Maximal Slices in Stationary Spacetimes with Ergoregions

Summary

Abstract

We describe some recent results (obtained in collaboration with Piotr Chruściel) which establish existence of a maximal slice in a class of stationary spacetimes which contain an ergoregion but no black or white hole. No use of Einstein's equation or energy conditions is made in the proof. The result enables one to prove that all stationary solutions to the Einstein-Yang-Mills equations which have vanishing electric charge and do not contain a black or white hole must be static. Similar results for the case where a black and white hole with bifurcate horizon are present are briefly described.

A maximal slice in a spacetime (M, g_ab) is a closed, embedded, spacelike, submanifold of co-dimension one whose trace, of extrinsic curvature vanishes. The issue of whether maximal slices exist in certain classes of spacetimes in general relativity has arisen in many analyses. One of the most prominent early examples of the relevance of this issue occurs in the positive energy argument given by Dieter Brill in collaboration with Deser [5], where the existence of a maximal slice in asymptotically flat spacetimes was needed in order to assure positivity of the “kinetic terms” in the Hamiltonian constraint equation. The existence and properties of maximal slices has remained a strong research interest of Brill, and he has made a number of important contributions to the subject.