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.
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.
Nevertheless, some discussions may be helpful in clarifying coordinate relativistic causality. To discuss the Friedmann model, one must first accept essentially by faith that the mass distribution of the whole universe is homogeneous and isotropic. One must decide also modeling a galaxy as a particle is consistent with the normal understanding of Einstein's equivalence principle. Then, in Cartesian coordinates,
ds2 = d(2 - 2((){dx2 + dy2 + dz2}, (A1)
the Robertson-Walker geometry, is believed to be appropriate. Then, the Einstein equation (1) with source energy tensor T(( = u(u(+ P(u(u( - g(() leads to the following general evolution equations [17]:
3 = -4((( + 3P) (A2)
and
32/2 = 8(( - 3k/2, (A3)
where ( is the mass density, and P is the pressure. For different values of k, there are different types of solutions: k = +1 for the 3-sphere, k = 0 for the flat space, and k = -1 for the hyperboloid. For k = -1, 2(() is unbounded [17] and is therefore incompatible with the equivalence principle [11].
The rate of change of R (the distance between two isotropic observers at time () is
v= HR , (A4)
where H(() =/is identified with Hubble's constant. This means, however, the constant is time-dependent. Note, however, the observed red shifts may not be due to the Doppler effect alone [11,54,56].
However, within the above constraint, a model-independent feature of (() is
(()(( ( ( = 0; (A5a)
and
((()n(() = constant, where n ( 3 (A5b)
On the other hand, ds2 = 0 could imply that the light speed in the x-direction would be
(A6)
Thus, (A5a) and (A6) lead to a result that the light speed could be larger than c. Thus, it seems, either that coordinate relativistic causality could be violated or metric (A1) would be invalid.
Nevertheless, one must be careful because things are not that simple. For ds2 = 0 leads to a light speed in vacuum. However, in the Friedmann model, when a(() is very small, according to (A5b), not only there is no vacuum but the mass density ((() would be too large for the light to go through. Thus, the argument that leads to (A6) breaks down. Moreover, to justify the Robertson-Walker geometry, the effects of gravitational radiation should have been shown to be negligible at least for the assumed early universe. The existence of gravitational radiation, as pointed out by Lorentz and Wheeler [1], is due to the theory of relativity. Thus, it is also not clear that Friedmann's solution must be deduced from general relativity.
In reality, a galaxy is not a particle, the mass distribution is not homogeneous, and a light speed has nothing to do with Friedmann's modeling. Thus, it is clearly unsuitable for a discussion on fundamental questions. Now, it should be clear also that the Big Bang theory, though can be related to (1), depends on too many dubious assumptions (see also [32,54]) for the claim of being a consequence of general relativity. (Also, in view of the idealizations, the possibility of deriving eqs. (A2) and (A3) from another equation cannot be rule out.) Nevertheless, this discussion illustrates also the importance of the equivalence principle.
Acknowledgments
This paper is dedicated to Professor J. E. Hogarth of Queen's University, Kingston, Ontario, Canada, who conjectured in 1953 the nonexistence of dynamic solutions for the 1915 Einstein equation. The author wishes to express his appreciation to Professor Xin Yu for the hospitality of the Hong Kong Polytechnic University where substantial of this work was done in 1995. The author gratefully acknowledges stimulating discussions with Dr. H. C. Chan, Professor C. Au, Professor J. E. Hogarth, Professor S. A. Lamb, Professor P. Morrison, and Professor H. Nicolai. The author wishes to thank the referees for valuable comments and pointing out useful literature; and Ms. P. Ma for the French abstract. The author is indebted to Mr. David P. Chan and Mr. Richard C. Y. Hui for their supports and hospitality while in Hong Kong. This publication is supported by Innotec Design, Inc., U.S.A.
ENDNOTES
1) Some authors prefer, different from Einstein, to define K = 8( (c-4 [55]. Then, the four velocity u( would be defined as cdx(/ds, where ds2 = g(( dx(dx( such that equation (1) remains the same.
2) The time-tested assumption that phenomena can be explained in terms of identifiable causes is called the principle of causality. This is the basis of relevance for all scientific investigations. The principle of causality implies that any parameter in a physical solution must be related to some physical causes.
3) This explicit reinterpretation of Einstein's equivalence principle (based on Pauli's misinterpretation that Einstein objected [57]) as just the signature of Lorentz metric was advocated by Synge [58] earlier and Friedman9) currently. Recently, it has been proven that such a reduction is inconsistent with Einstein's own interpretation and physical principles [11,35,57] as well as in disagreement with experiments including the Michelson-Morley experiment [59]. However, the advocates disregard all these inconsistencies because, owing to their inadequate understanding of physics at the fundamental level, they believe that a coordinate system (including its metric) has no physical meaning [60]. (Moreover, following the step of Fock [61], Ohanian, and Ruffini openly declared in their book [55], which is endorsed by Wheeler, that both of Einstein's equivalence principle and the principle of relativity are invalid.) Nevertheless, this seemingly exceedingly ingenious defense collapses because the observed gravitational red shifts unequivocally imply that their interpretation is invalid in physics.
4) A dynamic metric solution in gravity is related to the dynamics of its source matter. A dynamic source, according to relativity, would generate gravitational radiation [1]. For the perihelion of Mercury and the deflection of light, the metric is a static solution although solutions of the test particles are calculated. It was believed that the influence of a test particle to the metric could also be calculated with (1). However, as suspected by Gullstrand [40,41] and conjectured by Hogarth6) [31], the truth is the opposite.
5) K. Kuchar [62] claimed to have proved that the initial condition of Einstein's equation (1) can be approximated by the initial condition of the linear equation (3) by using a power series expansion. Note, however, that the only valid case of such a power series expansion is a non-dynamic solution (see Sections 2-4). Thus, he has proven only that the properties are true in an unintended void set. Such a basic mistake is essentially repeated 20 years later by Christodoulou and Klainerman [27] for claiming the existence of bounded radiative solutions (see Section 6). Nevertheless, the Editorial Board of Quantum and Classical Gravity [63], unlike the book review [64] and the Editor of GRG [65], considered these invalid claims as "proofs". Moreover, a solution relating to a dynamic source by an equation alone, as suggested by Klainerman and Nicolò [66], is insufficient because such a solution may still violate other physical requirements (see Section 5).
6) Hogarth conjectured that, for an exact solution of the two-particle problem, the energy-momentum tensor did not vanish in the surrounding space and would represent the energy of gravitational radiation.
7) The possibility of having an anti-gravity coupling was formally mentioned by Pauli [12]. In a different way, such a possibility was actually first mentioned by Einstein [67] in 1921. He wrote in "Geometry and Experience," "But, if the universe is finite, there is a second deviation from Newtonian theory, which, in the language of Newtonian theory, may be expressed thus: the gravitational field is such as if it were produced, not only by the ponderable masses, but in addition by a mass-density of negative sign, distributed uniformly through out space." He also firmly believed in such a possibility. However, it was not recognized that an anti-gravity coupling is crucial for a dynamic solution [9,13]. On the other hand, Hawking and Penrose [6,17] had implicitly assumed, in their singularity theorems, the impossibility of an anti-gravity coupling. A rather common erroneous ground to reject the existence an antigravity coupling is due to a misinterpretation of the equivalence of mass and energy in the energy-mass conservation law E = mc2 [68]. For instance, Fock [61] claimed, "We saw that to any energy E one should ascribe a mass m = E/c2 and to every mass one should ascribe an energy E = mc2." However, this is inconsistent with general relativity with a tensor field. According to Einstein [69], only the latter is valid. Einstein stated, "Now we can reverse the relation and say that an increase of E in the amount of energy must be accompanied by an increase of E/c2 in the mass. I can easily supply energy to the mass - for instance, if I heat it by ten degrees." He also wrote "For a mass increase to be measurable, the change of energy per mass unit must be enormously large." The key word in Einstein's statements is "increase". Thus, E/c2 is related to an increment of mass to massive matter. However, this does not mean that in general any kind of energy E has a related mass E/c2, as Fock claimed. He also remarked, "Also, the law permits us to calculate in advance, from precisely determined atomic weights, just how much energy will be released with any atomic disintegration we have in mind. The law says nothing, of course, as to whether-or how - the disintegration reaction can be brought about."
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