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Black hole uniqueness theorems in higher dimensional spacetimes

Stefan Hollands, Akihiro Ishibashi

TL;DR

The review addresses whether higher‑dimensional stationary black holes are uniquely determined by a finite set of charges, extending the four‑dimensional Kerr–Newman paradigm. It surveys the structural theorems—rigidity, topology, and topological censorship—and the Weyl–Papapetrou form, then develops the nonlinear sigma‑model framework and associated divergence identities that underpin uniqueness proofs in higher dimensions. Rotating solutions require strong symmetry (e.g., $U(1)^{D-3}$) to enable Weyl‑Papapetrou reductions and Mazur/Bunting type arguments; static cases are better understood, often yielding Schwarzschild‑Tangherlini as the unique solution. The review also discusses extensions to Einstein–Maxwell–dilaton, higher‑form fields, and supersymmetric theories, including near‑horizon geometry classifications and challenges in extremal and less symmetric settings, highlighting open problems and directions for future work. Overall, higher‑D uniqueness theorems are established in restricted symmetry classes, with much remaining to fully chart the landscape of possible black holes in diverse theories and boundary conditions.

Abstract

We review uniqueness theorems as well as other general results about higher dimensional black hole spacetimes. This includes in particular theorems about the topology of higher dimensional spacetimes, theorems about their symmetries (rigidity theorem), and the classification of supersymmetric black holes. We outline the basic ideas underlying the proofs of these statements, and we also indicate ways to generalize some of these results to more general contexts, such as more complicated theories.

Black hole uniqueness theorems in higher dimensional spacetimes

TL;DR

The review addresses whether higher‑dimensional stationary black holes are uniquely determined by a finite set of charges, extending the four‑dimensional Kerr–Newman paradigm. It surveys the structural theorems—rigidity, topology, and topological censorship—and the Weyl–Papapetrou form, then develops the nonlinear sigma‑model framework and associated divergence identities that underpin uniqueness proofs in higher dimensions. Rotating solutions require strong symmetry (e.g., ) to enable Weyl‑Papapetrou reductions and Mazur/Bunting type arguments; static cases are better understood, often yielding Schwarzschild‑Tangherlini as the unique solution. The review also discusses extensions to Einstein–Maxwell–dilaton, higher‑form fields, and supersymmetric theories, including near‑horizon geometry classifications and challenges in extremal and less symmetric settings, highlighting open problems and directions for future work. Overall, higher‑D uniqueness theorems are established in restricted symmetry classes, with much remaining to fully chart the landscape of possible black holes in diverse theories and boundary conditions.

Abstract

We review uniqueness theorems as well as other general results about higher dimensional black hole spacetimes. This includes in particular theorems about the topology of higher dimensional spacetimes, theorems about their symmetries (rigidity theorem), and the classification of supersymmetric black holes. We outline the basic ideas underlying the proofs of these statements, and we also indicate ways to generalize some of these results to more general contexts, such as more complicated theories.

Paper Structure

This paper contains 17 sections, 13 theorems, 93 equations, 3 figures.

Table of Contents

  1. Introduction
  2. General structure of uniqueness proof and history in $D=4$
  3. Higher dimensions
  4. Asymptotic conditions
  5. Rigidity theorem
  6. Topology theorems
  7. Weyl-Papapetrou coordinates and orbit space analysis
  8. Non-linear sigma-model reduction
  9. Divergence identities
  10. Uniqueness theorems for rotating stationary black holes
  11. Uniqueness for non-rotating (static) black holes
  12. Extremal black holes and their near horizon geometries
  13. Other Theories in $D \ge 4$
  14. Rotating stationary black holes
  15. Static black holes
  16. ...and 2 more sections

Key Result

Theorem 1

("Rigidity theorem") Let $(\mathscr{M},g)$ be an asymptotically flat, analytic stationary black hole solution to the vacuum Einstein equations. Assume further that the event horizon, $\mathscr{H}$, of the black hole is analytic and is topologically $\mathbb{R} \times B$, with $B$ compact and connect

Figures (3)

  • Figure 1: The numbers $(p,q)$ may be viewed as winding numbers associated with the generators of the 2-torus generated by the two axial Killing fields. In such a torus, an $U(1)$-orbit winds around the first $S^1$ generator $n$-times as it goes $p$-times around the other $S^1$-direction. Here ($qn \equiv 1 \, {\rm mod} \ p$). The figure shows the situation for $p=3$, $n=7$.
  • Figure 2: The invariant $p= |{\rm det} \, (\underline{v}_{h-1}, \underline{v}_{h+1})|$ characterizes the different horizon topologies.
  • Figure 3: The invariant $p= |{\rm det} \, (\underline{v}_{0}, \underline{v}_{N})|$ characterizes the different possible asymptotic behaviors.

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • Lemma 1
  • Theorem 11
  • ...and 3 more