uivalence 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]".
2. Einstein 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, t 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 ree 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 physical free fall [9,16]. In other words, given the mathematical existence of local Minkowski space co-moving along a time-like geodesic, the crucial physical question for the satisfaction of the equivalence principle is whether the geodesic represents a physical free fall.
Einstein [28] pointed out, s far as the prepositions of mathematics refers to reality, they are not certain; and as far as they are certain, they do not refer to reality." Thus, an application of a mathematical theorem should be carefully examined although ne cannot really argue wi
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