Gravitating macroscopic media in general relativity

Abstract

The problem of construction of a continuous (macroscopic) matter model for a given point-like (microscopic) matter distribution in general relativity is formulated. The existing approaches are briefly reviewed and a physical analogy with the similar problem in classical macroscopic electrodynamics is pointed out. The procedure by Szekeres in the linearized general relativity on Minkowski background to construct a tensor of gravitational quadruple polarization by applying Kaufman's method of molecular moments for derivation of the polarization tensor in macroscopic electrodynamics and to derive an averaged field operator by utilizing an analogy between the linearized Bianchi identities and Maxwell equations, is analyzed. It is shown that the procedure has some inconsistencies, in particular, (1) it has only provided the terms linear in perturbations for the averaged field operator which do not contribute into the dynamics of the averaged field, and (2) the analogy between electromagnetism and gravitation does break upon averaging. A macroscopic gravity approach in the perturbation theory up to the second order on a particular background space-time taken to be a smooth weak gravitational field is applied to write down a system of macroscopic field equations: Isaacson's equations with a source incorporating the quadruple gravitational polarization tensor, Isaacson's energy-momentum tensor of gravitational waves and energy-momentum tensor of gravitational molecules and corresponding equations of motion. A suitable set of material relations which relate all the tensors is proposed.