On the structure of solutions to the static vacuum Einstein equations

TL;DR

This paper characterizes the asymptotic behavior of solutions to the static vacuum Einstein equations with boundaries, showing they have finitely many ends that are either asymptotically flat or parabolic, impacting black hole uniqueness theorems.

Contribution

It provides a complete characterization of the asymptotic behavior of static vacuum Einstein solutions with boundaries, including new insights into Weyl metrics and implications for black hole uniqueness.

Findings

Solutions have finitely many ends, each asymptotically flat or parabolic.

Examples demonstrate both asymptotic behaviors.

Results enable dropping the AF assumption in black hole uniqueness theorems.

Abstract

A complete characterization is obtained of the asymptotic behavior of solutions of the static vacuum Einstein equations which have a (pseudo)-compact horizon or boundary and are complete away from the boundary. It is proved that the time-symmetric space-like hypersurface has only finitely many ends, each of which is either asymptotically flat (AF) or parabolic, as in the (static) Kasner metric. Examples are given with both types of behavior, together with an extensive discussion and new characterization of Weyl metrics. The asymptotics result allows one in most circumstances to drop the AF assumption from the static black hole uniqueness theorems and replace it with just a completeness assumption.