Uniqueness and Non-uniqueness in the Einstein Constraints

TL;DR

This paper investigates the uniqueness of solutions to the Einstein conformal thin sandwich equations, revealing conditions under which multiple solutions, including black hole solutions, can exist.

Contribution

It demonstrates the existence of multiple solutions to the CTS equations under certain conditions, highlighting fundamental properties of general relativity.

Findings

Only one solution for perturbed Minkowski space

Two distinct solutions when lapse is determined by a fifth elliptic equation

Existence of solutions containing black holes

Abstract

The conformal thin sandwich (CTS) equations are a set of four of the Einstein equations, which generalize the Laplace-Poisson equation of Newton's theory. We examine numerically solutions of the CTS equations describing perturbed Minkowski space, and find only one solution. However, we find {\em two} distinct solutions, one even containing a black hole, when the lapse is determined by a fifth elliptic equation through specification of the mean curvature. While the relationship of the two systems and their solutions is a fundamental property of general relativity, this fairly simple example of an elliptic system with non-unique solutions is also of broader interest.