On the Rigidity Theorem for Spacetimes with a Stationary Event Horizon or a Compact Cauchy Horizon

TL;DR

This paper generalizes existing theorems to show that in certain electrovac spacetimes with either a stationary black hole horizon or a compact Cauchy horizon, a Killing vector field exists near the horizon, even without analyticity.

Contribution

It extends Hawking's and Isenberg-Moncrief's rigidity theorems to smooth, non-analytic spacetimes with specific horizon structures.

Findings

Existence of a Killing vector field near the horizon

Generalization of rigidity theorems to non-analytic cases

Applicable to both black hole and cosmological horizons

Abstract

We consider smooth electrovac spacetimes which represent either (A) an asymptotically flat, stationary black hole or (B) a cosmological spacetime with a compact Cauchy horizon ruled by closed null geodesics. The black hole event horizon or, respectively, the compact Cauchy horizon of these spacetimes is assumed to be a smooth null hypersurface which is non-degenerate in the sense that its null geodesic generators are geodesically incomplete in one direction. In both cases, it is shown that there exists a Killing vector field in a one-sided neighborhood of the horizon which is normal to the horizon. We thereby generalize theorems of Hawking (for case (A)) and Isenberg and Moncrief (for case (B)) to the non-analytic case.