A two-mass expanding exact space-time solution

TL;DR

This paper explores exact solutions in general relativity for a universe with two masses on a 3-sphere, revealing that static regions cannot be directly joined and that an expanding solution involves a Kantowski-Sachs region, with implications for embedding local static configurations in an expanding universe.

Contribution

It introduces a novel exact space-time solution with two masses on a 3-sphere, demonstrating the impossibility of static regions being glued and proposing an expanding solution with a Kantowski-Sachs region.

Findings

Static regions around masses cannot be glued via Israel junction conditions.

A non-static solution exists with two static regions matching an expanding Kantowski-Sachs region.

The solution differs from Schwarzschild and has implications for cosmological models.

Abstract

In order to understand how locally static configurations around gravitationally bound bodies can be embedded in an expanding universe, we investigate the solutions of general relativity describing a space-time whose spatial sections have the topology of a 3-sphere with two identical masses at the poles. We show that Israel junction conditions imply that two spherically symmetric static regions around the masses cannot be glued together. If one is interested in an exterior solution, this prevents the geometry around the masses to be of the Schwarzschild type and leads to the introduction of a cosmological constant. The study of the extension of the Kottler space-time shows that there exists a non-static solution consisting of two static regions surrounding the masses that match a Kantowski-Sachs expanding region on the cosmological horizon. The comparison with a Swiss-Cheese construction is also discussed.