The derivation of mechanics from the law of gravitation in relativity theory

Abstract

The paper gives a general review of an investigation on some problems on motion in the relativity theory. We begin by discussing the motion of a single particle in a weak gravitational field and obtain both the linear and angular equations of motion; this is followed by brief accounts of the problem of two bodies and that of a rotating rod. In each case considered the equations of motion arise as conditions of integrability of the relativity equations for empty space. Only one of these problems has previously been treated by this method, this being the case of two bodies not connected by a material tension. However, this investigation, which was carried out by Einstein, Infeld and Hoffmann, introduces difficult ideas relating to the use of certain spatial surface integrals; in the present paper we avoid the use of these integrals. An essential feature of the work is that we take a first-order solution for the particular problem considered and substitute this solution in the quadratic terms of G μv and then build up a second-order solution. This method can be employed when we discuss the interior of matter; and, as an example, we conclude the paper by investigating the case of a rotating mass of liquid of constant density.