The Equivalence Principle, the Covariance Principle
and
the Question of Self-Consistency in General Relativity
C. Y. Lo
Applied and Pure Research Institute
17 Newcastle Drive, Nashua, NH 03060, USA
September 2001


Abstract
The equivalence principle, which states the local equivalence between acceleration and gravity, requires that a free falling observer must result in a co-moving local Minkowski space. On the other hand, covariance principle assumes any Gaussian system to be valid as a space-time coordinate system. Given the mathematical existence of the co-moving local Minkowski space along a time-like geodesic in a Lorentz manifold, a crucial question for a satisfaction of the equivalence principle is whether the geodesic represents a physical free fall. For instance, a geodesic of a non-constant metric is unphysical if the acceleration on a resting observer does not exist. This analysis is modeled after Einstein£r illustration of the equivalence principle with the calculation of light bending. To justify his calculation rigorously, it is necessary to derive the Maxwell-Newton Approximation with physical principles that lead to general relativity. It is shown, as expected, that the Galilean transformation is incompatible with the equivalence principle. Thus, general mathematical covariance must be restricted by physical requirements. Moreover, it is shown through an example that a Lorentz manifold may not necessarily be diffeomorphic to a physical space-time. Also observation supports that a spacetime coordinate system has meaning in physics. On the other hand, Pauli£r version leads to the incorrect speculation that in general relativity space-time coordinates have no physical meaning
1. Introduction.
Currently, a major problem in general relativity is that any Riemannian geometry with the proper metric signature would be accepted as a valid solution of Einstein£r equation of 1915, and many unphysical solutions were accepted [1]. This is, in part, due to the fact that the nature of the source term has been obscure since the beginning [2,3]. Moreover, the mathematical existence of a solution is often not accompanied with understanding in terms of physics [1,4,5]. Consequently, the adequacy of a source term, for a given physical situation, is often not clear [6-9]. Pauli [10] considered that ¥Phe theory of relativity to be an example showing how a fundamental scientific discovery, sometimes even against the resistance of its creator, gives birth to further fruitful developments, following its own autonomous course." Thus, in spite of observational confirmations of Einstein£r predictions, one should examine whether theoretical self-consistency is satisfied. To this end, one may first examine the consistency among physical ¥Lrinciples" which lead to general relativity.
The foundation of general relativity consists of a) the covariance principle, b) the equivalence principle, and c) the field equation whose source term is subjected to modification [3,7,8]. Einstein£r equivalence principle is the most crucial for general relativity [10-13]. In this paper, the consistency between the equivalence principle and the covariance principle will be examined theoretically, in particular through examples. Moreover, the consistency between the equivalence principle and Einstein£r field equation of 1915 is also discussed.
The principle of covariance [2] states that ¤‘he general laws of nature are to be expressed by equations which hold good for all systems of coordinates, that is, are covariant with respect to any substitutions whatever (generally covariant)." The covariance principle can be considered as consisting of two features: 1) the mathematical formulation in terms of Riemannian geometry and 2) the general validity of any Gaussian coordinate system as a space-time coordinate system in physics. Feature 1) was eloquently established by Einstein, but feature 2) remains an unverified conjecture. In disagreement with Einstein [2], Eddington [11] pointed out that ¥Opace is not a lot of points close together; it is a lot of distances interlocked." Einstein accepted Eddington£r criticism and no longer advocated the invalid arguments in his book, ¤‘he Meaning of Relativity" of 1921. Einstein also praised Eddington£r book of 1923 to be the finest presentation of the subject ever written
Moreover, in contrast to the belief of some theorists [14,15], it has never been established that the equivalence of all frames of reference requires the equivalence of all coordinate systems [9]. On the other hand, it has been pointed out that, because of the equivalence principle, the mathematical covariance must be restricted [8,9,16].
Moreover, Kretschmann [17] pointed out that the postulate of general covariance does not make any assertions about the physical content of the physical laws, but only about their mathematical formulation, and Einstein entirely concurred with his view. Pauli [10] pointed out further, ¤‘he generally covariant formulation of the physical laws acquires a physical content only through the principle of equivalence...." Nevertheless, Einstein [2] argued that "... there is no immediate reason for preferring certain systems of coordinates to others, that is to say, we arrive at the requirement of general co-variance."
Thus, Einstein£r covariance principle is only an interim conjecture. Apparently, he could mean only to a mathematical coordinate system for calculation since his equivalence principle, among others, is an immediate reason for preferring certain systems of coordinates in physics (‰Ñ 5 & 6). Note that a mathematical general covariance requires, as Hawking declared [18], the indistinguishability between the time-coordinate and a space-coordinate. On the other hand, the equivalence principle is related to the Minkowski space, which requires a distinction between the time-coordinate and a space-coordinate. Hence, the mathematical general covariance is inherently inconsistent with the equivalence principle.
Although the equivalence principle does not determine the space-time coordinates, it does reject physically unrealizable coordinate systems [9]. Whereas in special relativity the Minkowski metric limits the coordinate transformations, among inertial frames of reference, to the Lorentz-Poincar¨¦ transformations; in general relativity the equivalence principle limits the physical coordinate transformations to be among valid space-time coordinate systems, which are in principle physically realizable. Thus, the role of the Minkowski metric is extended by the equivalence principle even to where gravity is present.
Mathematically, however, the equivalence principle can be incompatible with a solution of Einstein£r equation, even if it is a Lorentz manifold (whose space-time metric has the same signature as that of the Minkowski space). It has been proven that coordinate relativistic causality can be violated for some Lorentz manifolds [9,16]. Unfortunately, due to inadequate physical understanding, some relativists [19-23] believe that a proper metric signature would imply a satisfaction of the equivalence principle. The misconception that, in a Lorentz manifold, a ¥Bree fall" would automatically result in a local Minkowski space [20,23], has deep-rooted physical misunderstandings from believing in the general mathematical covariance in physics.
Although the equivalence principle for a physical space-time1) is clearly stated, the conditions for its satisfaction in a Lorentz manifold have been misleadingly over simplified. Thus, it is necessary to clarify first, in terms of physics, the meaning of the equivalence principle and its satisfaction (¡ì2 & ¡ì3). The crucial condition for a satisfaction of the equivalence principle is that the geodesic represents a physical free fall. The mathematical existence of local Minkowski spaces means only mathematical compatibility of the theory of general relativity to Riemannian geometry. Then, it becomes possible to demonstrate meaningfully through detailed examples that diffeomorphic coordinate systems may not be equivalent in physics (¡ì5 & ¡ì 6). Moreover, to avoid prejudice due to theoretical preferences, these demonstrations are based on theoretical inconsistency.
To this end, Einstein£r illustration of the equivalence principle in his calculation of the light bending is used as a model for this analysis. However, in his calculation, there are related theoretical problems that must be addressed. First, the notion of gauge used in his calculation is actually not generally valid [9] as will be shown in this paper. Also, it is known that validity of the 1915 Einstein equation is questionable [7,8,24-26]. For a complete theoretical analysis, these issues should, of course, be addressed thoroughly. Nevertheless, for the validity of Einstein£r calculation on the light bending [2], it is sufficient to justify the linear field equation as a valid approximation. For this purpose, the Maxwell-Newton Approximation (i.e., the linear field equation) is derived directly from the physical principles that lead to general relativity (¡ì4).
Moreover, there are intrinsically unphysical Lorentz manifolds none of which is diffeomorphic [21] to a physical space-time (¡ì7). Thus, to accept a Lorentz manifold as valid in physics, it is necessary to verify the equivalence principle with a space-time coordinate system for physical interpretations. Then, for the purpose of calculation only, any diffeomorphism can be used to obtain new coordinates. It is only in this sense that a coordinate system for a physical space-time can be arbitrary.
In this paper, the requirement of a general covariance among all conceivable mathematical coordinate systems [2] will be further confirmed to be an over-extended demand [9]. (Note that Eddington [11] did not accept the gauge related to general mathematical covariance.) Analysis shows that a satisfaction of the equivalence principle restricted covariance (‰Ñ 3-5). After this necessary rectification, some currently accepted well-known Lorentz manifolds would be exposed as unphysical (¡ì7). But, general relativity as a physical theory is unaffected [9]. It is hoped that this clarification would help ¥Burther fruitful developments, following its own autonomous course [10]".
2. Einstein£r Equivalence Principle, Free Fall, and Physical Space-Time Coordinates
Initially based on the observation that the (passive) gravitational mass and inertial mass are equivalent, Einstein proposed the equivalence of uniform acceleration and gravity. In 1916, this proposal is extended to the local equivalence of acceleration and gravity [2] because gravity is in general not uniform. Thus, if gravity is represented by the space-time metric, the geodesic is the motion of a particle under the influence of gravity. Then, for an observer in a free fall, the local metric is locally constant. To be consistent with special relativity, such a local metric is required to be locally a Minkowski space [2].
Thus, a central problem in general relativity is whether the geodesic represents a physical free fall. However, validity of this global property is realized locally through a satisfaction of the equivalence principle. Moreover, Eddington [11] observed that special relativity should apply only to phenomena unrelated to the second order derivatives of the metric. Thus, Einstein [27] added a crucial phrase, ¤žt least to a first approximation" on the indistinguishability between gravity and acceleration.
The equivalence principle requires that a free fall physically result in a co-moving local Minkowski space2) [3]. However, in a Lorentz manifold, although a local Minkowski space exists in a ¥Bree fall" along a geodesic, the formation of such co-moving local Minkowski spaces may not be valid in physics since the geodesic may not represent a physica

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