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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.
Nevertheless, in cosmology, there are time-dependent solutions that do not involve a coordinate system chosen a priori, nor gravitational radiation. However, one should note also that all the cosmological models are based on idealizations that have not been established beyond reasonable doubt [32,54]. For this reason alone, such examples are unsuitable for our discussion on a fundamental problem of realistic situations. However, some discussions on this subject are needed, since it is claimed that the big bang theory is based on general relativity [32,55].
It is generally assumed [55] "that the energy-momentum tensor in the universe today is that of a uniform gas with zero pressure. The galaxies may be regarded as the 'particles' out of which this gas is made, and since the velocities of the galaxies do not deviate much from uniform expansion, we can neglect the 'pressure' of the gas of galaxies. ..." The Friedmann models assumed homogeneous, isotropic models of the universe with mass density but with zero pressure. A difficult in cosmology is that many usual physical requirements, on which a judgment of physical validity depends, are probably not applicable. |
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