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Mathematically, however, the equivalence principle can be incompatible with a solution of Einstein 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 ree 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 (§2 & §3). 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 (§5 & § 6). Moreover, to avoid prejudice due to theoretical preferences, these demonstrations are based on theoretical inconsistency.
To this end, Einstein 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 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 (§4).
Moreover, there are intrinsically unphysical Lorentz manifolds none of which is diffeomorphic [21] to a physical space-time (§7). 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 (壯 3-5). After this necessary rectification, some currently accepted well-known Lorentz manifolds would be exposed as unphysical (§7). But, general relativity as a physical theory is unaffected [9]. It is hoped that this clarification would help urther fruitful developments, following its own autonomous course [10]". |
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