Spherical Symmetry and Mass-Energy in General Relativity. II. Particular Cases

Abstract

A solution of Einstein's field equations for the motion of a spherically symmetric distribution of perfect fluid is investigated in an isotropic comoving coordinate system. It is shown that this solution includes all solutions discussed by McVittie in which the density depends on both the radial and time coordinates. Necessary and sufficient conditions for the solution to be singularity-free and for the density and pressure to be nonnegative and monotonically decreasing from the center of the material outwards to an outer boundary are found. The material is surrounded by empty space. Examples of both oscillating and ``bouncing'' solutions are produced. It is shown that the outer boundary of the material never penetrates the Schwarzschild radius in all singularity-free solutions.