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In physics, black hole thermodynamics is essentially the theoretical study of energy and entropy at the boundary regions of black holes. It is generally recognized that this is a special field of research, having been created within the last 30 years, entirely centered around the thermodynamics of black holes.

Contents [hide] • •

1 Black Hole Entropy 2 The Laws of Black hole mechanics o 2.1 Statement of the Laws  2.1.1 The Zeroth Law  2.1.2 The First Law  2.1.3 The Second Law  2.1.4 The Third Law o 2.2 Discussion of the Laws  2.2.1 The Zeroth Law  2.2.2 The First Law  2.2.3 The Second Law  2.2.4 The Third Law o 2.3 Interpretation of the Laws 3 Problems o 3.1 Problem one o 3.2 Problem two o 3.3 References o 3.4 See also o


Black Hole Entropy

3.5 External links

Black hole entropy is the entropy carried by a black hole. If black holes carried no entropy, it would be possible to violate the second law of thermodynamics by throwing mass into the black hole. The only way to satisfy the second law is to admit that the black holes have entropy whose increase more than compensates for the decrease of the entropy carried by the object that was swallowed. Starting from some theorems proved by Stephen Hawking, Jacob Bekenstein conjectured that the black hole entropy was proportional to the area of its event horizon. Later, Stephen Hawking showed that black holes emit thermal Hawking radiation corresponding to a certain temperature (Hawking temperature). Using some arguments rooted in thermodynamics, Hawking was also able to calculate the entropy that the black hole must carry. The result confirmed Bekenstein's conjecture:

where k is Boltzmann's constant, and is the Planck length. The black hole entropy is proportional to its area A. The fact that the black hole entropy is also the maximal entropy that can be squeezed within a fixed volume was the main observation that led to the holographic principle. The subscript BH either stands for "black hole" or "Bekenstein-Hawking". Until 1995, no one was able to make a controlled calculation of black hole entropy based on statistical mechanics, i.e. on counting the number of actual microstates. The situation changed in 1995 when Andrew Strominger and Cumrun Vafa calculated the right Bekenstein-Hawking entropy of a supersymmetric black hole in string theory, using methods based on D-branes. Their calculation was followed by many similar computations of entropy of large classes of other extremal and near-extremal black holes, and the result always agreed with the Bekenstein-Hawking formula. Loop quantum gravity, viewed as the main competitor of string theory, also offered a slightly more heuristic calculation of the black hole entropy. Unfortunately, the numerical coefficient 1 / 4 was not reproduced correctly. The multiplicative discrepancy is called the Immirzi parameter.


The Laws of Black hole mechanics

The four laws of black hole mechanics are physical properties that black holes are believed to satisfy. The laws, analogous to the laws of thermodynamics, were discovered by Brandon Carter, Stephen Hawking and James Bardeen. [edit]

Statement of the Laws The laws of black hole mechanics are expressed in geometrized units. [edit]

The Zeroth Law The horizon has constant surface gravity for a stationary black hole κ. [edit]

The First Law We have

, where M is the mass, A is the horizon area, Ω is the angular velocity, J is the angular momentum, Φ is the electrostatic potential, κ is the surface gravity and Q is the electric charge. [edit]

The Second Law The horizon area is, assuming the weak energy condition, a non-decreasing function of time, [edit]

The Third Law It is not possible to form a black hole with vanishing surface gravity. κ=0 is not possible to achieve. [edit]

Discussion of the Laws [edit]

The Zeroth Law The zeroth law is analogous to the zeroth law of thermodynamics which states that the temperature is constant throughout a body in thermal equilibrium. It suggests that the surface gravity is analogous to temperature. T constant for thermal equilbrium for a normal system is analogous to Îş constant over the horizon of a stationary black hole [edit]

The First Law The left hand side, , is the change in mass/energy. Although the first term does not have an immediately obvious physical interpretation, the second and third terms on the right hand side represent changes in energy due to rotation and electromagnetism. Analogously, the first law of thermodynamics is a statement of energy conservation, which contains on its right hand side the term . [edit]

The Second Law The second law is the statement of Hawking's area theorem. Analogously, the second law of thermodynamics states that the entropy of a closed system is a non-decreasing function of time, suggesting a link between entropy and the area of a black hole horizon. However,this version violates the second law of thermodynamics by matter losing (its) entropy as it falls in, giving a decrease in entropy. Generalised second law introduced as total entropy = black hole entropy + outside entropy [edit]

The Third Law Extremal black holes have vanishing surface gravity. Stating that Îş cannot go to zero is analogous to the third law of thermodynamics which, in its weak formulation, states that it is impossible to reach absolute zero temperature in a physical process. The strong version of the third law of thermodynamics, which states that as the temperature approaches zero, the entropy also approaches zero, does not have an analogue for black holes. [edit]

Interpretation of the Laws The four laws of black hole mechanics suggest that one should identify the surface gravity of a black hole with temperature and the area of the event horizon with entropy, at least up to some multiplicative constants. If one only considers black holes classically, then they have zero temperature and, by the no hair theorem, infinite entropy, and the laws of black hole mechanics remain an analogy. However, when quantum mechanical effects are taken into account, one finds that black holes emit thermal radiation (Hawking radiation) at temperature

. From the first law of black hole mechanics, this determines the multiplicative constant of the Bekenstein-Hawking entropy which is

. [edit]

Problems [edit]

Problem one Classical thermodynamics states as its second principle that entropy is an always increasing function in a closed system - and the universe is a closed system, as nothing can escape it. So we ask; what happens to the information when a particle falls inside a black hole? Remember that only three parameters are required to fully describe a black hole: its mass, its electrical charge, and its angular momentum. But, in order to describe a physical system, we need other information, especially entropy, which is a measurement of unavailable energy; losing this information would be a violation of the Second Law of Thermodynamics. We imagine a black hole as the singularity in the center surrounded by a spherical event horizon. We know that when a black hole is created by a collapsing neutron star that the neutrons are crushed out of existence, they cease to be neutrons. We have seen that all matter has a wave aspect, and quantum mechanics describes the behavior of these waves. So, we shall think about representing the mass-energy inside the event horizon as waves. Now, what kind of waves are possible inside the black hole? The answer is standing waves, waves that "fit" inside the black hole with a node at the event horizon. We know that the energy represented by a particular wave state is related to the frequency and amplitude of its oscillation, higher frequency waves contain more energy.

Assume that the total mass-energy inside the event horizon is fixed. So, we have various standing waves, each with a certain amount of energy, and the sum of the energy of all these waves equals the total mass-energy of the black hole. There are a large number of ways that the total mass-energy can distribute itself among the standing waves. We could have it in only a few high energy waves or a larger number of low energy waves. It turns out that all the possible standing wave states are equally probable. Thus, we can calculate the probability of a particular combination of waves containing the total mass-energy of the black hole the same way we calculate the probability of getting various combinations for dice. Just as for the dice, the state with the most total combinations will be the most probable state. Entropy is just a measure of the probability, and can be expressed as:

where A is the surface area of the sphere contained within the event horizon of the black hole, k is Boltzmann's constant, is the reduced form of Planck's constant, c is the speed of light, and G is the gravitational constant. Thus we can calculate the entropy of a black hole which solves our first problem. However entropy measures the heat divided by the absolute temperature.(In this context "heat" is just the total mass-energy of the black hole.) If we know the total mass-energy, or heat of the black hole, and we also know the entropy, then we can calculate a temperature for the black hole. The laws of thermodynamics predict that all bodies with temperature above absolute-zero must radiate energy in the form of electromagnetic waves. This result, however, leads to a paradox. Because of the intense gravitational field created by the black hole, nothing can escape from within the event horizon. So how is it possible for a black hole to emit radiation? This paradox leads to problem 2. [edit]

Problem two Any body with a temperature above absolute zero will radiate energy. And we have just seen that a black hole has a non-zero temperature. Thus thermodynamics says it will radiate energy and evaporate. We can calculate the rate of radiation for a given temperature from classical thermodynamics. We can also use the following formula to calculate the black hole's temperature:

So, how is this possible? Nothing can get across the event horizon, so how can the black hole radiate? The answer is via virtual pair production. Consider a virtual electron-positron pair produced just outside the event horizon. Once the pair is created, the intense curvature of spacetime of the black hole can put energy into the pair. Thus the pair can become non-virtual; the electron does not fall back into the hole. There are many possible fates for the pair. Consider one of them: the positron falls into the black hole and the electron escapes. According to Feynman's view we can describe this as follows: The electron crosses the event horizon travelling backwards in time, scatters, and then radiates away from the black hole travelling forwards in time. Using the field of physics that calculates virtual pair production etc., called Quantum Electrodynamics, we can calculate the rate at which these electrons etc. will be radiating away from the black hole. The result is the same as the rate of radiation that we calculate using classical thermodynamics. The fact that we can get the radiation rate in two independent ways, from classical Thermodynamics or from Quantum Electrodynamics, is another indicator of the strength attributed to this theory; that black holes radiate their energy away and evaporate. Since this radiation's discovery is attributed to Hawking in the main, it is called Hawking radiation. We can calculate the rate of radiation for a black hole, and it generally comes about to give the prediction that the black hole will evaporate completely in 1064 years time! This is more than the predicted age of the universe, so unfortunately, from the traditional model of black hole formation, we shall never be able to witness the complete evaporation of a black hole take place. However, the possible existence of a black hole smaller than 3 solar masses can be thought about, enabling the black hole to evaporate within the lifetime of our universe. This primordial black hole, as it were, would have to be created by the external application of extreme pressures onto a mass, so as to emulate the situation of a neutron star collapsing-this kind of situation would probably be feasible after the initial creation of the universe, when extreme pressures and heat were commonplace before such processes as reionisation occurred. As yet though, none of these black holes have been found-which may be due to the fact that they simply do not exist, or that the event horizon is so close to the central mass that more precise instruments will be needed to detect the effects of black holes on a smaller external environment. So, as Stephen Hawking realised, "We can apply all of Thermodynamics to a black hole." [edit]

References • • • • • •

J. M. Bardeen, B. Carter and S. W. Hawking, "The four laws of black hole mechanics", Commun. Math. Phys. 31, 161 (1973). J. D. Bekenstein, "Black holes and entropy", Phys. Rev. D 7, 2333 (1973). S. W. Hawking, "Black hole explosions?", Nature 248, 30 (1974). S. W. Hawking, "Particle creation by black holes", Commun. Math. Phys. 43, 199 (1975). S. W. Hawking and G. F. R. Ellis, "The large-scale structure of space-time", Cambridge University Press (1973). S. W. Hawking, "The Nature of Space and Time", (1994) [1]


See also •

Hawking radiation


External links • •

Black Hole Thermodynamics Black hole entropy on

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