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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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