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Uniqueness of stationary, electro-vacuum black holes revisited

Piotr T. Chruściel

TL;DR

This paper surveys the global structure of stationary electro-vacuum black holes, presenting a corrected rigidity theorem and a static-degenerate uniqueness result, and canvasses the landscape of stationary-axisymmetric non-degenerate black holes with multiple horizons. It combines topology, isometry-group structure, and horizon regularity to refine the classification program, including the role of bifurcation surfaces and the Majumdar–Papapetrou family as a canonical degenerate limit. It also cautions against assuming horizon differentiability and highlights several key open problems that shape future work in black hole uniqueness. Overall, the work advances a coherent framework for understanding which stationary black holes can exist under various regularity and topological hypotheses and where the remaining gaps lie.

Abstract

In recent years there has been some progress in the understanding of the global structure of stationary black hole space-times. In this paper we review some new results concerning the structure of stationary black hole space-times. In particular we prove a corrected version of the ``black hole rigidity theorem'', and we prove a uniqueness theorem for static black holes with degenerate connected horizons. This paper is an expanded version of a lecture given at the Journées relativistes in Ascona, May 1996.

Uniqueness of stationary, electro-vacuum black holes revisited

TL;DR

This paper surveys the global structure of stationary electro-vacuum black holes, presenting a corrected rigidity theorem and a static-degenerate uniqueness result, and canvasses the landscape of stationary-axisymmetric non-degenerate black holes with multiple horizons. It combines topology, isometry-group structure, and horizon regularity to refine the classification program, including the role of bifurcation surfaces and the Majumdar–Papapetrou family as a canonical degenerate limit. It also cautions against assuming horizon differentiability and highlights several key open problems that shape future work in black hole uniqueness. Overall, the work advances a coherent framework for understanding which stationary black holes can exist under various regularity and topological hypotheses and where the remaining gaps lie.

Abstract

In recent years there has been some progress in the understanding of the global structure of stationary black hole space-times. In this paper we review some new results concerning the structure of stationary black hole space-times. In particular we prove a corrected version of the ``black hole rigidity theorem'', and we prove a uniqueness theorem for static black holes with degenerate connected horizons. This paper is an expanded version of a lecture given at the Journées relativistes in Ascona, May 1996.

Paper Structure

This paper contains 12 sections, 19 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. The topology of black hole space--times
  3. The structure of isometry groups of asymptotically flat space--times.
  4. Killing vectors vs. isometry groups
  5. The rigidity theorem
  6. Bifurcation surfaces
  7. Uniqueness of static, degenerate black holes
  8. Uniqueness of stationary, axisymmetric, non--degenerate black holes
  9. Differentiability of event horizons
  10. Conclusions
  11. Analyticity of isometries
  12. A comment to the classification theorem of non--singular MP black holes.

Key Result

Theorem 2.1

Consider a space--time which is asymptotically flat at null infinity and which has a $\hbox{${\cal J}$ {$(\ $}}$ which satisfies the regularityThe regularity condition of $\hbox{${\cal J}$ {$(\ $}}$ can be replaced by other conditions, cf. e.g.Galloway:fitopology. While the regularity condition gi Then every globally hyperbolic domain of outer communications is simply connected.

Theorems & Definitions (19)

  • Theorem 2.1: G. Galloway, 96
  • Theorem 2.2: P.C. & R. Wald, 94
  • Proposition 2.3
  • Theorem 3.1: P.C. & R. Beig, 96
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 5.1
  • Lemma 5.2
  • Theorem 6.1: I. Rácz & R. Wald, 96
  • Theorem 7.1: M. Heusler, 96
  • ...and 9 more