The Gravitational Equations and the Problem of Motion

Introduction. In this paper we investigate the fundamentally simple question of the extent to which the relativistic equations of gravitation determine the motion of ponderable bodies. Previous attacks on this problem' have been based upon gravitational equations in which some specific energy-momentum tensor for matter has been assumed. Such energy-momentum tensors, however, must be regarded as purely temporary and more or less phenomenological devices for representing the structure of matter, and their entry into the equations makes it impossible to determine how far the results obtained are independent of the particular assumption made concerning the constitution of matter. Actually, the only equations of gravitation which follow without ambiguity from the fundamental assumptions of the general theory of relativity are the equations for empty space, and it is important to know whether they alone are capable of determining the motion of bodies. The answer to this question is not at all obvious. It is possible to find examples in classical physics leading to either answer, yes or no. For instance, in the ordinary Maxwell equations for empty space, in which electrical particles are regarded as point singularities of the field, the motion of these singularities is not determined by the linear field equations. On the other hand, the well-known theory of Helmholtz on the motion of vortices in a non-viscous fluid gives an instance where the motion of line singularities is actually determined by partial differential equations alone, which are there non-linear. We shall show in this paper that the gravitational equations for empty space are in fact sufficient to determine the motion of matter represented as point singularities of the field. The gravitational equations are non-linear, and, because of the necessary freedom of choice of the coordinate system, are such that four differential relations exist between them so that they form an overdetermined system of equations. The overdetermination is responsible for the existence of equations of motion, and the non linear character for the existence of terms expressing the interaction of moving bodies. Two essential steps lead to the determination of the motion.