The gravitational contribution to the stress-energy tensor of a medium in general relativity

Abstract

The macroscopic stress-energy tensor of an astronomical “medium” such as a galaxy of stars is determined by the field equation of general relativity from the small-scale variations in mass and velocity. In the weak-field, slow-motion approximation, in which the gravitational fields of the stars are Newtonian, it is found that the contribution by the small-scale gravitational fields to the macroscopic density and stress are, respectively, the Newtonian gravitational energy density and the Newtonian gravitational stress tensor. This result is based on the general-relativity field equation, not conservation laws, although the general-relativity field equation has the well-known property of being consistent with conservation laws.