 118: The value of the product GM for the Earth is 4.0 ë 1014 m3 ~s2 .

Challenge 227, page 118: All points can be reached for general inclinations; but when shooting horizontally in one given direction, only points on the first half of the circumference can be reached. Challenge 229, page 119: On the moon, the gravitational acceleration is 1.6 m~s2 , about one sixth of the value on Earth. The surface values for the gravitational acceleration for the planets can be found on many internet sites. Challenge 230, page 119: The Atwood machine is the answer: two almost equal weights con-

nected by a string hanging from a well-oiled wheel. The heavier one falls very slowly. Can you determine the acceleration as a function of the two masses? Challenge 231, page 119: You should absolutely try to understand the origin of this expres-

sion. It allows to understand many essential concepts of mechanics. The idea is that for small amplitudes, the acceleration of a pendulum of length l is due to gravity. Drawing a force diagram for a pendulum at a general angle α shows that

d2 α = −mд sin α dt 2 d2 α l 2 = −д sin α . dt

ml

(869)

For the mentioned small amplitudes (below 15°) we can approximate this to l

d2 α = −дα . dt 2

(870)

This is the equation for a harmonic oscillation (i.e., a sinusoidal oscillation). The resulting motion is: α(t) = A sin(ωt + φ) . (871)

Copyright © Christoph Schiller November 1997–May 2006

ma = −mд sin α

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

the values?

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The amplitude A and the phase φ depend on the initial conditions; however, the oscillation frequency is given by the length of the pendulum and the acceleration of gravity (check it!): ¾ l . (872) ω= д

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006

(For arbitrary amplitudes, the formula is much more complex; see the internet or special mechanics books for more details.) Challenge 232, page 119: Walking speed is proportional to l~T, which makes it proportional to l 1~2 . Challenge 234, page 120: Cavendish suspended a horizontal handle with a long metal wire. He then approached a large mass to the handle, avoiding any air currents, and measured how much the handle rotated. Challenge 235, page 120: The acceleration due to gravity is a = Gm~r 2  5 nm~s2 for a mass of 75 kg. For a fly with mass m fly = 0.1 g landing on a person with a speed of v fly = 1 cm~s and deforming the skin (without energy loss) by d = 0.3 mm, a person would be accelerated by a = (v 2 ~d)(m fly ~m) = 0.4 µm~s2 . The energy loss of the inelastic collision reduces this value at least by a factor of ten. Challenge 237, page 122: The easiest way to see this is to picture gravity as a flux emanating form a sphere. This gives a 1~r d−1 dependence for the force and thus a 1~r d−2 dependence of the potential. Challenge 239, page 123: Since the paths of free fall are ellipses, which are curves lying in a plane, this is obvious. Challenge 241, page 124: The low gravitational acceleration of the Moon, 1.6 m~s2 , implies that gas molecules at usual temperatures can escape its attraction. Challenge 242, page 126: A flash of light is sent to the Moon, where several Cat’s-eyes have been deposited by the Lunakhod and Apollo missions. The measurement precision of the time a flash take to go and come back is sufficient to measure the Moon’s distance change. For more details, see challenge 546. Challenge 249, page 129: This is a resonance effect, in the same way that a small vibration of a string can lead to large oscillation of the air and sound box in a guitar. Challenge 251, page 131: The total angular momentum of the Earth and the Moon must remain constant. Challenge 257, page 136: The centre of mass of a broom falls with the usual acceleration; the end thus falls faster. Challenge 258, page 136: Just use energy conservation for the two masses of the jumper and the string. For more details, including the comparison of experimental measurements and theory, see N. Dubel aar & R. Brantjes, De valversnelling bij bungee-jumping, Nederlands tijdschrift voor natuurkunde 69, pp. 316–318, October 2003. Challenge 259, page 136: About 1 ton. Challenge 260, page 136: About 5 g. Challenge 261, page 137: Your weight is roughly constant; thus the Earth must be round. On a flat Earth, the weight would change from place to place. Challenge 262, page 137: Nobody ever claimed that the centre of mass is the same as the centre of gravity! The attraction of the Moon is negligible on the surface of the Earth. Challenge 264, page 137: That is the mass of the Earth. Just turn the table on its head.

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Challenge 267, page 137: The Moon will be about 1.25 times as far as it is now. The Sun then will slow down the Earth–Moon system rotation, this time due to the much smaller tidal friction from the Sun’s deformation. As a result, the Moon will return to smaller and smaller distances to Earth. However, the Sun will have become a red giant by then, after having swallowed both the Earth and the Moon. º Challenge 269, page 137: As Galileo determined, for a swing (half a period) the ratio is 2 ~pi. (See challenge 231). But not more than two, maybe three decimals of π can be determined this way.

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Challenge 270, page 138: Momentum conservation is not a hindrance, as any tennis racket has

the same effect on the tennis ball. Challenge 271, page 138: In fact, in velocity space, elliptic, parabolic and hyperbolic motions Ref. 98

are all described by circles. In all cases, the hodograph is a circle.

Challenge 273, page 138: Due to the plateau, the effective mass of the Earth is larger. Challenge 274, page 138: The choice is clear once you notice that there is no section of the orbit which is concave towards the Sun. Can you show this? Challenge 275, page 139: It would be a black hole; no light could escape. Black holes are discussed in detail in the chapter on general relativity. Challenge 276, page 139: A handle of two bodies.

Challenge 279, page 139: Using a maximal jumping height of h =0.5 m on Earth and an estim-

ated asteroid density of ρ =3 Mg~m3 , we get a maximum radius of R 2 = 3дh~4πGρ  703 m.

Challenge 280, page 140: For each pair of opposite shell elements (drawn in yellow), the two attractions compensate.

Challenge 281, page 141: There is no practical way; if the masses on the shell could move, along

the surface (in the same way that charges can move in a metal) this might be possible, provided that enough mass is available. Challenge 282, page 141: Capture of a fluid body if possible if it is split by tidal forces.

Challenge 285, page 141: The centre of mass of the solar system can be as far as twice the radius

from the centre of the Sun; it thus can be outside the Sun. Challenge 286, page 142: First, during northern summer time the Earth moves faster around

the Sun than during northern winter time. Second, shallow Sun’s orbits on the sky give longer days because of light from when the Sun is below the horizon. Challenge 287, page 142: Apart from the visibility of the moon, no such effect has ever been detected. Gravity effects, electrical effects, magnetic effects, changes in cosmic rays seem all to be independent of the phase. The locking of the menstrual cycle to the moon phase is a visual effect. Challenge 288, page 142: Distances were difficult to measure.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 283, page 141: The tunnel would be an elongated ellipse in the plane of the Equator, reaching from one point of the Equator to the point at the antipodes. The time of revolution would not change, compared to a non-rotating Earth. See A.J. Simonson, Falling down a hole through the Earth, Mathematics Magazine 77, pp. 171–188, June 2004.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Challenge 272, page 138: This question is old (it was already asked in Newton’s times) and deep. One reason is that stars are kept apart by rotation around the galaxy. The other is that galaxies are kept apart by the momentum they got in the big bang. Without the big bang, all stars would have collapsed together. In this sense, the big bang can be deduced from the attraction of gravitation and the immobile sky at night. We shall find out later that the darkness of the night sky gives a second argument for the big bang.

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Challenge 289, page 142: See the mentioned reference. Challenge 290, page 142: True.

money =

(873)

which shows that the less you know, the more money you make. That is why scientists have low salaries. Challenge 310, page 151: True? Challenge 313, page 151: From dv~dt = д − v 2 (1~2c w Aρ~m) and using the abbreviation c = 1~2c w Aρ, we can solve for v(t) by putting all terms containing the» variable v on one » side, all terms with t on the other, and integrating on both sides. We get v(t) = дm~c tanh c д~m t. Challenge 315, page 153: The phase space has 3N position coordinates and 3N momentum coordinates. Challenge 316, page 153: The light mill is an example. Challenge 317, page 153: Electric charge. Challenge 318, page 153: If you have found reasons to answer yes, you overlooked something. Just go into more details and check whether the concepts you used apply to the universe. Also define carefully what you mean by ‘universe’. Challenge 320, page 155: A system showing energy or matter motion faster than light would imply that for such systems there are observers for which the order between cause and effect are reversed. A space-time diagram (and a bit of exercise from the section on special relativity) shows this.

Copyright © Christoph Schiller November 1997–May 2006

Page 573

work , knowledge

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Challenge 294, page 143: Never. The Moon points always towards the Earth. The Earth changes position a bit, due to the ellipticity of the Moon’s orbit. Obviously, the Earth shows phases. Challenge 296, page 143: What counts is local verticality; with respect to it, the river always flows downhill. Challenge 297, page 143: There are no such bodies, as the chapter of general relativity will show. Challenge 299, page 146: The oscillation is a purely sinusoidal, or harmonic oscillation, as the restoring force increases linearly » with distance from the centre of the Earth. The period T for a homogeneous Earth is T = 2π R 3 ~GM = 84 min. Challenge 300, page 146: The period is the same for all such tunnels and thus in particular it is the same as the 84 min valid also for the pole to pole tunnel. See for example, R.H. Romer, The answer is forty-two – many mechanics problems, only one answer, Physics Teacher 41, pp. 286– 290, May 2003. Challenge 301, page 146: There is no simple answer: the speed depends on the latitude and on other parameters. Challenge 303, page 148: In reality muscles keep an object above ground by continuously lifting and dropping it; that requires energy and work. Challenge 304, page 148: The electricity consumption of a rising escalator indeed increases when the person on it walks upwards. By how much? Challenge 308, page 149: The lack of static friction would avoid that the fluid stays attached to the body; the so-called boundary layer would not exist. One then would have to wing effect. Challenge 305, page 148: Knowledge is power. Time is money. Now, power is defined as work per time. Inserting the previous equations and transforming them yields

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Copyright © Christoph Schiller November 1997–May 2006

Challenge 321, page 155: If reproducibility would not exist, we would have difficulties in checking observations; also reading the clock is an observation. The connection between reproducibility and time shall become important in the third part of our adventure. Challenge 322, page 156: Even if surprises were only rare, each surprise would make it impossible to define time just before and just after it. Challenge 325, page 157: Of course; moral laws are summaries of what others think or will do about personal actions. Challenge 326, page 157: Space-time is defined using matter; matter is defined using spacetime. Challenge 327, page 157: Fact is that physics has been based on a circular definition for hundreds of years. Thus it is possible to build even an exact science on sand. Nevertheless, the elimination of the circularity is an important aim. Challenge 328, page 161: For example, speed inside materials is slowed, but between atoms, light still travels with vacuum speed. Challenge 331, page 174: Figure 405 shows the most credible reconstruction of a southpointing carriage. Challenge 332, page 175: The water is drawn up along the sides of the spinning egg. The fastest way to empty a bottle of water is to spin the water while emptying it. Challenge 333, page 175: The right way is the one where the chimney falls like a V, not like an inverted V. See challenge 257 on falling brooms for inspiration on how to deduce the answer. Challenge 341, page 181: In one dimension, the expression F = ma can be written as ˙ 21 m x˙ 2 )] = 0. This can be −dV ~dx = md2 x~dt 2 . This can be rewritten as d(−V )~dx − d~dt[d~dx( 1 1 2 2 ˙ 2 m x˙ − V (x))] = 0, which is Lagrange’s equation expanded to ∂~∂x( 2 m x˙ − V (x)) − d~[∂~∂ x( for this case. Challenge 343, page 182: Do not despair. Up to now, nobody has been able to imagine a universe (that is not necessarily the same as a ‘world’) different from the one we know. So far, such attempts have always led to logical inconsistencies.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

F I G U R E 405 The south-pointing carriage

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c 1 sin α 1 = . c2 α2

(874)

The particular speed ratio between air (or vacuum, which is almost the same) and a material gives the index of refraction n: c 1 sin α 1 n= = (875) c0 α0

Challenge 355, page 188: Gases are mainly made of vacuum. Their index of refraction is near

to one.

Copyright © Christoph Schiller November 1997–May 2006

Page 1046

Challenge 356, page 188: Diamonds also sparkle because they work as prisms; different colours have different indices of refraction. Thus their sparkle is also due to their dispersion; therefore it is a mix of all colours of the rainbow. Challenge 357, page 188: The principle for the growth of trees is simply the minimum of potential energy, since the kinetic energy is negligible. The growth of vessels inside animal bodies is minimized for transport energy; that is again a minimum principle. The refraction of light is the path of shortest time; thus it minimizes change as well, if we imagine light as moving entities moving without any potential energy involved. Challenge 358, page 188: Special relativity requires that an invariant measure of the action exist. It is presented later in the walk. Challenge 359, page 189: The universe is not a physical system. This issue will be discussed in detail later on. Challenge ??, page ??: Physical fields are properties which vary from point to point. Parities are observables (in the wide sense of the term) but not fields. Challenge 360, page 189: We talk to a person because we know that somebody understands us. Thus we assume that she somehow sees the same things we do. That means that observation is partly viewpoint-independent. Thus nature is symmetric. Challenge 361, page 190: Memory works because we recognize situations. This is possible because situations over time are similar. Memory would not have evolved without this reproducibility. Challenge 362, page 191: Taste differences are not fundamental, but due to different viewpoints and – mainly – to different experiences of the observers. The same holds for feelings and judgements, as every psychologist will confirm.

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Challenge 345, page 182: The two are equivalent since the equations of motion follow from the principle of minimum action and at the same time the principle of minimum action follows from the equations of motion. Challenge 347, page 183: For gravity, all three systems exist: rotation in galaxies, pressure in planets and the Pauli pressure in stars. Against the strong interaction, the Pauli principle acts in nuclei and neutron stars; in neutron stars maybe also rotation and pressure complement the Pauli pressure. But for the electromagnetic interaction there are no composites other than our everyday matter, which is organized by the Pauli principle alone. Challenge 349, page 187: Angular momentum is the change with respect to angle, whereas rotational energy is again the change with respect to time, as all energy is. Challenge 350, page 187: Not in this way. A small change can have a large effect, as every switch shows. But a small change in the brain must be communicated outside, and that will happen roughly with a 1~r 2 dependence. That makes the effects so small, that even with the most sensitive switches – which for thoughts do not exist anyway – no effects can be realized. Challenge 354, page 188: The relation is

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Challenge 363, page 192: The integers under addition form a group. Does a painter’s set of oil colours with the operation of mixing form a group? Challenge 364, page 192: There is only one symmetry operation: a rotation about π around the

central point.

Challenge 370, page 196: Scalar is the magnitude of any vector; thus the speed, defined as v =

SvS, is a scalar, whereas the velocity v is not. Thus the length of any vector (or pseudovector), such as force, acceleration, magnetic field, or electric field, is a scalar, whereas the vector itself is not a scalar.

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Challenge 373, page 197: The charge distribution of an extended body can be seen as a sum of a

charge, a charge dipole, a charge quadrupole, a charge octupole, etc. The quadrupole is described by a tensor. Compare: The inertia against motion of an extended body can be seen as sum of a mass, a mass dipole, a mass quadrupole, a mass octupole, etc. The mass quadrupole is described by the moment of inertia. Challenge 380, page 203: An oscillation has a period in time, i.e. a discrete time translation

symmetry. A wave has both discrete time and discrete space translation symmetry. Challenge 381, page 203: Motion reversal is a symmetry for any closed system; despite the observations of daily life, the statements of thermodynamics and the opinion of several famous physicists (who form a minority though) all ideally closed systems are reversible. Challenge 389, page 207: The potential energy is due to the ‘bending’ of the medium; a simple

displacement produces no bending and thus contains no energy. Only the gradient captures the bending idea. Challenge 391, page 208: The phase changes by π. Challenge 393, page 209: Waves can be damped to extremely low intensities. If this is not pos-

sible, the observation is not a wave. Challenge 394, page 209: Page 560 tells how to observe diffraction and interference with your

naked fingers. Challenge 399, page 214: If the distances to the loudspeaker is a few metres, and the distance to the orchestra is 20 m, as for people with enough money, the listener at home hears it first. Challenge 401, page 214: An ellipse (as for planets around the Sun) with the fixed point as

centre (in contrast to planets, where the Sun is in a focus of the ellipse). Challenge 403, page 215: The sound of thunder or of car traffic gets lower and lower in freChallenge 405, page 215: Neither; both possibilities are against the properties of water: in sur-

face waves, the water molecules move in circles. Challenge 406, page 216: Swimmers are able to cover 100 m in 48 s, or slightly better than 2 m~s. With a body length of about 1.9 m, the critical speed is 1.7 m~s. That is why short distance swimming depends on training; for longer distances the technique plays a larger role, as the critical speed has not been attained yet. The formula also predicts that on the 1500 m distance, a 2 m tall swimmer has a potential advantage of over 45 s on one with body height of 1.8 m. In addition, longer swimmers have an additional advantage: they swim shorter distances (why?). It is thus predicted that successful long-distance swimmers will get taller and taller over time. This is a pity for a sport that so far could claim to have had champions of all sizes and body shapes, in contrast to many other sports.

Copyright © Christoph Schiller November 1997–May 2006

quency with increasing distance.

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Challenge 377, page 199: The conserved charge for rotation invariance is angular momentum.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006

Challenge 408, page 217: To reduce noise reflection and thus hall effects. They effectively diffuse the arriving wave fronts. Challenge 410, page 217: Waves in a river are never elliptical; they remain circular. Challenge 411, page 217: The lens is a cushion of material that is ‘transparent’ to sound. The speed of sound is faster in the cushion than in the air, in contrast to a glass lens, where the speed of light is slower in the glass. The shape is thus different: the cushion must look like a biconcave lens. Challenge 413, page 217: The Sun is always at a different position than the one we observe it to be. What is the difference, measured in angular diameters of the Sun? Challenge 414, page 217: The 3  3  3 cube has a rigid system of three perpendicular axes, on which a square can rotate at each of the 6 ends. The other squares are attaches to pieces moving around theses axes. The 4  4  4 cube is different though; just find out. The limit on the segment number seems to be 6, so far. A 7  7  7 cube requires varying shapes for the segments. But more than 5  5  5 is not found in shops. However, the website http://www.oinkleburger.com/Cube/ applet/ allows to play with virtual cubes up to 100  100  100 and more. Challenge 416, page 218: An overview of systems being tested at present can be found in K.U. Graw, Energiereservoir Ozean, Physik in unserer Zeit 33, pp. 82–88, Februar 2002. See also Oceans of electricity – new technologies convert the motion of waves into watts, Science News 159, pp. 234–236, April 2001. Challenge 417, page 218: In everyday life, the assumption is usually justified, since each spot can be approximately represented by an atom, and atoms can be followed. The assumption is questionable in situations such as turbulence, where not all spots can be assigned to atoms, and most of all, in the case of motion of the vacuum itself. In other words, for gravity waves, and in particular for the quantum theory of gravity waves, the assumption is not justified. Challenge 419, page 219: There are many. One would be that the transmission and thus reflection coefficient for waves would almost be independent of wavelength. Challenge 420, page 220: A drop with a diameter of 3 mm would cover a surface of 7.1 m2 with a 2 nm film. Challenge 421, page 221: For jumps of an animal of mass m the necessary energy E is given as E = mдh, and the work available to a muscle is roughly speaking proportional to its mass W  m. Thus one gets that the height h is independent of the mass of the animal. In other words, the specific mechanical energyof animals is around 1.5  0.7 J~kg. 4 Challenge 423, page 222: The critical height for a column of material is given by h crit = β E m , where β  1.9 is the constant determined by the calculation when a column buckles 4π д ρ2 under its own weight. Challenge 425, page 223: One possibility is to describe see particles as extended objects, such as clouds; another is given in the third part of the text. Challenge 427, page 226: Throwing the stone makes the level fall, throwing the water or the piece of wood leaves it unchanged. Challenge 428, page 226: No metal wire allows to build such a long wire. Only the idea of carbon nanotubes has raised the hope again; some dream of wire material based on them, stronger than any material known so far. However, no such material is known yet. The system faces many dangers, such as fabrication defects, lightning, storms, meteorites and space debris. All would lead to the breaking of the wires – if such wires will ever exist. But the biggest of all dangers is the lack of cash to build it. Challenge 432, page 226: The pumps worked in suction; but air pressure only allows 10 m of height difference for such systems.

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Challenge 434, page 227: This argument is comprehensible only when one remembers that ‘twice the amount’ means ‘twice as many molecules’. Challenge 435, page 227: The alcohol is frozen and the chocolate is put around it. Challenge 436, page 227: I suggested in an old edition that a machine should be based on the same machines that throw the clay pigeons used in the sports of trap shooting and skeet. In the meantime, Lydéric Bocquet and Christophe Clanet have built such a machine, but using s different design; a picture can be found on the website lpmcn.univ-lyon1.fr/%7Elbocquet/. Challenge 437, page 227: The third component of air is the noble gas argon, making up about 1 %. The rest is made up by carbon dioxide, water vapour and other gases. Are these percentages volume or weight percentages?

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Challenge 438, page 227: It uses the air pressure created by the water flowing downwards. Challenge 439, page 227: Yes. The bulb will not resist two such cars though. Challenge 442, page 228: None.

mercury. Challenge 444, page 228: The pressure destroys the lung. Challenge 446, page 229: Either they fell on inclined snowy mountain sides, or they fell into

high trees, or other soft structures. The record was over 7 km of survived free fall. Challenge 447, page 229: The blood pressure in the feet of a standing human is about 27 kPa,

double the pressure at the heart.

Challenge 448, page 229: Calculation gives N = J~ j = 0.0001 m3 ~s~(7 µm2 0.0005 m~s), or

about 6 ë 109 ; in reality, the number is much larger, as most capillaries are closed at a given instant. The reddening of the face shows what happens when all small blood vessels are opened at the same time.

Challenge 449, page 229: The soap flows down the bulb, making it thicker at the bottom and thinner at the top, until it bursts. Challenge 450, page 229: A medium-large earthquake would be generated. Challenge 451, page 229: A stalactite contains a thin channel along its axis through which the water flows, whereas a stalagmite is massive throughout. Challenge 453, page 230: About 1 part in a thousand. Challenge 454, page 230: For this to happen, friction would have to exist on the microscopic Challenge 455, page 230: The longer funnel is empty before the short one. (If you do not believe

it, try it out.) In the case that the amount of water in the funnel outlet can be neglected, one can use energy conservation for the fluid motion. This yields the famous Bernoulli equation p~ρ + дh + v 2 ~2 = const, where p is pressure, ρ the density of water, and д is 9.81 m~s2 . Therefore, the speed v is higher for greater lengths h of the thin, straight part of the funnel: the longer funnel empties first. But this is strange: the formula gives a simple free fall relation, as the pressure is the same above and below and disappears from the calculation. The expression for the speed is thus independent of whether a tube is present or not. The real reason for the faster emptying of the tube is thus that a tube forces more water to flow out that a lack of tube. Without tube, the diameter of the water flow diminishes during fall. With tube, it stay constant. This difference leads to the faster emptying.

Copyright © Christoph Schiller November 1997–May 2006

scale and energy would have to disappear.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Challenge 443, page 228: He brought the ropes into the cabin by passing them through liquid

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Challenge 456, page 230: The eyes of fish are positioned in such a way that the pressure reduc-

In other words, one has

6πηa `d 2 e . (878) 4T t All quantities on the right can be measured, thus allowing to determine the Boltzmann constant k. Since the ideal gas relation shows that the ideal gas constant R is related to the Boltzmann k=

Copyright © Christoph Schiller November 1997–May 2006

where k is the Boltzmann constant and T the temperature. The relation is deduced by studying the motion of a particle with drag force −µv that is subject to random hits. The linear drag coefficient µ of a sphere of radius a is given by µ = 6πηa . (877)

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Page 1046

tion by the flow is compensated by the pressure increase of the stall. By the way, their heart is positioned in such a way that it is helped by the underpressure. Challenge 458, page 230: Glass shatters, glass is elastic, glass shows transverse sound waves, glass does not flow (in contrast to what many books state), not even on scale of centuries, glass molecules are fixed in space, glass is crystalline at small distances, a glass pane supported at the ends does not hang through. Challenge 459, page 231: This feat has been achieved for lower mountains, such as the Monte Bianco in the Alps. At present however, there is no way to safely hover at the high altitudes of the Himalayas. Challenge 461, page 231: The iron core of the Earth formed in the way described by collecting the iron from colliding asteroids. However, the Earth was more liquid at that time. The iron will most probably not sink. In addition, there is no known way to make the measurement probe described. Challenge 462, page 231: Press the handkerchief in the glass, and lower the glass into the water with the opening first, while keeping the opening horizontal. This method is also used to lower people below the sea. The paper ball in the bottle will fly towards you. Blowing into a funnel will keep the ping-pong ball tightly into place, and the more so the stronger you blow. Blowing through a funnel towards a candle will make it lean towards you. Challenge 465, page 237: In 5000 million years, the present method will stop, and the Sun will become a red giant. abut it will burn for many more years after that. Challenge 469, page 240: We will find out later that the universe is not a physical system; thus the concept of entropy does not apply to it. Thus the universe is neither isolated nor closed. Challenge 472, page 241: The answer depends on the size of the balloons, as the pressure is not a monotonous function of the size. If the smaller balloon is not too small, the smaller balloon wins. Challenge 474, page 241: Measure the area of contact between tires and street (all four) and then multiply by 200 kPa, the usual tire pressure. You get the weight of the car. Challenge 478, page 243: If the average square displacement is proportional to time, the matter is made of smallest particles. This was confirmed by the experiments of Jean Perrin. The next step is to deduce the number number of these particles form the proportionality constant. This constant, defined by `d 2 e = 4Dt, is called the diffusion constant (the factor 4 is valid for random motion in two dimensions). The diffusion constant can be determined by watching the motion of a particle under the microscope. We study a Brownian particle of radius a. In two dimensions, its square displacement is given by 4kT `d 2 e t, (876) µ

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constant by R = N A k, the Avogadro constant N A that gives the number of molecules in a mole is also found in this way. Challenge 485, page 249: Yes, the effect is easily noticeable.

Challenge 487, page 250: Hot air is less dense and thus wants to rise. Challenge 490, page 250: The air had to be dry. Challenge 491, page 250: In general, it is impossible to draw a line through three points. Challenge 492, page 250: No, as a water molecule is heavier than that. However, if the water is allowed to be dirty, it is possible. What happens if the quantum of action is taken into account?

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Challenge 488, page 250: Keep the paper wet. Challenge 493, page 251: The danger is not due to the amount of energy, but due to the time in which it is available. Challenge 494, page 251: The internet is full of solutions.

Challenge 499, page 252: For such small animals the body temperature would fall too low. They

could not eat fast enough to get the energy needed to keep themselves warm. Challenge 508, page 253: It is about 10−9 that of the Earth. Challenge 510, page 253: The thickness of the folds in the brain, the bubbles in the lung, the

density of blood vessels and the size of biological cells. Challenge 511, page 254: The mercury vapour above the liquid gets saturated. Challenge 512, page 254: A dedicated NASA project

studies this question. Figure 406 gives an example comparison. You can find more details on their website. Challenge 513, page 254: The risks due to storms and the financial risks are too large. Challenge 514, page 254: The vortex in the tube is

F I G U R E 406 A candle on Earth and in

microgravity (NASA)

Challenge 515, page 254: Egg white hardens at 70°C, egg white at 65 to 68°C. Cook an egg at the latter temperature, and the feat is possible. Challenge 519, page 255: This is also true for the shape human bodies, the brain control of

human motion, the growth of flowers, the waves of the sea, the formation of clouds, the processes leading to volcano eruptions, etc. Challenge 524, page 259: There are many more butterflies than tornadoes. In addition, the belief in the butterfly effect completely neglects an aspect of nature that is essential for selforganization: friction and dissipation. The butterfly effect, assumed it did exist, requires that dissipation is neglected. There is no experimental basis for the effect, it has never been observed.

Copyright © Christoph Schiller November 1997–May 2006

cold in the middle and hot at its outside; the air from the middle is sent to one end and the air from the outside to the other. The heating of the outside is due to the work that the air rotating inside has to do on the air outside to get a rotation that eats up angular momentum. For a detailed explanation, see the beautiful text by Mark P. Silverman, And Yet it Moves: Strange Systems and Subtle Questions in Physics, Cambridge University Press, 1993, p. 221.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Challenge 496, page 252: Only if it is a closed system. Is the universe closed? Is it a system? This is discussed in the third part of the mountain ascent.

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Challenge 534, page 264: All three statements are hogwash. A drag coefficient implies that the

cross area of the car is known to the same precision. This is actually extremely difficult to measure and to keep constant. In fact, the value 0.375 for the Ford Escort was a cheat, as many other measurements showed. The fuel consumption is even more ridiculous, as it implies that fuel volumes and distances can be measured to that same precision. Opinion polls are taken by phoning at most 2000 people; due to the difficulties in selecting the right representative sample, that gives a precision of at most 3 %. Challenge 535, page 264: No. Nature does not allow more than about 20 digits of precision, as we will discover later in our walk. That is not sufficient for a standard book. The question whether such a number can be part of its own book thus disappears.

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Challenge 537, page 265: Every measurement is a comparison with a standard; every compar-

ison requires light or some other electromagnetic field. This is also the case for time measurements.

Challenge 539, page 265: Angle measurements have the same properties as length or time measurements. Challenge 540, page 275: A cone or a hyperboloid also look straight from all directions,

provided the positioning is correct. One thus needs not only to turn the object, but also to displace it. The best method to check planarity is to use interference between an arriving and a departing coherent beam of light. If the fringes are straight, the surface is planar. (How do you ensure the wavefront of the light beam is planar?) Challenge 541, page 276: A fraction of infinity is still infinite. Challenge 542, page 276: The time at which the Moon Io enters the shadow in the second meas-

urement occurs about 1000 s later than predicted from the first measurement. Since the Earth is about 3 ë 1011 m further away from Jupiter and Io, we get the usual value for the speed of light. Challenge 543, page 277: To compensate for the aberration, the telescope has to be inclined along the direction of motion of the Earth; to compensate for parallaxis, against the motion. Challenge 544, page 277: Otherwise the velocity sum would be larger than c. Challenge 545, page 277: The drawing shows it. Observer, Moon and Sun form a triangle. When the Moon is half full, the angle at the Moon is a right angle. Thus the distance ration can be determined, though not easily, as the angle at the observer is very near a right angle as well. Challenge 546, page 278: There are Cat’s-eyes on the Moon deposited there during the Apollo

Challenge 547, page 278: Fizeau used a mirror about 8.6 km away. As the picture shows, he only had to count the teeth of his cog-wheel and measure its rotation speed when the light goes in one direction through one tooth and comes back to the next.

Challenge 548, page 279: The time must be shorter than T = l~c, in other words, shorter than

30 ps; it was a gas shutter, not a solid one. It was triggered by a red light pulse (show in the photographed) timed by the one to be photographed; for certain materials, such as the used gas, strong light can lead to bleaching, so that they become transparent. For more details about the shutter and its neat trigger technique, see the paper by the authors.

Copyright © Christoph Schiller November 1997–May 2006

and Lunakhod missions. They are used to reflect laser 35 ps light pulses sent there through telescopes. The timing of the round trip then gives the distance to the Moon. Of course, absolute distance is not know to high precision, but the variations are. The thickness of the atmosphere is the largest source of error. See the http://www.csr.utexas.edu/mlrs and http://ilrs.gsfc.nasa.gov websites.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Challenge 538, page 265: Every mass measurement is a comparison with a standard; every comparison requires light or some other electromagnetic field.

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1258

Challenge 549, page 279: Just take a photograph of a lightning while moving the camera horizontally. You will see that a lightning is made of several discharges; the whole shows that lightning is much slower than light. If lightning moved only nearly as fast as light itself, the Doppler effect would change it colour depending on the angle at which we look at it, compared to its direction of motion. A nearby lightning would change colour from top to bottom. Challenge 550, page 280: The fastest lamps were subatomic particles, such as muons, which

decay by emitting a photon, thus a tiny flash of light. However, also some stars emit fasts jets of matter, which move with speeds comparable to that of light.

Ref. 258

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Challenge 551, page 281: The speed of neutrinos is the same as that of light to 9 decimal digits, since neutrinos and light were observed to arrive together, within 12 seconds of each other, after a trip of 170 000 light years from a supernova explosion. Challenge 553, page 283: This is best discussed by showing that other possibilities make no

sense. c(k 2 − 1)T~2; the time coordinate is (k 2 + 1)T~2. Their ratio must be v. Solving for k gives the result.

Challenge 556, page 284: The motion of radio waves, infrared, ultraviolet and gamma rays is also unstoppable. Another past suspect, the neutrino, has been found to have mass and to be thus in principle stoppable. The motion of gravity is also unstoppable. Challenge 558, page 286: λ R ~λ S = γ.

Challenge 559, page 286: To change from bright red (650 nm) to green (550 nm), v = 0.166c is

necessary.

Challenge 560, page 286: People measure the shift of spectral lines, such as the shift of the soPage 753

called Lyman-α line of hydrogen, that is emitted (or absorbed) when a free electron is captured (or ejected) by a proton. It is one of the famous Fraunhofer lines. Challenge 561, page 286: The speeds are given by

v~c =

(z + 1)2 − 1 (z + 1)2 + 1

(879)

Challenge 562, page 286: No Doppler effect is seen for a distant observer at rest with respect to

the large mass. In other cases there obviously is a Doppler effect, but it is not due to the deflection. Challenge 563, page 287: Sound speed is not invariant of the speed of observers. As a result,

the Doppler effect for sound even confirms – within measurement differences – that time is the same for observers moving against each other. » 2 ë 30~511 or v  0.3c.

Challenge 566, page 288: Inside colour television tubes (they use higher voltages than black

and white ones), electrons are described by v~c 

Challenge 567, page 289: If you can imagine this, publish it. Readers will be delighted to hear

the story.

Copyright © Christoph Schiller November 1997–May 2006

which implies v(z = −0.1) = 31 Mm~s = 0.1c towards the observer and v(z = 5) = 284 Mm~s = 0.95c away from the observer. A red-shift of 6 implies a speed of 0.96c; such speeds appear because, as we will see in the section of general relativity, far away objects recede from us. And high red-shifts are observed only for objects which are extremely far from Earth, and the faster the further they are away. For a red-shift of 6 that is a distance of several thousand million light years.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Challenge 554, page 283: The spatial coordinate of the event at which the light is reflected is

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1259

Challenge 569, page 289: The connection between observer invariance and limit property Page 1068

seems to be generally valid in nature, as shown in section 36.. However, a complete and airtight argument is not yet at hand. If you have one, publish it! Challenge 572, page 291: If the speed of light is the same for all observers, no observer can pretend to be more at rest than another (as long as space-time is flat), because there is no observation from electrodynamics, mechanics or another part of physics that allows to make the statement. Challenge 575, page 293: Redrawing Figure 137 on page 283 for the other observer makes the

point.

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Challenge 576, page 293: The human value is achieved in particle accelerators; the value in

nature is found in cosmic rays of the highest energies. Challenge 577, page 294: The set of events behaves like a manifold, because it behaves like a four-dimensional space: it has infinitely many points around any given starting point, and distances behave as we are used to, limits behave as we are used to. It differs by one added dimension, and by the sign in the definition of distance; thus, properly speaking, it is a Riemannian manifold.

Challenge 579, page 296: The light cone remains unchanged; thus causal connection as well. Challenge 580, page 296: In such a case, the division of space-time around an inertial observer into future, past and elsewhere would not hold any more, and the future could influence the past (as seen from another observer). Challenge 583, page 299: The ratio predicted by naive reasoning is (1~2)( 6.4~2.2) = 0.13.

Challenge 584, page 299: The time dilation factor for v = 0.9952c is 10.2, giving a proper time of 0.62 µs; thus the ratio predicted by special relativity is (1~2)( 0.62~2.2) = 0.82.

Challenge 585, page 299: Send a light signal from the first clock to the second clock and back. Take the middle time between the departure and arrival, and then compare it with the time at the reflection. Repeat this a few times. See also Figure 137. Challenge 588, page 301: Hint: think about different directions of sight. Challenge 589, page 301: Not with present experimental methods. Challenge 591, page 301: Hint: be careful with the definition of ‘rigidity’. Challenge 593, page 302: The light cannot stay on at any speed, if the glider is shorter than the

Challenge 594, page 302: Yes, the rope breaks; in accelerated cars, distance changes, as shown

later on in the text. Challenge 595, page 302: The submarine will sink. The fast submarine will even be heavier, as

his kinetic energy adds to his weight. The contraction effect would make it lighter, as the captain

Copyright © Christoph Schiller November 1997–May 2006

gap. This is strange, because the bar does not light the lamp even at high speeds, even though in the frame of the bar there is contact at both ends. The reason is that in this case there is not enough time to send the signal to the battery that contact is made, so that the current cannot start flowing. Assume that current flows with speed u, which is of the order of c. Then, as Dirk Van de Moortel showed, the lamp will go off if the glider length l glider and the gap length l gap obey l glider ~l gap < γ(u + v)~u. See also the cited reference. Why are the debates often heated? Some people will (falsely) pretend that the problem is unphysical; other will say that Maxwell’s equations are needed. Still others will say that the problem is absurd, because for larger lengths of the glider, the on/off answer depends on the precise speed value. However, this actually is the case in this situation.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Challenge 578, page 295: Infinity is obvious, as is openness. Thus the topology equivalence can be shown by imagining that the manifold is made of rubber and wrapped around a sphere.

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says, but by a smaller amount. The total weight – counting upwards as positive – is given by F = −mд(γ − 1~γ). Challenge 596, page 302: A relativistic submarine would instantly melt due to friction with

the water. If not, it would fly of the planet because it moves faster than the escape velocity. And produce several other disasters.

Challenge 597, page 305: The question confuses observation of Lorentz contraction and its measurement. A relativistic pearl necklace does get shorter, but the shortening can only be measured, not photographed. The measured sizes of the pearls are flattened ellipsoids relativistic speeds. The observed necklace consists of overlapping spheres.

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Challenge 598, page 305: The website offers a prize for a film checking this issue. Challenge 601, page 305: Yes, ageing in a valley is slowed compared to mountain tops. How-

ever, the proper sensation of time is not changed. The reason for the appearance of grey hair is not known; if the timing is genetic, the proper time at which it happens is the same in either location.

Challenge 603, page 307: Just use plain geometry to show this. Challenge 604, page 307: Most interestingly, the horizon can easily move faster than light, if you move your head appropriately, as can the end of the rainbow. Challenge 607, page 311: Relativity makes the arguments of challenge 130 watertight. Challenge 612, page 313: The lower collision in Figure 161 shows the result directly, from energy conservation. For the upper collision the result also follows, if one starts from momentum conservation γmv = ΓMV and energy conservation (дamma + 1)m = ΓM.

Challenge 614, page 314: Annihilation of matter and antimatter.

Challenge 621, page 317: Just turn the left side of Figure 164 a bit in anti-clockwise direction.

Challenge 622, page 318: In collisions between relativistic charges, part of the energy is radi-

ated away as light, so that the particles effectively lose energy. Challenge 623, page 319: Probably not, as all relations among physical quantities are known

now. However, you might check for yourself; one might never know. It is worth to mention that the maximum force in nature was discovered (in this text) after remaining hidden for over 80 years. Challenge 624, page 321: Write down the four-vectors U ′ and U and then extract v ′ as function Challenge 625, page 321: Any motion with light speed.

Challenge 626, page 322: b 0 = 0, b i = γ 2 a i .

Challenge 629, page 323: For ultrarelativistic particles, like for massless particles, one has E =

pc.

Challenge 630, page 323: Hint: evaluate P1 and P2 in the rest frame of one particle.

Challenge 631, page 324: Use the definition f = dp~dt and the relation KU = 0 = fv − dE~dt valid for rest-mass preserving forces.

Challenge 659, page 333: The energy contained in the fuel must be comparable to the rest mass of the motorbike, multiplied by c 2 . Since fuel contains much more mass than energy, that gives a big problem.

Copyright © Christoph Schiller November 1997–May 2006

of v and the relative coordinate speed V . Then rename the variables.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Challenge 602, page 306: There is no way to put an observer at the specified points. Proper velocity can only be defined for observers, i.e., for entities which can carry a clock. That is not the case for images.

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1261

Challenge 661, page 334: Constant acceleration and gravity are similar in their effects, as discussed in the section on general relativity. Challenge 667, page 336: Yes, it is true. Challenge 668, page 336: It is flat, like a plane. Challenge 670, page 337: Yes; however, the effect is minimal and depends on the position of

the Sun. In fact, what is white at one height is not white at another. Challenge 672, page 337: Locally, light always moves with speed c.

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Challenge 673, page 338: Away from Earth, д decreases; it is effectively zero over most of the

distance.

Challenge 675, page 339: See the cited reference. The factor 2 was forgotten there; can you de-

duce it? Challenge 678, page 340: Though there are many publications pretending to study the issue, there are also enough physicists who notice the impossibility. Measuring a variation of the speed of light is not much far from measuring the one way speed of light: it is not possible. However, the debates on the topic are heated; the issue will take long to be put to rest. Challenge 679, page 349: The inverse square law of gravity does not comply with the maximum

speed principle; it is not clear how it changes when one changes to a moving observer. Challenge 680, page 353: Take a surface moving with the speed of light, or a surface defined with a precision smaller than the Planck length.

Challenge 684, page 364: If so, publish it; then send it to the author. Challenge 685, page 366: For example, it is possible to imagine a surface that has such an in-

tricate shape that it will pass all atoms of the universe at almost the speed of light. Such a surface is not physical, as it is impossible to imagine observers on all its points that move in that way all at the same time. Challenge 693, page 378: They are accelerated upwards. Challenge 694, page 378: In everyday life, (a) the surface of the Earth can be taken to be flat, (b) the vertical curvature effects are negligible, and (c) the lateral length effects are negligible. Challenge 698, page 379: For a powerful bus, the acceleration is 2 m~s2 ; in 100 m of accelera-

tion, this makes a relative frequency change of 2.2 ë 10−15 .

Copyright © Christoph Schiller November 1997–May 2006

Challenge 681, page 357: Also shadows do not remain parallel in curved surfaces. forgetting this leads to strange mistakes: many arguments allegedly ‘showing’ that men have never been on the moon neglect this fact when they discuss the photographs taken there.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Challenge 674, page 339: Light is necessary to determine distance and to synchronize clocks; thus there is no way to measure the speed of light from one point to another alone. The reverse motion needs to be included. However, some statements on the one-way speed of light can still be made (see http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html). All experiments on the one-way speed of light performed so far are consistent with an isotropic value that is equal to the two-way velocity. However, no experiment is able to rule out a group of theories in which the one-way speed of light is anisotropic and thus different from the two-way speed. All theories from this group have the property that the round-trip speed of light is isotropic in any inertial frame, but the one-way speed is isotropic only in a preferred ‘ether’ frame. In all of these theories, in all inertial frames, the effects of slow clock transport exactly compensate the effects of the anisotropic one-way speed of light. All these theories are experimentally indistinguishable from special relativity. In practice, therefore, the one-way speed of light has been measured and is constant. But a small option remains.

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Challenge 699, page 379: Yes, light absorption and emission are always lossless conversions of

Challenge 733, page 392: They use a spring scale, and measure the oscillation time. From it

Challenge 753, page 401: Since there is no negative mass, gravitoelectric fields cannot be neut-

ralized. In contrast, electric fields can be neutralized around a metallic conductor with a Faraday cage. Challenge 766, page 408: One needs to measure the timing of pulses which cross the Earth at different gravitational wave detectors on Earth. Challenge 782, page 415: No; a line cannot have intrinsic curvature. A torus is indeed intrinsically curved; it cannot be cut open to a flat sheet of paper. Challenge 804, page 423: The trace of the Einstein tensor is the negative of the Ricci scalar; it is thus the negative of the trace of the Ricci tensor.

Copyright © Christoph Schiller November 1997–May 2006

they deduce their mass. Challenge 734, page 392: The apple hits the wall after about half an hour. Challenge 738, page 393: With ħ as smallest angular momentum one get about 100 Tm. Challenge 739, page 393: No. The diffraction of the beams does not allow it. Also quantum theory makes this impossible; bound states of massless particles, such as photons, are not stable. Challenge 741, page 394: The orbital radius is 4.2 Earth radii; that makes c. 38 µs every day. Challenge 742, page 395: To be honest, the experiments are not consistent. They assume that some other property of nature is constant – such as atomic size – which in fact also depends on G. More on this issue on page 501. Challenge 743, page 395: Of course other spatial dimensions could exist which can be detected only with the help of measurement apparatuses. For example, hidden dimensions could appear at energies not accessible in everyday life.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

energy into mass. Challenge 702, page 379: For a beam of light, in both cases the situation is described by an environment in which masses ‘fall’ against the direction of motion. If the Earth and the train walls were not visible – for example if they were hidden by mist – there would not be any way to determine by experiment which situation is which. Or again, if an observer would be enclosed in a box, he could not distinguish between constant acceleration or constant gravity. (Important: this impossibility only applies if the observer has negligible size!) Challenge 708, page 381: Both fall towards the centre of the Earth. Orbiting particles are also in free fall; their relative distance changes as well, as explained in the text. Challenge 711, page 383: Such a graph would need four or even 5 dimensions. Challenge 713, page 385: The energy due to the rotation can be neglected compared with all other energies in the problem. Challenge 723, page 390: Different nucleons, different nuclei, different atoms and different molecules have different percentages of binding energies relative to the total mass. Challenge 725, page 391: In free fall, the bottle and the water remain at rest with respect to each other. Challenge 726, page 391: Let the device fall. The elastic rubber then is strong enough to pull the ball into the cup. See M.T. Westra, Einsteins verjaardagscadeau, Nederlands tijdschrift voor natuurkunde 69, p. 109, April 2003. The original device also had a spring connected in series to the rubber. Challenge 727, page 391: Apart the chairs and tables already mentioned, important antigravity devices are suspenders, belts and plastic bags.

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1263

Challenge 821, page 434: Indeed, in general relativity gravitational energy cannot be localized

in space, in contrast to what one expects and requires from an interaction. Challenge 833, page 442: There is a good chance that some weak form of a jet exists; but a detection will not be easy. Challenge 840, page 454: The rabbit observes that all other rabbits seem to move away from

him. Challenge 846, page 458: Stand in a forest in winter, and try to see the horizon. If the forest is very deep, you hit tree trunks in all directions. If the forest is finite in depth, you have chance to see the horizon.

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Challenge 867, page 475: Flattening due to rotation requires other masses to provide the background against which the rotation takes place.

Challenge 901, page 487: The description says it all. A visual impression can be found in the

room on black holes in the ‘Deutsches Museum’ in München. Challenge 910, page 493: Any device that uses mirrors requires electrodynamics; without elec-

trodynamics, mirrors are impossible. Challenge 912, page 496: The hollow Earth theory is correct if usual distance are consistently 2 changed to r he = R Earth ~r. This implies a quantum of action that decreases towards the centre of the hollow sphere. Then there is no way to prefer one description over the other, except for reasons of simplicity. Challenge 919, page 521: The liquid drops have to detach from the flow exactly inside the metal counter-electrodes. Opel simply earthed the metal piece they had built into the cars without any contact to the rest of the car. Challenge 920, page 522: A lot of noise while the metal pendulum banged wildly between the

two fixed bells. Challenge 923, page 524: The field at a distance of 1 m from an electron is 1.4 nV~m. Challenge 924, page 525: A simple geometrical effect: anything flowing out homogeneously Challenge 925, page 525: One has F = αħcN A2 ~4R 2 = 3 ë 1012 N, an enormous force, corres-

ponding to the weight of 300 million tons. It shows the enormous forces that keep matter together. Obviously, there is no way to keep 1 g of positive charge together, as the repulsive forces among the charges would be even larger.

Challenge 926, page 525: To show the full equivalence of Coulomb’s and Gauss’s ‘laws’, first show that it holds for a single point charge. Then expand the result for more than one point charge. That gives Gauss’s ‘law’ in integral form, as given just before this challenge. To deduce the integral form of Gauss’s ‘law’ for a single point charge, one has to integrate over the closed surface. The essential point here is to note that the integration can be carried out for an inverse square dependence only. This dependence allows to transform the scalar product between the local field and the area element into a normal product between the charge and the

Copyright © Christoph Schiller November 1997–May 2006

from a sphere diminishes with the square of the distance.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Challenge 897, page 487: This happens in the same way that the static electric field comes out of a charge. In both cases, the transverse fields do not get out, but the longitudinal fields do. Quantum theory provides the deeper reason. Real radiation particles, which are responsible for free, transverse fields, cannot leave a black hole because of the escape velocity. However, virtual particles can, as their speed is not bound by the speed of light. All static, longitudinal fields are produced by virtual particles. In addition, there is a second reason. Classical field can come out of a black hole because for an outside observer everything making it up is continuously falling, and nothing has actually crossed the horizon. The field sources thus are not yet out of reach.

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1264 solid angle Ω:

E dA =

qdA cos θ qdΩ = . 4πε 0 r 2 4πε 0

(880)

In case that the surface is closed the integration is then straightforward. To deduce the differential form of (the static) Gauss’s ‘law’, namely ∇E =

ρ , ε0

(881)

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make use of the definition of the charge density ρ and of the purely mathematical relation

closedsurface

E dA =

enclosedvolume

∇E dV ,

(882)

∇B−

1 ∂E = µ0 j c 2 ∂t

(883)

Copyright © Christoph Schiller November 1997–May 2006

whereas the expression mentioned in the text misses the term ∂E . ∂t For another way to state the difference, see R ichard P. Feynman, R.B. Leighton & M. Sands, The Feynman Lectures on Physics, volume II, Addison Wesley, p. 21-1, 1977. Challenge 939, page 535: Only boosts with relativistic speeds mix magnetic and electric fields to an appreciable amount. Challenge 941, page 535: The dual field ‡F is defined on page 546. Challenge 942, page 536: Scalar products of four vectors are always, by construction, Lorentz invariant quantities. Challenge 948, page 539: Usually, the cables of high voltage lines are too warm to be comfortable. Challenge 949, page 539: Move them to form a T shape. Challenge 950, page 539: For four and more switches, on uses inverters; an inverter is a switch with two inputs and two outputs which in one position, connects first and second input to first and second output respectively, and in the other position connects the first input to the second output and vice versa. (There are other possibilities, though; wires can be saved using electromagnetic relay switches.) For three switches, there is a simpler solution than with inverters. Challenge 952, page 540: It is possible; however, the systems so far are not small and are dangerous for human health. The idea to collect solar power in deep space and then beam it to the Earth as microwaves has often been aired. Finances and dangers have blocked it so far.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

This mathematical relation, valid for any vector field E, is called Gauss’s theorem. It simply states that the flux is the volume integral of the divergence. To deduce the full form of Gauss’s law, including the time-derivative of the magnetic field, include relativistic effects by changing viewpoint to a moving observer. Challenge 928, page 526: No; batteries only separate charges and pump them around. Challenge 929, page 526: Uncharged bodies can attract each other if they are made of charged constituents neutralizing each other, and if the charges are constrained in their mobility. The charge fluctuations then lead to attraction. Most molecules interact among each other in this way; such forces are also at the basis of surface tension in liquids and thus of droplet formation. Challenge 931, page 527: The ratio q~m of electrons and that of the free charges inside metals is not exactly the same. Challenge 938, page 534: The correct version of Ampère’s ‘law’ is

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Challenge 953, page 540: Glue two mirrors together at a right angle. Or watch yourself on TV

using a video camera. Challenge 954, page 540: This is again an example of combined triboluminescence and triboelectricity. See also the websites http://scienceworld.wolfram.com/physics/Triboluminescence. html and http://www.geocities.com/RainForest/9911/tribo.htm. Challenge 956, page 541: Pepper is lighter than salt, and thus reacts to the spoon before the

salt does. Challenge 957, page 542: For a wavelength of 546.1 nm (standard green), that is a bit over 18

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wavelengths. Challenge 958, page 542: The angular size of the Sun is too large; diffraction plays no role here. Challenge 959, page 542: Just use a high speed camera. Challenge 960, page 543: The current flows perpendicularly to the magnetic field and is thus

deflected. It pulls the whole magnet with it. Challenge 964, page 543: The most simple equivalent to a coil is a rotating mass being put into

rotation by the flowing water. A transformer would then be made of two such masses connected through their axis. Challenge 967, page 545: The charged layer has the effect that almost only ions of one charge pass the channels. As a result, charges are separated on the two sides of the liquid, and a current is generated. Challenge 972, page 548: Some momentum is carried away by the electromagnetic field. Challenge 973, page 548: Field lines and equipotential surfaces are always orthogonal to each

other. Thus a field line cannot cross an equipotential surface twice. Challenge 984, page 554: Just draw a current through a coil with its magnetic field, then draw the mirror image of the current and redraw the magnetic field. Challenge 985, page 555: Other asymmetries in nature include the helicity of the DNA mo-

lecules making up the chromosomes and many other molecules in living systems, the right hand preference of most humans, the asymmetry of fish species which usually stay flat on the bottom of the seas. Challenge 986, page 555: This is not possible at all using gravitational or electromagnetic systems or effects. The only way is to use the weak nuclear interaction, as shown in the chapter on the nucleus.

Challenge 988, page 556: The image flips up: a 90 degree rotation turns the image by 180 de-

grees. Challenge 989, page 556: Imagine E and B as the unite vectors of two axes in complex space. Then any rotation of these axes is also a generalized duality symmetry. Challenge 992, page 561: In every case of interference, the energy is redistributed into other directions. This is the general rule; sometimes it is quite tricky to discover this other direction. Challenge 993, page 561: The author regularly sees about 7 lines; assuming that the distance is around 20 µm, this makes about 3 µm per line. The wavelength must be smaller than this value and the frequency thus larger than 100 THz. The actual values for various colours are given in the table of the electromagnetic spectrum.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 987, page 556: The Lagrangian does not change if one of the three coordinates is changed by its negative value.

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Challenge 961, page 543: Light makes seven turns of the Earth in one second.

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1 1 1 + = . do di f

(884)

Copyright © Christoph Schiller November 1997–May 2006

It is valid for diverging and converging lenses, as long as their own thickness is negligible. The strength of a lens can thus be measured with the quantity 1~ f . The unit 1 m−1 is called a diopter; it is used especially for reading glasses. Converging lenses have positive, diverging lenses negative values. Challenge 1005, page 569: A light microscope is basically made of two converging lenses. One lens – or lens system – produces an enlarged real image and the second one produces an enlarged virtual image of the previous real image. Figure 407 also shows that microscopes always turn images upside down. Due to the wavelength of light, light microscopes have a maximum resolution of about 1 µm. Note that the magnification of microscopes is unlimited; what is limited is their resolution. This is exactly the same behaviour shown by digital images. The resolution is simply the size of the smallest possible pixel that makes sense. Challenge 1007, page 570: The dispersion at the lens leads to different apparent image positions, as shown in Figure 408. For more details on the dispersion in the human eye and the ways of using it to create three-dimensional effects, see the article by C. Ucke & R. Wolf, Durch Farbe in die dritte Dimension, Physik in unserer Zeit 30, pp. 50–53, 1999. Challenge 1008, page 570: The 1 mm beam would return 1000 times as wide as the 1 m beam. A perfect 1 m-wide beam of green light would be 209 m wide on the Moon; can you deduce this result from the (important) formula that involves distance, wavelength, initial diameter and final diameter? Try to guess this beautiful formula first, and then deduce it. In reality, the values are a few times larger than the theoretical minimum thus calculated. See the http://www.csr.utexas. edu/mlrs and http://ilrs.gsfc.nasa.gov websites. Challenge 1009, page 570: The answer should lie between one or two dozen kilometres, assuming ideal atmospheric circumstances. Challenge 1014, page 572: A surface of 1 m2 perpendicular to the light receives about 1 kW of radiation. It generates the same pressure as the weight of about 0.3 mg of matter. That generates 3 µPa for black surfaces, and the double for mirrors.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Challenge 995, page 562: He noted that when a prism produces a rainbow, a thermometer placed in the region after the colour red shows a temperature rise. Challenge 996, page 562: Light reflected form a water surface is partly polarized. Mirages are not. Challenge 998, page 563: Drawing them properly requires four dimensions; and there is no analogy with two-dimensional waves. It is not easy to picture them. The direction of oscillation of the fields rotates as the wave advances. The oscillation direction thus forms a spiral. Picturing the rest of the wave is not impossible, but not easy. Challenge 1001, page 567: Such an observer would experience a wavy but static field, which cannot exist, as the equations for the electromagnetic field show. Challenge 1002, page 567: You would never die. Could you reach the end of the universe? Challenge 1003, page 568: Syrup shows an even more beautiful effect in the following setting. Take a long transparent tube closed at one end and fill it with syrup. Shine a red helium–neon laser into the tube from the bottom. Then introduce a linear polarizer into the beam: the light seen in the tube will form a spiral. By rotating the polarizer you can make the spiral advance or retract. This effect, called the optical activity of sugar, is due to the ability of sugar to rotate light polarization and to a special property of plants: they make only one of the two mirror forms of sugar. Challenge 1004, page 568: The thin lens formula is

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focus

ocular real intermediate image

focus

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objective

object

F I G U R E 407 Two converging lenses make a microscope

Challenge 1016, page 573: The shine side gets twice the momentum transfer as the black side,

Copyright © Christoph Schiller November 1997–May 2006

and thus should be pushed backwards. Challenge 1019, page 574: A polarizer can do this. Challenge 1022, page 574: The interference patterns change when colours are changed. Rainbows also appear because different colours are due to different frequencies. Challenge 1024, page 575: The full rainbow is round like a circle. You can produce one with a garden hose, if you keep the hose in your hand while you stand on a chair, with your back to the evening Sun. (Well, one small part is missing; can you imagine which part?) The circle is due to the spherical shape of droplets. If the droplets were of different shape, and if they were all aligned, the rainbow would have a different shape than a simple circle. Challenge 1028, page 577: Film a distant supernova explosion and check whether it happens at the same time for each colour separately. Challenge 1030, page 579: The first part of the forerunner is a feature with the shortest possible effective wavelength; thus it is given by taking the limit for infinite frequency. Challenge 1031, page 579: The light is pulsed; thus it is the energy velocity. Challenge 1032, page 579: Inside matter, the energy is transferred to atoms, then back to light, then to the next atoms, etc. That takes time and slows down the propagation. Challenge 1034, page 581: This is true even in general relativity, when the bending of the vacuum is studied. Challenge 1035, page 582: Not really; a Cat’s-eye uses two reflections at the sides of a cube. A living cat’s eye has a large number of reflections. The end effect is the same though: light returns back to the direction it came from.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

virtual image

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Eye lens dispersion

apparent blue position

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apparent red position real position

Challenge 1036, page 583: There is a blind spot in the eye; that is a region in which images are

not perceived. The brain than assumes that the image at that place is the same than at its borders. If a spot falls exactly inside it, it disappears. Challenge 1038, page 585: The eye and brain surely do not switch the up and the down direc-

tion at a certain age. Challenge 1039, page 586: The eye and vision system subtract patterns that are constant in

time. Challenge 1041, page 587: In fact, there is no way that a hologram of a person can walk around

Challenge 1042, page 590: See challenge 566. Challenge 1043, page 590: The electrons move slowly, but the speed of electrical signals is given

by the time at which the electrons move. Imagine long queue of cars (representing electrons) waiting in front of a red traffic light. All drivers look at the light. As soon as it turns green, everybody starts driving. Even though the driving speed might be only 10 m~s, the speed of traffic flow onset was that of light. It is this latter speed which is the speed of electrical signals. Water pipes tell the same story. A long hose provides water almost in the same instant as the tap is opened, even if the water takes a long time to arrive from the tap to the end of the hose. The speed with which the water reacts is gives by the speed for pressure waves in water. Also for water hoses the signal speed, roughly given by the sound speed in water, is much higher than the speed of the water flow.

Copyright © Christoph Schiller November 1997–May 2006

and frighten a real person. A hologram is always transparent; one can always see the background through the hologram. A hologram thus always gives an impression similar to what moving pictures usually show as ghosts.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

F I G U R E 408 The relation between the colour effect and the lens

dispersion

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Challenge 1044, page 591: One can measure current fluctuations, or measure smallest charges,

showing that they are always multiples of the same unit. The latter method was used by Millikan.

Challenge 1047, page 591: Earth’s potential would be U = −q~(4πε o R) = 60 MV, where the number of electrons in water must be taken into account.

Challenge 1049, page 592: Almost no light passes; the intensity of the little light that is trans-

mitted depends exponentially on the ratio between wavelength and hole diameter. One also says that after the hole there is an evanescent wave. Challenge 1051, page 592: The angular momentum was put into the system when it was

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formed. If we bring a point charge from infinity along a straight line to its final position close to a magnetic dipole, the magnetic force acting on the charge is not directed along the line of motion. It therefore creates a non-vanishing torque about the origin. See J.M. Aguirregabiria & A. Hernandez, The Feynman paradox revisited, European Journal of Physics 2, pp. 168–170, 1981.

Challenge 1054, page 593: There is always a measurement error when measuring field values,

even when measuring a ‘vanishing’ electromagnetic field.

Challenge 1055, page 593: The green surface seen at a low high angle is larger than when seen vertically, where the soil is also seen; the soil is covered by the green grass in low angle observation. Challenge 1058, page 593: The charges in a metal rearrange in a way that the field inside re-

mains vanishing. This makes cars and aeroplanes safe against lightning. Of course, if the outside field varies so quickly that the rearrangement cannot follow, fields can enter the Faraday cage. (By the way, also fields with long wavelengths penetrate metals; remote controls regularly use frequencies of 25 kHz to achieve this.) However, one should wait a bit before stepping out of a car after lightning has hit, as the car is on rubber wheels with low conduction; waiting gives the charge time to flow into the ground. For gravity and solid cages, mass rearrangement is not possible, so that there is no gravity shield. Challenge 1063, page 595: Of course not, as the group velocity is not limited by special relativity. The energy velocity is limited, but is not changed in this experiments. Challenge 1065, page 595: The Prussian explorer Alexander von Humboldt extensively checked this myth in the nineteenth century. He visited many mine pits and asked countless mine workers in Mexico, Peru and Siberia about their experiences. He also asked numerous chimney-sweeps. Neither him nor anybody else had ever seen the stars during the day. Challenge 1068, page 596: The charging stops because a negatively charged satellite repels elec-

trons and thus stops any electron collecting mechanism. Electrons are captured more frequently than ions because it is easier for them than for ions to have an inelastic collision with the satellite, due to their larger speed at a given temperature. Challenge 1069, page 596: Any loss mechanism will explain the loss of energy, such as elec-

trical resistance or electromagnetic radiation. After a fraction of a second, the energy will be lost. This little problem is often discussed on the internet. Challenge 1071, page 597: Show that even though the radial magnetic field of a spherical wave

is vanishing by definition, Maxwell’s equations would require it to be different from zero. Since electromagnetic waves are transversal, it is also sufficient to show that it is impossible to comb a hairy sphere without having a (double) vortex or two simple vortices. Despite these statements,

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1066, page 596: The number of photons times the quantum of action ħ.

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Challenge 1053, page 593: Leakage currents change the picture. The long term voltage ratio is given by the leakage resistance ratio V1 ~V2 = R 1 ~R 2 , as can be easily verified in experiments.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006

quantum theory changes the picture somewhat: the emission probability of a photon from an excited atom in a degenerate state is spherically symmetric exactly. Challenge 1072, page 598: The human body is slightly conducting and changes the shape of the field and thus effectively short circuits it. Usually, the field cannot be used to generate energy, as the currents involved are much too small. (Lightning bolts are a different story, of course. They are due – very indirectly – to the field of the Earth, but they are too irregular to be used consistently. Franklin’s lightning rod is such an example.) Challenge 1076, page 602: This should be possible in the near future; but both the experiment, which will probably measure brain magnetic field details, and the precise check of its seriousness will not be simple. Challenge 1083, page 605: Any new one is worth a publication. Challenge 1084, page 608: Sound energy is also possible, as is mechanical work. Challenge 1085, page 610: Space-time deformation is not related to electricity; at least at everyday energies. Near Planck energies, this might be different, but nothing has been predicted yet. Challenge 1087, page 611: Ideal absorption is blackness (though it can be redness or whiteness at higher temperatures). Challenge 1088, page 611: Indeed, the Sun emits about 4 ë 1026 W from its mass of 2 ë 1030 kg, about 0.2 mW~kg. The adult human body (at rest) emits about 100 W (you can check this in bed at night), thus about 1.2 W~kg per ton. This is about 6000 times more than the Sun. Challenge 1089, page 612: The average temperature of the Earth is thus 287 K. The energy from the Sun is proportional to the fourth power of the temperature. The energy is spread (roughly) over half the Earth’s surface. The same energy, at the Sun’s surface, comes from a much smaller 2 4 surface, given by the same angle as the Earth subtends there. We thus have E  2πR Earth TEarth = 4 2 2 TSun R Earth α , where α is half the angle subtended by the Sun. As a result, the temperature of the 4 Sun is estimated to be TSun = (TEarth ~α 2 )0.25 = 4 kK. Challenge 1096, page 613: At high temperature, all bodies approach black bodies. The colour is more important than other colour effects. The oven and the objects have the same temperature. Thus they cannot be distinguished from each other. To do so nevertheless, illuminate the scene with powerful light and then take a picture with small sensitivity. Thus one always needs bright light to take pictures of what happens inside fires. Challenge 1101, page 633: The issue is: is the ‘universe’ a concept? More about this issue in the third part of the text. Challenge 1103, page 636: When thinking, physical energy, momentum and angular momentum are conserved, and thermodynamic entropy is not destroyed. Any experiment that this would not be so would point to unknown processes. However, there is no evidence for this. Challenge 1104, page 636: The best method cannot be much shorter than what is needed to describe 1 in 6000 million, or 33 bits. The Dutch and UK post code systems (including the letters NL or UK) are not far from this value and thus can claim to be very efficient. Challenge 1105, page 636: For complex systems, when the unknowns are numerous, the advance is thus simply given by the increase in answers. For the universe as a whole, the number of open issues is quite low, as shown on page 959; here there has not been much advance in the last years. But the advance is clearly measurable in this case as well. Challenge 1106, page 637: Is it possible to use the term ‘complete’ when describing nature? Challenge 1109, page 638: There are many baths in series: thermal baths in each light-sensitive cell of the eyes, thermal baths inside the nerves towards the brain and thermal baths inside brain cells.

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Challenge 1111, page 638: Yes. Challenge 1114, page 644: Physicists claim that the properties of objects, of space-time and of

— — — — —

P1 cuts the cake into N pieces. P2 to PN choose a piece. P1 keeps the last part. P2...PN assemble their parts back into one. Then P2...PN repeat the algorithm for one person less.

The problem is much more complex if the reassembly is not allowed. A just method (in finite many steps) for 3 people, using nine steps, was published in 1944 by Steinhaus, and a fully satisfactory method in the 1960s by John Conway. A fully satisfactory method for four persons was found only in 1995; it has 20 steps. Challenge 1120, page 648: (x, y) = ˜x, ˜x, y. Challenge 1121, page 649: Hint: show that any countable list of reals misses at least one number. This was proven for the first time by Cantor. His way was to write the list in decimal expansion and then find a number that is surely not in the list. Second hint: his world-famous trick is called the diagonal argument. Challenge 1122, page 649: Hint: all reals are limits of series of rationals. Challenge 1124, page 651: Yes, provided division by zero is not allowed. Challenge 1125, page 651: There are infinitely many of them. But the smallest is already quite

, 1 = ˜g , 2 = ˜˜g etc. Challenge 1127, page 655: Subtraction is easy. Addition is not commutative only for cases when infinite numbers are involved: ω + 2 x 2 + ω. Challenge 1128, page 655: Examples are 1 − ε or 1 − 4ε 2 − 3ε 3 . Challenge 1129, page 656: The answer is 57; the cited reference gives the details. Challenge 1126, page 651: 0 = g

22

44

Challenge 1130, page 657: 22 and 44 . Challenge 1132, page 658: This is not an easy question. The first nontrivial numbers are 7, 23, 47, 59, 167 and 179. See Robert Matthews, Maximally periodic reciprocals, Bulletin of the Institute of Mathematics and its Applications 28, pp. 147–148, 1992. Matthews shows that a number n for which 1~n generates the maximum of n − 1 decimal digits in the decimal expansion is a

Copyright © Christoph Schiller November 1997–May 2006

large.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

interactions form the smallest list possible. However, this list is longer than the one found by linguists! The reason is that physicists have found primitives that do not appear in everyday life. In a sense, the aim of physicists is limited by list of unexplained questions of nature, given on page 959. Challenge 1115, page 645: Neither has a defined content, clearly stated limits or a domain of application. Challenge 1116, page 645: Impossible! That would not be a concept, as it has no content. The solution to the issue must be and will be different. Challenge 1117, page 647: To neither. This paradox shows that such a ‘set of all sets’ does not exist. Challenge 1118, page 647: The most famous is the class of all sets that do not contain themselves. This is not a set, but a class. Challenge 1119, page 648: Dividing cakes is difficult. A simple method that solves many – but not all – problems among N persons P1...PN is the following:

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special sort of prime number that can be deduced from the so-called Sophie Germain primes S; one must have n = 2S + 1, where both S and 2S + 1 must be prime and where S mod 20 must be 3, 9, or 11. Thus the first numbers n are 7, 23, 47, 59, 167 and 179, corresponding to values for S of 3, 11, 23, 29, 83 and 89. In 1992, the largest known S that meets the criteria was S = (39051 ë 26002 ) − 1 ,

(885)

a 1812-digit long Sophie Germain prime number that is 3 mod 20. It was discovered by Wilfred Keller. This Sophie Germain prime leads to a prime n with a decimal expansion that is around 101812 digits long before it starts repeating itself. Read your favourite book on number theory to find out more. Interestingly, the solution to this challenge is also connected to that of challenge 1125. Can you find out more?

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Challenge 1133, page 658: Klein did not belong to either group. As a result, some of his nastier students concluded that he was not a mathematician at all.

is thus contradictory and has to be rejected. Challenge 1135, page 658: See the http://members.shaw.ca/hdhcubes/cube_basics.htm web

page for more information on magic cubes.

Challenge 1136, page 658: Such an expression is derived with the intermediate result (1−22 )−1 . The handling of divergent series seems absurd, but mathematicians know how to give the expression a defined content. (See Godfrey H. Hardy, Divergent Series, Oxford University Press, 1949.) Physicists often use similar expressions without thinking about them, in quantum field theory. Challenge 1137, page 668: ‘All Cretans lie’ is false, since the opposite, namely ‘some Cretans say the truth’ is true in the case given. The trap is that the opposite of the original sentence is usually, but falsely, assumed to be ‘all Cretans say the truth’. Challenge 1138, page 668: The statement cannot be false, due to the first half and the ‘or’ con-

struction. Since it is true, the second half must be true and you are an angel. Challenge 1147, page 669: The light bulb story seems to be correct. The bulb is very weak, so that the wire is not evaporating. Challenge 1149, page 674: Only induction allows to make use of similarities and thus to define

concepts. Page 1046

Challenge 1151, page 676: Yes, as we shall find out. Challenge 1153, page 677: Lack of internal contradictions means that a concept is valid as a

thinking tool; as we use our thoughts to describe nature, mathematical existence is a specialized version of physical existence, as thinking is itself a natural process. Indeed, mathematical concepts are also useful for the description of the working of computers and the like. Another way to make the point is to stress that all mathematical concepts are built from sets and relations, or some suitable generalizations of them. These basic building blocks are taken from our physical environment. Sometimes the idea is expressed differently; many mathematicians have acknowledged that certain mathematical concepts, such as natural numbers, are taken directly from experience. Challenge 1154, page 677: Examples are Achilles, Odysseus, Mickey Mouse, the gods of polytheism and spirits.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1152, page 677: Yes, as observation implies interaction.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Challenge 1134, page 658: A barber cannot belong to either group; the definition of the barber

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r

(886)

This is about 160 nm; indeed, this guessed equation is simply π times the Bohr radius. Challenge 1169, page 707: Due to the quantum of action, atoms in all people, be they giants or dwarfs, have the same size. That giants do not exist was shown already by Galilei. The argument is based on the given strength of materials, which thus implies that atoms are the same everywhere. That dwarfs cannot exist is due to the same reason; nature is not able to make people smaller than usual (except in the womb) as this would require smaller atoms. Challenge 1181, page 713: Also photons are indistinguishable. See page 732. Challenge 1184, page 716: The total angular momentum counts, including the orbital angular momentum. The orbital angular momentum L is given, using the radius and the linear momentum, L = r  p. Challenge 1185, page 716: Yes, we could have! Challenge 1207, page 735: The quantum of action implies that two subsequent observations always differ. Thus the surface of a liquid cannot be at rest.

Copyright © Christoph Schiller November 1997–May 2006

Page 221

ħ 2 4πε 0 me 2

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Challenge 1156, page 679: Torricelli made vacuum in a U-shaped glass tube, using mercury, the same liquid metal used in thermometers. Can you imagine how? A more difficult question: where did he get mercury from? Challenge 1157, page 680: Stating that something is infinite can be allowed, if the statement is falsifiable. An example is the statement ‘There are infinitely many mosquitoes.’ Other statements are not falsifiable, such as ‘The universe continue without limit behind the horizon.’ Such a statement is a belief, not a fact. Challenge 1158, page 682: They are not sets either and thus not collections of points. Challenge 1159, page 682: There is still no possibility to interact with all matter and energy, as this includes oneself. Challenge 1160, page 688: No. There is only a generalization encompassing the two. Challenge 1161, page 689: An explanation of the universe is not possible, as the term explanation require the possibility to talk about systems outside the one under consideration. The universe is not part of a larger set. Challenge 1162, page 689: Both can in fact be seen as two sides of the same argument: there is no other choice;there is only one possibility. The rest of nature shows that it has to be that way, as everything depends on everything. Challenge 1163, page 704: Classical physics fails in explaining any material property, such as colour or softness. Material properties result from nature’s interactions; they are inevitably quantum. Explanations always require particles and their quantum properties. Challenge 1164, page 705: Classical physics allows any observable to change smoothly with time. There is no minimum value for any observable physical quantity. » Challenge 1166, page 706: The simplest length is 2Għ~c 3 . The factor 2 is obviously not fixed; it is explained later on. Including it, this length is the smallest length measurable in nature. Challenge 1167, page 706: The electron charge is special to the electromagnetic interactions; it does not take into account the nuclear interactions. It is also unclear why the length should be of importance for neutral systems or for the vacuum. On the other hand, it turns out that the differences are not too fundamental, as the electron charge is related to he quantum of action by º e = 4πε 0 αcħ . Challenge 1168, page 707: On purely dimensional grounds, the radius of an atom must be

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Challenge 1218, page 751: Use ∆E < E and a ∆t < c.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006

Challenge 1222, page 751: The difficulties to see hydrogen atoms are due to their small size and their small number of electrons. As a result, hydrogen atoms produce only weak contrasts in X-ray images. For the same reasons it is difficult to image them using electrons; the Bohr radius of hydrogen is only slightly larger than the electron Compton wavelength. Challenge 1226, page 752: r = 86 pm, thus T = 12 eV. That compares to the actual value of 13.6 eV. The trick for the derivation of the formula is to use < ψSr 2x Sψ A= 31 < ψSrrSψ A, a relation valid for states with no orbital angular momentum. It is valid for all coordinates and also for the three momentum observables, as long as the system is non-relativistic. Challenge 1227, page 752: The fields are crated by neutrons or protons, which have a smaller Compton wavelength. Challenge 1241, page 763: A change of physical units such that ħ = c = e = 1 would change the value of ε 0 in such a way that 4πε o = 1~α = 137.036... Challenge 1242, page 771: Point particles cannot be marked; nearby point particles cannot be distinguished, due to the quantum of action. Challenge 1248, page 774: For a large number of particles, the interaction energy will introduce errors. For very large numbers, the gravitational binding energy will do so as well. Challenge 1250, page 775: Two write two particles on paper, one has to distinguish them, even if the distinction is arbitrary. Challenge 1254, page 779: Twins differ in the way their intestines are folded, in the lines of their hands and other skin folds. Sometimes, but not always, features like black points on the skin are mirror inverted on the two twins. Challenge 1260, page 785: Three. Challenge 1261, page 785: Angels can be distinguished by name, can talk and can sing; thus they are made of a large number of fermions. In fact, many angels are human sized, so that they do not even fit on the tip of a pin. Challenge 1269, page 789: Ghosts, like angels, can be distinguished by name, can talk and can be seen; thus they contain fermions. However, they can pass through walls and they are transparent; thus they cannot be made of fermions, but must be images, made of bosons. That is a contradiction. Challenge 1271, page 796: The loss of non-diagonal elements leads to an increase in the diagonal elements, and thus of entropy. Challenge 1274, page 801: The energy speed is given by the advancement of the outer two tails; that speed is never larger than the speed of light. Challenge 1275, page 804: A photograph requires illumination; illumination is a macroscopic electromagnetic field; a macroscopic field is a bath; a bath implies decoherence; decoherence destroys superpositions. Challenge 1278, page 805: Such a computer requires clear phase relations between components; such phase relations are extremely sensitive to outside disturbances. At present, they do not hold longer than a microsecond, whereas long computer programs require minutes and hours to run. Challenge 1282, page 812: Any other bath also does the trick, such as the atmosphere, sound vibrations, electromagnetic fields, etc. Challenge 1283, page 812: The Moon is in contact with baths like the solar wind, falling meteorites, the electromagnetic background radiation of the deep universe, the neutrino flux from the Sun, cosmic radiation, etc.

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Challenge 1284, page 813: Spatially periodic potentials have the property. Decoherence then

leads to momentum diagonalisation. Challenge 1285, page 816: A virus is an example. It has no own metabolism. (By the way, the

ability of some viruses to form crystals is not a proof that they are not living beings, in contrast to what is often said.) Challenge 1286, page 817: The navigation systems used by flies are an example. Challenge 1287, page 820: The thermal energy kT is about 4 zJ and a typical relaxation time is

0.1 ps.

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Challenge 1288, page 822: This is not possible at present. If you know a way, publish it. It would

help a sad single mother who has to live without financial help from the father, despite a lawsuit, as it was yet impossible to decide which of the two candidates is the right one. Challenge 1289, page 822: Also identical twins count as different persons and have different

Challenge 1290, page 822: Life’s chemicals are synthesized inside the body; the asymmetry has been inherited along the generations. The common asymmetry thus shows that all life has a common origin. Challenge 1291, page 823: Well, men are more similar to chimpanzees than to women. More seriously, the above data, even though often quoted, are wrong. Newer measurements by Roy Britten in 2002 have shown that the difference in genome between humans and chimpanzees is about 5 % (See R.J. Britten, Divergence between samples of chimpanzee and human DNA sequences is 5 %, counting indels, Proceedings of the National Academy of Sciences 99, pp. 13633–13635, 15th of October, 2002.) In addition, though the difference between man and woman is smaller than one whole chromosome, the large size of the X chromosome, compared with the small size of the Y chromosome, implies that men have about 3 % less genetic material than women. However, all men have an X chromosome as well. That explains that still other measurements suggest that all humans share a pool of at least 99.9 % of common genes. Challenge 1310, page 836: All detectors of light can be called relativistic, as light moves with

maximal speed. Touch sensors are not relativistic following the usual sense of the word, as the speeds involved are too small. The energies are small compared to the rest energies; this is the case even if the signal energies are attributed to electrons only.

Challenge 1293, page 824: Since all the atoms we are made of originate from outer space, the answer is yes. But if one means that biological cells came to Earth from space, the answer is no, as cells do not like vacuum. The same is true for DNA. Challenge 1292, page 824: The first steps are not known yet. Challenge 1295, page 824: Chemical processes, including diffusion and reaction rates, are

strongly temperature dependent. They affect the speed of motion of the individual and thus its chance of survival. Keeping temperature in the correct range is thus important for evolved life forms. Challenge 1296, page 826: Haven’t you tried yet? Physics is an experimental science.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1311, page 836: The noise is due to the photoacoustic effect; the periodic light periodically heats the air in the jam glass at the blackened surface and thus produces sound. See M. Euler, Kann man Licht hören?, Physik in unserer Zeit 32, pp. 180–182, 2001.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

fates. Imprinting in the womb is different, so that their temperament will be different. The birth experience will be different; this is the most intense experience of every human, strongly determining his fears and thus his character. A person with an old father is also quite different from that with a young father. If the womb is not that of his biological mother, a further distinction of the earliest and most intensive experiences is given.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006

Challenge 1298, page 828: Radioactive dating methods can be said to be based on the nuclear interactions, even though the detection is again electromagnetic. Challenge 1315, page 842: With a combination of the methods of Table 62 it is possible; but whether there will ever be an organization willing to pay for this to happen is another question. Challenge 1317, page 844: For example, a heavy mountain will push down the Earth’s crust into the mantle, makes it melt on the bottom side, and thus lowers the position of the top. Challenge 1318, page 844: These developments are just starting; the results are still far from the original one is trying to copy, as they have to fulfil a second condition, in addition to being a ‘copy’ of original feathers or of latex: the copy has to be cheaper than the original. That is often a much tougher request than the first. Challenge 1320, page 845: Since the height of the potential is always finite, walls can always be overcome by tunnelling. Challenge 1321, page 845: The lid of a box can never be at rest, as is required for a tight closure, but is always in motion, due to the quantum of action. Challenge 1326, page 850: The one somebody else has thrown away. Energy costs about 10 cents~kWh. For new lamps, the fluorescence lamp is the best for the environment, even though it is the least friendly to the eye and the brain, due to its flickering. Challenge 1327, page 853: This old dream depends on the precise conditions. How flexible does the display have to be? What lifetime should it have? The newspaper like display is many years away and maybe not even possible. Challenge 1328, page 853: The challenge here is to find a cheap way to deflect laser beams in a controlled way. Cheap lasers are already available. Challenge 1329, page 853: There is only speculation on the answer; the tendency of most researchers is to say no. Challenge 1330, page 853: No, as it is impossible because of momentum conservation, because of the no-cloning theorem. Challenge 1332, page 854: The author predicts that mass-produced goods using this technology (at least 1 million pieces sold) will not be available before 2025. Challenge 1333, page 854: Maybe, but for extremely high prices. Challenge 1334, page 856: For example, you could change gravity between two mirrors. Challenge 1335, page 856: As usual in such statements, either group or phase velocity is cited, but not the corresponding energy velocity, which is always below c. Challenge 1336, page 858: Echoes do not work once the speed of sound is reached and do not work well when it is approached. Both the speed of light and that of sound have a finite value. Moving with a mirror still gives a mirror image. This means that the speed of light cannot be reached. If it cannot be reached, it must be the same for all observers. Challenge 1337, page 858: Mirrors do not usually work for matter; in addition, if they did, matter would require much higher acceleration values. Challenge 1340, page 860: The overhang can have any value whatsoever. There is no limit. Taking the indeterminacy principle into account introduces a limit as the last brick or card must not allow the centre of gravity, through its indeterminacy, to be over the edge of the table. Challenge 1341, page 861: A larger charge would lead to field that spontaneously generate electron positron pairs, the electron would fall into the nucleus and reduce its charge by one unit. Challenge 1344, page 862: The Hall effect results from the deviation of electrons in a metal due to an applied magnetic field. Therefore it depends on their speed. One gets values around 1 mm. Inside atoms, one can use Bohr’s atomic model as approximation.

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Challenge 1345, page 862: The usual way to pack oranges on a table is the densest way to pack

spheres. Challenge 1346, page 863: Just use a paper drawing. Draw a polygon and draw it again at latter times, taking into account how the sides grow over time. You will see by yourself how the faster growing sides disappear over time. Challenge 1347, page 864: The steps are due to the particle nature of electricity and all other

moving entities. Challenge 1348, page 864: Mud is a suspension of sand; sand is not transparent, even if made

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of clear quartz, because of the scattering of light at the irregular surface of its grains. A suspension cannot be transparent if the index of refraction of the liquid and the suspended particles is different. It is never transparent if the particles, as in most sand types, are themselves not transparent. Challenge 1349, page 864: No. Bound states of massless particles are always unstable. Challenge 1350, page 864: The first answer is probably no, as composed systems cannot be

Challenge 1351, page 865: Methods to move on perfect ice from mechanics:

— if the ice is perfectly flat, rest is possible only in one point – otherwise you oscillate around that point, as shown in challenge 21; — do nothing, just wait that the higher centrifugal acceleration at body height pulls you away; — to rotate yourself, just rotate your arm above your head; — throw a shoe or any other object away; — breathe in vertically, breathing out (or talking) horizontally (or vice versa); — wait to be moved by the centrifugal acceleration due to the rotation of the Earth (and its oblateness); — jump vertically repeatedly: the Coriolis acceleration will lead to horizontal motion; — wait to be moved by the Sun or the Moon, like the tides are; — ‘swim’ in the air using hands and feet; — wait to be hit by a bird, a flying wasp, inclined rain, wind, lava, earthquake, plate tectonics, or any other macroscopic object (all objects pushing count only as one solution); — wait to be moved by the change in gravity due to convection in Earth’s mantle; — wait to be moved by the gravitation of some comet passing by; — counts only for kids: spit, sneeze, cough, fart, pee; or move your ears and use them as wings. Challenge 1352, page 865: Methods to move on perfect ice using thermodynamics and electro-

dynamics: — — — — — —

use the radio/TV stations nearby to push you around; use your portable phone and a mirror; switch on a pocket lam, letting the light push you; wait to be pushed around by Brownian motion in air; heat up one side of your body: black body radiation will push you; heat up one side of your body, e.g. by muscle work: the changing airflow or the evaporation will push you; — wait for one part of the body to be cooler than the other and for the corresponding black body radiation effects;

Copyright © Christoph Schiller November 1997–May 2006

Note that gluing your tongue is not possible on perfect ice.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

smaller than their own compton wavelength; only elementary systems can. However, the universe is not a system, as it has no environment. As such, its length is not a precisely defined concept, as an environment is needed to measure and to define it. (In addition, gravity must be taken into account in those domains.) Thus the answer is: in those domains, the question makes no sense.

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— wait for the magnetic field of the Earth to pull on some ferromagnetic or paramagnetic metal piece in your clothing or in your body; — wait to be pushed by the light pressure, i.e. by the photons, from the Sun or from the stars, maybe using a pocket mirror to increase the efficiency; — rub some polymer object to charge it electrically and then move it in circles, thus creating a magnetic field that interacts with the one of the Earth. Note that perfect frictionless surfaces do not melt. Challenge 1353, page 865: Methods to move on perfect ice using quantum effects:

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— wait for your wave function to spread out and collapse at the end of the ice surface; — wait for the pieces of metal in the clothing to attract to the metal in the surrounding through the Casimir effect; — wait to be pushed around by radioactive decays in your body. Challenge 1354, page 865: Methods to move on perfect ice using general relativity:

move an arm to emit gravitational radiation; deviate the cosmic background radiation with a pocket mirror; wait to be pushed by gravitational radiation from star collapses; wait to the universe to contract.

Challenge 1355, page 865: Methods to move on perfect ice using materials science, geophysics,

astrophysics: — — — — — —

be pushed by the radio waves emitted by thunderstorms and absorbed in painful human joints; wait to be pushed around by cosmic rays; wait to be pushed around by the solar wind; wait to be pushed around by solar neutrinos; wait to be pushed by the transformation of the Sun into a red giant; wait to be hit by a meteorite.

Challenge 1356, page 865: A method to move on perfect ice using selforganisation, chaos theory, and biophysics:

— wait that the currents in the brain interact with the magnetic field of the Earth by controlling your thoughts. Challenge 1357, page 865: Methods to move on perfect ice using quantum gravity, supersym-

metry, and string theory:

Challenge 1361, page 870: This is easy only if the black hole size is inserted into the entropy

bound by Bekenstein. A simple deduction of the black hole entropy that includes the factor 1/4 is not yet at hand. Challenge 1362, page 871: An entropy limit implies an information limit; only a given information can be present in a given region of nature. This results in a memory limit. Challenge 1363, page 871: In natural units, the expression for entropy is S = A~4 = 0.25A. If

each Planck area carried one bit (degree of freedom), the entropy would be S = ln W = ln(2 A ) = A ln 2 = 0.693A. This quite near the exact value.

Copyright © Christoph Schiller November 1997–May 2006

— accelerate your pocket mirror with your hand; — deviate the Unruh radiation of the Earth with a pocket mirror; — wait for proton decay to push you through the recoil.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

— — — —

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Challenge 1367, page 875: The universe has about 1022 stars; the Sun has a luminosity of about

1026 W; the total luminosity of the visible matter in the universe is thus about 1048 W. A gamma ray burster emits up to 3 ë 1047 W. Challenge 1373, page 877: They are carried away by the gravitational radiation. Challenge 1385, page 900: Two stacked foils show the same effect as one foil of the same total thickness. Thus the surface plays no role. Challenge 1387, page 902: The electron is held back by the positive charge of the nucleus, if the

number of protons in the nucleus is sufficient, as is the case for those nuclei we are made of.

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Challenge 1389, page 909: The number is small compared with the number of cells. However,

it is possible that the decays are related to human ageing. Challenge 1392, page 911: The radioactivity necessary to keep the Earth warm is low; lava is

only slightly more radioactive than usual soil. Challenge 1394, page 919: By counting decays and counting atoms to sufficient precision.

strongly repels the protons. Challenge 1396, page 927: Touching something requires getting near it; getting near means a small time and position indeterminacy; this implies a small wavelength of the probe that is used for touching; this implies a large energy. Challenge 1397, page 935: Building a nuclear weapon is not difficult. University students can

do it, and even have done so once, in the 1980s. The problem is getting or making the nuclear material. That requires either an extensive criminal activity or an vast technical effort, with numerous large factories, extensive development, coordination of many technological activities. Most importantly, such a project requires a large financial investment, which poor countries cannot afford. The problems are thus not technical, but financial. Challenge 1400, page 952: Most macroscopic matter properties fall in this class, such as the change of water density with temperature. Challenge 1403, page 963: Before the speculation can be fully tested, the relation between particles and black holes has to be clarified first. Challenge 1404, page 964: Never expect a correct solution for personal choices. Do what you yourself think and feel is correct. Challenge 1407, page 970: A mass of 100 kg and a speed of 8 m~s require 43 m2 of wing surface. Challenge 1410, page 980: The infinite sum is not defined for numbers; however, it is defined for a knotted string.

iton at a time. This is another point speaking against the existence of gravitons. Challenge 1419, page 989: Lattices are not isotropic, lattices are not Lorentz invariant. Challenge 1421, page 991: Large raindrops are pancakes with a massive border bulge. When

the size increases, e.g. when a large drop falls through vapour, the drop splits, as the central membrane is then torn apart. Challenge 1422, page 991: It is a drawing; if it is interpreted as an image of a three-dimensional

object, it either does not exist, or is not closed, or is an optical illusion of a torus. Challenge 1423, page 991: See T. Fink & Y. Mao, The 85 Ways to Tie a Tie, Broadway Books,

2000.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1413, page 982: This is a simple but hard question. Find out. Challenge 1383, page 882: No system is known in nature which emits or absorbs only one grav-

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Challenge 1395, page 922: The nuclei of nitrogen and carbon have a high electric charge which

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Challenge 1424, page 991: See T. Cl arke, Laces high, Nature Science Update 5th of Decem-

ber, 2002, or http://www.nature.com/nsu/021202/021202-4.html. Challenge 1426, page 1001: The other scale is the horizon of the universe, as we will see shortly. Challenge 1427, page 1003: Sloppily speaking, such a clock is not able to move its hands in

such a way to guarantee precise time reading. Challenge 1430, page 1016: The final energy E produced by a proton accelerator increases with

Challenge 1432, page 1017: Not really, as the mass error is equal to the mass only in the Planck

case. Challenge 1433, page 1017: It is improbable that such deviations can be found, as they are

masked by the appearance of quantum gravity effects. However, if you do think that you have a prediction for a deviation, publish it. Challenge 1435, page 1017: There is no gravitation at those energies and there are no particles. There is thus no paradox. ¼ Challenge 1437, page 1018: The Planck acceleration is given by a Pl = c 7 ~ħG = 5.6 ë 1051 m~s2 . Challenge 1438, page 1019: All mentioned options could be valid at the same time. The issue is not closed and clear thinking about it is not easy.

Challenge 1439, page 1019: The energy is the unification energy, about 800 times smaller than

the Planck energy. Challenge 1440, page 1020: This is told in detail in the section on maximum force starting on

page 1068. Challenge 1441, page 1025: Good! Publish it. Challenge 1443, page 1033: The cosmic background radiation is a clock in the widest sense of

the term. Challenge 1465, page 1047: For the description of nature this is a contradiction. Nevertheless, the term ‘universe’, ‘set of all sets’ and other mathematical terms, as well as many religious concepts are of this type. Challenge 1468, page 1048: For concept of ‘universe’. Challenge 1472, page 1050: Augustine and many theologians have defined ‘god’ in exactly this way. (See also Thomas Aquinas, Summa contra gentiles, 1, 30.) They claim that it possible to say what ‘god’ is not, but that it is not possible to say what it is. (This statement is also part of the official roman catholic catechism: see part one, section one, chapter one, IV, 43.)

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1442, page 1032: See the table on page 184.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

its radius R roughly as E  R 1.2 ; as an example, CERN’s SPS achieves about 450 GeV for a radius of 740 m. Thus we would get a radius of more than 100 000 light years (larger than our galaxy) for a Planck energy accelerator. An accelerator achieving Planck energy is impossible. A unification energy accelerator would be about 1000 times smaller. Nature has no accelerator of this power, but gets near it. The maximum measured value of cosmic rays, 1022 eV, is about one thousandth of the unification energy. The mechanisms of acceleration are obscure. Black holes are no sources for unification energy particles, due to their gravitational potential. But also the cosmic horizon is not the source, for some yet unclear reasons. This issue is still a topic of research. » Challenge 1431, page 1016: The Planck energy is E Pl = ħc 5 ~G = 2.0 GJ. Car fuel delivers about 43 MJ~kg. Thus the Planck energy corresponds to the energy of 47 kg of car fuel, about a tankful.

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Many legal scholars would also propose a different concept that fits the definition – namely ‘administration’. It is difficult to say what it is, but easy to say what it is not. More seriously, the properties common to the universe and to ‘god’ suggest the conclusion that the both are the same. Indeed, the analogy between the two concepts can be expanded to a proof. (This is left to the reader.) In fact, this might be the most interesting of all proofs of the existence of gods. This proof certainly lacks all the problems that the more common ‘proofs’ have. Despite its interest, the present proof is not found in any book on the topic. The reason is obvious: the result of the proof, the equivalence of ‘god’ and the universe, is a heresy for most religions. If one is ready to explore the analogy nevertheless, one finds that a statement like ‘god created the universe’ translates as ‘the universe implies the universe´. The original statement is thus not a lie any more, but is promoted to a tautology. Similar changes appear for many other – but not all – statements using the term ‘god’. Enjoy the exploration.

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Challenge 1473, page 1050: If you find one, publish it! And send it to the author as well. Challenge 1475, page 1051: If you find one, publish it and send it to the present author as well.

Challenge 1480, page 1058: Just measure the maximum water surface the oil drop can cover,

by looking at the surface under a small angle. Challenge 1481, page 1058: Keep the fingers less than 1 cm from your eye. Challenge 1482, page 1065: As vacuum and matter cannot be distinguished, both share the same properties. In particular, both scatter strongly at high energies. Challenge 1483, page 1069: Take ∆ f ∆t E 1 and substitute ∆l = c~∆ f and ∆a = c~∆t.

Challenge 1505, page 1111: The number of spatial dimensions must be given first, in order to talk about spheres.

Challenge 1506, page 1114: This is a challenge to you to find out and publish; it is fun, may bring success and would yield an independent check of the results of the section. Challenge 1511, page 1125: The lid of a box must obey the indeterminacy relation. It cannot

be at perfect rest with respect to the rest of the box. Challenge 1512, page 1125: Of course not, as there are no infinite quantities in nature. The question is whether the detector would be as large as the universe or smaller. What is the answer? Challenge 1517, page 1126: Yes, as nature’s inherent measurement errors cannot clearly distin-

guish between them. Challenge 1518, page 1126: Of course.

Challenge 1519, page 1126: We still have the chance to find the best approximate concepts possible. There is no reason to give up. Challenge 1520, page 1126: A few thoughts. beginning of the big bang does not exist, but is given by that piece of continuous entity which is encountered when going backwards in time as much as possible. This has several implications.

— Going backwards in time as far as possible – towards the ‘beginning’ of time – is the same as zooming to smallest distances: we find a single strand of the amoeba. — In other words, we speculate that the whole world is one single piece, knotted, branched and fluctuating.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1516, page 1126: No. Time is continuous only if either quantum theory and point particles or general relativity and point masses are assumed. The argument shows that only the combination of both theories with continuity is impossible.

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Challenge 1479, page 1057: Any change in rotation speed of the Earth would change the sea

level.

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— Going far away into space – to the border of the universe – is like taking a snapshot with a short shutter time: strands everywhere. — Whenever we sloppily say that extended entities are ‘infinite’ in size, we only mean that they reach the horizon of the universe. In summary, no starting point of the big bang exists, because time does not exist there. For the same reason, no initial conditions for particles or space-time exist. In addition, this shows there was no creation involved, since without time and without possibility of choice, the term ‘creation’ makes no sense.

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Challenge 1521, page 1126: The equivalence follows from the fact that all these processes re-

quire Planck energy, Planck measurement precision, Planck curvature, and Planck shutter time. Challenge 1500, page 1092: The system limits cannot be chosen in other ways; after the limits

have been corrected, the limits given here should still apply.

Challenge 1531, page 1162: Do not forget the relativistic time dilation. Challenge 1533, page 1163: Since the temperature of the triple point of water is fixed, the tem-

perature of the boiling point is fixed as well. Historically, the value of the triple point has not been well chosen. Challenge 1534, page 1163: Probably the quantity with the biggest variation is mass, where a

prefix for 1 eV~c2 would be useful, as would be one for the total mass in the universe, which is about 1090 times larger. Challenge 1535, page 1164: The formula with n − 1 is a better fit. Why? Challenge 1538, page 1167: No, only properties of parts of the universe. The universe itself has no properties, as shown on page 1050. Challenge 1539, page 1168: The slowdown goes quadratically with time, because every new

slowdown adds to the old one! Challenge 1540, page 1170: The double of that number, the number made of the sequence of all even numbers, etc. Challenge 1532, page 1162: About 10 µg. Challenge 1542, page 1173: This could be solved with a trick similar to those used in the irrationality of each of the two terms of the sum, but nobody has found one. Challenge 1543, page 1173: There are still many discoveries to be made in modern mathem-

Challenge 1544, page 1177: The gauge coupling constants determine the size of atoms, the strength of chemical bonds and thus the size of all things. Challenge 1545, page 1191: Covalent bonds tend to produce full shells; this is a smaller change

on the right side of the periodic table. a b ‘. −b a Challenge 1551, page 1197: Use Cantor’s diagonal argument, as in challenge 1121.

Challenge 1548, page 1196: SzS is the determinant of the matrix z = Œ

Challenge 1555, page 1199: Any rotation by an angle 2π is described by −1. Only a rotation by

4π is described by +1; quaternions indeed describe spinors.

Copyright © Christoph Schiller November 1997–May 2006

atics, especially in topology, number theory and algebraic geometry. Mathematics has a good future.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Challenge 1527, page 1158: Planck limits can be exceeded for extensive observables for which many particle systems can exceed single particle limits, such as mass, momentum, energy or electrical resistance.

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Challenge 1557, page 1201: Just check the result component by component. See also the men-

x>R

In simple terms: the maximum value taken by the absolute of the function is its norm. It is also called ‘sup’-norm. Since it contains a supremum, this norm is only defined on the subspace of bounded continuous functions on a space X, or, if X is compact, on the space of all continuous functions (because a continuous function on a compact space must be bounded). Challenge 1575, page 1213: Take out your head, then pull one side of your pullover over the corresponding arm, continue pulling it over the over arm; then pull the other side, under the first, to the other arm as well. Put your head back in. Your pullover (or your trousers) will be inside out. Challenge 1579, page 1217: The transformation from one manifold to another with different

topology can be done with a tiny change, at a so-called singular point. Since nature shows a minimum action, such a tiny change cannot be avoided. Challenge 1580, page 1219: The product M † M is Hermitean, and has positive eigenvalues. Thus H is uniquely defined and Hermitean. U is unitary because U † U is the unit matrix.

So far, out of 1580 challenges, 745 solutions are given; in addition, 241 solutions are too easy to be included. Another 594 solutions need to be written; let the author know which one you want most.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

tioned reference. Challenge 1559, page 1204: For a Gaussian integer n + im to be prime, the integer n 2 + m 2 must be prime, and in addition, a condition on n mod 3 must be satisfied; which one and why? Challenge 1563, page 1205: The metric is regular, positive definite and obeys the triangle inequality. Challenge 1567, page 1207: The solution is the set of all two by two matrices, as each two by two matrix specifies a linear transformation, if one defines a transformed point as the product of the point and this matrix. (Only multiplication with a fixed matrix can give a linear transformation.) Can you recognize from a matrix whether it is a rotation, a reflection, a dilation, a shear, or a stretch along two axes? What are the remaining possibilities? Challenge 1570, page 1208: The (simplest) product of two functions is taken by point-by-point multiplication. Challenge 1571, page 1208: The norm Y f Y of a real function f is defined as the supremum of its absolute value: Y f Y = sup S f (x)S . (887)

Copyright © Christoph Schiller November 1997–May 2006

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Appendix G

LI ST OF I LLUSTR ATIONS

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An example of motion observed in nature Motion Mountain and the trail to be followed Illusions of motion: look at the figure on the left and slightly move the page, or look at the white dot at the centre of the figure on the right and move your head back and forward How much water is required to make a bucket hang vertically? At what angle does the pulled reel change direction of motion? (© Luca Gastaldi) A time line of scientific and political personalities in antiquity (the last letter of the name is aligned with the year of death) An example of transport Transport, growth and transformation (© Philip Plisson) A block and tackle and a differential pulley Galileo Galilei A typical path followed by a stone thrown through the air Two proofs of the three-dimensionality of space: a knot and the inner ear of a mammal René Descartes A curvemeter or odometer A fractal: a self-similar curve of infinite length (far right), and its construction A polyhedron with one of its dihedral angles (© Luca Gastaldi) A photograph of the Earth – seen from the direction of the Sun A model illustrating the hollow Earth theory, showing how day and night appear (© Helmut Diehl) Leaving a parking space The definition of plane and solid angles How the apparent size of the Moon and the Sun changes How the apparent size of the Moon changes during its orbit (© Anthony Ayiomamitis) A vernier/nonius/clavius Anticrepuscular rays (© Peggy Peterson) Two ways to test that the time of free fall does not depend on horizontal velocity Various types of graphs describing the same path of a thrown stone Three superimposed images of a frass pellet shot away by a caterpillar (© Stanley Caveney)

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Derivatives Gottfried Leibniz Orion (in natural colours) and Betelgeuse How an object can rotate continuously without tangling up the connection to a second object Legs and ‘wheels’ in living beings In which direction does the bicycle turn? Collisions define mass The standard kilogram (© BIPM) Is this dangerous? Antoine Lavoisier Christiaan Huygens What happens? Robert Mayer Angular momentum and the two versions of the right-hand rule How a snake turns itself around its axis Can the ape reach the banana? The velocities and unit vectors for a rolling wheel A simulated photograph of a rolling wheel with spokes The measured motion of a walking human (© Ray McCoy) The parallaxis – not drawn to scale Earth’s deviation from spherical shape due to its rotation The deviations of free fall towards the east and towards the Equator due to the rotation of the Earth The turning motion of a pendulum showing the rotation of the Earth Showing the rotation of the Earth through the rotation of an axis Demonstrating the rotation of the Earth with water The gyroscope The precession and the nutation of the Earth’s axis The continental plates are the objects of tectonic motion Changes in the Earth’s motion around the Sun The motion of the Sun around the galaxy Is it safe to let the cork go? A simple model for continents and mountains A well-known toy An elastic collision that seems not to obey energy conservation The centre of mass defines stability How does the ladder fall? Observation of sonoluminescence and a diagram of the experimental set-up How does the ball move when the jar is accelerated? The famous Celtic stone and a version made with a spoon How does the ape move? A long exposure of the stars at night - over the Mauna Kea telescope in Hawaii (Gemini) A basilisk lizard (Basiliscus basiliscus) running on water, showing how the propulsing leg pushes into the water (© TERRA) A water strider (© Charles Lewallen) A physicist’s and an artist’s view of the fall of the Moon: a diagram by Christiaan Huygens (not to scale) and a marble statue by Auguste Rodin

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The measurements that lead to the definition of the metre (© Ken Alder) The potential and the gradient The shape of the Earth, with exaggerated height scale (© GeoForschungsZentrum Potsdam) The motion of a planet around the Sun, showing its semimajor axis d, which is also the spatial average of its distance from the Sun The change of the moon during the month, showing its libration (© Martin Elsässer) Maps (not photographs) of the near side (left) and far side (right) of the moon, showing how often the latter saved the Earth from meteorite impacts (courtesy USGS) The possible orbits due to universal gravity The two stable Lagrangian points Tidal deformations due to gravity The origin of tides A spectacular result of tides: volcanism on Io (NASA) Particles falling side by side approach over time Masses bend light Brooms fall more rapidly than stones (© Luca Gastaldi) The starting situation for a bungee jumper An honest balance? Which of the two Moon paths is correct? The analemma over Delphi, between January and December 2002 (© Anthony Ayiomamitis) The vanishing of gravitational force inside a spherical shell of matter A solar eclipse (11 August 1999, photographed from the Russian Mir station) Shapes and air/water resistance What shape of rail allows the black stone to glide most rapidly from point A to the lower point B? Can motion be described in a manner common to all observers? What happens when one rope is cut? How to draw a straight line with a compass: fix point F, put a pencil into joint P and move C with a compass along a circle A south-pointing carriage How and where does a falling brick chimney break? Why do hot-air balloons stay inflated? How can you measure the weight of a bicycle rider using only a ruler? What determines the number of petals in a daisy? Defining a total effect as an accumulation (addition, or integral) of small effects over time Joseph Lagrange The minimum of a curve has vanishing slope Aggregates in nature Refraction of light is due to travel-time optimization Forget-me-not, also called Myosotis (Boraginaceae) (© Markku Savela) A Hispano–Arabic ornament from the Governor’s Palace in Sevilla The simplest oscillation Decomposing a general wave or signal into harmonic waves The formation of gravity waves on water

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The six main properties of the motion of waves The electrical signals measured in a nerve A solitary water wave followed by a motor boat, reconstructing the discovery by Scott Russel (© Dugald Duncan) Solitons are stable against encounters Shadows and refraction Floors and mountains as fractals (© Paul Martz) Atoms exist: rotating an aluminium rod leads to brightness oscillations Atomic steps in broken gallium arsenide crystals can be seen under a light microscope The principle, and a simple realization, of an atomic force microscope The atoms on the surface of a silicon crystal mapped with an atomic force microscope The result of moving helium atoms on a metallic surface (© IBM) What is your personal stone-skipping record? Heron’s fountain Which funnel is faster? The basic idea of statistical mechanics about gases Which balloon wins? Daniel Bernoulli Example paths for particles in Brownian motion and their displacement distribution The fire pump Can you boil water in this paper cup? The invisible loudspeaker The Wirbelrohr or Ranque–Hilsch vortex tube Examples of self-organization for sand Oscillons formed by shaken bronze balls; horizontal size is about 2 cm (© Paul Umbanhowar) Magic numbers: 21 spheres, when swirled in a dish, behave differently from non-magic numbers, like 23, of spheres (redrawn from photographs, © Karsten Kötter) Examples of different types of motion in configuration space Sensitivity to initial conditions The wavy surface of icicles Water pearls A braiding water stream (© Vakhtang Putkaradze) Rømer’s method of measuring the speed of light The rain method of measuring the speed of light Fizeau’s set-up to measure the speed of light (© AG Didaktik und Geschichte der Physik, Universität Oldenburg) A photograph of a light pulse moving from right to left through a bottle with milky water, marked in millimetres (© Tom Mattick) A consequence of the finiteness of the speed of light Albert Einstein A drawing containing most of special relativity Moving clocks go slow The set-up for the observation of the Doppler effect Lucky Luke

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Experimental values (dots) for the electron velocity v as function of their kinetic energy T, compared with the prediction of Galilean physics (blue) and that of special relativity (red) How to deduce the composition of velocities The result, the schematics and the cryostat set-up for the most precise Michelson–Morley experiment performed to date (© Stephan Schiller) Two inertial observers and a beam of light Space-time diagrams for light seen from two different observers using coordinates (t, x) and (τ, ξ) A space-time diagram for a moving object T seen from an inertial observer O in the case of one and two spatial dimensions The twin paradox More muons than expected arrive at the ground because fast travel keeps them young The observations of the pilot and the barn owner The observations of the trap digger and of the snowboarder, as (misleadingly) published in the literature Does the conducting glider keep the lamp lit at large speeds? What happens to the rope? Flying through twelve vertical columns (shown in the two uppermost images) with 0.9 times the speed of light as visualized by Nicolai Mokros and Norbert Dragon, showing the effect of speed and position on distortions (© Nicolai Mokros) Flying through three straight and vertical columns with 0.9 times the speed of light as visualized by Daniel Weiskopf: on the left with the original colours; in the middle including the Doppler effect; and on the right including brightness effects, thus showing what an observer would actually see (© Daniel Weiskopf) What a researcher standing and one running rapidly through a corridor observe (ignoring colour effects) (© Daniel Weiskopf) For the athlete on the left, the judge moving in the opposite direction sees both feet off the ground at certain times, but not for the athlete on the right A simple example of motion that is faster than light Another example of faster-than-light motion Hypothetical space-time diagram for tachyon observation If O’s stick is parallel to R’s and R’s is parallel to G’s, then O’s stick and G’s stick are not An inelastic collision of two identical particles seen from two different inertial frames of reference A useful rule for playing non-relativistic snooker The dimensions of detectors in particle accelerators are based on the relativistic snooker angle rule Space-time diagram of a collision for two observers There is no way to define a relativistic centre of mass The space-time diagram of a moving object T Energy–momentum is tangent to the world line On the definition of relative velocity Observers on a rotating object The simplest situation for an inertial and an accelerated observer

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The hyperbolic motion of an rectilinearly, uniformly accelerating observer Ω The definitions necessary to deduce the composition behaviour of accelerations Hyperbolic motion and event horizons Clocks and the measurement of the speed of light as two-way velocity The mountain attempt to exceed the maximum mass flow value Inside an accelerating train or bus The necessity of blue- and red-shift of light: why trees are greener at the bottom Tidal effects: what bodies feel when falling The mattress model of space: the path of a light beam and of a satellite near a spherical mass All paths of flying stones have the same curvature in space-time A puzzle The Thirring and the Thirring–Lense effects The LAGEOS satellites: metal spheres with a diameter of 60 cm, a mass of 407 kg, and covered with 426 retroreflectors The reality of gravitomagnetism A Gedanken experiment showing the necessity of gravity waves Effects on a circular or spherical body due to a plane gravitational wave moving in a direction perpendicular to the page Detection of gravitational waves Comparison between measured time delay for the periastron of the binary pulsar PSR 1913+16 and the prediction due to energy loss by gravitational radiation Calculating the bending of light by a mass Time delay in radio signals – one of the experiments by Irwin Shapiro The orbit around a central body in general relativity The geodesic effect Positive, vanishing and negative curvature in two dimensions The maximum and minimum curvature of a curved surface Curvature (in two dimensions) and geodesic behaviour The Andromeda nebula M31, our neighbour galaxy (and the 31st member of the Messier object listing) (NASA) How our galaxy looks in the infrared (NASA) The elliptical galaxy NGC 205 (the 205th member of the New Galactic Catalogue) (NASA) The colliding galaxies M51 and M110 (NASA) The X-rays in the night sky, between 1 and 30 MeV (NASA) Rotating clouds emitting jets along their axis; top row: a composite image (visible and infrared) of the galaxy 0313-192, the galaxy 3C296, and the Vela pulsar; bottom row: the star in formation HH30, the star in formation DG Tauri B, and a black hole jet from the galaxy M87 (NASA) The universe is full of galaxies – this photograph shows the Perseus cluster (NASA) An atlas of our cosmic environment: illustrations at scales up to 12.5, 50, 250, 5 000, 50 000, 500 000, 5 million, 100 million, 1 000 million and 14 000 million light years (© Richard Powell, http://www.anzwers.org/free/universe) The relation between star distance and star velocity

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The Hertzsprung–Russell diagram (© Richard Powell) The ranges for the Ω parameters and their consequences The evolution of the universe’s scale a for different values of its mass density The long-term evolution of the universe’s scale factor a for various parameters The fluctuations of the cosmic background radiation (WMAP/NASA) The absorption of the atmosphere (NASA) How one star can lead to several images The Zwicky-Einstein ring B1938+666, seen in the radio spectrum (left) and in the optical domain (right) (NASA) Multiple blue images of a galaxy formed by the yellow cluster CL0024+1654 (NASA) The light cones in the equatorial plane around a non-rotating black hole, seen from above Motions of uncharged objects around a non-rotating black hole – for different impact parameters and initial velocities Motions of light passing near a non-rotating black hole The ergosphere of a rotating black hole Motion of some light rays from a dense body to an observer A ‘hole’ in space A model of the hollow Earth theory (© Helmut Diehl) Objects surrounded by fields: amber, lodestone and mobile phone How to amaze kids Lightning: a picture taken with a moving camera, showing its multiple strokes (© Steven Horsburgh) A simple Kelvin generator Franklin’s personal lightning rod Consequences of the flow of electricity The magentotactic bacterium Magnetobacterium bavaricum with its magnetosomes (© Marianne Hanzlik) An old and a newer version of an electric motor An electrical current always produces a magnetic field Current makes a metal rods rotate The two basic types of magnetic material behaviour (tested in an inhomogeneous field): diamagnetism and paramagnetism The relativistic aspect of magnetism The schematics, the realization and the operation of a Tesla coil, including spark and corona discharges (© Robert Billon) Lifting a light object – covered with aluminium foil – using high a tension discharge (© Jean-Louis Naudin at http://www.jlnlabs.org) The magnetic field due to the tides (© Stefan Maus) A unipolar motor The simplest motor (© Stefan Kluge) The correspondence of electronics and water flow The first of Maxwell’s equations The second of Maxwell’s equations Charged particles after a collision Vector potentials for selected situations Which one is the original landscape? (NOAA)

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A plane, monochromatic and linearly polarized electromagnetic wave, with the fields as described by the field equations of electrodynamics The first transmitter (left) and receiver (right) of electromagnetic (micro-) waves Heinrich Hertz The primary and secondary rainbow, and the supernumerary bows below the primary bow (© Antonio Martos and Wolfgang Hinz) The light power transmitted through a slit as function of its width Sugar water bends light The real image produced by a converging lens and the virtual image produced by a diverging lens Refraction as the basis of the telescope – shown here in the original Dutch design Watching this graphic at higher magnification shows the dispersion of the lens in the human eye: the letters float at different depths In certain materials, light beams can spiral around each other Masses bend light Reflection at air interfaces is the basis of the Fata Morgana The last mirror of the solar furnace at Odeillo, in the French Pyrenees (© Gerhard Weinrebe) Levitating a small glass bead with a laser A commercial light mill turns against the light Light can rotate objects Proving that white light is a mixture of colours Umbrellas decompose white light Milk and water simulate the evening sky (© Antonio Martos) The definition of important velocities in wave phenomena Positive and negative indices of refraction The path of light for the dew on grass that is responsible for the aureole A limitation of the eye What is the angle between adjacent horizontal lines? The Lingelbach lattice: do you see white, grey, or black dots? An example of an infrared photograph, slightly mixed with a colour image (© Serge Augustin) A high quality photograph of a live human retina, including a measured (false colour) indication of the sensitivity of each cone cell (© Austin Roorda) The recording and the observation of a hologram Sub-wavelength optical microscopy using stimulated emission depletion (© MPI für biophysikalische Chemie/Stefan Hell) Capacitors in series Small neon lamps on a high voltage cable How natural colours (top) change for three types of colour blind: deutan, protan and tritan (© Michael Douma) Cumulonimbus clouds from ground and from space (NASA) The structure of our planet Trapping a metal sphere using a variable speed drill and a plastic saddle Floating ‘magic’ nowadays available in toy shops Bodies inside a oven at room temperature (left) and red hot (right) Ludwig Wittgenstein

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Devices for the definition of sets (left) and of relations (right) The surreal numbers in conventional and in bit notation An example of a quantum system (© Ata Masafumi) An artistic impression of a water molecule Hills are never high enough Leaving enclosures Identical objects with crossing paths Transformation through reaction How do train windows manage to show two superimposed images? A particle and a screen with two nearby slits Illumination by pure-colour light Observation of photons Various types of light An interferometer How to measure photon statistics: with an electronic coincidence counter and the variation by varying the geometrical position of a detector The kinetic energy of electrons emitted in the photoelectric effect Light crossing light Interference and the description of light with arrows (at one particular instant of time) Light reflected by a mirror and the corresponding arrows (at one particular instant of time) The light reflected by a badly placed mirror and by a grating If light were made of little stones, they would move faster inside water Diffraction and photons A falling pencil Steps in the flow of electricity in metals Matter diffracts and interferes Trying to measure position and momentum On the quantization of angular momentum The Stern–Gerlach experiment Erwin Schrödinger Climbing a hill The spectrum of daylight: a stacked section of the rainbow (© Nigel Sharp (NOAO), FTS, NSO, KPNO, AURA, NSF) The energy levels of hydrogen Paul Dirac Identical objects with crossing paths Two photons and interference Particles as localized excitations An argument showing why rotations by 4π are equivalent to no rotation at all A belt visualizing two spin 1/2 particles Another model for two spin 1/2 particles The human arm as spin 1/2 model The extended belt trick, modelling a spin 1/2 particle: independently of the number of bands attached, the two situations can be transformed into each other, either by rotating the central object by 4π or by keeping the central object fixed and moving the bands around it Some visualizations of spin representations

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Equivalence of exchange and rotation in space-time Belts in space-time: rotation and antiparticles Artist’s impression of a macroscopic superposition Quantum mechanical motion: an electron wave function (actually its module squared) from the moment it passes a slit until it hits a screen Bohm’s Gedanken experiment A system showing probabilistic behaviour The concepts used in the description of measurements A quantum machine (© Elmar Bartel) Myosin and actin: the building bricks of a simple linear molecular motor Two types of Brownian motors: switching potential (left) and tilting potential (right) A modern version of the evolutionary tree The different speed of the eye’s colour sensors, the cones, lead to a strange effect when this picture (in colour version) is shaken right to left in weak light Atoms and dangling bonds An unusual form of the periodic table of the elements Several ways to picture DNA Nuclear magnetic resonance shows that vortices in superfluid 3 He-B are quantized. An electron hologram Ships in a swell What is the maximum possible value of h/l? Some snow flakes (© Furukawa Yoshinori) QED as perturbation theory in space-time The weakness of gravitation A Gedanken experiment allowing to deduce the existence of black hole radiation A selection of gamma ray bursters observed in the sky Sagittal images of the head and the spine – used with permission from Joseph P. Hornak, The Basics of MRI, http://www.cis.rit.edu/htbooks/mri, Copyright 2003 Henri Becquerel The origin of human life (© Willibrord Weijmar Schultz) Marie Curie All known nuclides with their lifetimes and main decay modes (data from http://www.nndc.bnl.gov/nudat2) An electroscope (or electrometer) (© Harald Chmela) and its charged (left) and uncharged state (right) Viktor Heß A Geiger–Müller counter An aurora borealis produced by charged particles in the night sky An aurora australis on Earth seen from space (in the X-ray domain) and one on Saturn The lava sea in the volcano Erta Ale in Ethiopia (© Marco Fulle) Various nuclear shapes – fixed (left) and oscillating (right), shown realistically as clouds (above) and simplified as geometric shapes (below)

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Photographs of the Sun at wavelengths of 30.4 nm (in the extreme ultraviolet, left) and around 677 nm (visible light, right, at a different date), by the SOHO mission (ESA and NASA) A simplified drawing of the Joint European Torus in operation at Culham, showing the large toroidal chamber and the magnets for the plasma confinement (© EFDA-JET) A selection of mesons and baryons and their classification as bound states of quarks The spectrum of the excited states of proton and neutron The essence of the QCD Lagrangian The behaviour of the three coupling constants with energy for the standard model (left) and for the minimal supersymmetric model (right) (© Dmitri Kazakov) A simplified history of the description of motion in physics, by giving the limits to motion included in each description A flying fruit fly, tethered to a string Vortices around a butterfly wing (© Robert Srygley/Adrian Thomas) Two large wing types The wings of a few types of insects smaller than 1 mm (thrips, Encarsia, Anagrus, Dicomorpha) (HortNET) A swimming scallop (here from the genus Chlamys) (© Dave Colwell) Ciliated and flagellate motion A well-known ability of cats A way to turn a sphere inside out, with intermediate steps ordered clockwise (© John Sullivan) The knot diagrams for the simplest prime knots (© Robert Scharein) Crossing types in knots The Reidemeister moves and the flype The diagrams for the simplest links with two and three components (© Robert Scharein) A hagfish tied into a knot How to simulate order for long ropes The mutually perpendicular tangent e, normal n, torsion w and velocity v of a vortex in a rotating fluid Motion of a vortex: the fundamental helical solution and a moving helical ‘wave packet’ The two pure dislocation types: edge and screw dislocations Swimming on a curved surface using two discs A large raindrop falling downwards Is this possible? ‘Tekenen’ by Maurits Escher, 1948 – a metaphor for the way in which ‘particles’ and ‘space-time’ are usually defined: each with the help of the other (© M.C. Escher Heirs) Andrei Sakharov A Gedanken experiment showing that at Planck scales, matter and vacuum cannot be distinguished Coupling constants and their spread as a function of energy Measurement errors as a function of measurement energy Trees and galaxies

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Picture credits

Copyright © Christoph Schiller November 1997–May 2006

The mountain photograph on the cover is courtesy and copyright by Dave Thompson (http:// www.daveontrek.co.uk). The basilisk running over water, on the back cover and on page 115, is courtesy and copyright by the Belgian group TERRA vzw and found on their website http://www. terra.vzw.org. The copyrights of the figures on page 28 and on page 34 cannot be traced. The figures on pages 31, 55 and 135 were made especially for this text and are copyright by Luca Gastaldi. The famous photograph of the Les Poulains and its lighthouse by Philip Plisson on page 35 is courtesy and copyright by Pechêurs d’Images; see the websites http://www.plisson.com and http:// www.pecheurs-d-images.com. It is also found in Plisson’s magnus opus La Mer, a stunning book of photographs of the sea. The hollow Earth figure on pages 58 and 496 is courtesy of Helmut Diel and was drawn by Isolde Diel. The Moon photographs on page 62 and the analemma on page 140 are courtesy and copyright by Anthony Ayiomamitis; the story of the photographs is told on his website at http://www.perseus.gr. The anticrepuscular photograph on page 64 is courtesy and copyright by Peggy Peterson. The firing caterpillar figure of page 66 is courtesy and copyright of Stanley Caveney. The photograph of the standard kilogram on page 76 is courtesy and copyright by the Bureau International des Poids et Mesures. The measured graph of the walking human on page 93 is courtesy and copyright of Ray McCoy. Figure 61 on page 113 is courtesy and copyright of the Australian Gemini project at http://www.ausgo.unsw.edu.au. The water strider photograph on page 115 is courtesy and copyright by Charles Lewallen. The figure on the triangulation of the

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385 1039 The speed and distance of remote galaxies 386 1064 A Gedanken experiment showing that at Planck scales, matter and vacuum cannot be distinguished 387 1105 Measurement requires matter and radiation 388 1122 A possible model for a spin 1/2 particle 389 1125 Planck effects make the energy axis an approximation 390 1164 A precision experiment and its measurement distribution 391 1196 A property of triangles easily provable with complex numbers 393 1199 The hand and the quaternions 392 1200 Combinations of rotations 394 1215 Examples of orientable and non-orientable manifolds of two dimensions: a disc, a Möbius strip, a sphere and a Klein bottle 395 1215 Compact (left) and noncompact (right) manifolds of various dimensions 396 1216 Simply connected (left), multiply connected (centre) and disconnected (right) manifolds of one (above) and two (below) dimensions 397 1217 Examples of homeomorphic pairs of manifolds 398 1217 The first four two-dimensional compact connected orientable manifolds: 0-, 1-, 2- and 3-tori 399 1236 A simple way to measure bullet speeds 400 1236 Leaving a parking space – the outer turning radius 401 1237 A simple drawing proving Pythagoras’ theorem 402 1237 The trajectory of the middle point between the two ends of the hands of a clock 403 1238 The angles defined by the hands against the sky, when the arms are extended 404 1243 Deducing the expression for the Sagnac effect 405 1250 The south-pointing carriage 406 1256 A candle on Earth and in microgravity (NASA) 407 1267 Two converging lenses make a microscope 408 1268 The relation between the colour effect and the lens dispersion

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006

meridian of Paris on page 120 is copyright and courtesy of Ken Alder and found on his website http://www.kenalder.com. The geoid of page 122 is courtesy and copyright by the GFZ Potsdam, to be found at http://www.gfz-potsdam.de. The beautiful film of the lunation on page 125 was calculated from real astronomical data and is copyrighted by Martin Elsässer; it is used here with his permission. It can be found on his website http:://www.mondatlas.de/lunation.html. The moon maps on page 126 are courtesy of the USGS Astrogeology Research Program, http:://astrogeology. usgs.gov, in particular Mark Rosek and Trent Hare. The figure of myosotis on page 189 is courtesy and copyright by Markku Savela. The figure of the soliton in the water canal on page 212 is copyright and courtesy of Dugald Duncan and taken from his website on http://www.ma.hw.ac. uk/solitons/soliton1.html. The fractal mountain on page 219 is courtesy and copyright by Paul Martz, who explains on his website http://www.gameprogrammer.com/fractal.html how to program such images. The figure of helium atoms on metal on page 225 is copyright and courtesy of IBM. The oscillon picture on page 257 is courtesy and copyright by Paul Umbanhowar. The drawing of swirled spheres on page 257 is courtesy and copyright by Karsten Kötter. The fluid flowing over an inclined plate on page 261 is courtesy and copyright by Vakhtang Putkaradze. The photograph of the reconstruction of Fizeau’s experiment is copyright by AG Didaktik und Geschichte der Physik, Universität Oldenburg, and courtesy of Jan Frercks, Peter von Heering and Daniel Osewold. The photograph of a light pulse on page 279 is courtesy and copyright of Tom Mattick. The data and images of the Michelson–Morely experiment on page 291 are copyright and courtesy of Stephan Schiller. The relativistic images of the travel through the simplified Stonehenge on page 303 are copyright of Nicolai Mokros and courtesy of Norbert Dragon. The relativistic views on page 304 and 304 are courtesy and copyright of Daniel Weiskopf. The figures of galaxies on pages 438, 439, 440, 440, 441, 441, 445, 459, 469 and 469 are courtesy of NASA. The maps of the universe on page 446 and the Hertzsprung-Russell diagram on page 450 are courtesy and copyright of Richard Powell, and taken from his website http://www.anzwers.org/free/universe. The photograph of lightning on page 519 is copyright Steven Horsburgh (see http://www.horsburgh. com) and used with his permission. The photograph of M. bavaricum on page 530 is copyright by Marianne Hanzlik and is courtesy of Nicolai Petersen. The pictures of the Tesla coil on page 538 are courtesy and copyright of Robert Billon, and found on his website http://f3wm.free.fr. The photograph of a lifter on page 541 is courtesy and copyright of Jean-Louis Naudin; more information can be found on his website http://www.jlnlabs.org. The ocean figure on page 542 is courtesy of Stefan Maus, and taken from his http://www.gfz-potsdam.de/pb2/pb23/SatMag/ ocean_tides.html website. The simple motor photograph on page 543 is courtesy and copyright of Stefan Kluge. The picture of the rainbow on page 555 is from the NOAA website. The secondary rainbow picture on page 561 is courtesy and copyright of Antonio Martos. The supernumerary rainbow picture on page 561 is courtesy and copyright of Wolfgang Hinz and from his website http://www.meteoros.de. The solar furnace photograph on page 572 is courtesy and copyright of Gerhard Weinrebe. The picture of milky water on page 576 was made for this text and is copyright by Antonio Martos. The infrared photograph on page 585 is copyright and courtesy of Serge Augustin. The pictures of retinas on page 586 are courtesy and copyright of Austin Roorda. The microscope picture on page 588 is copyright and courtesy of Stefan Hell. The pictures showing colour blindness on page 596 are courtesy and copyright of Michael Douma, from his splendid website at http://webexhibits.org/causesofcolor/2.html. The cloud photographs on page 597 are courtesy of NASA. The photograph of on page 704 is a present to the author by Ata Masafumi, who owns the copyright. The pictures of snow flakes on page 818 are courtesy and copyright of Elmar Bartel. The pictures of snow flakes on page 863 are courtesy and copyright of Furukawa Yoshinori. The MRI images of the head and the spine on page 898 are courtesy and copyright of Joseph Hornak and taken from his website http://www.cis.rit.edu/htbooks/mri/. The MRI image on page 900 of humans making love is courtesy and copyright of Willibrord Wejimar Schultz. The

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G list of illustrations

1297

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

nuclide chart on page 903 is used with permission of Alejandro Sonzogni of the National Nuclear Data Centre, using data extracted from the NuDat database. The electroscope photograph on page 904 is courtesy and copyright of Harald Chmela and taken from his website http://www. hcrs.at. The figure of the Erta Ale volcano on page 911 is courtesy and copyright of Marco Fulle and part of the photograph collection at http://www.stromboli.net. The drawing of the JET reactor on page 924 is courtesy and copyright of EFDA-JET. The picture of the butterfly in a wind tunnel on page 970 is courtesy and copyright of Robert Srygley and Adrian Thomas. The pictures of feathers insects on page 972 are material used with kind permission of HortNET, a product of The Horticulture and Food Research Institute of New Zealand. The picture of the scallop on page 973 is courtesy and copyright of Dave Colwell. The picture of the eversion of the sphere on page 977 is courtesy and copyright of John Sullivan from the Optiverse website, on http://new.math. uiuc.edu/optiverse/. The knot and link diagrams on pages 978 and 979 are courtesy and copyright of Robert Scharein and taken from his website on http://www.knotplot.com. The copyright of the figure on page 999 belongs to the M.C. Escher Heirs, c/o Cordon Art, Baarn, The Netherlands, who kindly gave permission for its use. All the drawings not explicitly mentioned here are copyright © 1997 – 2006 by Christoph Schiller.

Copyright © Christoph Schiller November 1997–May 2006

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Appendix H

LI ST OF TABLES

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25

200

26 27 28 29 30 31 32

204 206 232 233 236 238 244

33

247

Content of books about motion found in a public library Family tree of the basic physical concepts Properties of everyday – or Galilean – velocity Some measured velocity values Selected time measurements Properties of Galilean time Properties of Galilean space Some measured distance values The exponential notation: how to write small and large numbers Some measured acceleration values Some measured momentum values Some measured mass values Properties of Galilean mass Some measured energy values Some measured power values Some measured rotation frequencies Correspondence between linear and rotational motion An unexplained property of nature: the Titius–Bode rule The orbital periods known to the Babylonians Some measured force values Selected processes and devices changing the motion of bodies Some action values for changes either observed or imagined Some major aggregates observed in nature Correspondences between the symmetries of an ornament, a flower and nature as a whole The symmetries of relativity and quantum theory with their properties; also the complete list of logical inductions used in the two fields Some mechanical frequency values found in nature Some wave velocities Steel types, properties and uses Extensive quantities in nature, i.e. quantities that flow and accumulate Some temperature values Some measured entropy values Some typical entropy values per particle at standard temperature and pressure as multiples of the Boltzmann constant Some minimum flow values found in nature

Copyright © Christoph Schiller November 1997–May 2006

33 38 43 44 46 48 51 52 59 70 79 81 81 85 87 89 89 144 145 147 150 177 184 195

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

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51 52 53 54 55 56

72

73 74 75

Sand patterns in the sea and on land Properties of the motion of light The expected spectrum of gravitational waves Some observations about the universe A short history of the universe The colour of the stars Types of tests of general relativity Searches for magnetic monopoles, i.e., for magnetic charges Some observed magnetic fields Properties of classical electric charge Values of electrical charge observed in nature Some observed electric fields Some observed electric current values Voltage values observed in nature The electromagnetic spectrum Experimental properties of (flat) vacuum and of the ‘aether’ Selected matter properties related to electromagnetism, showing among other things the role it plays in the constitution of matter; at the same time a short overview of atomic, solid state, fluid and business physics 616 Examples of disastrous motion of possible future importance 643 The semantic primitives, following Anna Wierzbicka 647 The defining properties of a set – the ZFC axioms 652 Some large numbers 662 The ‘scientific method’ 709 Some small systems in motion and the observed action values for their changes 782 Particle spin as representation of the rotation group 797 Common and less common baths with their main properties 819 Motion and motors in living beings 823 Approximate number of living species 841 The types of rocks and stones 843 Signals penetrating mountains and other matter 844 Matter at lowest temperatures 850 A selection of lamps 874 The principles of thermodynamics and those of horizon mechanics 901 The main types of radioactivity and rays emitted by matter 905 The main types of cosmic radiation 916 Some radioactivity measurements 917 Human exposure to radioactivity and the corresponding doses 920 Natural isotopes used in radiometric dating 951 Some comparisons between classical physics, quantum theory and experiment 959 Everything quantum field theory and general relativity do not explain; in other words, a list of the only experimental data and criteria available for tests of the unified description of motion 962 A selection of the consequences of changing the properties of nature 1000 The size, Schwarzschild radius and Compton wavelength of some objects appearing in nature 1050 Physical statements about the universe

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57 58 59 60 61 62 63 64 65 66 67 68 69 70 71

256 280 402 442 451 467 499 519 520 523 524 524 528 538 564 581 605

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34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

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1300 76 77

78 79 80

93 94 95

1052 Properties of nature at maximal, everyday and minimal scales 1062 Everything quantum field theory and general relativity do not explain; in other words, a list of the only experimental data and criteria available for tests of the unified description of motion 1063 A small selection of the consequences when changing aspects of nature 1107 Effects of various camera shutter times on photographs 1144 The ancient and classical Greek alphabets, and the correspondence with Latin and Indian digits 1145 The beginning of the Hebrew abjad 1156 The derived SI units of measurement 1156 The SI unit prefixes 1157 Planck’s (uncorrected) natural units 1165 Basic physical constants 1166 Derived physical constants 1167 Astrophysical constants 1168 Astronomical constants 1174 The elementary particles 1175 Elementary particle properties 1177 Properties of selected composites 1181 The periodic table of the elements known in 2006, with their atomic numbers 1181 The elements, with their atomic number, average mass, atomic radius and main properties 1220 Properties of the most important real and complex Lie groups 1228 The structure of the Arxiv preprint archive 1229 Some interesting servers on the world-wide web

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81 81 81 83 84 85 86 87 88 89 90 91 92

h list of tables

Copyright © Christoph Schiller November 1997–May 2006

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Appendix I

NAME I NDEX

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A

Page numbers in italic typeface refer to pages where the person is presented in more detail.

Abbott

Anderson, J.D. 348 Anderson, J.L. 508 Anderson, James A. 697 Anderson, M.H. 767 Anderson, R. 165 Andreotti, B. 272 Andres, M.R. 885 Andrews, M.R. 767 Aniçin, I.V. 168 Anton, A. 766 Antonini, P. 343, 345 Aquinas, Thomas 159, 680, 698, 785 Arago, Dominique-François 530, 571 Arago, François 571 Araki, T. 889 Archimedes 77 Ardenne, Manfred von 629 Aripov, Otanazar 695 Aristarchos 164, 342 Aristarchos of Samos 94, 100, 277 Aristarchus of Samos 164 Aristotle 45, 49, 118, 163, 342, 676, 679, 966, 967, 1110, 1113, 1132 Arlt, J. 623 Armstrong, J.T. 163 Armstrong, Neil 623 Arndt, M. 767, 884 Arnowitt 433 Aronson, Jeff 22 Aronson, Jeff K. 1171 Arseneau, Donald 22 Ascher, Marcia 651

Copyright © Christoph Schiller November 1997–May 2006

Alber, G. 884 Albert, David Z. 885 Albertus Magnus 941 Alcock, Nathaniel W. 1193 Alcubierre, M. 505 Alder, Ken 168, 1296 Aleksandrov, D.V. 1192 Alexander, R. McN. 167 Alexandrov, P.S. 1054, 1099 Ali, A. 1054 Alighieri, Dante 116 Allen, L. 623 Allen, Les 623 Allen, Woody 445 Almeida, C. de 627 Alpher, R.A. 946 Alsdorf, D. 170 Alspector, J. 342 Altman, S.L. 1226 Alvarez Diez, C. 892 Alvarez, E. 1134 Alväger, T. 342 Amaldi, U. 946, 947, 1031 Amati, D. 1028 Amelino–Camelia, G. 1027, 1029, 1030, 1097, 1098 Amenomori, M. 946 Ampère, André-Marie 531, 536 Amundsen, Roald 116, 167 An, K. 269, 891 Anaxagoras 683, 695 Anders, S. 168 Anderson 426 Anderson, Carl 759, 904 Anderson, I.M. 508

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A Abbott, T.A. 627 Abdalla, M.C.B. 1054 Abe, E. 629 Abe, F. 162, 946 Abramowicz, M.A. 513 Abrikosova, I.I. 890 Acef, O. 622 Ackermann, Peter 1171 Adams, Douglas 685, 964, 1047 Adelberger, E. 514 Adelberger, E.G. 168 Adenauer, Konrad 135, 368 Adler, C.G. 348 Aedini, J. 766 Aetius 75, 164, 438, 509 Aflatooni, K. 769 Aguirregabiria, J.M. 626, 1269 Aharonov, Y. 768 Ahlgren, A. 161, 888 Ahluwalia, D.V. 1027 Ahmad, Q.R. 344 Ahrens, C. Donald 170 Aichelburg, P.C. 512 Aigler, M. 55 Aitken, M.J. 629 Ajdari, A. 629 Akama, K. 1029 Akerboom, F. 623 Åkerman, N. 170 Al-Dayeh, M. 627 al-Farisi, Kamal al-Din 576 al-Hadi al-Hush, Ramadan 695 Alanus de Insulis 476

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name index

1302

A Ashburner

Bell, J.S. 886 Bell, John 810 Bellini, Giovanni 625 Benbrook, J.R. 627 Bender, P.L. 508 Benjamin, J. 701 Benka, S. 164 Bennet, C.L. 166, 512 Bennet, S.C. 947 Bennett, C.H. 630, 697, 1067 Bennett, Charles H. 271 Bentlye, W.A. 272 Benzenberg, Johann Friedrich 95 Berg, E. 627 Berg, H.C. 992 Bergmann, L. 700 Bergquist, J. 1171 Bering, E. 627 Bering, E.A. 627 Berkeley, George 678 Berlekamp, E.R. 699 Berlin, Brent 624 Berlin, Isaiah 1122 Bernard, C. 946 Bernoulli, Daniel 240 Bernoulli, J. 1148 Bernreuther, W. 1029 Bernstein, Aaron 567 Berry, M.V. 628, 792, 1133 Berry, Michael 792, 976 Bertotti, B. 508 Bertulani, C.A. 1192 Besnard, P. 887 Bessel 100, 753 Bessel, Friedrich 888 Bessel, Friedrich Wilhelm 101, 458 Besso, Michele 319 Beth, R. 623 Bethe, H. 946 Bethe, Hans 915 Bettelheim, Bruno 696 Bettermann, D. 628 Beutelspacher, Albrecht 814 Beuys, Joseph 635, 692, 701 Bevis, M. 170 Beyer, Lothar 22 Bhalotra, S.R. 623

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Copyright © Christoph Schiller November 1997–May 2006

B Babinet, Jacques 1155 Babloyantz, A. 272 Bachem 380 Bader, Richard F.W. 764 Baez, John 22 Baggett, N. 342 Bagnoli, Franco 22 Bailey, J. 345 Bailey, J.M. 342 Balachandran, A.P. 1026 Balashov, Yuri I. 701 Balibar, Françoise 764 Balmer, Johann 754 Banach, Stefan 54, 55, 220 Banavar, J.R. 267 Banday, A.J. 512

Bandler, R. 159 Bandler, Richard 157, 159, 632, 661 Barber, B.P. 166 Barberi Gnecco, Bruno 22 Barbour, John 162 Barbour, Julian 512 Bardeen, J. 890 Bardeen, John 847, 884 Barnett, S.J. 620, 621 Barnett, S.M. 767 Barnhill, M.V. 620 Barrow, J.D. 512 Barrow, John D. 701, 967 Bartel, Elmar 22, 818, 1296 Bartocci, Umberto 318, 319 Barton, G. 890 Bartussek, R. 886 Barwise, J. 700 Basieux, Pierre 698 Basri, G. 509 Bass, Thomas A. 166 Batelaan, H. 769 Bateman, H. 344 Batty, R.S. 268 Baudelaire, Charles 32 Bauerecker, S. 628 Baumann, K. 885 Bautista, Ferdinand 22 Baylor, D.A. 765 Beale, I.L. 622 Beaty, William 22 Beauvoir, B. de 622 Bechinger, C. 891 Becker, A. 510 Becker, J. 170 Becquerel, Henri 899 Bedford, D. 507 Beeksma, Herman 22 Beenakker, C.W.J. 622, 886, 1098 Behroozi, C.H. 161, 343 Bekenstein, J.D. 513, 892, 1099, 1133 Bekenstein, Jakob 255, 485, 870 Belfort, François 22 Belic, D. 1172, 1192 Belinfante, F.J. 792

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Ashburner, J. 697 Ashby, M.F. 269 Ashcroft, N. 895 Ashcroft, Neil 612 Ashkin, A. 623 Ashkin, Arthur 573 Ashtekar, A. 374, 515, 1026, 1029, 1098, 1133, 1134 Asimov, I. 967 Askin, A. 623 Aspect, A. 885 Aspect, Alain 803 Aspelmeyer, M. 766 Aspinwall, P. 1028 Asterix 226 Astumian, R.D. 886 Ata Masafumi 22, 1296 Audoin, C. 1171 Augel, Barbara and Edgar 22 Augustin, Serge 585, 1296 Augustine 511, 1280 Augustine of Hippo 537 Ausloos, M. 268 Autumn, K. 889 Avogadro, Amadeo 224 Avron, J.E. 627 Awschalom, D.D. 885 Axel, Richard 887 Axelrod, R. 158 Ayers, Donald M. 1153 Ayiomamitis, Anthony 1295

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name index

B Bharucha

Brezger, B. 884 Briand, J.P. 864, 892 Briatore 380 Briatore, L. 504 Briggs, F. 512 Bright, William 1152 Brightwell, G. 1028 Brillouin 754 Brillouin, L. 271 Brillouin, Louis 624 Brillouin, Léon 245 Bringhurst, Robert 887 Britten, R.J. 1275 Britten, Roy 1275 Broch, Henri 699 Brock, J.B. 625 Brody, A.L. 623 Broglie, L. de 767 Broglie, Louis de 715, 738 Bronstein, M. 512 Bronstein, Matvey 476 Brooks, Mel 237 Brookshear, J. Glenn 697 Brouwer, Luitzen 50 Brown, B.L. 623 Brown, J.H. 267 Brown, Peter 373 Brown, Robert 242 Bruce, Tom 22 Brumberg 717 Brumberg, E.M. 765 Brune, M. 884 Brunetti, M. 946 Brunner, H. 622 Brush, S. 166 Bryant, D.A. 892 Bub, J. 765 Buchanan, Mark 628 Buck, Linda 887 Buckley, Michael 159 Buddakian, R. 629 Budney, Ryan 22 Bunn, E.F. 512 Bunsen, Robert 753 Bunyan, Paul 269 Burbidge, G. 511 Burbridge, E. Margaret 701 Burbridge, G.R. 701 Buridan, Jean 118, 167

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Copyright © Christoph Schiller November 1997–May 2006

Bonner, Yelena 1012 Bonnor 408 Bonnor, W.B. 507, 512 Boone, Roggie 22 Bordé, Ch.J. 892 Borelli, G. 159 Borgne, J.-F. Le 768 Born, M. 891 Born, Max 623, 707, 710, 765 Bose, S.N. 791 Bose, Satyenra Nath 775 Boss, Alan P. 267 Bouchiat, C.C. 946 Bouchiat, M.A. 946 Boughn, S.P. 346 Bour, L. 623 Bourbaki, N. 1148 Bousso, R. 892, 1054, 1098, 1134 Bouwmeester, D. 791 Bowden, F.P. 170 Bower, B. 267 Bowlby, John 700 Boyce, K.R. 346 Boyda, E.K. 623 Boyer, T.H. 890 Boyer, Timothy 856 Brackenbury, John 972 Bradley 277 Bradley, C.C. 767 Bradley, James 101, 101, 276 Braginsky, V.B. 169, 506, 507 Brahe, Tycho 116, 116 Brahm, A. de 1149 Brahm, D.E. 514 Brahmagupta 1194 Brandes, John 22 Brandt, E.H. 628 Brantjes, R. 1247 Brattain, Walter 847 Brault, J.W. 504 Braxmeier, C. 343 Bray, H.L. 509 Brebner, Douglas 22 Brecher, K. 342 Brehme, R.W. 348 Brendel, J. 766 Brennan, Richard 159 Brewer, Sydney G. 342

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Bharucha, C.F. 888 Bianco, C.L. 893, 1098 Bierce, Ambrose 831 Biggar, Mark 22 Bilaniuk, O.M. 346 Bilanuk, O.M.P. 346 Bilger, H.R. 165 Billon, Robert 538, 1296 Bimonte, G. 1026 Biraben, F. 622 Bird, D.J. 1171 Birkhoff 429 Bischoff, Bernard 1152 Bjorkholm, J.E. 623 Björk, G. 766 Blaauwgeers, R. 889 Bladel, Jean van 343 Blagden, Charles 250 Blair, D.G. 1097, 1172 Blair, David 505 Blanché, Robert 699 Blandford, R. 513, 1026 Blandford, R.D. 506 Blau, Stephen 297 Blau, Steven K. 345 Bliss, G.W. 628 Block, S.M. 886 Blumensath, Achim 22 Boamfa, M.I. 628 Bocquet, Lydéric 269 Bode, Johann Elert 144 Boer, W. de 947, 1031 Boersma, S.L. 890 Boethius 1232 Bohm 1047, 1133 Bohm, D. 768 Bohm, David 885 Bohr 734, 1080 Bohr, Aage 848 Bohr, N. 764, 766, 767, 1097 Bohr, Niels 246, 281, 656, 705, 739, 747, 793, 793, 812, 837, 884, 1069, 1182 Bollinger, J.J. 767 Boltzmann, Ludwig 181, 242 Bombelli, L. 515, 1028, 1133 Bombelli, Luca 22, 1026 Bondi, H. 504 Bondi, Hermann 343

1303

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name index

1304

B Burke

Burke, D.L. 890 Burša, Milan 162 Busch, Paul 765 Busche, Detlev 161 Butikov, E.I. 766 Butler, Samuel 632 Butoli, André 620 Buzek, V. 791 Bäßler, S. 514 Böhm, A. 946 Börner, G. 510 Böhncke, Klaus 22

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Copyright © Christoph Schiller November 1997–May 2006

Ciufolini, I. 506, 508 Ciufolini, Ignazio 398, 506 Clairon, A. 622 Clancy, E.P. 504 Clancy, Tom 268 Clark, R. 946 Clarke 161 Clarke, T. 1280 Clauser, J.F. 886 Clausius 870 Clausius, Rudolph 237, 242, 471, 472, 870 Clavius, Christophonius 63 Cleobulus 52 Clerk Maxwell, James 294, 967 Clery, D. 889 Clifton, R. 765 Close, F. 946 Codling, K. 1171 Coehoorn, Reinder 629 Cohen, M.H. 346 Cohen, P. J. 698 Cohen, Paul 698 Cohen, Paul J. 647 Cohen, Philip 626 Cohen-Tannoudji, C. 764 Cohen-Tannoudji, G. 271 Cohen-Tannoudji, Gilles 246 Colazingari, Elena 22 Colella, R. 892 Coleman–McGhee, Jerdome 269 Colladon, Daniel 228 Collins, D. 791 Collins, J.J. 159 Columbus 474 Colwell, Dave 973, 1297 Comins, Neil F. 168 Commins, E.D. 1029 Compton, A.H. 166, 765 Compton, Arthur 98, 720 Conde, J. 1134 Conkling, J.A. 64 Conroy, R.S. 623 Conti, Andrea 22 Conway, J. 512 Conway, J.H. 699 Conway, John 653, 1271

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

C Caianiello, E.R. 374, 1099 Cajori, Florian 1152 Caldeira, A.O. 884 Calzadilla, A. 267 Campbell, D. 272 Campbell, D.K. 272 Cantor, Georg 649 Caplan, S.R. 993 Caps, H. 270 Caraway, E.L. 627 Carducci, Giosuè 1364 Carilli, C.L. 512 Carlip, S. 892 Carlip, Steve 22, 373, 374, 511, 1068 Carlyle, Thomas 104 Carmona, Humberto 628 Carneiro, S. 511 Carnot, Sadi 86 Carr, Jim 22 Carroll, Lewis 699 Carroll, Lewis, or Charles Lutwidge Dogson 246 Carruthers, P. 767 Carter 482 Carter, Brandon 701 Cartesius 51 Casati, Roberto 164 Casimir, H.B.G. 890 Casimir, Hendrik 855 Cassini, Givanni 276 Cassius Dio 145 Castagnino, M. 515 Cauchy, Augustin-Louis 192 Cavendish, Henry 120

Caveney, S. 162 Caveney, Stanley 66, 1295 Caves, C.M. 507, 895 Cayley, Arthur 192, 1202, 1203 Ceasar, Gaius Julius 1149 Cecchini, S. 946 Celsius, Anders 1163 Chahravarty, S. 885 Chaineux, J. 1152 Chaitin, Gregory J. 699 Chan, M.A. 165 Chan, W.P. 889 Chandler, David G. 159 Chandler, Seth 100 Chandrasekhar, Subramanyan 690 Chang, I. 766 Chang, J. 1192 Chang, P.Z. 628 Chantell, M. 946 Chaplin, Charlie 555 Chapman, M.S. 769 Charlemagne 1149 Charpak, G. 1171 Chen, B. 1098 Chen, P. 893 Cheseaux, Jean Philippe Loÿs de 459 Chiao, R. 624 Chiao, R.Y. 624, 625 Chiba, D. 629 Childs, J.J. 269, 891 Chinnapared, R. 513 Chmela, Harald 904, 1297 Chomsky, Noam 161, 667 Choquet-Bruhat, Yvonne 508 Christ, N.H. 1028 Christiansen, W.A. 1030 Christlieb, N. 168 Christodoulou, D. 513 Christou, A.A. 169 Chu, S. 623, 624, 892 Chuang, I.L. 625 Chudnovsky, D.V. 1173 Chudnovsky, G.V. 1173 Chung, K.Y. 892 Ciafaloni, M. 1028 Cicero 736 Cicero, Marcus Tullius 691

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name index

C Conway

Descartes, R. 1148 Descartes, René 700 Descartes, René 51, 633, 1238 Deser 433 Deshpande, V.K. 346 Desloge, E.A. 348 Desloge, Edward A. 165, 348 Desmet, S. 620 Destexhe, A. 272 DeTemple, D.W. 1173 Detweiler, S. 513 Deutschmann, Matthias 1232 Dewdney, Alexander 63 Dewdney, Alexander K. 162 DeWitt, C. 886 DeWitt-Morette, Cecile 508 de Picciotto, R. 889 De Pretto, O. 347 Dholakia, K. 623 Diana, princess of Wales 665 Dicke, R.H. 169, 506 Dicke, Robert H. 767 Dickey, J.M. 509 Diehl, Helmut 58, 496 Dieks, D. 791 Dieks, Dennis 778 Diel, Helmut 1295 Diel, Isolde 1295 Diemer, T. 515 Dietl, T. 629 Dietrich, F. 628 Dietrich von Freiberg 624 Dietze, H. 344, 994 DiFilippo, F. 346 DiFilippo, Frank 22 Dill, L.M. 268 Dillard-Bleick, Margaret 508 Diner, S. 1026, 1054, 1067, 1097, 1132 Dio, Cassius 170 Diophantus of Alexandria 1196 Dirac 672, 688, 762, 783, 869 Dirac, P.A.M. 766, 851, 891, 895 Dirac, Paul 203, 729, 757, 891, 1148 Dirr, Ulrich 22 Disney, Walt 821

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Copyright © Christoph Schiller November 1997–May 2006

D Daffertshofer, A. 791 Dahlman, John 22 Dale Stewart, T. 620 Dalì, Salvador 832 Dalibard, J. 885

Dalton, John 596, 902 Dalton, K. 509 Dam, H. van 1035 Dam, H. van 1030, 1054 Dambier, G. 1192 Damour 875 Damour, T. 514, 893 Damour, Thibault 498 Danecek, Petr 22 Daniels, Peter T. 1152 Dante Alighieri 116 Darius, J. 164 Darley, Vincent 22 Darre, Daniel 22 Darwin 663, 666, 677 Darwin, Charles 696, 888 Das, A. 1028 Dasari, R.R. 269 Davidson, C. 505 Davies, Paul 871 Davis, Chandler 695 Davis, K.B. 767 Davis, S.D. 268 Davson, H. 698 Davy, Humphry 531 de Bruyn, A.G. 512 de Gennes, P.-G. 1133 de Groot, D.G. 630, 888 de Haas, W.J. 621 de Vega, H.J. 1026 de Vries 213 de Witt, B.S. 1026 Deaver, B.S. 1165 Decker, Rick 697 Degen, Carl Ferdinand 1203 Dehmelt, H. 269, 628, 1192 Dehmelt, Hans 269, 627, 768, 1029, 1192 Dehn, Max 55, 980 Dekker, J.P. 630, 888 Della Scala, S. 699 DeMille, D. 1029 Demko, T.M. 165 Democritus 223, 667, 676, 679, 700, 966, 1060, 1115 Denardo, B. 890 DeRaad, L.L. 620 Derjaguin, B.W. 890 Desari, R.R. 891

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Conway, John H. 699 Cooper, Heather 167 Cooper, J.N. 159 Cooper, L.N. 890 Cooper, Leon N. 847 Copernicus, Nicolaus 278, 701 Copperfield, David 604, 629 Corballis, M.C. 622 Corbin, V. 627 Cordero, N.A. 508 Coriolis, Gustave-Gaspard 84, 95 Cornell, E.A. 161 Cornell, Eric 848 Coron, N. 1192 Corovic, Dejan 22 Cosman, E.C. 622 Costa, S.S. 346 Costa–Krämer, J.L. 848, 889 Costabel, Pierre 220 Costella, J.P. 346, 347 Costella, John P. 768 Couch, E. 513 Couder, Y. 698 Coule, D.H. 1030 Coulomb, Charles-Augustin de 525 Courtial, J. 767 Cox, A.N. 509 Crane, H. Richard 165 Craven, J.D. 169 Crescimanno, M. 620 Crespi, Roberto 22 Crinion, J.T. 697 Crookes, William 573 Crowe, Michael J. 165, 342 Crutchfield, J. 272 Crystal, David 641, 1152 Cser, L. 891 Curie, Marie 847, 899, 1183 Curie, Pierre 899, 1183 Curtis, L.J. 768

1305

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name index

1306

D Dittus

644 Empedocles 276 Emsley, John 271 Enders, A. 625 Engels, F. 765 Engels, Friedrich 154, 716, 737 Engemann, S. 269 Engl, Walter L. 622 Englert, Berthold-Georg 764 Enhag, Per 1193 Enquist, B.J. 267 Enss, Christian 270 Epicurus 692, 693, 717, 719 Epikuros 692 Erdős, Paul 1194 Erdös, Paul 55 Erlykin, A.D. 626 Erschow, A. 792 Ertmer, W. 628 Escher, Maurits 999 Eshelby, J. 344, 994 Etchemendy, J. 700 Etienne, B. 889 Euclid 43, 1007 Eukleides 43 Euler, H. 768, 891 Euler, Hans 759 Euler, L. 1148 Euler, Leonhard 100, 146, 147, 166, 416, 658, 1198 Euler, M. 1275 Evans, J. 169 Everitt, C.W. 1165 Everitt, F. 514 Exton, A. 513 F Fabeni, P. 625 Facchi, P. 767, 1132 Facchi, Paolo 888 Faestermann, T. 510 Fairbank, W.M. 620 Fairbanks, J.D. 1165 Fairhust, S. 374 Faisst, H. 272 Falco, E.E. 511 Falk, G. 169, 504 Faller, James E. 623 Fang Lizhi 695

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E Eőtvős, R. von 169, 506 Eőtvős, Roland von 97, 169, 395, 506 Earman, John 162 Ebbinghaus, H.-D. 1226 Ebstein, R.P. 701 Eccles, John 701 Eckhardt, B. 272 Eckstein, G. 344 Economou, E.N. 625 Eddington, A.S. 505 Eddington, Arthur 82, 681, 700, 1060, 1067, 1115, 1133 Edelmann, H. 168 Edge, Ron 628 Edwards, R. 620 Ehlers, J. 511 Ehrenfest, P. 347 Ehrenreich, H. 1098 Ehrenstein, W.H. 626 Eidelman, S. 1175 Eigler, D.M. 626 Einstein 754 Einstein, A. 270, 507, 511, 621, 765, 791, 885 Einstein, Albert 31, 129, 132, 133, 182, 196, 198, 203, 242, 243, 281, 282, 295, 310, 315, 318, 343, 347, 377, 379, 382, 391, 397, 427, 429, 453, 494, 499, 502, 503, 555, 636, 656, 695, 712, 714, 726, 727, 750, 775, 850, 888, 996, 1048, 1184 Einstein, Eduard 382 Ekman, Walfrid 165 Ekstrom, C.R. 769 Ellis, D.G. 768 Ellis, G.F.R. 505, 511, 1054 Ellis, J. 1030, 1054 Els, Danie 22 Elsevier, Louis 1235 Elswijk, H.B. 269, 620 Elswijk, Herman B. 22 Elsässer, Martin 125, 1296 Eltsov, V.B. 889 Emde Boas, P. Van 791 Emerson, Ralph Waldo 214,

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Dittus, H. 514 Diu, B. 764 DiVicenzo, D.P. 885 Dixon, Bernard 162, 992 Dmitriyev, V.P. 992 Dobra, Ciprian 22 Dogarin, A. 624 Dolto, Françoise 887 Doorne, C.W.H. van 272 Doplicher, S. 1028 Doppler, Christian 285 Dorbolo, S. 270 Dorbolo, Stéphane 232 Dorda, G. 890 Dorfi, E.A. 510 Doroshkevich 449 Doroshkevich, A.G. 510 Dorsey, A.T. 885 Douady, S. 698 Dougherty, R. 162 Douglas, R. 697 Douma, Michael 596, 1296 Doyle, Arthur Conan 219 Dragon, Norbert 302, 303, 1296 Dresden, M. 700 Droste, J. 385, 505 Drude, Paul 591 Drumond, P.D. 895 Drury, L.O’C. 892 Dubelaar, N. 1247 Dubrulle, B. 170 Dudek, J. 946 Duguay 278 Duguay, M.A. 342 Duhem, Pierre 699 Dumont, Jean-Paul 158, 164, 273, 342, 509, 510, 700, 967, 1132 Duncan, Dugald 212, 1296 Duran, E. 792 Durfee, D.S. 767 Durieux, M. 270 Durkin, L.S. 946 Dusenbery, David 163, 993 Dutton, Z. 161, 343 Dwyer, J.R. 627 Dyson, F.W. 505 Dziedzic, J.M. 623

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name index

F Faraday

Freimund, D.L. 769 French, A.P. 348, 626 French, Robert M. 162 Frenkel, J. 344, 993 Frenzel 108 Frercks, J. 342 Frercks, Jan 278, 1296 Fresnel, Augustin Jean 561, 571 Friedberg, R. 1028 Friedman, A. 510 Friedman, David 147 Friedman, J.I. 945 Friedman, J.L. 1026 Friedman, J.R. 885 Friedmann, A. 510 Friedmann, Aleksander Aleksandrowitsch 454 Frieze, M.E.J. 624 Frisch, D.H. 345 Fritsch, G. 268 Fritzsch, H. 1029 Frova, Andrea 162 Fukuda, Y. 344 Full, R.J. 889 Full, Robert 845 Fulle, Marco 1297 Fulling, S.A. 892 Fulling, Stephen 871 Fumagalli, Giuseppe 164, 267, 700, 764 Furrie, Pat 22 Furukawa Yoshinori 863, 1296 Furukawa, Y. 892 Fölsing, Albrecht 343 Füllerkrug, M. 627 Fürstenau, H. 947, 1031 G Gabor, Dennis 587 Gabrielse, G. 1192 Gabuzda, D.C. 346 Gadelka, A.L. 1054 Gadolin, Johan 1184 Gaensler, B.M. 509 Gaessler, W. 1030 Galajda, P. 623 Galilei, Galileo 42, 48, 62, 64,

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Fisher, M.P.A. 885 Fitzgerald, George F. 294 Fizeau, Hippolyte 278 Flach, S. 272 Flachsel, Erwein 159 Flambaum, V.V. 894 Flato, M. 1226 Flavell, J.H. 634 Fließbach, Torsten 504 Flindt, Rainer 161 Flint, H.T. 1028 Floyd, R.M. 513 fluoride, deuterium 589 Foldy, L.L. 750, 768 Fonseca, E.J.S. 766 Foong, S.K. 894 Ford 883 Ford, L.H. 894 Foreman, M. 162 Forsberg, B. 170 Forssman, Friedrich 887 Fortes, L.P. 170 Fortey, Richard 696 Fortson, E.N. 1031 Foster, James 267, 503 Foteinopoulou, S. 625 Foucault, Jean Bernard Léon 96 Fowler, E.C. 342 Fowler, L.A. 507 Fowler, W.A. 701 Fowler, Willy 690, 701 Fox, D.W. 893 Frackowiak, R.S. 697 Fraenkel, Adolf/Abraham 647 Francis, G.K. 993, 1226 Francis, George 994 Frank, F.C. 344, 993 Frank, Louis A. 169 Franke-Arnold, S. 767 Franklin, Benjamin 220, 522 Fraser, Alistair B. 622 Frasinski, L.J. 1171 Fraunhofer, Joseph 561, 753 Fray, S. 892 Fredenhagen, K. 1028 Frederick, Ann 887 Fredman, R.A. 442 Fredriksson, S. 1029

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Faraday, Michael 531, 534, 536, 541, 542 Faragó, B. 891 Farinati, Claudio 22 Farley, F.J.M. 342 Farmer, J. 272 Farrant, Penelope 164 Fasching, G. 438 Fatio de Duillier, Nicolas 142 Faust 1050 Fayeulle, Serge 170 Fearing, R. 889 Febbraio, M. 887 Fehr, C. 267 Fekete, E. 169, 506 Feld, M.S. 269 Fereira, P.G. 512 Fermani, Antonio 22 Fermi, E. 886 Fermi, Enrico 775, 913, 1184 Fernandez–Nieves, A. 167 Few, A.A. 627 Feyerabend, Paul 667, 686 Feynman, R.P. 509 Feynman, Richard 431, 813, 858, 927 Feynman, Richard P. 670 Feynman, Richard (‘Dick’) Phillips 729 Feynman, Richard P. 158, 508, 886, 1026, 1264 Fibonacci 1146 Field, M.S. 891 Figer, D. 509 Figueroa, D. 267 Filippov, T. 268 Fink, Hans–Werner 862, 891 Fink, Hans-Werner 269 Fink, T. 1279 Finkelstein, D. 627, 1028 Finkelstein, David 1009 Finkenzeller, Klaus 22 Finne, A.P. 889 Firestone, R.B. 792, 1192 Firpić, D.K. 168 Fischbach, E. 765 Fischer, M.C. 767, 888 Fischer, Ulrike 22

1307

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name index

1308

G Galileo

Goss Levi, Barbara 886 Gottfried, Kurt 885, 945 Goudsmit, Samuel 749 Gould, Andrew 410 Gould, P.L. 768 Gould, R. 792 Gould, Stephen J. 696, 1192 Gour 859 Gour, G. 891 Gousmit, S. 768 Gracovetsky, Serge 93, 164 Graham, Geoff 887 Graham, Neill 886 Grandjean, F. 620 Graner, F. 170 Grant, E. 625 Grassmann, Hermann Günther 1204 Graves, John 1203 Graw, K.-U. 1253 Gray, C.G. 267, 767, 1242 Grebe 380 Greegor, R.B. 625 Green, A.J. 509 Green, M.B. 1026 Greenberg, O.W. 789, 791, 792 Greenberger, D.M. 792 Greenbergest, T.V. 792 Greene, Brian 1082, 1098, 1133 Greenland, P.T. 1192 Greenler, R. 623 Greenside, Henry 1230 Greenstein, J.L. 504 Gregori, A. 1134 Gregori, Andrea 1124 Gregory, R. 1029 Greiner, J. 515, 893 Greiner, Jochen 22, 875 Griessen, R. 630, 888, 889 Griffith, W.C. 1031 Griffiths, D.J. 627 Grimaldi, Francesco 215 Grimes, Barbara 642 Grinder, John 157, 159 Grinstein, G. 885 Gross, B. 622, 1192 Gross, D.J. 1028 Gross, David 932 Gross, R.S. 166

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Gerlach, Walter 742 Geroch, R. 514 Geroch, Robert 493, 872 Ghahramani, Z. 267 Ghavimi–Alagha, B. 167 Ghirardi, G.C. 892 Gibbons, G.W. 374, 505, 893, 1097 Gibbons, Gary 349, 373, 387, 1071 Gibbs, Josiah Willard 772 Gibbs, P. 1027, 1133 Gibbs, Phil 623, 1018, 1116 Gide, André 422 Giessibl, F. 889 Gilbert, William 518 Gilles, G.T. 622 Gilligan, James 170 Giltner, D.M. 344 Giovannini, G. 946 Gisin, N. 766, 791 Giulini, D. 510 Glasheen, J. W. 167 Glassey, Olivier 22 Glattli, D.C. 889 Glatzmaier, G.A. 621 Glauber 731 Glauber, R.J. 766 Gleiter, H. 629 Gmelin 1193 God, A. 946 Goenner 407 Goenner, Hubert 504 Gold, Tommy 209 Goldhaber, A.S. 620 Goldman, V.J. 889 Goldsmith, D. 627 Goldstein, E. Bruce 160 Goldstein, Herbert 267 Goles, E. 272 Golestanian, R. 891 Golubitsky, M. 159 Gonshor, H. 699 González-Herráez, M. 624 González, Antonio 22 Good, R.H. 348 Gordon, A. 627 Goriely, A. 269 Gossard, A.C. 890

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65, 86, 94, 119, 134, 137, 220, 221, 223, 276, 569, 679, 684, 695, 706, 708, 1235, 1240 Galileo 94 Galle, Johann Gottfried 753 Galli, M. 946 Gallivan, B.C. 698 Galois, Evariste 192 Galvani, Luigi 528 Gamow, G. 510, 946 Gamow, George 449, 910 Gans, R. 628 Garanderie, A. de la 887 Garay, L. 1027 Garay, L.J. 1030 Garcia, N. 889 Garcia–Molchales, P. 889 Garcia-Ruiz, J.M. 698 Gardner, Martin 700, 791 Garg, A. 885 Garlaschelli, L. 699 Garret, Don 1132 Garrett, A. 164 Garwin, R.L. 1171 Gasiorowicz, Stephen 768 Gaspard, Pierre 243, 271 Gasparoni, S. 766 Gasse Tyson, Neil de 169 Gastaldi, Luca 22, 1295 Gauss, Carl-Friedrich 418 Gauthier, N. 268 Gauß, Carl-Friedrich 982 Gauß, Carl-Friedrich 1204 Ge, M.-L. 1026 Gearhart, R. 344 Gearloose, Ratchet 821 Gehrels, N. 513, 1026 Geiger, Hans 899 Geim, A.K. 628 Geim, Andre 628 Gelb, M. 159 Gelbaum, Bernard R. 162, 1224 Gell-Mann, M. 884 Gell-Mann, Murray 793, 813, 926 Gennes, P.-G. de 892 Georgi, Renate 22, 701 Gerlach, W. 766

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name index

G Grossmann

Hawking 370 Hawking, S. 1054 Hawking, S.W. 893, 1027, 1133 Hawking, Stephen 464, 485, 489, 494, 511, 872 Hawking, Stephen W. 511 Hayasaki, Y. 891 Hayashi, H. 889 Hayes, Allan 22 Hays, J.D. 166 Hayward, S.A. 374 Heß, Viktor 904 Heath, T. 1028 Heath, Thomas 164 Heaviside 545, 1201 Heber, U. 168 Heckel, B. 514 Heckel, B.R. 168 Heckenberg, N.R. 624 Heering, Peter von 1296 Heflinger, L.O. 628 Hegerfeldt, G. 886 Hegerfeldt, Gerhard 803 Heinzen, D.J. 767 Heisenberg 672, 1080 Heisenberg, W. 768, 891, 895, 1027 Heisenberg, Werner 246, 263, 569, 707, 708, 710, 739, 759, 902 Helden, A. van 167 Hell, S.W. 626 Hell, Stefan 589, 1296 Heller, E. 268 Hellinger, Bert 170, 663, 694, 887 Hellund, E.J. 1028 Hellwege, K.-H. 700 Helmholtz 561 Helmholtz, Hermann von 235, 583, 693 Helmond, Tom 22, 373 Helmont, Johan Baptista van 240 Hemley, R. 889 Henbest, Nigel 167 Henderson, Lawrence J. 271 Henderson, Paula 22 Henon, M. 170

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Copyright © Christoph Schiller November 1997–May 2006

H Haake, F. 884 Haas, Arthur Erich 707 Haas, Wander Johannes de 532 Haber, John 22 Hackermüller, L. 767, 884 Hadamard, Jacques 701 Hadley, M. 515 Haeckel, Ernst 661 Hafele 380 Hafele, J.C. 345, 504 Hagen, J.G. 165 Hagen, John 97 Hagiwara, K. 620 Hahn, Otto 913 Hajnal, A. 791 Haken, Hermann 850 Hakonen, P.J. 1171 Halberg, F. 161, 888 Hales, Tom 862 Haley, Stephen 22, 604 Haley, Stephen B. 629

Hall, D.B. 345 Hallam, Tony 630 Halley, Edmund 127 Halliday, A.N. 945 Halliwell, J.J. 162 Halvorson, H. 765 Hamblyn, Richard 627 Hamilton 1198 Hamilton, J. Dwayne 348 Hamilton, William 644, 1148 Hamilton, William Rowan 1197 Hammond, T.D. 769 Hanbury Brown, R. 765 Hanbury Brown, Robert 725 Hannout, M. 766 Hanns Ruder 302 Hanzlik, Marianne 530, 1296 Harari, H. 1029 Hardcastle, Martin 22 Hardin, C.L. 624 Hardy, G.H. 1226 Hardy, Godfrey H. 124, 663 Hardy, Godfrey H. 1272 Hare, Trent 1296 Harlen, V. 697 Haroche, S. 885, 886 Haroche, Serge 799 Harrington, D.L. 888 Harrington, R.F. 621 Harriot, T. 1148 Harris, S.E. 161, 343 Hart, J.C. 1226 Hart, Nathan H. 163 Hartle, J.B. 1133 Hartman, W.K. 168 Hartmann, D. 515, 893 Hartung, R.W. 792 Harvey, A. 345, 511 Hasenöhrl, F. 347 Hasenöhrl, Friedrich 319 Hashimoto, H. 992 Hatcher, W.S. 698 Hatfield, Brian 508 Hattori, T. 1029 Haubrich, D. 628 Hausch, T.W. 622, 1192 Hausherr, Tilman 22, 629 Haverkorn, M. 509

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Grossmann, A. 268 Grotz, K. 946 Gruber, C. 345 Gruber, Christian 301 Gruber, R.P. 514 Grünbaum, Branko 267 Grünberg, H.-H. von 891 Gschneidner, Karl A. 629 Guglielmini, Giovanni Battista 94 Guiragossian, Z.G.T. 344 Gunzig, E. 1026, 1054, 1067, 1097, 1132 Gurevich, A.V. 627 Gutfreund, Hanoch 343 Guth, A. 511 Guth, Alan 471 Gutierrez, G. 267 Gutiérrez-Medina, B. 767 Gutzwiller, Martin C. 168 Guy, R.K. 699 Gál, J. 622 Gödel, Kurt 667, 668, 698 Göklü, E. 343 Gácsi, Zoltán 22

1309

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name index

1310

H Heracles

Hotchner, Aaron E. 267 Houck, A.A. 625 Houdini, Harry 699 Houten, H. van 886 Howard, Luke 597 Howell, J.C. 791 Hoyle, C.D. 514 Hoyle, F. 511, 701 Hoyle, Fred 462, 511, 690, 923, 1061 Hoyt, S. 766 Hradil, Z. 767 Hsich, T. 889 Htun, Bo Bo 695 Hu, Z. 628 Hubble, Edwin 447 Huber, A. 622, 1192 Hudson, R.P. 270 Huff, D. 884 Hughes, David W. 267 Hughes, R.J. 509 Huiberts, J.N. 630, 888 Huisken, G. 374 Hulet, R.G. 767 Hulet, Rand 848 Hulse, Russel 507 Humboldt, Alexander von 1269 Humphreys, Roberta 166 Humphreys, W.J. 272 Hunklinger, Siegfried 270 Hunter, D.J. 163 Huppertz, H. 630 Hurwitz, Adolf 1203 Huxley 212 Huxley, A.F. 268 Huygens, Christiaan 78, 118, 276, 560 Hypatia 641, 667, 695 Hänggi, P. 886 Hänsch, T.W. 892 I Icke, Vincent 765 Ifrah, G. 1146 Illich, Ivan 691, 1227, 1228, 1229 Ilmanen, T. 374 Imae, Y. 163

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Hinshaw, G. 512 Hinz, Wolfgang 561, 1296 Hipparchos 100, 278 Hirano, M. 170 Hirshfeld, A.C. 993 Hirshfield, Stuart 697 Hirth, J.P. 344, 994 Hirzebruch, F. 1226 Hitler, Adolf 612 Hobbes, Thomas 827 Hodges, L. 169 Hodgkin 212 Hodgkin, A.L. 268 Hoeher, Sebastian 270 Hoekstra, Rolf F. 696 Hof, B. 272 Hoffer, Eric 1102 Hoffman, Donald D. 37, 697 Hogan, C.J. 967 Hohenstatt, M. 269, 628 Holdin, P. 993 Hollander, E. 269 Holst, Gilles 847 Holstein, B.R. 508 Holt, R.A. 886 Holzmüller, G. 507 Hones, Bill 628 Hong, C.K. 791 Hong, F.-L. 1171 Honner, John 1097 ’t Hooft, Gerard 662, 871, 932, 935, 942, 1009, 1028, 1054, 1120 Hooft, G.W. ’t 622 Hooke 124 Hooke, Robert 116, 203 Hopper, Arthur F. 163 Horace, in full Quintus Horatius Flaccus 152 Horgan, John 699, 946 Hornak, Joseph 1296 Hornak, Joseph P. 898 Hornberger, K. 884 Horne, M.A. 886 Horowitz, Gary T. 1133 Horsburgh, Steven 519, 520, 1296 Horváth, G. 622 Hoste, J. 993

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Heracles 439 Heraclitos of Ephesos 189 Heraclitus 30 Heraclitus of Ephesus 38, 450 Herbert, Wally 167 Hergerfeldt, G.C. 885 Hermes, H. 1226 Hernandez, A. 626, 1269 Hernandez, L. 1134 Herrmann, F. 270 Herrmann, Ludimar 583 Hersch, R. 698 Hersch, Reuben 647 Herschel, John 459 Herschel, William 103, 562 Hertz 545, 1201 Hertz, H.G. 164 Hertz, Heinrich 181, 350, 726, 757 Hertz, Heinrich Rudolf 152, 559 Hertz, J. 635 Hertzlinger, Joseph 22 Hesiod 447 Hess, Victor 745 Hestenes, D. 170, 345 Heumann, John 22 Hewett, J.A. 168 Hewitt, Leslie A. 158 Hewitt, Paul G. 158 Hewitt, Susan 967 Hey, J.D. 768 Hicks, K. 1192 Higashi, R. 1171 Higgs, P.W. 946 Higgs, Peter 936 Hilbert, David 182, 198, 422, 425, 656, 657, 667, 1049, 1127, 1194 Hilborn, R. 792 Hilborn, R.C. 1133 Hilgevoord, Jan 765 Hilico, L. 622 Hill, E.L. 1028 Hiller, R.A. 629 Hillery, M. 791 Hillman, Chris 22 Hillman, L. 1030 Hillman, L.W. 1030

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name index

I Imbrie

Imbrie, J. 166 Inaba Hideaki 846 Inaba, H. 889 Ingram, Jay 268 Innanen, Kimmo A. 168 Inverno, Ray d’ 374, 503, 1097 Irving, Washington 669 Irwin, Jim 623 Isham, C.J. 1026 Israel 482 Itano, W.M. 628, 767, 884 Itano, Wayno 747 Ivanov, Igor 22 Ives, H.E. 343 Ivry, R.B. 161, 888, 1067

Jetzer, P. 509 Jin, Deborah 848 Jin, Y. 889 Johansson, Mikael 22 Johnson, Ben 67, 163 Johnson, Samuel 503, 1142, 1225, 1227 Johnston, K.J. 163 Joliot, Frédéric 915 Joliot-Curie, Irène 915 Jones, Gareth 1172 Jones, H. 1148 Jones, P.D. 630 Jones, Quentin David 22 Jones, T.B. 628 Jones, Tony 1171 Jones, Vaughan 980 Jong, Marc de 22 Joos, E. 884, 894 Joos, Erich 884 Jordan, D.M. 627 Jordan, P. 947 Jordan, Pascual 707, 710, 939 Joseph Bohm, David 802 Joule, James 235 Joule, James Prescott 235 Joutel, F. 168 Joyce, James 801, 927 Jozefowski, L. 622 Ju, L. 1097, 1172 Julien, L. 622 Juszkiewicz, R. 512 Jürgens, H. 259 Jürgens, Hartmut 53

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Copyright © Christoph Schiller November 1997–May 2006

K K. Batchelor, George 992 Köppe, Thomas 22 Kac, Victor 1212 Kadanoff, L.P. 272 Kalbfleisch, G.R. 342 Kalckar, Jørgen 403 Kamerlingh Onnes, Heike 847 Kamerling Onnes, Heike 165 Kampen, N.G. van 271 Kanada Yasumasa 1173 Kanecko, R. 170 Kanno, S. 894 Kant, Immanuel 132, 437, 438,

448, 509, 645 Kantor, Yakov 1230 Kapitsa, Pyotr 847 Kapitza 762 Kapuścik, E. 344 Kara-Ivanov, M. 993 Kardar, M. 891 Karl, G. 267, 767, 1242 Karlhede, A. 515 Karliner, M. 1132 Karolyhazy, F. 1030, 1054 Kasparian, J. 891 Kasparov, Gary 817 Katori, H. 1171 Katsuura, K. 1029 Kauffman, L.H. 1226 Kauffman, Louis H. 993, 1226 Kaufmann, W.J. 442 Kawagishi, I. 163 Kawamura, M. 992 Kay 883 Kay, B.S. 894 Kay, Paul 624 Kayser, R. 511 Keating 380 Keating, R.E. 345 Keating, Richard E. 504 Keesing, R.G. 161, 1172 Keller 754 Keller, C. 767 Keller, Wilfred 1272 Kells, W. 1192 Kelu, Jonatan 22 Kelvin 235 Kemp, David 209 Kempf, A. 1028 Kempf, Achim 1098 Kempf, Jürgen 161 Kendall, H.W. 945 Kenderdine, M.A. 164 Kendrick, E. 170 Kennard, E.H. 159, 894, 1027 Kennedy, R.J. 343 Kenneth, O. 890 Kenny, T. 889 Kepler 124 Kepler, Johannes 116, 123, 459, 572, 862 Kerr, John 563

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J Jüstel, T. 891 Jackson, J.D. 620, 621 Jackson, John David 1171 Jackson, L. 696 Jacobi, Carl 181 Jacobs, J.P. 1031 Jacobson, J. 766 Jacobson, T. 374, 993, 1097 Jacobson, Ted 987 Jaekel, M.-T. 1027, 1097 Jaencke, Peter 700 Jahn, O. 268 Jalink, Kim 22 James, M.C. 169 Jamil, M. 22 Jammer, M. 167 Jammer, Max 764, 1097 Janssen, Jules 754 Jauch 1017 Jauch, W. 1029 Jeanloz, R. 621 Jeffreys, Harold 98 Jefimenko, Oleg D. 621 Jen, E. 272 Jenkins, Francis A. 342 Jennings, G.M. 268 Jensen, D. 993 Jensen, Hans 1152 Jensen, P. 700 Jeon, H. 620, 947 Jerauld, J. 627 Jerie, M. 515

1311

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name index

1312

K Kerr

Kuchiev, Michael 880 Kudo, S. 163 Kuerti, G. 625 Kumaradtya, K.K. 512 Kurizki, G. 624 Kurn, D.M. 767 Kuroda, Paul 918 Kurths, Jürgen 168 Kusch, K. 696 Kusch, S. 696 Kusner, R.B. 993 Kuzin, Pavel 22 Kuzmich, A. 624 Kuščer, I. 768 Kvale, E.P. 165 Kwait, P.G. 624 Kwiat, P.G. 625 Kwok, D.Y. 621 Kötter, K. 272 Kötter, Karsten 257, 1296 Künzle, H.P. 513 Küstner 166 Küstner, Friedrich 100 L La Caille 117 Lachièze-Rey, M. 511 Laer Kronig, R. de 947 Laer Kronig, Ralph de 939 Lafrance, R. 894 Lagrange 186 Lagrange, Joseph 1198 Lagrange, Joseph Louis 178 Lagrangia, Giuseppe Lodovico 178 Lakes, Rod S. 344 Lalande 117 Laloë, F. 764 Lamas-Linares, A. 791 Lamb 395 Lamb, Frederick 514 Lamb, Willis 857 Lambeck, K. 165 Lambert, Johann 418 Lambrecht, A. 890 Lamoreaux, S.K. 890, 1031 Lancaster, George 267 Landau, L. 509 Landauer, R. 697

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Kochen 810 Kochen, S. 886 Kockel, B. 768, 891 Koecher, M. 1226 Koeman, N.J. 630, 888 Kogut, A. 512 Kohlrausch, Friedrich 700 Kohshima, S. 888 Koltenbah, B.E.C. 625 Konishi, K. 1028 Konstantinov, M.Yu. 994 Kontorowa, T. 344, 993 Koolen, Anna 22 Koomans, A.A. 269, 620 Kopeikin, S.M. 507 Kopnin, N.B. 889 Korschinek, G. 510 Korteweg 213 Kostiuk, L.W. 621 Koul, R.K. 1133 Kovetz, A. 627 Kovtun, P. 1099 Kowalski, L. 626 Kozhekin, A.E. 624 Kramer, D. 629 Krehl, P. 269 Krehl, Peter 231 Kreimer, D. 1134 Kreimer, Dirk 867, 892, 1123 Krenn, G. 767 Kreuzer 385 Kreuzer, L.B. 505 Krexner, G. 891 Krider, E.P. 620 Krijn, Marcel 22 Krisher, T.P. 348 Krishnan, B. 374 Krogh, A. 635 Kronig, Ralph 749 Krotkow, R. 169, 506 Krumm, P. 507 Krusius, M. 889 Kruskal, M.D. 268 Kruskal, Martin 213, 653 Kryowonos, A. 766 Królikowski, Jarosław 22 Kröner, Ekkehart 374, 509 Kubala, Adrian 22 Kuchiev, M.Yu. 894

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Kerr, R.P. 513 Kerr, Roy 482 Kerswell, R. 272 Kesel, A.B. 889 Kettering, C.F. 620 Ketterle, W. 767 Ketterle, Wolfgang 848 Khan, S. 992 Kidd, R. 766 Kielpinski, D. 884 Killing, Wilhelm 1210 Kilmister, C.W. 347 Kim Song-Man 695 Kimble, H.J. 628 King, B.E. 884 Kippenhahn, R. 946 Kirchhoff, Gustav 753 Kiss, Joseph 22 Kistler 229 Kistler, S.F. 269 Kittinger 381, 504 Kittinger, Joseph 377 Kivshar, Y.S. 272 Kjellman, J. 342 Klapdor, H.V. 946 Klapdor-Kleingrothaus, H.V. 620 Klauder, J.R. 766 Klauder, John 503 Klebanov, I. 1132 Klein, Felix 658 Klein, Oskar 759 Kleinert, F. 700 Kleppner, Daniel 507 Klett, J.D. 994 Klich, I. 890 Klitzing, Klaus von 849 Kloor, H. 765 Klose, S. 515, 893 Kluge, Stefan 543, 1296 Kneib, J.-P. 768 Knie, K. 510 Knop, R.A. 508 Knuth, D. 698 Knuth, Donald 653 Knutsen, H. 510 Kobayashi, H. 888 Koch, G.W. 268 Koch, Robert 691

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name index

L L andauer

Lieu, R. 1030 Lie groups, non-linear 1219 Lifshitz, E. 509 Lifshitz, E.M. 890 Light, Michael 946 Lille, Alain de 476 Lilley, Sam 503 Lincoln, Abraham 695 Lindblad, G. 884 Linde, Johan 22 Linderholm, Carl E. 1194 Lindh, A.G. 506 Linfield, R.P. 346 Lingelbach, B. 626 Lingelbach, Elke 583 Lintel, H. van 345 Lintel, Harald van 22, 301 Lipperhey, Johannes 569 Lissauer, J.J. 168, 272 Lith-van Dis, J. van 271, 1098 Liu Gang 695 Liu, C. 161 Livingston, James D. 518 Llewellyn Thomas 749 Lloyd, S. 885, 1054 Lloyd, Seth 271, 1048 Lockyer, Joseph 754 Lodge, Oliver 98 Logan, R.T. 348 Loll, R. 1028 Lombard, Peter (the) 698 Lombardi, Luciano 22 Lombardo, F. 515 Longair, M. 510 Longo, M. 620, 947 Lorentz, H.A. 344, 345 Lorentz, Hendrik Antoon 1068 Lorentz, Hendrik A. 533 Lorentz, Hendrik Antoon 290, 294 Loschmidt, Joseph 220 Loss, D. 885 Lothe, J. 344, 994 Loudon, Rodney 765 Louisell, W.H. 767 Lovell, Jim 623 Lovász, P. 791 Low, R.J. 347

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Copyright © Christoph Schiller November 1997–May 2006

Leibfried, G. 344, 994 Leibniz 84, 124, 161, 176, 181, 1048 Leibniz, G. 1148 Leibniz, Gottfried 178 Leibniz, Gottfried Wilhelm 68, 1054 Leidenfrost, Johann Gottlieb 229 Leiderer, P. 891 Leighton, R.B. 158, 509, 1026, 1264 Lemaître, Georges A. 454 Lenin 531, 1008 Lennard, J. 1149 Lennerz, C. 1245 Lennie, P. 626 Lense, J. 506 Lense, Josef 397 Lenstra, J.K. 791 Leonardo da Vinci 740 Leonardo of Pisa 1146 Leone, F.C. 625 Lepak, J. 620 Leppmaier, Max 891 Lerner, L. 507 Lesage, G.-L. 169 Lesage, Georges-Louis 142 Leschiutta 380 Leschiutta, S. 504 Leucippus 679, 966 Leucippus of Elea 223 Levine, Peter A. 887 Levitov, L. 885 Levy, Silvio 993 Lewallen, Charles 115, 1295 Lewis, G.N. 346 Le Verrier, Urbain 129 Li, K. 625 Li, Y-Q. 1098 Li, Y.C. 627 Liang, Y. 889 Libby, Willard 910 Lichtenberg, Georg Christoph 42, 690, 1094 Lide, David R. 1152 Lie, Sophus 1209, 1218 Liebscher, Dierck-Ekkehard 343

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Landauer, Rolf 271 Landolt-Börnstein 700 Lang, K.R. 71 Lang, N.D. 626 Langacker, P. 946 Langel, R.A. 765 Laplace, Pierre 477 Laplace, Pierre Simon 129 Larousse, Pierre 699 Larraza, A. 890 Larsen, Jeffrey 166 Laskar, J. 169 Laskar, Jacques 143 Laskkar, J. 168 Lasota, J.P. 513 Latham, J. 627 Laue, Max von 325 Laugerette, F. 887 Laughlin, R.B. 889 Laughlin, Robert 849 Lavenda, B. 271 Lavenda, Bernard 1099 Lavoisier, Antoine 695 Lavoisier, Antoine-Laurent 77 Lawrence, Ernest 1185 Laws, Kenneth 272 Lawson, J.D. 945 Lawson, John 925 Lawvere, F. William 650 Le Bellac, M. 622 Leach, J. 767 Lebedev, P. 623 Lebedev, Peter/Pyotr 573 Leblanc, J. 1192 Lecar, M. 170 Ledingham, K.W.D. 947 Ledingham, Ken 940 Ledoux, Joseph 697 Lee Tsung-Dao 936 Lee, David M. 847 Lee, J. 1028, 1133 Lee, Raymond L. 622 Lee, T.D. 1028 Leeuwenhoek, Antoni van 569 Lega, Joceline 272 Legendre, A. 1148 Leggett, A.J. 884, 885 Lehmann, Paul 1149, 1152

1313

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name index

1314

L Lu

Mattheck, Claus 765 Matthews, Robert 169, 1271 Matthews, W.N. 347 Mattick 278 Mattick, A.T. 342 Mattick, Tom 1296 Maupertuis 94 Maus, S. 621 Maus, Stefan 542, 1296 Mavromatos, N.E. 1030, 1134 Maxwell 526, 672, 1201 Maxwell, Delle 993 Maxwell, James Clerk 545, 961 Mayer, A.R. 888 Mayer, Julius Robert 86, 235 Mayer, N.J. 696 Mayné, Fernand 22 Mayr, Peter 22 Mazur 482 Mazur, P.O. 513 McClure-Griffiths, N.M. 509 McCoy, Ray 93, 1295 McCrone, J. 888 McCullogh 625 McCuskey, S.W. 625 McDonald, K.T. 507 McDonald, Kirk T. 624 McGowan, R.W. 344 McGuire, Bill 630 McInerney, J.O. 887 McKellar, B.H.J. 346, 347 McKenna, P. 947 McLaughlin, William 158 McLean, H. 162 McMahob, T.A. 268 McMahon, T. A. 167 McMillen, T. 269 McNamara, Geoff 505 McQuarry, George 22 McTaggart 663 Mc Kellar, Bruce H.J. 768 Mead, C.A. 1027, 1097, 1132 Mechelli, A. 697 Meekhof, D.M. 884, 947 Meetz, Kurt 622 Meitner, Lise 913, 1185 Melo, F. 272 Mena Marugán, G.A. 511 Mende, P.F. 1028, 1134

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Copyright © Christoph Schiller November 1997–May 2006

M Maan, Jan Kees 628 MacCallum, M.A.H. 513 Macdonald, A. 514 MacDonald, G.J.F. 165 MacDougalls, Duncan 105 Mach, Ernst 79, 474 Macrobius 510 Maddox, J. 1133 Madelung, O. 700 Madison, K.W. 888 Maekawa, Y. 163 Maeterlink, Maurice 682 Maffei, Paolo 509 Maffi, Luisa 624 Magariyama, Y. 163 Maggiore, M. 1028 Magueijo, J. 1029, 1098 Magyar 852 Mahoney, Alan 22 Mahowald, M. 697 Mahrl, M. 272 Maiman, Theodore 850 Main, Peter 628 Mainwaring, S.R. 345 Mainzer, K. 1226

Majid, S. 1026, 1133 Major, S.A. 1134 Maldacena, J. 1134 Maleki, L. 348 Malik, J. 967 Malin, D. 71 Mallat, Stéphane 268 Malley, R. 170 Malus, Louis 562 Mandel 852 Mandel, L. 765, 766, 791 Mandelbaum, G. 1029 Mandelbrot, Benoit 219 Mandelbrot, Benoît 53 Mandelstam, Stanley 267 Manly, Peter 569 Mann, A. 890 Mann, A.K. 946 Mao, Y. 1279 Marciano, W.J. 946 Marcillac, P. de 1192 Marcus, Richard 159 Mariano, A. 767 Maricourt, Pierre de 529, 620 Mark, M.B. van der 622 Mark, Martin van der 251 Mark, Martin van der 22 Markow, T.A. 992 Markus, M. 272 Marmo, G. 1026 Marques, F. 1192 Marsaglia, G. 170 Marsh, N.D. 510 Martikainen, J.E. 1171 Martin, A. 267, 889 Martin, S. 630 Martos, Antonio 22, 561, 576, 1296 Martz, Paul 219, 1296 Marx, Groucho 1143 Marzke, R.F. 514 Mashhoon, B. 507 Mason, W.P. 344, 994 Masood-ul-Alam, A.K.M. 513 Massa, Corrado 373, 1068 Massar, S. 791 Masterson, B.P. 947 Matsas, G.E.A. 346 Matsukura, F. 629

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Lu, F. 621 Luca Bombelli 22 Lucrece 45, 223, 1132 Lucretius Carus, Titus 223, 1132 Ludvigsen, Malcolm 503 Lui, A.T.Y. 765 Luke, Lucky 226, 287 Luminet, J.-P. 511 Lundeen, J.S. 766 Lundmark 448 Lundmark, K. 509 Lundmark, Knut 447 Luo, J. 622 Lutes, G.F. 348 Lynch, D.K. 626 Lämmerzahl, C. 514, 892 Lévy-Leblond, J.-M. 622 Lévy-Leblond, Jean-Marc 91, 142, 158, 620, 764 Lüders, Klaus 158 Lühr, H. 621

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name index

M Mende

Munzner, Tamara 993 Muramoto, K. 163 Murata, Y. 170 Murdock, Ron 22 Mureika, J.R. 170 Murillo, Nadia 22 Murphy, R. 1148 Murray, C.D. 169 Murray, J.J. 344 Musil, Rober 458 Musiol, Gerhard 629 Muskens, O. 268 Muynck, Wim de 22 Myatt, C.J. 884 Myers, R.C. 894 Møller, Christian 347 Müller, G. 272 Müller, Gerhard 262 Müller, H. 343 Müller, J. 515 N Nabarro, Frank R.N. 344 Nagano, M. 515 Nagaoka Hantaro 747 Nagaoka, H. 766 Nahin, Paul J. 345 Nairz, O. 767 Nakahara, Mikio 1226 Nakazato, H. 888 Namba, K. 163 Namiki, M. 888 Namouni, F. 169 Namouni, Fathi 22 Nanopoulos, D.V. 1030 Nansen, Fridtjof 165 Napiwotzki, R. 168 Napoleon 129 Narlikar, J.V. 511 Nassau, K. 74 Natarajan, V. 346 Naudin, Jean-Louis 541, 1296 Nauenberg, M. 167, 946 Navier, Claude 263 Neddermeyer, Seth 904 Needham, Joseph 267 Neidhart, B. 628 Nelson, A.E. 168 Nemiroff, R.J. 506

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Copyright © Christoph Schiller November 1997–May 2006

Mitchell, M.W. 766 Mitskievic, N.V. 512 Mittelstaedt, H. 512 Mittelstaedt, M.-L. 512 Mlynek, J. 343 Moalic, J.-P. 1192 Mohazzabi, P. 169, 504 Mohideen, U. 890 Mohr, P.J. 1172, 1175 Mokros, Nicolai 302, 303, 1296 Molière 691 Monitz, E.J. 164 Monken, C.H. 766 Monroe, C. 884, 885 Montgomery, H. 621 Montgomery, R. 993 Montie, E.A. 622 Montmayeur, J.P. 887 Montonen, C. 1132 Montonen, Claus 556 Moody, Robert 1212 Mooij, J.E. 885 Moon, F.C. 628 Moore, C.P. 512 Moore, Henry 416 Moore, J.A. 267 Moortel, Dirk Van de 22, 1259 Moothoo, D.N. 623 Moreau de Maupertuis, Pierre Louis 94 Morin, B. 993 Morin, Bernard 976 Morinigo, Fernando B. 508 Morlet, J. 268 Morley, E.W. 345 Morley, Edward William 290 Morrow, P.R. 888 Morré, D.J 888 Moser, Lukas Fabian 22 Moses Maimonides 1001 Mottelson, Ben 848 Mozart 693, 827 Mozurkewich, D. 163 Mugnai, D. 625, 626 Mullarkey, M. 887 Muller, R.A. 268 Mulligan, J.F. 164 Munk, W.H. 165

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Mende, Paul 1124 Mendeleiev, Dimitri Ivanovitch 1185 Mendoza, E. 272 Menocal, P.B. de 510 Mensky, M.B. 764 Menten, K.M. 512 Merckx, Eddy 87 Mermin, D. 895 Mermin, David 581, 612 Merrit, John 22 Mertens, K. 272 Meschede, D. 628 Metha, A. 626 Mettrie, J. Offrey de la 260 Mewes, M.-O. 767 Meyenn, K. von 768 Meyer, D. 1028 Michaelson, P.F. 1165 Michel, Stanislav 83 Michell, J. 513 Michell, John 120, 477 Michelson 352 Michelson, A.A. 345 Michelson, Albert 98, 618 Michelson, Albert Abraham 290 Miescher, Friedrich 838 Mikiewicz, M. 946 Mikkola, Seppo 168 Milankovitch, Milutin 102 Millikan 905, 1269 Milonni, P.W. 765, 890 Milton, K.A. 620 Mineyev, A. 269 Minkowski, Hermann 294, 295 Minnaert, Marcel 576 Minnaert, Marcel G.J. 74 Mirabel, I.F. 161, 346 Misawa, H. 891 Mishra 334 Mishra, L. 348 Misner 433 Misner, C. 1097 Misner, C.W. 792, 1026 Misner, Charles 348 Misner, Charles W. 1152 Mitchell, Edgar 853

1315

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name index

1316

N Nesvizhevsky

Osterle, Fletcher 545 Otto, Rudolf 693, 1061 Ou, Z.Y. 791 Oughtred, W. 1148 Overhauser, A.W. 892 Ovidius, in full Publius Ovidius Naro 280, 833

O O’Connell, Sanjida 696 O’Doberty, J. 697 O’Keefe, R. 167 Oberdiek, Heiko 22 Oberquell, Brian 22 Occam, William of 1115 Ockham, William of 1115 Odysseus 1272 Oersted 531 Oersted, Hans Christian 531 Oey, M.S. 509 Offner, Carl 22 Offrey de la Mettrie, J. 260 Ohanian, H.C. 374 Ohanian, Hans C. 503, 504, 792, 1099, 1152 Ohm, Georg Simon 592 Ohno, H. 629 Ohno, Y. 629 Ohtani, K. 629 Okhapkin, M. 343, 345 Okun, Lev B. 347 Olbers, Wilhelm 458 Olive, D. 1132 Olive, David 556 Olmsted, John M.H. 162, 1224 Olum, K.D. 505 Omiya, T. 629 Oostrum, Piet van 22 Oppenheimer, J. 1028 Oppenheimer, R. 513 Oppenheimer, Robert 183, 478, 695 Orban, F. 620 Orlando, T.P. 885 Orlitzky, A. 791 Ormos, P. 623 Osewold, Daniel 1296 Osheroff, Douglas D. 847 Osserman, Bob 470

P Padgett, M. 767 Padgett, M.J. 623 Padgett, Miles 623 Padmanabhan, T. 1027, 1132 Paffuti, G. 1028 Page, Don 22, 878, 879 Page, Don N. 513, 894 Pagliarin, A. 946 Paine, Thomas 669 Pais, A. 268, 699 Palmer, John D. 161, 888 Palmer, R. 635 Pan, J.-W. 766 Panov, V.I. 169, 506 Papapetrou, A. 347 Papini, G. 768, 1099 Pappus 77 Parazzoli, C.G. 625 Park, David 158, 622 Parker, Barry 510 Parker, D.E. 630 Parkes, Malcolm B. 1152 Parlett, Beresford 36 Parmenides 31, 69 Parmenides of Elea 30 Parrott, S. 627 Pascal, Blaise 721 Pascazio, S. 767, 888, 1132 Pascazio, Saverio 22, 747, 888, 1068 Pasi, Enrico 22 Passilly-Degrace, P. 887 Paterson, Alan L.T. 162 Pati, J.C. 1029 Patris, B. 887 Paul, H. 766, 891 Paul, T. 268 Paul, W. 347 Paul, Wolfgang 902 Pauli, Wolfgang 310, 508, 664,

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Copyright © Christoph Schiller November 1997–May 2006

Novikov 449 Novikov, I.D. 510 Novikov, Igor 503, 512, 513, 893 Novikov, V.A. 267, 767, 1242 Nutsch, W. 993 Nuñes, Pedro 63

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Nesvizhevsky, V.V. 892 Neuhauser, W. 269, 628 Neukammer, J. 768 Neukirch, J. 1226 Neumaier, Heinrich 22, 373 Neumann, Dietrich 126, 168 Neumann, John von 794, 809 Neumann, János von 810 Neve, Erik de 977 New, M. 630 Newcomb, Simon 129, 169 Newman, E.T. 513 Newton 124, 161, 732, 1163 Newton, Isaac 42, 94, 117 Nez, F. 622 Ng 1035 Ng, Y.J. 1027, 1030, 1054, 1097 Nicolai, H. 508 Nicolis, Grégoire 271 Niederer, U. 197, 268 Nieminen, T.A. 624 Nienhuis, G. 768 Nieto, L.M. 508 Nieto, M.M. 767 Nietzsche, Friedrich 359 Nieuwpoort, Frans van 22 Nieuwstadt, F.T.M. 272 Nieves, F.J. de las 167 Nightingale, J.D. 267, 503 Nimtz, G. 625 Nimtz, Günter 580 Niobe 1186 Niot, I. 887 Nishida, N. 891 Niu, Q. 888 Nobel, Alfred 1186 Nobel, P. 269 Noecker, M.C. 947 Noether, Emmy 198 Nojiri, S. 992 Nollert, H.-P. 894 Nolte, John 160 Nonnius, Peter 63 Noppeney, U. 697 Norbury, John W. 886 Nordström, Gunnar 482 Nordtvedt, Kenneth 515 Norrby, L.J. 888 Norton, John D. 508

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name index

P Pauling

Popescu, S. 791 Popper, Karl 670, 699 Popper, Karl R. 701 Pound 380 Pound, R.V. 504 Povh, Bogdan 945 Povinelli, D.J. 690 Powell, Cecil 904 Powell, R. 887 Powell, Richard 446, 450, 1296 Poynting, John Henry 552 Pradl, O. 343 Prakash, A. 513 Prentis, Jeffrey J. 766 Prentiss, M. 623 Preparata, G. 893, 1098 Prescott, James W. 888 Press, Hans J. 887 Prestel, A. 1226 Preston, S. Tolver 347 Preston, Tolver 319 Preston-Thomas, H. 270 Pretto, Olinto De 318, 343 Prevedelli, M. 622, 1192 Prewitt, O. 884 Price, C.J. 697 Price, R.H. 82 Prigogine, Ilya 511, 830 Primas, L.E. 348 Prince, G.E. 515 Pritchard, Carol 22 Pritchard, D.E. 346, 768, 769 Pritchard, David 314, 752 Proença, Nuno 22 Prometheus 1187 Protagoras 1163 Prout, William 902 Provero, P. 1028 Pruppacher, H.R. 994 Pryce, M.H.L. 347 Pryce, Robert M. 167 Ptolemy 62 Purcell, E. 992 Putkaradze, V. 272 Putkaradze, Vakhtang 1296 Putterman, S.J. 166 Putterman, Seth J. 629 Pythagoras 214, 656, 1067

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Copyright © Christoph Schiller November 1997–May 2006

Petit, Jean-Pierre 1231 Pfenning 883 Pfenning, M.J. 894 Pfister, Herbert 512 Pflug, Hans 823 Phillips, A. 993 Phillips, Melba 695 Phillips, R.J. 168 Philpott, R.J. 348 Piaget, J. 634 Piaget, Jean 159, 634 Picasso, Pablo 555, 639, 685 Piccard, Auguste 228 Pikovsky, Arkady 168 Pimpinelli, A. 891 Pines, David 848 Pinker, Steven 161, 697 Pinkse, P. 769 Piper 408 Piper, M.S. 507 Piran, T. 1030 Pitnick, S. 992 Pittacus 1106 Planck, M. 1027, 1132 Planck, Max 244, 310, 326, 351, 612, 705, 722, 726, 750, 764, 1064 Plastino, A. 791 Plastino, A.R. 791 Plato 223, 263, 462, 887, 1035, 1060, 1116, 1133 Plinius, in full Gaius Plinius Secundus 158 Plisson, Philip 35, 1295 Plutarchus 173 Podolsky, B. 885 Pohl, Robert 158 Pohl, Robert O. 158 Poincaré, Henri 104, 282, 292, 293, 377, 393, 693, 726, 766 Poincaré, J.H. 347 Poinsot, Louis 97 Poisson, Denis 571 Poisson, Siméon-Denis 121 Polchinski, J. 1133 Polder, D. 890 Politzer, David 932 Polster, B. 270 Pompeius, Gnaeus 173

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710, 729, 749, 762, 769, 792 Pauling, Linus 888 Paulsen, C. 885 Paulus of Tarsus 667 Pavlis, E.C. 506 Pazzi, G.P. 625 Peş ić, P.D. 888 Peano, G. 1148 Pearson, T.J. 346 Peary, Robert 116, 167, 1246 Pecharsky, Vitalij 629 Peeters, Bert 22 Peirce, Benjamin 1207 Peirce, Charles 641, 667 Peitgen, H.-O. 259 Peitgen, Heinz-Otto 53 Pekár, V. 169, 506 Pello, R. 768 Pelt, J. 511 Pelt, Jaan 468 Pendry, J. 625 Pendry, John 579, 580 Peng, J.L. 628 Penrose 370 Penrose, R. 346, 374, 513, 514, 792 Penrose, Roger 484, 489, 512, 699 Penzias, Arno 449 Penzias, Arno A. 510 Pepper, M. 890 Perc, M. 164 Peredo, M. 765 Perelman, Yakov 158, 217 Peres, A. 512, 1028 Peres, Asher 764 Perini, Romano 22 Perkins, D.H. 345 Perlman, E. 512 Perot 380 Perrin, J. 270, 766 Perrin, Jean 243, 271, 747 Perros, Georges 827 Persius 683 Peşić, P.D. 510 Peters, A. 343, 892 Petersen, Nicolai 1296 Peterson, Peggy 64, 1295 Petit, J.-P. 993

1317

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1318

P Py ykkö

Pyykkö, P. 888 Pěč, Karel 162 Pádua, de 727 Pádua, S. de 766 Päsler, Max 267 Pérez–Mercader, J. 162

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Copyright © Christoph Schiller November 1997–May 2006

Roberts, M. 891 Roberts, P.H. 621 Robertson, H.P. 454, 895 Robinson 482 Robinson, D.C. 513 Robinson, John Mansley 160, 267 Robutel, P. 168 Rodgers, P. 628 Rodin, Auguste 118 Rodríguez, L.F. 161 Rodrigues, W.A. 626 Rodriguez, L.F. 346 Roger, G. 885 Rohrbach, Klaus 166 Rohrlich, F. 628 Roll, P.G. 169, 506 Romalis, M.V. 1031 Roman 883 Roman, T.A. 894 Romanowicz, B. 621 Romer, R.H. 1249 Roorda, A. 626 Roorda, Austin 585, 586, 1296 Rooy, T.L. van 269, 620 Rosek, Mark 1296 Rosen, N. 885, 1028 Rosenblum, Michael 168 Rosenfeld 734, 1045 Rosenfeld, L. 766, 1054 Ross Macdonald, Malcolm 696 Ross, S.B. 1029 Rossi, B. 345 Rosu, H. 893 Rothbart, G.B. 344 Rothenstein, B. 344 Rothman, T. 1054 Rothmann, T. 511 Rottmann, K. 348 Roukema, B.F. 511 Rovelli, C. 1026, 1028, 1134 Roy, A. 890 Rozental, I.L. 515 Ruben, Gary 22 Rubinstein, J. 627 Rubinsztein-Dunlop, H. 624 Rucker, Rudy 649, 650, 698, 1226

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Rector, J.H. 630, 888 Redondi, Pietro 42, 220 Rees, W.G. 158 Refsdal, S. 511 Regan, B.C. 1029 Regiomontanus, J. 1148 Řeháček, J. 767 Q Reichert, J. 622, 1192 Quinlan, G.D. 169 Reichl, Linda 270, 271, 630, 884 R Reimers, D. 168 R.N. Nabarro, Frank 994 Reissner, Hans 482 Röngten, Conrad 899 Remmert, R. 1226 Radzikowski 883 Rempe, G. 769 Radzikowski, M. 894 Renaud, S. 1027, 1097 Ragazzoni, R. 1030 Renselle, Doug 22 Rahtz, Sebastian 22 Reppisch, Michael 22 Raimond, J.-M. 886 Research Council Staff, National 627 Raizen, M.G. 767, 888 Raizen, Mark 747 Revzen, M. 890 Rakov, V.A. 627 Reynolds, Osborne 623 Ramaccini, F. 699 Reznik, B. 1028 Ramakrishna, S.A. 625 Rhodes, C. 892 Ramanujan, Srinivasa 124 Ricci-Cubastro, Gregorio 422 Ramberg 778 Richard, J. 768 Ramberg, E. 791 Richards, P.G. 627 Rambo, K.J. 627 Richardson, Lewis Fray 53 Ramsey, N.F. 1029 Richardson, Robert C. 847 Randi, James 35, 699, 700, 967 Richtmeyer, F.K. 159 Randjbar-Daemi, S. 1054 Ridgeway, S.L. 628 Ranfagni, A. 625, 626 Rieflin, E. 163 Rankl, Wolfgang 22 Rieke, F. 765 Rao, S.M. 888 Riemann, Bernhard 435, 563 Rappmann, R. 697 Rigor, I.G. 630 Rassoul, H.K. 627 Rikken, Geert 629 Rawlins, D. 167 Rimini, A. 892 Rawlinson, A.A. 346, 347 Rinaldo, A. 267 Raymer, M.G. 895, 1027 Rindler, W. 345, 792 Raymer, Michael 1002 Rindler, Wolfgang 343, 374, Raymond Smullyan 667 503, 508, 622, 1097, 1099 Raymond, D.J. 348 Rindt, Jochen 43 Raymond, David 1230 Rinnooy Kan, A.H.G. 791 Readhead, A.C.S. 346 Rith, Klaus 945 Reball, Siegfried 629 Ritter, Johann Wilhelm 562 Rebka 380 Ritz 342 Rebka, G.A. 504 Rivas, M. 626 Recami, E. 346 Rivas, Martin 22 Recorde, R. 1148 Robbins, J.M. 792, 1133 Recorde, Robert 1147 Roberts, J.E. 1028

name index

Dvipsbugw


name index

R Ruder

Schleich, W.P. 884 Schlichting, H.-J. 1245 Schmidt, Herbert Kurt 345 Schmidt, Klaus T. 163 Schmidt, M. 893 Schmidt, T. 272 Schmidt-Nielsen, K. 271 Schmiedmayer, J. 769 Schneider, Jean 267 Schneider, M. 515 Schneider, P. 511 Schneider, Wolf 1153 Schoeck, Helmut 701 Scholz, Christoph 945 Schooley, J.F. 270 Schrödinger, Erwin 715, 745, 767, 839 Schramm, Herbert 514 Schramm, T. 511 Schrieffer, J. Robert 847 Schrieffer, J.R. 890 Schröder, Ulrich E. 504 Schröder, Ulrich E. 347 Schrödinger 578, 672 Schrödinger, Erwin 895, 1104 Schubert, Max 764 Schucking, E. 345, 511, 768 Schulte 108 Schultz, Charles 1125 Schultz, S. 625 Schunck, N. 946 Schuster, S.C. 992 Schutz, B.F. 507 Schutz, Bernard 503 Schwab, K. 892 Schwartz, Richard 159 Schwarz, J.H. 1026, 1132 Schwarzschild 379 Schwarzschild, Bertram 891 Schwarzschild, Karl 385 Schwenk, Jörg 814 Schwenk, K. 272 Schwenkel, D. 269 Schwinger, Julian 620, 670, 764 Schwob, C. 622 Schäfer, C. 700 Schäfer, G. 514 Schön, M. 1027, 1132

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Copyright © Christoph Schiller November 1997–May 2006

S Saa, A. 627 Sackett, C.A. 767, 884 Sacks, Oliver 697 Sade, Donatien de 152, 969 Sagan, Carl 1054 Sagan, Hans 1224 Sagnac, Georges 98 Sahl, Mort 691 Saitou, T. 889 Sakar, S. 1030 Sakharov, A.D. 1029, 1098, 1132 Sakharov, Andrei 695, 925, 1012, 1078, 1079 Salam, A. 1029 Salam, Abdus 765 Salamo, G. 623 Salditt, T. 626 Salecker 1127

Salecker, H. 1027, 1054, 1134 Salingaros, N. 621 Sally, P. 1226 Salmonson, J.D. 893 Samarski, Wassily 1188 Saminadayar, L. 889 Samuel, Stuart 507 Sanchez, N.G. 1134 Sanctorius 80 Sands, M. 158, 509, 1026, 1264 Sanger, Frederick 847 Santander, M. 508 Santorio Santorio 80 San Suu Kyi, Aung 170 Sarazin, X. 946 Sassen, K. 624 Sastry, G.P. 346 Satake Ichiro 1217 Sauer, J. 1245 Saupe, D. 259 Saupe, Dietmar 53 Saussure, Ferdinand de 641 Savela, Markku 189, 1296 Scarcelli, G. 344 Schadwinkel, H. 628 Schaefer, B.E. 343, 344, 1030 Schaerer, D. 768 Schanuel, Stephen H. 650 Scharein, Robert 978, 979, 1297 Scharmann, Arthur 514 Scharnhorst, K. 890 Schata, P. 697 Scheer, Elke 626 Schelby, R.A. 625 Schiff, L.I. 620 Schild, A. 1028 Schiller, Britta 22 Schiller, C. 269, 374, 620, 1026, 1054, 1067, 1097, 1098, 1132 Schiller, Christoph 374, 1297 Schiller, Friedrich 717 Schiller, Isabella 22 Schiller, P. 344, 994 Schiller, Peter 22 Schiller, S. 343 Schiller, Stephan 1296 Schilthuizen, Menno 696 Schlegel, K. 627

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Ruder, H. 515 Rudolff, C. 1148 Rudolff, Christoff 1152 Ruff, G.A. 768 Ruffini 875 Ruffini, R. 374, 505, 513, 893, 1098 Ruffini, Remo 503, 504, 1099, 1152 Ruga, Spurius Carvilius 1143 Rugel, G. 510 Ruggieri, R. 626 Ruggiero, M.L. 347, 507 Ruppel, W. 169, 504 Rusby, R.L. 270, 1172 Rusby, Richard 1172 Ruschewitz, F. 628 Russel, J.S. 268 Russell, Bertrand 326 Russell, Betrand 180 Russell, John Scott 212 Russo, Lucio 158 Rutherford, Ernest 36, 899, 901, 1188 Rybicki, G.R. 346 Rydberg, Johannes 754 Rüffer, U. 993 Rømer, Ole 276

1319

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name index

1320

S

Smale, S. 993 Smale, Stephen 976 Smirnov, B.M. 627 Smith, D.R. 625 Smith, David 625 Smith, J.B. 345 Smith, J.F. 885 Smith, S.P. 623 Smith, Steven B. 1147 Smolin, L. 967, 1028, 1029, 1098 Smoluchowski, Marian von 243 Smoluchowski, Marian von 242 Snider, J.L. 504 Snow 778 Snow, G.A. 791 Snyder, H. 513 Snyder, H.S. 1028 Snyder, Hartland 478 Socrates 263 Soffel, M. 515 Soffel, Michael H. 508 Soffer, B.H. 626 Soldner 409, 410 Soldner, J. 169, 505 Soldner, Johann 133, 390 Solomatin, Vitaliy 22 Solomon, R. 1226 Sommerfeld, Arnold 578, 624, 762 Son, D.T. 1099 Sonett, C.P. 165 Song, K.-Y. 624 Song, X.D. 627 Sonoda, D.H. 512 Sonzogni, Alejandro 1297 Sorabji, R. 1029 Sorkin, R.D. 1026, 1028, 1133 Sossinsky, Alexei 993 Soukoulis, C.M. 625 Sparenberg, Anja 629 Sparnaay, M.J. 890 Sparnaay, Marcus 856 Specker 810 Specker, E.P. 886 Spence, J.C.H. 269 Spencer, R. 161, 888, 1067

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Copyright © Christoph Schiller November 1997–May 2006

Sheldon, E. 345 Sheldon, Eric 22 Sheng, Z. 270 Shephard, G.C. 267 Shepp, L. 791 Sherrill, B.M. 1192 Shih, M. 623 Shih, Y. 344 Shimony, A. 886 Shinjo, K. 170 Shirham, King 41 Shockley, William 847 Short, J. 1172 Shulman, Polly 698 Shupe, M.A. 1029 Sierra, Bert 22 Sigwarth, J.B. 169 Silk, J. 512 Sillett, S.C. 268 Silverman, M. 163 Silverman, M.P. 768 Silverman, Mark 762 Silverman, Mark P. 169, 768, 895, 1256 Simon, C. 791 Simon, Julia 22 Simon, M.D. 628 Simon, M.I. 163 Simon, R.S. 346 Simoni, A. 1026 Simonson, A.J. 1248 Simplicius 264, 1113 Simpson, A.A. 886 Simpson, N.B. 623 Singh, C. 164 Singh, T.P. 514 Singhal, R.P. 947 Singleton, D. 626 Singleton, Douglas 22 Sirlin, A. 946 Sitter, W. de 508 Sitter, Willem de 414, 458 Sivardière, Jean 622 Skrbek, L. 889 Slabber, André 22 Slater, Elizabeth M. 569 Slater, Henry S. 569 Slavnov, A.A. 1133 Sloterdijk, Peter 921

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Schönenberger, C. 269, 620 Schörner, E. 1245 Sciama, D.W. 512 Sciama, Dennis 475, 512 Scott, G.G. 620 Schönenberger Scott, Jonathan 22 Scott, W.T. 628 Scriven 229 Scriven, L.E. 269 Seaborg, Glenn 1188 Searle, Anthony 302 Seeger, A. 344, 994 Seeger, J. 626 Segev 571 Segev, M. 623 Segev, Mordechai 623 Segrè, Gino 270 Seidelmann, P. Kenneth 1172 Seidl, T. 889 Seielstad, G.A. 346 Selig, Carl 343 Semon, Mark D. 621 Sen, A. 1134 Send, W. 992 Send, Wolfgang 992 Seneca, Lucius Annaeus 491 Serena, P.A. 889 Sessa ben Zahir 41 Sexl, R.U. 512 Sexl, Roman 56, 495 Sexl, Roman U. 345, 514 Shackleton, N.J. 166 Shakespeare, William 35, 846, 1112 Shalyt-Margolin, A.E. 271, 1098 Shankland, R.S. 625 Shapere, A. 993 Shapiro, A.H. 165 Shapiro, Arnold 976 Shapiro, Asher 96 Shapiro, I.I. 507, 508 Shapiro, Irwin I. 410 Sharkov, I. 891 Sharma, Natthi L. 164 Shaw, George Bernard 662, 735 Shaw, R. 345 Shea, J.H. 504

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name index

S Spicer

T Tőrők, Gy. 891 Tabor, D. 170 Tait, P.G. 1148 Tajima, T. 893 Takamoto, M. 1171 Takita, A. 891 Talleyrand 119 Tamm, Igor 925

Tamman, Gustav 230 Tanaka, K. 1028 Tangen, K. 509 Tanielian, M. 625 Tantalos 1186 Tarko, Vlad 22 Tarski, Alfred 55, 220, 699 Tartaglia, A. 507 Tartaglia, Niccolò 42 Tauber, G.E. 627 Taylor, B.N. 1172, 1175 Taylor, Edwin F. 343, 346, 348, 374, 503, 506, 513 Taylor, G.J. 168 Taylor, J.H. 507, 508, 1162 Taylor, John R. 621, 1172 Taylor, Joseph 407, 507 Taylor, R.E. 945 Taylor, W.R. 993 Taylor, William 981 Teeter Dobbs, Betty Jo 161 Tegelaar, Paul 22 Tegmark, M. 508, 884, 886 Telegdi, V.L. 271 Teller, Edward 915 Tennekes, Henk 992 Terence, in full Publius Terentius Afer 816, 1101 Terentius Afer, Publius 816, 1101 Terletskii, Y.P. 624 Terrell, J. 346 Tesla, Nikola 537 Thaler, Jon 22 Thales of Miletus 518 Theodoricus Teutonicus de Vriberg 576 Thidé, Bo 548 Thierry, Paul 694 Thies, Ingo 22 Thirring, H. 506 Thirring, Hans 397 Thistlethwaite, M. 993 Thober, D.S. 626 Thomas Aquinas 652, 1280 Thomas, A.L.R. 992 Thomas, Adrian 970, 1297 Thomas, L. 768 Thomas, Llewellyn 310

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Copyright © Christoph Schiller November 1997–May 2006

Stromberg, G. 509 Stromberg, Gustaf 447 Strominger, A. 1133 Strunk, William 1153 Strunz, W.T. 884 Styer, D. 624, 885 Su, B. 889 Su, Y. 169, 506 Subitzky, Edward 967 Suchocki, John 158 Sudarshan, E.C. 346 Sudarshan, E.C.G. 346, 766 Sugamoto, A. 992 Sugiyama, S. 163 Sullivan, John 977, 1297 Sulloway, Frank J. 663 Sun, X.L. 627 Sundaram, B. 888 Supplee, J.M. 346 Surdin, Vladimir 22, 626 Surry, D. 162 Susskind, L. 1054, 1132 Susskind, Leonard 1106 Sussman, G.J. 169 Suzuki, M. 1029 Svensmark, H. 510 Svozil, K. 890 Swackhamer, G. 170 Swagten, Henk 629 Swenson, C.A. 270 Swift, G. 271 Swinney, H.L. 272 Swope, Martha 272 Synge, J.L. 347 Szczesny, Gerhard 700 Szilard, L. 271, 697, 1098 Szilard, Leo 245, 638, 639, 1079 Szuszkiewicz, E. 513

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Spicer, G. 992 Spiderman 226 Spieker, H. 625 Spinoza, B. 1132 Spinoza, Baruch 1132 Spiropulu, M. 168 Sreedhar, V.V. 268 Sriramkumar 859 Sriramkumar, L. 891 Srygley, R.B. 992 Srygley, Robert 970, 1297 Stachel, John 503 Starinets, A.O. 1099 Stark, Johannes 286 Stasiak, A. 993 Stearns, Stephen C. 696 Stedman, G.E. 165, 345, 1172 Stegeman, George 623 Steinberg, A.M. 624, 625, 766 Steiner, Kurt 269 Steinhaus 1271 Stengl, Ingrid 161 Stephan, S. 345 Stephenson, G. 347 Stephenson, G.J. 346 Stephenson, Richard 165 Stern, O. 766 Stern, Otto 742 Stettbacher, J.K. 696 Steur, P.P.M. 270 Stewart, I. 159, 698, 993 Stewart, Ian 55, 699, 1028 Stilwell, G.R. 343 Stocke, J.T. 512 Stodolsky, Leo 344 Stoehlker, Thomas 891 Stokes, Georges Gabriel 263 Stone, Michael 1133 Stoney, G.J. 514, 1171 Stoney, George 527 Stoney, George Johnston 1159 Stong, C.L. 163 Stormer, H.L. 889 Story, Don 22 Stowe, Timothy 889 Strassman, Fritz 913 Strauch, F. 628 Strauch, S. 945 Straumann, N. 510

1321

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name index

1322

T Thompson

Trevorrow, Andrew 22 Trout, Kilgore 470 Trower, W.P. 620 Truesdell, C. 270 Truesdell, Clifford 42 Truesdell, Clifford A. 161 Tsagas, C.G. 628 Tsai, W.Y. 620 Tsang, W.W. 170 Tschichold, J. 1152 Tschichold, Jan 887 Tsuboi, Chuji 162 Tsubota, M. 889 Tsui, D.C. 889 Tu, L.-C. 622 Tucholski, Kurt 190 Tuckermann, R. 628 Tuppen, Lawrence 22 Turatto, M. 1030 Turchette, Q.A. 884 Turnbull, D. 1098 Turner, M.S. 166, 512 Twain, Mark 92, 662, 671, 691 Twamley, J. 792 Twiss, R.Q. 765 Twiss, Richard 725 Tyler, R.H. 621 Tyler Bonner, J. 268

V Vafa, C. 1133 Valanju, A.P. 625 Valanju, P.M. 625 Valencia, A. 344 Valentin, Karl 828, 840 Valsiner, Jaan 696 Valsinger, Jaan 696 van Druten, N.J. 767 Vanadis 1190 Vancea, I.V. 1054 Vandewalle, N. 270 Vanier, J. 1171 Vannoni, Paul 22 Van Dam, H. 621, 1027, 1030, 1054, 1097 Van Den Broeck, Chris 505 Vardi, Ilan 791 Vavilov 717 Vavilov, S.I. 765 Veer, René van der 696 Veneziano, G. 1026, 1028 Vergilius 687 Vergilius, Publius 396 Vermeil, H. 508 Vermeulen, R. 512 Verne, Jules 576 Vernier, Pierre 63 Veselago, V.G. 624, 625 Veselago, Victor 579 Vessot 380 Vessot, R.F.C. 504 Vestergaard Hau, L. 161, 343 Vico, Giambattista 811 Vigotsky, Lev 634, 696 Villain, J. 891 Vincent, D.E. 167 Visser, Matt 893 Viswanath, R.N. 629 Vitali, Giuseppe 53 Viviani, Vincenzo 96

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Copyright © Christoph Schiller November 1997–May 2006

U Ucke, C. 1245, 1266 Udem, T. 1192 Udem, Th. 622 Ueberholz, B. 628 Uffink, J. 271, 1098 Uffink, Jos 1099 Uglum, J. 1132 Uguzzoni, Arnaldo 22 Uhlenbeck, G.E. 768 Uhleneck, George 749 Ulam, Stanislaw 255 Ulfbeck, Ole 403 Ullman, Berthold Louis 1152 Uman, M.A. 627 Umbanhowar, P.B. 272 Umbanhowar, Paul 256, 1296 Unruh, W.G. 504, 893, 1028, 1029, 1134 Unruh, William 378, 871, 872

Unwin, S.C. 346 Unzicker, A. 994 Upright, Craig 22 Uranus 1190 Urban, M. 946 Ursin, R. 766 Ustinov, Peter 555

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Thompson, C. 509 Thompson, Dave 1295 Thompson, R.C. 347 Thomson (Kelvin), William 235, 521 Thomson, Joseph John 527 Thomson, W. 1134 Thor 1189 Thorndike, E.M. 343 Thorne, J.A. Wheeler, K.S. 1026 Thorne, K. 1097 Thorne, K.S. 507, 792 Thorne, Kip 348 Thorne, Kip S. 503, 1152 Thorp, Edward O. 166 Thurston, William 1217 Thurston, William P. 977 Thévenaz, L. 624 Tian, L. 885 Tiggelen, Bart van 629 Tillich, Paul 1032 Tino, G. 792 Tino, G.M. 1133 Tipler, Frank J. 701, 967 Tisserand, F. 507 Titius, Johann Daniel 144 Tittel, Wolfgang 766 Tollett, J.J. 767 Tolman, R.C. 346 Tolman, Richard 512 Tolman, Richard C. 620 Tomonaga 670, 927 Topper, D. 167 Torge, Wolfgang 162 Torre 426 Torre, A.C. de la 621 Torre, C.G. 508 Torrence, R. 513 Torricelli, Evangelista 679 Toschek, P.E. 269, 628 Townsend, P.K. 1027, 1097 Tozaki, K.-I. 889 Tran Thanh Van, J. 947 Travis, J. 622 Treder, H.-J. 1054 Trefethen, L.M. 165 Tregubovich, A.Ya. 271, 1098 Treille, D. 946

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name index

V Voigt

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Copyright © Christoph Schiller November 1997–May 2006

Westerweel, J. 272 Westphal, V. 626 Westra, M.T. 1262 Weyl, Hermann 49, 781 Wheatstone, Charles 527 Wheeler 482 Wheeler, J.A. 505, 792, 886, 1026, 1097 Wheeler, John 493, 514, 882, 1233 Wheeler, John A. 343, 346, 348, 374, 477, 503, 505, 507, 513, 884, 1152 Wheeler, John Archibald 477 Whewell, William 541 White, E.B. 1153 White, Harvey E. 342 White, M. 166, 512 Whitehead, Alfred North 673 Whitney, A.R. 346 Whitney, C.A. 71 Whittaker, Edmund T. 625 Widmer-Schnidrig, R. 268 Widom, A. 766 Wiechert, Johann Emil 527 Wiegert, Paul A. 168 Wieman, Carl 848 Wiemann, C.E. 947 Wien, Wilhelm 613 Wiens, R. 945 Wierda, Gerben 22 Wierzbicka, Anna 22, 161, 642, 643, 660, 698 Wigner, E. 792 Wigner, E.P. 621, 1027, 1054, 1134 Wigner, Eugene 656, 699, 780, 1127 Wigner, Eugene P. 765 Wijk, Mike van 22 Wijngarden, R.J. 630, 888 Wikell, Göran 699 Wilczek, F. 945, 946, 993, 1135 Wilczek, Frank 928, 932, 933 Wilde, Oscar 635, 675, 825, 1033, 1034 Wiley, Jack 270 Wilk, S.R. 626 Wilkie, Bernard 164

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Waser, A. 1226 Washizu, M. 628 Watson, A.A. 515 Wearden, John 888 Webb, R.A. 885 Weber, G. 1133 Weber, Gerhard 764 Weber, R.L. 272 Weber, T. 892 Wedin, H. 272 Weekes, T.C. 946 Weeks, J. 993 Wegener, Alfred 100, 166 Wehinger, S. 769 Wehner, R. 622 Weierstall, U. 269 Weierstrass, K. 1148 W Weijmar Schultz, W.C.M. 945 Wagner, William G. 508 Weijmar Schultz, Willibrord Wagon, Steve 55, 162 898 Wal, C.H. van der 885 Weil, André 1148 Wald 883 Weinberg, Steven 503, 510, Wald, George 247 894, 1026, 1029, 1044 Wald, R.M. 514, 893, 894, 1029, Weinrebe, Gerhard 1296 1134 Weis, A. 1029 Wald, Robert 872 Weisberg, J.M. 507 Wald, Robert M. 503 Weiskopf, Daniel 302, 304, Waldhauser, F. 627 305, 1296 Waleffe, F. 272 Weiss, M. 162 Walgraef, Daniel 272 Weiss, Martha 22 Walker, A.G. 454 Weisskopf, V.F. 889, 946 Walker, Gabriele 510 Weisskopf, Victor 693, 827 Walker, J. 624 Weisskopf, Victor F. 945 Walker, Jearl 159 Weissmüller, J. 629 Walker, R.C. 346 Weitz, M. 622, 1192 Wallin, I. 342 Weitz, Martin 892 Wallis, J. 1148 Weitzmann, Chaim 397 Wallner, A. 510 Wejimar Schultz, Willibrord Walser, R.M. 625 1296 Walter, H. 628 Wells, M. 170 Walter, Henrik 171 Weninger, K. 629 Walther, P. 766 Weninger, K.R. 166 Wambsganss, J. 511 Wernecke, M.E. 887 Wampler, E. Joseph 623 Werner, S.A. 892 Wang Juntao 695 Wertheim, Margaret 1067, Wang, L.J. 624 1134 Wang, Y. 508 Wesson, John 945 Wang, Z.J. 992 Wesson, Paul 459, 510 Warkentin, John 22 West, G.B. 267 Voigt, Woldemar 294 Voit, A. 623 Volin, Leo 22 Vollmer, G. 697 Vollmer, M. 624 Volovik, G.E. 889 Volta, Alessandro 539 Voltaire 146, 182, 684, 694, 701, 953, 1067, 1163 von Klitzing, K. 890 Vorobieff, P. 272 Vos–Andreae, J. 767 Voss, Herbert 22 Voss, R.F. 885 Vuorinen, R.T. 1171 Völz, Horst 1171

1323

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name index

1324

W Wilkinson

Wollaston, William 753 Wolpert, Lewis 159 Wolsky, A.M. 769 Wong, S. 624 Wood, B. 697 Wood, C.S. 947 Woodhouse, Nick M.J. 343 Woods, P.M. 509 Wootters 778 Wootters, W.K. 791 Wouthuysen, S.A. 750, 768 Wright, B. 627 Wright, E.M. 1226 Wright, K. 888 Wright, Steven 502 Wu, C. 628 Wu, T.T. 622 Wulfila 1145 Wussing, H. 1054, 1099 Wynands, R. 628 Würschum, R. 629 X Xavier, A.L. 626 Xu Liangying 695 Xue, S.-S. 893, 1098

Z Zürn, W. 268 Zabusky, N.J. 268 Zabusky, Norman 213

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Copyright © Christoph Schiller November 1997–May 2006

Y Yamafuji, K. 270 Yamamoto, H. 891 Yamamoto, Y. 766 Yamane, T. 623 Yandell, Ben H. 1054, 1099 Yang Chen Ning 936 Yang, C.N. 622 Yang, J. 621 Yao, E. 767 Yazdani, A. 626, 769 Yearian, M.R. 344 Yoshida, K. 895 Young, James A. 163 Young, Thomas 84, 561 Yourgray, Wolfgang 267 Yukawa Hideki 927

Zaccone, Rick 22 Zakharian, A. 165 Zalm, Peer 22 Zander, Hans Conrad 699 Zanker, J. 160 Zbinden, H. 766 Zedler, Michael 22 Zeeman, Pieter 290 Zeh, H.D. 884, 1029 Zeh, Hans Dieter 161, 794, 884 Zeilinger, A. 766, 767, 884, 1171 Zeller, Eduard 700 Zeno of Elea 28, 30, 60, 69, 833, 835, 1006, 1096, 1105, 1107, 1109, 1113 Zensus, J.A. 346 Zermelo, Ernst 647 Zetsche, Frank 945 Zeus 439 Zhang 395 Zhang, J. 627 Zhang, Yuan Zhong 344 Zhao, C. 1097, 1172 Zhou, H. 992 Ziegler, G.M. 55 Zimmer, P. 629 Zimmerman, E.J. 888, 1027, 1054, 1127, 1134 Zimmermann, H.W. 271 Zimmermann, Herbert 246 Zouw, G. van der 767 Zuber, K. 620 Zuckerman, G. 1226 Zurek 778 Zurek, W.H. 697, 791 Zurek, Wojciech 638 Zurek, Wojciech H. 162, 884 Zuse, Konrad 810 Zwaxh, W. 889 Zweck, Josef 166, 629 Zweig, G. 945 Zweig, George 927 Zwerger, W. 885 Zwicky, F. 511 Zwicky, Fritz 468 Zybin, K.P. 627

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Wilkinson, S.R. 888 Will, C. 343, 374, 506 Will, C.M. 348, 514 Willberg, Hans Peter 887 William of Occam 1115 Williams, D.R. 626 Williams, David 585 Williams, David R. 626 Williams, Earle R. 627 Williams, G.E. 165 Williams, H.H. 946 Williams, R. 505 Wilson, B. 268 Wilson, Charles 900 Wilson, J.R. 893 Wilson, Robert 449 Wilson, Robert W. 510 Wiltschko, R. 620 Wiltschko, W. 620 Wineland, D.J. 628, 767, 884 Winterberg, F. 1132 Wirtz 448 Wirtz, C. 509 Wirtz, Carl 447 Wisdom, J. 170, 994 Wisdom, Jack 989 Wise, N.W. 894, 1165 Witt, Bryce de 886 Witte, H. 625 Witteborn, F.C. 620 Witten, E. 1026, 1134 Witten, Ed 1123 Witten, Edward 982 Wittgenstein, Ludwig 23, 30, 36, 39, 49, 71, 88, 154, 157, 248, 631, 641, 646, 656, 657, 659, 660, 668, 673, 675, 687, 700 Wittke, James P. 767 Wodsworth, B. 946 Woerdman, J.P. 768 Woit, Peter 1135 Wolf, C. 1029 Wolf, E. 1132 Wolf, Emil 623 Wolf, R. 1266 Wolfendale, A.W. 626 Wolfenstätter, Klaus-Dieter 814

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Dvipsbug Zybin

Z

name index

1325

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Copyright © Christoph Schiller November 1997–May 2006


Appendix J

SUBJEC T I NDEX

Dvipsbugw Page numbers in italic typeface refer to pages where the keyword is defined or presented in detail. The subject index thus contains a glossary.

Copyright © Christoph Schiller November 1997–May 2006

A @ (at sign) 1149 α-rays 275 a (year) 45 a (year) 453

a posteriori 646 accretion discs 442 a priori 646 accumulability 523 a priori concepts 634 accumulation 233 abacus 1147 accuracy 264, 1164 Abelian, or commutative 192 accuracy, maximal 1036 aberration 94, 101, 277, 302 Acinonyx jubatus 44 abjad 1146 acoustical thermometry 270 Abrikosova 856 acoustoelectric effect 605 absolute value 1196 acoustomagnetic effect 606 absorption 608 acoustooptic effect 610 abstraction 650 actin 820 abugida 1146 actinium 1181 Academia del Cimento 65 actinoids 1180 acausal 296, 801 action 178, 195, 952 acausality 155 action, limit to 1085 accelerating frames 331 action, physical 176 acceleration 69, 345 action, principle of least 123 acceleration composition action, quantum of 245 theorem 334 actuators 150 acceleration, limit to 1087 adaptive optics 585 acceleration, maximum 751 addition 192, 1194 acceleration, proper 322, 330 additivity 43, 48, 51, 81, 523 acceleration, quantum limit additivity of area and volume 751 54 acceleration, relativistic adenosine triphosphate 164, behaviour 330 820 acceleration, tidal 131 adjoint representation 1210 acceleration, uniform 332 adjunct 777 accelerations, highest 108 ADM mass 433, 1044 accelerator mass spectroscopy adventures, future 616 919 Aeneis 687 accelerometer 1234 aerodynamics 970 accents, in Greek language aeroplane, model 969 1145 aeroplanes 79 accretion 488 aether 581, 625, 666, 1005

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Symbols n-th harmonic generation 610 ( 1148 ) 1148 *-algebra 1208 + 1148 − 1148 ë 1148  1148  1148 < 1148 = 1148 A 1148 DNA 709, 745 EPR experiments 779 TNT energy content 1167 ftp 1229 Time magazine 382 x 1148 º 1148 ∆ 1148 ª 1148 ∇ 1148 4-acceleration 321 4-coordinates 294, 319, 321 4-jerk 322 4-momentum 322 4-vector 321 4-velocity 320

Dvipsbugw


subject index

A aether

anthropic principle 689, 1061 anti-atoms 1179 anti-bubbles 232 anti-bunching 725 anti-gravity devices 604 anti-Hermitean 194 anti-theft stickers 606 anti-unitary 194 anticommutator bracket 777 antigravity 128, 135 antigravity devices 135 antihydrogen 1178 antiknot 980 antimatter 82, 317, 432, 462 antimony 1182 Antiqua 1145 antisymmetry 194, 1209 anyons 789, 789 apes 923, 1061 aphelion 1169 apogee 1169 Apollo 414, 570, 1247, 1257 apparatus, classical 812 apparatus, irreversible 812 apple as battery 539 apple trees 1163 apple, standard 1163 apples 76, 396, 664, 1048 Aquarius 139 Ar 606 Arabic numbers 1146 Archaeozoicum 452 archean 452 arcs 522 Arcturus 186 argon 910, 1182 argument 650 Aries 139 Aristotelian view 148 Aristotle 1246 arm 783, 1200 Armillaria ostoyae 52, 81 arms, human 474 arrow 1109 arrow of time 49 arsenic 1182 artefact 74, 685, 1155 ash 1143 Ashtekar variables 501

Dvipsbugw

Copyright © Christoph Schiller November 1997–May 2006

alphasyllabaries 1146 Alps 123, 143 Altair 186, 467 aluminium 620, 1182 aluminium amalgam 840 amalgam 1186 Amazonas 100 Amazonas River 215 amber 517 americium 1182 amoeba 976 ampere 1154 amplitude 559 Ampère’s ‘law’ 534 analemma 139 anapole moment 947 AND gate, logical 638 andalusite 609 Andromeda nebula 438, 447 angel, are you one? 668 angels 33, 75, 652, 785, 953 angle 89 angle, plane 60 angle, solid 60 angular acceleration 89 angular frequency 559 angular momentum 88, 89, 109, 533, 574, 787, 952 angular momentum as a tensor 325 angular momentum conservation 123 angular momentum of light 573 angular momentum, extrinsic 90 angular momentum, indeterminacy relation 741 angular momentum, intrinsic 90 angular momentum, limit to 1085 angular velocity 88, 89 anholonomic constraints 181 animism 634 annihilation 462, 605 annihilation operator 777 anode 541 Antares 71, 186

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

aether and general relativity 358, 384 aether models 625 affine Lie algebra 1212 AFM, atomic force microscope 224 Africa collides with Europe 617 AgBr 608 AgCl 562, 609 age 456 age of universe 318, 1033 ageing 712 aggregates of matter 183 AgI 608 agoraphobics 455 air 465, 605, 1254 air cannot fill universe 462 air pressure 83 airbag sensors 529 akne 824 Al 606 albedo 572 Albert Einstein, teenager 567 alchemy 42 Aldebaran 71, 186, 467 aleph 1145 algebra 1207, 1208 algebra, alternative 1202 algebra, associated affine 1212 algebra, linear 1207 algebraic structure 651 algebraic surfaces 1231 algebraic system 651 Alice in Wonderland 246 alkali metals 1180 Allen, Woody 637 Alluvium 452 Alnilam 467 Alnitak 467 Alpha Centauri 186 alpha decay 449 alpha ray dating 919 alphabet 1144 alphabet, Greek 1143 alphabet, Hebrew 1146 alphabet, Latin 1142 alphabet, phonemic 1146 alphabets, syllabic 1146

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A aspects

Dvipsbugw

Copyright © Christoph Schiller November 1997–May 2006

aurora borealis 907 Barnett effect 532 aurum 1184 barometer light 605 austenitic steels 231 barycentre 141 autism 633, 696 barycentric coordinate time autistics 633, 696 506 average 181, 673, 798, 1019 barycentric dynamical time average curvature 423 393 Avogadro’s number 220, 1166 baryon number 914 awe 686, 964 baryon number density 1168 axiom of choice 647, 647 baryon table 1180 axioms 1195 base units 1154 axioms of set theory 647 basic units 1197 axioms, ZFC of set theory 647 Basiliscus basiliscus 114, 115 axis of the Earth, motion of basilisk 114 100 basis 1206, 1210 axis, Earth’s 102 bath, physical 638, 795 axle, impossibility in living bath-tub vortex 165 beings 1240 bathroom scale 105 azoicum 451 bathroom scales 230 bathtub vortex 96 B BaTiO3 607, 608 β-rays 275 bats 826 B1938+666 468 battery 539 Babylonia 1150 bce 1150 Babylonians 144 BCS theory 890 background 37, 38, 49, 295 beam, tractor 592 background radiation 449, beans 957 453, 458, 462 Beaufort 44 bacteria 74, 824, 829 beauty 32, 255, 258, 1177 bacterium 87 beauty quark 928 bacterium lifetime 823 beauty, origin of 255 badminton 844 becquerel 1156 badminton serve, record 44 beer 232, 610, 851 badminton smashes 1236 Beetle 416 bags as antigravity devices beetle, click 114 604 before the Common Era 1150 bags, plastic 1262 beginning of the universe 448 ball lightning 598 beginning of time 448, 1034 balloons 241 behaviour 33 Banach measure 54 beings, living 816 Banach–Tarski paradox or Bekenstein’s entropy bound 1089 theorem 55, 220, 857, 1006, 1008, 1057 Bekenstein-Hawking banana catching 91 temperature 873 bananas, knotted 982 belief systems 75 BaO2 606 beliefs 35, 664, 964 barber paradox 658 beliefs and Ockham 1115 barium 1182 beliefs in physical concepts Barlow’s wheel 542 660

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

aspects of nature 634 associative algebra 1207 associativity 192 astatine 1182 asteroid 139 asteroid hitting the Earth 617 asteroid, Trojan 128 asteroids 141, 186 Astrid, an atom 269 astrology 124, 139, 140, 955 astronomers, smallest known 126 astronomical unit 144 astronomy 1200 astronomy picture of the day 1231 astrophysics 1228 asymptotic freedom 932, 933 at-sign 1149 atmosphere of the Moon 253 atom 116 1178 atom formation 451 atom rotation 751 atomic 1162 atomic force microscope 224, 225, 1244 atomic force microscopes 152 atomic mass unit 1177 atomic number 1180 atomic radius 1191 atoms 24, 705 atoms and swimming 972 atoms are not indivisible 249 atoms are rare 908 atoms, hollow 864 atoms, manipulating single 224 atoms, matter is not made of 908 ATP 83, 164, 820, 1179 ATP consumption of molecular motors 820 ATP synthase 820 atto 1156 Atwood machine 1246 Au 606 Auger effect 610 aureole 582 aurora australis 907

subject index

Dvipsbugw


subject index

B Bell’s

entropy in 245 boost 200, 294, 353, 353 boosts, concatenation of 309 boredom as sign of truth 666 boring physics 644 boron 1182 Bose–Einstein condensates 736, 769, 1091 Bose-Einstein condensate 848 bosonisation 1123 bosons 732, 775, 777 bottle 82, 108 bottle, full 175 bottom quark 928, 1176 bottomness 1177 boundaries 75 boundary conditions 553 boundary layer 1249 boundary of space-time 1041 box tightness 845 boxes 339 boxes, limits to 1105 bradyons 317 Bragg reflection 610 braid 789, 980 braid pattern 262 braid symmetry 1121 brain 252, 602, 631, 652, 749, 827 brain and Moon 689 brain and physics paradox 265 brain size of whales 827 brain stem 37 brain’s energy consumption 640 brain’s interval timer 831 brain, clock in 831 Brans-Dicke ‘theory’ 430 bread 221, 1235 breaking 712 breathings 1145 breaths 653 bremsstrahlung 609 brick tower, infinitely high 361 brilliant 652 bromine 1182 brooms 136, 1250 brown dwarfs 186, 443, 467, 467

Dvipsbugw

Copyright © Christoph Schiller November 1997–May 2006

black body radiation constant 611, 874 black hole 358, 408, 466, 478, 505, 1000 black hole collisions 489 black hole entropy 1118 black hole halo 487 black hole radiation 512 black hole, entropy of 485 black hole, extremal 482 black hole, Kerr 482 black hole, primordial 488 black hole, rotating 482 black hole, Schwarzschild 482 black hole, stellar 488 black holes 154, 350, 385, 503, 963 black holes do not exist 486 black paint 459 black vortex 483 black-body radiation 465 black-hole temperature 873 blackness 608 blasphemies 684, 735 blinks 653 block and tackle 41 blood pressure 229 blood supply 73 blue 565 blue colour of the sea 762 blue shift 286 blue whale nerve cell 1179 board divers 91 boat 82, 277 Bode’s rule 144 bodies, rigid 174, 862 body 37 body, rigid 339 body, solid 339 Bohr radius 755, 1167 bohrium 1182 Boltzmann constant 241, 242, 242, 245, 248, 310, 798, 1165 Boltzmann’s constant 612 bomb 314 bombs 875 bones, seeing own 918 books 1142 books, information and

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Bell’s equation 810 Bell’s inequality 810 Bellatrix 467 bells 785 belt 1200 belt trick 783, 1199 belts and spin 1122 Bennett–Brassard protocol 779 BeppoSAX satellite 875 berkelium 1182 Bernoulli’s ‘law’ 230 Bernoulli’s principle 241 beryllium 1182 Bessel functions 101 Betelgeuse 71, 72, 186, 467, 617 beth 1145 bets, how to win 97 Bi 607 bible 668, 672 bicycle riding 76 bicycle weight 241 big bang 458, 462, 463 big bang was not a singularity 370 big brother 649 billiards 79, 312 bimorphs 150 binary pulsars 394, 413 biographies of mathematicians 1231 biological evolution 34, 256 biology 150, 1177 bioluminescence 609 biphoton 727 BIPM 672, 1154, 1155 bird appearance 452 bird speed 44 birds 24, 539, 562, 923 birefringence 562, 609 BiSb 607 BiSeTe 605 bismuth 46, 1178, 1182, 1192 Bi2 Te3 605 bits 636 bits to entropy conversion 1167 black body 611 black body radiation 611, 611

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subject index

1330

B Brownian

causality and maximum speed 296 causality of motion 155 cause 691 cause and effect 295 cavitation 108 Cayley algebra 1202 Cayley numbers 1202 Cd 606 CdS 605, 610 CeB6 606 CeF3 607 celestrocentric system 1236 cell 1179 cell motility 820 cell, voltaic 539 cells 652 Celsius temperature scale 234 Celtic wiggle stone 111 cementite 232 cenozoic 452 censorship, cosmic 369 centi 1156 centre 1209 centre of gravity 139 centre of mass 90, 139, 317 centre, quaternion 1200 centrifugal acceleration 109, 118 centrifugal effect 486 centrifugal force 122 Cepheids 935 Čerenkov effect 610 Čerenkov radiation 282 cerium 1183 CERN 938 CERN 345 Cetus 139 chain 62 chair as time machine 299 chalkogens 1180 challenge level 23, 30 challenge solution number 1283 challenge, toughest of science 558 challenges 21, 23, 30, 32, 36–51, 53–63, 65–75, 77, 79–86, 88–93, 95–98, 101,

Dvipsbugw

Copyright © Christoph Schiller November 1997–May 2006

C C*-algebra 1208 c. 1150 C14 dating 630 cable, eliminating power 540 CaCO 3 562 cadmium 1183 Caenozoicum 452 caesium 46, 1183 caesium and parity non-conservation 936 CaF2 609 calcite 562 calcium 1183 calculating prodigies 1147 calculus of variations 180 calendar 1149, 1150 calendar, modern 1150 californium 1183 calorie 1163 Calpodes ethlius 66 Cambrian 452 camera 60 camera obscura 1107 camera, holy 626 Canary islands 616

Cancer 139 candela 1154 candle 41, 254, 598, 850, 1160 canoeing 115 Canopus 186, 467 cans of beans 957 cans of peas 112 cans of ravioli 112 capacitor 539, 596 Capella 186 Capricornus 139 capture of light 481 capture, gravitational 481 capture, in universal gravity 141 car parking 57 car theft 748 car weight 241 carbohydrates, radicals in 981 carbon 1178, 1183 Carboniferous 452 cardinality 649 cardinals 1204 cardinals, inaccessible 649 carriage, south-pointing 174, 267 cars 264 cars on highways 739 Cartan algebra 1211 Cartan metric tensor 1210 Cartan superalgebra 1212 Cartesian 51 Cartesian product 648 cartoon physics, ‘laws’ of 75 Casimir effect 611, 856, 952, 997 Casimir effect, dynamical 859 cat 82, 582 catastrophes 616 categories 1047 category 645, 650 catenary 1237 caterpillars 66 cathode 541 cathode rays 275, 527 cats 825 causal 155 causal connection 296 causality 809

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Brownian motion 243 Brownian motors 821 browser 1228 brute force approach 360 bubble chamber 900 bubbles 229 bucket 32 bucket experiment, Newton’s 104, 474 bulge 777 bullet speed 44 bullet speed measurement 57 Bureau International des Poids et Mesures 1154 Burgers vector 987 bus, best seat in 305 buses 379 bushbabies 1