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Chapter 32
distinguish a uniform gravitational field from an accelerated frame. However, the two are not equivalent since they obviously depend on the direction of acceleration relative to the gravitating body and the distance from the gravitating body since the gravitational force is a central force. (In the latter case, only a line of a massive body may be exactly radial, not the entire mass.) And, this assumption leads to conflicts with special relativity. The success of Einstein’s gravity equation can be traced to a successful solution which arises from assumptions and approximations whereby the form of the solution ultimately conflicts with the properties of the original equation; no solution is consistent with the experimental data in the case of the possible cosmological solutions of Einstein’s general relativity. Furthermore, Einstein’s general relativity is a partial theory in that it deals with matter on the scale of celestial objects, but not on an atomic scale. And, it fails on the cosmological scale. All gravitating bodies are composed of matter and are collections of atoms that are composed of fundamental particles such as electrons, which are leptons, and quarks, which make up protons and neutrons. Gravity originates from the fundamental particles. The Einstein’s theory has as its foundation that gravity is a force unique from electromagnetism. The magnetic force was unified with the Coulomb force by Maxwell. Lorentz derived the transformations named after him which formalize the origin of the magnetic force as a relativistic correction of the Coulomb force. The unification of electricity and magnetism by Maxwell permitted him to derive a wave equation that predicted the propagation of electromagnetic waves at the speed of light. Maxwell’s wave equation defines a four-dimensional spacetime and the speed of light as a maximum permitted according to the permeability and permittivity of spacetime. Minkowski originated the concept of a four-dimensional spacetime formally expressed as the Minkowski tensor [2]. The Minkowski tensor corresponds to the electromagnetic wave equation derived by Maxwell and can be derived from it [3]. Special relativity is implicit in the wave equation of electromagnetic waves that travel at the speed of light. As given in the Relativity section and the Equivalence of Inertial and Gravitational Masses Due to Absolute Space and Absolute Light Velocity section, the generalization of this metric to mass as well as charge requiring application of Lorentz transformations comprises the theory of special relativity invented by Poincaré in 1904 [4-6]1. The Lorentz transformations quantify the measurement of the increase in mass, length contraction, and time dilation in the direction of constant relative motion of separate inertial frames due to the finite maximum speed of light. The goal of Einstein, who worked on special relativity, was to generalize it to accelerated frames of reference as well as inertial frames moving at constant relative velocity. But, gravity is not a force separable from electromagnetism. The true origin of gravity is the relativistic correction of spacetime itself as opposed to the relativistic correction of mass, length, and time of objects of inertial frames in constant relative motion. The production of a massive particle from a photon with zero rest mass traveling at the speed of light requires time dilation and length contraction of spacetime. The present theory of gravity also maintains the constant maximum speed of light for the propagation of any form of energy. (Recently the speed of gravity has been measured to be the speed of light [7].) And, the origin of the gravitational force is also a relativistic correction. In the metric which arises due to the presence of mass, spacetime itself must be relativistically corrected as a consequence of the presence of mass in order to that (i) the speed of light is constant and a maximum, (ii) the angular momentum of a photon, , is conserved, and (iii) the energy of the photon is conserved as mass. Spacetime must undergo time dilation and length contraction due to the production event. The event must be spacelike even though the photon of the particle production event travels at the speed of light and the particle must travel at a velocity less than the speed of light. The relativistically altered spacetime gives rise to a gravitational force between separated masses. Thus, the production of matter and its motion alters spacetime and the altered spacetime affects the motion of matter, which must follow geodesics. When speaking of the relativity of a frame of reference or simply of relativity, one usually means that there exist identical physical processes in different frames of reference. According to the generalized Galilean principle of relativity identical processes are possible in all inertial frames of reference related by Lorentz transformations. On the other hand, Lorentz transformations characterize the uniformity of Galilean spacetime. Using the four-dimensional coordinates x for describing the events and the world-line in spacetime the separation of proper time between two events x and x dx is (32.6) d 2 g dx dx where g is the metric tensor which determines the geometric character of spacetime. For different coordinate systems, the dx may not be the same, but the separation d 2 remains unchanged. The metric g for Euclidean space called the Minkowski tensor is
1 In 1900, Lorentz conjectured that gravitation could be attributed to actions that propagate with the velocity of light. Poincaré, in a paper in July 1905 (submitted days before Einstein’s special relativity paper), suggested that all forces should transform according to Lorentz transformations. In this case, he notes that Newton’s Law of Gravitation is not valid and proposed gravitational waves that propagated with the velocity of light. Specifically, Poincaré pointed out that all forces must propagate with the finite light velocity, that interaction implies a time delay, and it is mediated by field waves. Thus, Poincaré made for the first time the hypothesis of the existence of gravitational waves [4].