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The anti-gravity coupling of t(g)ab that explains the dynamical failure of eq. (1), is due to Einstein radiation formula [7]. However, Pauli [10] was the first to point out explicitly the possibility of such an antigravity coupling. Moreover, the existence of such a coupling is, in a way, implicitly suggested by the singularity theorems, which show that if all the couplings are of the same sign [21], the existence of unrealistic spacetime singularities would b inevitable. The need of an anti-gravity coupling was first discovered in calculating the gravity of an electromagnetic wave [6].
Moreover, it was Einstein and Rosen [35] who first discover that the 1915 equation may not have a propagating wave solution. In 1953 Hogarth [36] conjectured that this equation does not have a dynamic solution. A definitive indication of this is the non-existence of the plane-wave solution [8]. Note that the lane waves" proposed by Bondi, Pirani, and Robinson [37], are actually unbounded although they believe that a plane-wave is an idealization of a weak wave from a distant isolated source. These unbounded solutions satisfy the condition of planeness but are not related to any weak wave. Also, although Misner, Thorne & Wheeler [13] conclude correctly that the plane-waves are bounded, their equation for plane-waves actually has no bounded wave solution [8]. This illustrates that over confidence may lead to careless, and result in inconsistency.
In short, the theoretical framework of general relativity permit an additional term Y(1)ab ( 0 whose existence is required by the dynamic cases. The 1915 equation is only an over simplified special choice of Einstein. Note, however, such a choice is consistent with the equivalence principle is known only for the static case.
5. Validity of a Space-Time Metric and the Equivalence Principle
Einstein proposed that the equivalence principle is satisfied in a physical space-time1). In fact, the equivalence principle is satisfied, if and only if the space-time manifold is physically realizable, since a satisfaction of the equivalence principle requires that the geodesic represent a physical free fall. Thus, although defining a coordinate system for the purpose of calculation is only a mathematical step, choosing a space-time coordinate system requires physical considerations.
It will be shown that not all mathematical coordinate systems are equivalent in physics as claimed by Bergmann [14] and Liu [15]. For clarity, this will be illustrated with a few simple Lorentz metrics without gravity.
Example 1. To see the need of considering beyond the metric signature, consider the artificial metric,
ds2 = (2dt2 - dx2 - dy2 - dz2, (18a)
the time unit of t is second, the space unit is cm, and ( (( 2c, c = 3x1010 cm/sec). If the equivalence principle were valid, ds2 = 0 would imply the light speed to be (. Immediately, there is a contradiction, and thus the equivalence principle cannot be valid.
Nevertheless, one might argue that metric (18a) can be transformed to |
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