This chapter embarks on a study of the two-body problem in general relativity. In other words, it seeks to describe the motion of two compact, self-gravitating bodies which are far-separated and moving slowly. It limits the discussion to corrections proportional to v2 ~ m/R, the so-called post-Newtonian or 1PN corrections to Newton’s universal law of attraction. The chapter first examines the gravitational field, that is, the metric, created by the two bodies. It then derives the equations of motion, and finally the actual motion, that is, the post-Keplerian trajectories, which generalize the post-Keplerian geodesics obtained earlier in the chapter.
Hod claimed to have a method to deal with a simplified two-body problem. The basic error of Hod and the previous researchers is that they failed to see that in general relativity there is no bounded dynamic solution for a two-body problem. A common error is that the linearized equation is considered as always providing a valid approximation in mathematics. However, validity of the linearization is proven only for the static and the stable cases when the gravitational wave is not involved. In a dynamic problem when gravitational wave is involved, since it is proven in 1995 that there is no bounded dynamic solution, the process of linearization is not valid in mathematics. This is the difference between Einstein and Gullstrand who suspected that a dynamic solution does not exist. In fact, for the dynamic case, the Einstein equation and the linearized equation are essentially independent equations, and the perturbation approach is not valid. Note that the linearized equation is a linearization of the Lorentz-Levi-Einstein equation, which has bounded dynamic solutions but not the Einstein equation, which has no bounded dynamic solution. Because of inadequacy in non-linear mathematics, many had made errors without knowing them.
Einstein’s theory of general relativity affords an enormously successful description of gravity. The theory encodes the gravitational interaction in the metric, a tensor field on spacetime that satisfies partial differential equations known as the Einstein equations. This review introduces some of the fundamental concepts of numerical relativity—solving the Einstein equations on the computer—in simple terms. As a primary example, we consider the solution of the general relativistic two-body problem, which features prominently in the new field of gravitational wave astronomy.
Entropy theorems in classical mechanics, general relativity, and the gravitational two-body problem
