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

TL;DR

This paper extends rigidity theorems for spacetimes with stationary black holes or compact Cauchy horizons to include various matter fields, demonstrating the existence of symmetries and invariance properties in these complex systems.

Contribution

It generalizes existing rigidity theorems to broader matter models, establishing the presence of Killing vectors and invariance in diverse Einstein-matter systems.

Findings

Existence of a Killing vector field near horizons in multiple Einstein-matter systems.

Matter fields are shown to be invariant under the Killing symmetry.

Results strengthen the black hole rigidity and cosmic censorship conjectures.

Abstract

A rigidity theorem that applies to 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 was given in a recent work \cite{frw}. Here we enlarge the framework of the corresponding investigations by allowing the presence of other type of matter fields. In the first part the matter fields are involved merely implicitly via the assumption that the dominant energy condition is satisfied. In the second part Einstein-Klein-Gordon (EKG), Einstein-[non-Abelian] Higgs (E[nA]H), Einstein-[Maxwell]-Yang-Mills-dilaton (E[M]YMd) and Einstein-Yang-Mills-Higgs (EYMH) systems are studied. The black hole event horizon or, respectively, the compact Cauchy horizon of the considered spacetimes is assumed to be a smooth non-degenerate null hypersurface. It is proven that there exists a Killing vector field in a one-sided neighborhood of the horizon in EKG, E[nA]H, E[M]YMd and EYMH spacetimes. This Killing vector field is normal to the horizon, moreover, the associated matter fields are also shown to be invariant with respect to it. The presented results provide generalizations of the rigidity theorems of Hawking (for case A) and of Moncrief and Isenberg (for case B) and, in turn, they strengthen the validity of both the black hole rigidity scenario and the strong cosmic censor conjecture of classical general relativity.