On the structure of solutions to the static vacuum Einstein equations

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.