On completeness of orbits of Killing vector fields

TL;DR

This paper proves a theorem linking the completeness of Killing vector field orbits in certain space-times to properties near the Cauchy surface, strengthening black hole uniqueness results.

Contribution

It establishes a reduction of orbit completeness to orbit properties near the Cauchy surface, enhancing understanding of black hole uniqueness.

Findings

All Killing orbits are complete in maximal developments of asymptotically flat data.

All Killing orbits are complete in maximal developments of data on compact manifolds.

Strengthens black hole uniqueness theorems.

Abstract

A Theorem is proved which reduces the problem of completeness of orbits of Killing vector fields in maximal globally hyperbolic, say vacuum, space--times to some properties of the orbits near the Cauchy surface. In particular it is shown that all Killing orbits are complete in maximal developements of asymptotically flat Cauchy data, or of Cauchy data prescribed on a compact manifold. This result gives a significant strengthening of the uniqueness theorems for black holes.