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Rigidity of marginally trapped surfaces and the topology of black holes

Gregory J. Galloway

TL;DR

This work extends Hawking’s topology results to higher dimensions by proving a rigidity statement for marginally outer trapped surfaces (MOTS) under the dominant energy condition. The authors analyze the stability operator $L$ governing variations of null expansion, relate its principal eigenvalue to the possibility of positive scalar curvature, and construct a local foliation by MOTS near a non-positively curved MOTS via an inverse function argument. The main contribution shows that outermost MOTS must be of positive scalar curvature type, eliminating exceptional horizon topologies (e.g., toroidal horizons) in stationary and dynamical black hole spacetimes. The results provide a spacetime analogue to rigidity phenomena for minimal surfaces and yield strong topological constraints on higher-dimensional black hole horizons, consistent with observed examples like $S^2 \times S^1$ in five dimensions.

Abstract

In a recent paper (gr-qc/0509107) the author and Rick Schoen obtained a generalization to higher dimensions of a classical result of Hawking concerning the topology of black holes. It was proved that, apart from certain exceptional circumstances, cross sections of the event horizon, in the stationary case, and 'weakly outermost' marginally outer trapped surfaces, in the general case, in black hole spacetimes obeying the dominant energy condition, are of positive Yamabe type. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology $S^2 \times S^1$. In the present paper, we rule out for 'outermost' marginally outer trapped surfaces, in particular, for cross sections of the event horizon in stationary black hole spacetimes, the possibility of any such exceptional circumstances (which might have permitted, e.g., toroidal cross sections). This follows from the main result, which is a rigidity result for marginally outer trapped surfaces that are not of positive Yamabe type.

Rigidity of marginally trapped surfaces and the topology of black holes

TL;DR

This work extends Hawking’s topology results to higher dimensions by proving a rigidity statement for marginally outer trapped surfaces (MOTS) under the dominant energy condition. The authors analyze the stability operator governing variations of null expansion, relate its principal eigenvalue to the possibility of positive scalar curvature, and construct a local foliation by MOTS near a non-positively curved MOTS via an inverse function argument. The main contribution shows that outermost MOTS must be of positive scalar curvature type, eliminating exceptional horizon topologies (e.g., toroidal horizons) in stationary and dynamical black hole spacetimes. The results provide a spacetime analogue to rigidity phenomena for minimal surfaces and yield strong topological constraints on higher-dimensional black hole horizons, consistent with observed examples like in five dimensions.

Abstract

In a recent paper (gr-qc/0509107) the author and Rick Schoen obtained a generalization to higher dimensions of a classical result of Hawking concerning the topology of black holes. It was proved that, apart from certain exceptional circumstances, cross sections of the event horizon, in the stationary case, and 'weakly outermost' marginally outer trapped surfaces, in the general case, in black hole spacetimes obeying the dominant energy condition, are of positive Yamabe type. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology . In the present paper, we rule out for 'outermost' marginally outer trapped surfaces, in particular, for cross sections of the event horizon in stationary black hole spacetimes, the possibility of any such exceptional circumstances (which might have permitted, e.g., toroidal cross sections). This follows from the main result, which is a rigidity result for marginally outer trapped surfaces that are not of positive Yamabe type.

Paper Structure

This paper contains 3 sections, 8 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Analytic and geometric preliminaries
  3. Proof of Theorem \ref{['rigid']}

Key Result

Theorem 1.1

Let $(M^{n+1},g)$, $n\ge 3$, be a spacetime satisfying the dominant energy condition, and let $\Sigma^{n-1}$ be an outermost MOTS in a spacelike hypersurface $V^n$. Then $\Sigma^{n-1}$ is of positive Yamabe type, i.e., admits a metric of positive scalar curvature.

Theorems & Definitions (9)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3.1