A point mass in an isotropic universe: III. The region $R\leq 2m$

TL;DR

This paper investigates the nature of the scalar curvature singularity at R=2m in McVittie's solution, showing it is gravitationally weak and exploring the structure of the region R≤2m without relying on asymptotic analysis.

Contribution

It provides a characterization of McVittie's solution near the singularity without using asymptotic methods and demonstrates the singularity's weak gravitational nature.

Findings

The singularity at R=2m is gravitationally weak.

The structure of the region R≤2m is characterized without asymptotic assumptions.

The singularity is space-like and precedes other events in the expanding universe case.

Abstract

McVittie's solution of Einstein's field equations, representing a point mass embedded into an isotropic universe, possesses a scalar curvature singularity at proper radius $R=2m$. The singularity is space-like and precedes, in the expanding case, all other events in the space-time. It is shown here that this singularity is gravitationally weak, and the possible structure of the region $R\leq 2m$ is investigated. A characterization of this solution which does not involve asymptotics is given.