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Their claim shows an inadequate understanding of the equivalence principle. To satisfy this principle requires that a time-like geodesic must represent a physical free fall. This means that all (not just some) physical requirements are necessarily satisfied. Thus, the equivalence principle may not be satisfied in a Lorentz Manifold [11,35], which implies only the necessary condition of the mathematical existence of a co-moving local Minkowski space along a time-like geodesic. It will be shown that manifold (12) cannot satisfy coordinate relativistic causality. Moreover, as pointed out earlier, an unbounded wave is unphysical.
Weber and Wheeler's arguments for unboundedness are complicated, and they agreed with Fierz's analysis, based on (13), that ( is a strictly positive where ( = 0 [37]. However, it is possible to see that there is no physical wave solution in a simpler way. Gravitational red shifts imply that gtt ( 1 [2]; and
-g(( ( gtt , -g((/(2 ( gtt , and -gzz ( gtt , (15a)
are implies by coordinate relativistic causality. Thus, according to these constraints, from metric (12) one has
exp(2() ( 1 and exp (2() ( exp(4(). (15b)
Equation (15) implies that gtt ( 1 and that ( ( 0. However, this also means that the condition ( > 0 cannot be met. Thus, this shows again that there is no physical wave solution for G(( = 0.
Weber and Wheeler are probably the earliest to show the unboundedness of a wave solution for G(( = 0. Nevertheless, due to their inadequate understanding of the equivalence principle, they did not reach a valid conclusion. It is ironic that they therefore criticized Rosen who come to a valid conclusion, though with dubious reasoning.
2. Robinson and Trautman [38] dealt with a metric of spherical "gravitational waves" for G(( = 0. However, their metric has the same problem of unboundedness and having no dynamic source connection. This confirms further that the cause of this problem is intrinsically physical in nature. Their metric has the following form:
ds2 = 2d(d( + (K - 2H( - 2m/()d(2 - (2p-2{[d( + ((q/(()d(]2 + [d( +((q/(()d(]2}, (16a)
where m is a function of ( only, p and q are functions of (, (, and (,
H = p-1(p/(( + p(2p-1q/(((( - pq (2p-1/(((( , (16b)
and K is the Gaussian curvature of the surface ( = 1, ( = constant,
K = p2((2/((2 + (2/((2)ln p. (16c)
For this metric, the empty-space condition G(( = 0 reduces to
(2q/((2 + (2q/((2 = 0, and (2K/((2 + (2K/((2 = 4p-2((/(( - 3H)m. (17)
To see this metric has no dynamic connection, let us examine their special case as follows:
ds2 = 2d(d( - 2Hd(2 - d(2 - d(2, and (H/(( = (2H/((2 + (2H/((2 = 0. (18)
This is a plane-fronted "wave" [39] derived from metric (16) by specializing
p = 1 + ((2 + (2)K(()/4. (19a)
substituting
( = (-2 + (-1, ( = (, ( = (2, ( = (2, q = (4, (19b)
where ( is constant, and taking the limit as ( tends to zero [38]. Although (18) is a Lorentz metric, there is a singularity on every wave front where the homogeneity conditions
(3H/((3 = (3H/((3 = 0. (20)
are violated [38]. Obviously, this is also incompatible with Einstein's notion of weak gravity [2]. A problem in current theory is its rather insensitivity toward theoretical self-consistency [9,13,35,40-42]. |
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