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It seems, the principle of causality2) (i.e., phenomena can be explained in terms of identifiable causes) [9,10] would be qualified as Wheeler's utterly simple idea. Being a physicist, his notion of beauty should be based on compelling and inevitability, but would not be based on some perceived mathematical ideas. It will be shown that the principle of causality is useful in examining validity of accepted "wave" solutions.
According to the principle of causality, a wave solution must be related to a dynamic source, and therefore is not just a time-dependent metric. A time-dependent solution, which can be obtained simply by a coordinate transformation, may not be related to a dynamic source8) [33]. Even in electrodynamics, satisfying the vacuum equation can be insufficient. For instance, the electromagnetic potential solution A0[exp(t - z)2] (A0 is a constant), is not valid in physics because one cannot relate such a solution to a dynamic source. Thus, as shown in Section 4, a solution free of singularities may not be physically valid.
A major problem in general relativity is that the equivalence principle has not been understood adequately [11,34]. Since a Lorentz manifold was mistaken as always valid, physical principles were often not considered. For instance, the principle of causality was neglected such that a gravitational wave was not considered as related to a dynamic source, which may not be just the source term in the field equation [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. |
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