ON INCOMPATIBILITY OF GRAVITATIONAL RADIATION WITH THE..(三)

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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].
Within the theoretical framework of general relativity, however, the gravitational field of a radiating asymptotically Minkowskian system is given by the Maxwell-Newton Approximation [13]. With the need of rectifying the 1915 Einstein equation established, the exact form of t(g)(( in the equation of 1995 update [13] is an important problem since a dynamic solution that gives an approximation for the perihelion of Mercury remains unsolved [41]. Moreover, the update equation shows that the singularity theorems prove only the breaking down of theories of the Wheeler-Hawking school3), but not general relativity (see Section 4). Experimentally, the Maxwell-Newton Approximation would be further tested by the Gravity Probe-B gyroscopes [53] on the precessions. This analysis suggests that further confirmation of this Approximation and thus the equivalence principle is expected.

Appendix: Dynamic Space-Time, Space-Time Coordinate System, and the Big Bang Theory
The equivalence principle, in a certain sense, is a non-local property, since its physics is whether the geodesic represents a physical free fall [11]. Thus, one must consider beyond the mathematical tangent space, that is, mathematical local Minkowski spaces. To determine whether a manifold solution can be diffeomorphic to a physical space is a difficult problem and physical requirements are needed [10].
In physics, the frame of reference is often chosen to be best for the problem. If a valid physical solution cannot be found, the difficult is usually not due to the coordinates. In addition, as a practical approximate means, a Galilean transformation can be used in some class of problems. Thus, that a certain coordinate system is useful for some calculations does not mean that the coordinate system is, in principle, realizable.
For a practical problem, in spite of talks about coordinates cannot be chosen a priori, general relativity is actually not an exception11). For instance, in the Schwarzschild static solution, the frame of reference is chosen a priori and the radial r is (x2 y2 z2)1/2. This frame of reference is used to access the amount of light bending. In the problem of light bending, the total field (space-time metric) should be time-dependent, but r as a variable would be the same if the frame of reference does not change.

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