|
|
Einstein [2] proposed the linearized theory for a weak radiating gravitational field. But, Bondi [24] commented, "it is never entirely clear whether solutions derived by the usual method of linear approximation necessarily correspond in every case to exact solutions, or whether there might be spurious linear solutions which are not in any sense approximations to exact ones." Thus, in calculating gravitational waves from the Einstein equation, problems are considered as due to the method rather than inherent in the equations.
Physically, it is natural to continue assuming Einstein's notion of weak gravity is valid. (Boundedness, though a physical requirement, may not be mathematically compatible to a nonlinear field equation. But, no one except perhaps Gullstrand [40,41], expected the nonexistence of dynamic solutions.) The complexity of the Einstein equation makes it very difficult to have a close form. Thus, it is necessary that a method of expansion should be used to examine the problem of weak gravity, if one expects such an expansion to be valid.
A factor which contributes to this faith is that ((G(( ( 0 implies ((T(m)(( = 0, the energy-momentum conservation law. However, this is only necessary but not sufficient for a dynamic solution. Although the 1915 equation gives an excellent description of planetary motion, including the advance of the perihelion of Mercury, this is essentially a test-particle theory, in which the reaction of the test particle is neglected. Thus, the so obtained solutions are not dynamic solutions. As pointed out by Gullstrand [41,45] such a solution may not be obtainable as a limit of a dynamic solution. Nevertheless, Einstein, Infeld, and Hoffmann [22] incorrectly assumed the existence of bounded dynamic solution and deduced the geodesic equation from the 1915 equation. Recently, Feymann [23] made the same incorrect assumption that a physical requirement would be unconditionally applicable to a mathematical equation.
The nonlinear nature of Einstein equation certainly gives surprises. In 1959, Fock [14] pointed out that, in harmonic coordinates, there are divergent logarithmic deviations from expected linearized behavior of the radiation. After the discovery that some vacuum solutions are not logarithmic divergent [15], the inadequacy of Einstein's equation was not recognized. Instead, the method of calculation was mistaken as the problem.
To avoid the appearance of logarithms, Bondi et al. [24] and Sachs [46] introduced a new approach to gravitational radiation theory. They used a special type of coordinate system, and instead of assuming an asymptotic expansion in the gravitational coupling constant (, they assume the existence of an asymptotic expansion in inverse power of the distance r (from the origin where the isolated source is located in r ( a, which is a positive constant). The approach of Bondi-Sachs was clarified by the geometrical 'conformal' reformulation of Penrose [47].
However, this approach is unsatisfactory, "because it rests on a set of assumptions that have not been shown to be satisfied by a sufficiently general solution of the inhomogeneous Einstein field equation [48]." In other words, this approach provides only a definition of a class of space-times that one would like to associate to radiative isolated systems, neither the global consistency nor the physical appropriateness of this definition has been proven. Moreover, perturbation calculations have given some hints of inconsistency between the Bondi-Sachs-Penrose definition and some approximate solution of the field equation. Not less important, it seems a priori difficult to relate to the source located within r ( a [48]. |
|
