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= c(1- () + v, or = -c(1- () + v, where ( = (28b)
according to metric (28a). Thus, coordinate relativistic causality is violated for sufficiently large r. In other words, point 1) of the equivalence principle cannot be satisfied and metric (28a) is not realizable.
This illustrates that the equivalence principle is a requirement for a valid physical space-time coordinate system.
7. Restriction of Covariance, and Intrinsically Unphysical Lorentz Manifolds
Einstein proposed that the equivalence principle is satisfied in a physical space-time1). Moreover, the equivalence principle is satisfied only in a physical space-time since the existence of a local Minkowski space has been proven by mathematics and a satisfaction of the equivalence principle requires sufficient satisfactions of all physical conditions. For example, when coordinate relativistic causality is not satisfied, the equivalence principle is proven directly to be not valid for this manifold. The current confusion was due to that the equivalence principle has not been understood correctly from the viewpoint of physics.
However, one may still wonder whether a Lorentz manifold is always diffeomorphic to a physical space. If this were true, then the metric signature would be essentially equivalent to the equivalence principle. But, there are Lorentz manifolds any of which cannot be diffeomorphic to a physical space. In view of this, such misunderstanding of relativity must be rectified. Since the belief that a Lorentz manifold were diffeomorphic to a physical space, has never been proven; the burden of proof is on such believers. Nevertheless, it is desirable to give an example of an intrinsically unphysical Lorentz manifold. This can even be a solution of Einstein equation if it fails a physical requirement, which is independent of a coordinate system [8,9,16].
For instance, an accepted solution of metric for an electromagnetic plane wave [38] is
ds2 = du dv + H du2 - dxi dxj , where H = hij(u)xixj, hii(u) 0, hij = hji , (29)
u = t - z, v = t + z. This is a Lorentz manifold since its eigen values are H ( (H2 + 1)1/2, -1, and -1. However, since the condition 1 ( (1 + H)/(1 - H) may not be valid, metric (29) does not satisfy coordinate relativistic causality and therefore the equivalence principle. Moreover, since H can be arbitrarily large, metric (29) is incompatible with Einstein notion of weak gravity4) and the correspondence principle. Also, in the light bending experiment, the gravitational effect of the light is implicitly assumed to be negligible. Thus, metric (29) cannot be valid in physics.
Nevertheless, to show that metric (29) cannot be diffeomorphic to a physical space, needs more work. The gravitational force (related to (ztt = (1/2)((hijxixj)/(t has arbitrary parameters (the coordinate origin). This arbitrariness in the metric violates the principle of causality (i.e., the causes of phenomena are identifiable) [8,11]. Thus, the manifold (29) cannot be diffeomorphic to a physical space since a diffeomorphism cannot eliminate the parameters, which violate the principle of causality. |
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