Uniqueness and Non-uniqueness in the Einstein Constraints

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.

4 pages, 4 figures; abstract and introduction rewritten

References in corpus (2)

• Excision boundary conditions for black hole initial data
• Initial data for Einstein's equations with superposed gravitational waves

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