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There are two other main classes of approach: 1) the post-Newtonian approaches (1/c expansions) and the post-Minkowskian approaches (K expansions). The post-Newtonian approaches are fraught with serious internal consistency problems [48] because they often lead, in higher approximations, to divergent integrals. The post-Minkowskian approach is an extension of the linearization, one may expect that there are some problems related to divergent logarithmic deviations [14]. Moreover, it has unexpectedly been found that perturbative calculations on radiation actually depend on the approach chosen [49]. Mathematically, this non-uniqueness shows, in disagreement with (3), that a dynamic solution of (1) is unbounded.
Based on the binary pulsar experiments, it is proven that the Einstein equation does not have any dynamic solution even for weak gravity [13]. Mathematically, however, the proof that is aimed directly to the nonexistence of a dynamic solution is independent of the experimental supports for (3). This long process is, in part, due to theoretical consistency were inadequately considered [9,10,13,35]. Moreover, it was not recognized that boundedness of a wave is crucial for its association with a dynamic source. These inadequacies allowed acceptance of unphysical "time-dependent" solutions as physical waves (Sections 3 & 5).
Although non-linearity of the 1915 Einstein equation was new, in view of impressive observational confirmations, it seemed natural to assume that gravitational waves would be produced. Moreover, gravitational radiation is often considered as due to the acceleration in a geodesic alone [50-52]. It is remarkable that in 1936 Einstein and Rosen [4] are the first to discover this problem of excluding the gravitational wave. However, without clear experimental evidence, it was difficult to make an appropriate modification.
From studying the gravity of electromagnetic waves, it was also clear that Einstein equation must be modified [11,18]. However, the Hulse and Taylor binary pulsar experiments, which confirm Hogarth's 1953 conjecture6) [31,35], are indispensable for verifying the necessity of the anti-gravity coupling in general relativity [10,13]. In addition to experimental supports, the Maxwell-Newton Approximation can be derived from physical principles, and the equivalence principle also implies boundedness of a normalized metric in general relativity [11]. A perturbative approach cannot be fully established for (1) simply because there are no bounded dynamic solutions10), which must, owing to radiation, be associated with an anti-gravity coupling.
Nevertheless, Christodoulou and Klainerman [27] claimed to have constructed bounded gravitational (unverified) waves. Obviously, their claim is incompatible with the findings of others. Furthermore, their presumed solutions are incompatible with Einstein's radiation formula and are unrelated to dynamic sources [10,11]. Thus, they simply have mistaken5) an unphysical assumption (which does not satisfy physical requirements) as a wave [28]. |
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