XLIII.—On the Gravitational Mass of a System of Particles

Abstract

SummaryIn classical mechanics the mass of a system of gravitating particles can be denned to be the mass of an equivalent particle which gives the same field at great distances, or alternatively the mass can be defined by means of Gauss' Theorem. Reference to the former procedure was made by Eddington and Clark (1938) in a discussion on the problem of n bodies. The relativistic extension of Gauss' Theorem has been investigated by Whittaker (1935) for a particular form of the line-element and for more general fields by Ruse (1935). The latter, treating the problem from a purely geometrical point of view, expressed the integral of the normal component of the gravitational force as the sum of two volume integrals. The physical significance of one of these integrals was quite obvious but the meaning of the other was far from clear. In this paper the terms in Ruse's result are examined as far as the order m2 in the case of a fundamental observer at rest and the 1938 discussion modified to bring the two investigations into line. It is concluded that the surface integral of the normal component of the gravitational force taken over an infinite sphere is –4π × the energy of the system.