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As shown, the difficulty comes from the assumption of boundedness (Section 3), which allows the existence of a bounded first-order approximation, which in turn implies that a time-average of the radiative part of G(2)(( is non-zero (7(. The present method has an advantage over Fock's approach to obtaining logarithmic divergence [13,14( for being simple and clear.
In short, according to Einstein's radiation formula, a time average of G(2)(t is non-zero and of O(K2/r2) [13(. Although (3) implies G(1)(t is of order K2, its terms of O(1/r2) can have a zero time average because G(1)(t is linear on the metric elements. Thus, (1') cannot be satisfied. Nevertheless, a static metric can satisfy (1), since both G(1)(( and G(2)(( are of O(K2/r4) in vacuum. Thus, that a gravitational wave carries energy-momentum does not follow from the fact that G(2)(( can be identified with a gravitational energy-stress (8,17(. Just as G(( , G(2)(( should be considered only as a geometric part. Note that G(t = -KT(m)(t are constraints on the initial data.
In conclusion, in disagreement with the physical requirement, assuming the existence of dynamic solutions of weak gravity for (1) [14,15,19-24( is invalid. This means that the calculations [25,26( on the binary pulsar experiments should, in principle, be re-addressed [12(. This explains also that an attempt by Christodoulou and Klainerman [26( to construct bounded "dynamic" solutions for G(( = 0 fails to relate to a dynamic source and to be compatible with (3) [28] although their solutions do not imply that a gravitational wave carries energy-momentum.
For a problem such as scattering, although the motion of the particles is not periodic, the problem remains. This will be explained (see Section 4) in terms of the 1995 update of the Einstein equation, due to the necessary existence of gravitational energy-momentum tensor term with an antigravity coupling in the source. To establish the 1995 update equation, the supports of binary pulsar experiments for (3) are needed [13].
3. Gravitational Radiations, Boundedness of Plane-Waves, and the Maxwell-Newton Approximation
An additional piece of evidence is that there is no plane-wave solution for (1). A plane-wave is a spatial-local idealization of a weak wave from a distant source. The plane-wave propagating in the z-direction is a physical model although its total energy is infinite [8,10]. According to (3), one can substitute (t - R) with (t - z) and the other dependence on r can be neglected because r is very large. This results in(((xi, t) becoming a bounded periodic function of (t - z). Since the Maxwell-Newton Approximation provides the first-order, the exact plane-wave as an idealization is a bounded periodic function. Since the dependence of 1/r is neglected, one considers essentially terms of O(1/r2) in G(2)((. In fact, the non-existence of bounded plane-wave for G(( = 0, was proven directly in 1991 [9,18].
In short, Einstein & Rosen [4,29] is essentially right, i.e., there are no wave solutions for R(( = 0. The fact that the existing "wave" solutions are unbounded also confirms the nonexistence of dynamic solutions. The failure to extend from the linearized behavior of the radiation is due to the fact that there is no bounded physical wave solution for (1) and thus this failure is independent of the method used. |
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