n [8,35].
Since the principle of causality was not understood adequately, solutions with arbitrary nonphysical parameters were accepted as valid [34]. Similarly, Misner, Thorne & Wheeler [5], assumed that the metric due to an electromagnetic plane-wave is invariant with respect to a rotation whose axis is in the direction of propagation. Consequently, in addition to the fact that the polarization is incorrect, Misner et al. were not aware of that, in disagreement with what they stated, such a metric cannot be bounded. Such unbounded solutions disagree with experiments [10,11].
Among the existing so-called wave solutions, not only Einstein's equivalence principle but the principle of causality is not satisfied because they cannot be related to a dynamic source. (However, a source term in an equation, though related to, may not necessarily represent the physical cause [9,34].) Here, examples of accepted "gravitational waves" are shown as actually invalid in physics.
1. Let us examine the cylindrical waves of Einstein & Rosen [29]. In cylindrical coordinates, (, (, and z,

ds2 = exp(2( - 2()(dT2 - d(2) - (2exp(-2()d(2 - exp(2()dz2 (12)

where T is the time coordinate, and ( and ( are functions of ( and T. They satisfy the equations

((( + (1/()(( - (TT = 0, (( = ([((2 + (T2], and (T = 2((((T. (13)

Rosen [36] consider the energy-stress tensor T(( that has cylindrical symmetry. He found that

T44 + t44 = 0, and T4l + t4l = 0 (14)

where t(( is Einstein's gravitational pseudotensor, t4l is momentum in the radial direction.
However, Weber & Wheeler [37] argued that these results are meaningless since t(( is not a tensor. They further pointed out that the wave is unbounded and therefore the energy is undefined. Moreover, they claimed metric (12) satisfying the equivalence principle and speculated that the energy flux is non-zero.
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 (,

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