On Gauss' theorem and the concept of mass in general relativity

Abstract

The present communication is concerned with the extension to general relativity theory of the well-known theorem of Gauss on the Newtonian potential, viz., that the total flux of gravitational force through a simple closed surface is equal to (-4π) x the total gravitating mass contained within the surface: and to various questions which arise in connection with this. In the extended theorem, which is found in 2, the Newtonian concept of "gravitating mass" is naturally replaced by that of the energy-tensor, which does not in general consist solely of the "material" energy-tensor, and need not involve any "matter" at all. This new feature is illustrated in 3 by an example in which the "gravitating mass" is simply an electrostatic field. In 4 a theorem of "energy" is obtained which is required later, and which enables us to make precise the concept of the "potential energy" of an infinitesimal particle in a statical field in general relativity; this "potential energy" is shown to be the product of two factors, one depending on the particle alone (which may be called its "potential mass") and the other depending solely on its position. It is shown in 5 that the definition of "potential mass" introduced in 4 enables us to express the generalized Gauss' theorem of 2, in the case when the energy-tensor is due to actual masses, by a simple statement practically identical with the original Gauss' theorem of Newtonian theory. Finally in 6 it is shown that the electrostatical form of Gauss' theorem in Newtonian physics, viz., that the total strength of the tubes of force issuing from a closed surface is equal to the total electric charge within the surface, can also be extended to General Relativity, but that this extension is different in character from the gravitational theorem of 2.