A point mass in an isotropic universe. Existence, uniqueness and basic properties

TL;DR

This paper investigates the conditions for point mass solutions in open Robertson-Walker universes, proving existence, analyzing properties, and comparing solutions like McVittie's, with implications for cosmological models.

Contribution

It establishes criteria for point mass solutions in open RW universes, proves existence for the negative curvature case, and analyzes their properties and limits.

Findings

McVittie's solution satisfies criteria for k=0 but not for k=-1

Existence of a new solution for k=-1 is proven

Solutions exhibit specific behaviors at null infinity and singularities

Abstract

Criteria which a space-time must satisfy to represent a point mass embedded in an open Robertson--Walker (RW) universe are given. It is shown that McVittie's solution in the case $k=0$ satisfies these criteria, but does not in the case $k=-1$. Existence of a solution for the case $k=-1$ is proven and its representation in terms of an elliptic integral is given. The following properties of this and McVittie's $k=0$ solution are studied; uniqueness, behaviour at future null infinity, recovery of the RW and Schwarzschild limits, compliance with energy conditions and the occurence of singularities. Existence of solutions representing more general spherical objects embedded in a RW universe is also proven.