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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 pace is not a lot of points close together; it is a lot of distances interlocked." Einstein accepted Eddington criticism and no longer advocated the invalid arguments in his book, he Meaning of Relativity" of 1921. Einstein also praised Eddington 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 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 (壯 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-Poincaré 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. |
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