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A uniqueness theorem for stationary Kaluza-Klein black holes

Stefan Hollands, Stoytcho Yazadjiev

TL;DR

This work proves a uniqueness theorem for stationary vacuum black holes in $D\ge4$ with $D-3$ axial Killing fields, in asymptotically Kaluza-Klein spacetimes, showing that the spacetime exterior is uniquely determined by its angular momenta and an interval structure consisting of moduli $\{l_i\}$ and integer vectors $\{\underline{a}_i\}$. The proof reduces the Einstein equations to a two-dimensional orbit space $\hat M$ and employs the Mazur identity to compare two candidate solutions, concluding they must be isometric if their interval structure and angular momenta agree. The interval data determine the horizon topology (e.g., $S^3$, $S^2\times S^1$, Lens spaces $L(p,q)$) and the exterior topology, while the results extend higher-dimensional uniqueness beyond the classical four-dimensional Kerr/ Kerr-Newman cases. The paper also discusses model spaces, constructions from orbit data, and the limitations/assumptions (e.g., analyticity, causality) of the theorem, highlighting open questions about the existence of solutions for arbitrary interval structures.

Abstract

We prove a uniqueness theorem for stationary $D$-dimensional Kaluza-Klein black holes with $D-2$ Killing fields, generating the symmetry group ${\mathbb R} \times U(1)^{D-3}$. It is shown that the topology and metric of such black holes is uniquely determined by the angular momenta and certain other invariants consisting of a number of real moduli, as well as integer vectors subject to certain constraints.

A uniqueness theorem for stationary Kaluza-Klein black holes

TL;DR

This work proves a uniqueness theorem for stationary vacuum black holes in with axial Killing fields, in asymptotically Kaluza-Klein spacetimes, showing that the spacetime exterior is uniquely determined by its angular momenta and an interval structure consisting of moduli and integer vectors . The proof reduces the Einstein equations to a two-dimensional orbit space and employs the Mazur identity to compare two candidate solutions, concluding they must be isometric if their interval structure and angular momenta agree. The interval data determine the horizon topology (e.g., , , Lens spaces ) and the exterior topology, while the results extend higher-dimensional uniqueness beyond the classical four-dimensional Kerr/ Kerr-Newman cases. The paper also discusses model spaces, constructions from orbit data, and the limitations/assumptions (e.g., analyticity, causality) of the theorem, highlighting open questions about the existence of solutions for arbitrary interval structures.

Abstract

We prove a uniqueness theorem for stationary -dimensional Kaluza-Klein black holes with Killing fields, generating the symmetry group . It is shown that the topology and metric of such black holes is uniquely determined by the angular momenta and certain other invariants consisting of a number of real moduli, as well as integer vectors subject to certain constraints.

Paper Structure

This paper contains 18 sections, 13 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Description of the problem, assumptions, notations
  3. Analysis of the orbit space
  4. Manifolds with torus actions
  5. Remarks:
  6. The fundamental group of $\Sigma$
  7. The fundamental group of $\Sigma$
  8. The orbit space of the domain of outer communication
  9. Model spaces, examples
  10. Example 1:
  11. Example 2:
  12. Example 3:
  13. Example 4:
  14. Stationary vacuum black holes in $D$ dimensions
  15. Asymptotic form of the metric
  16. ...and 3 more sections

Key Result

Lemma 1

Let $(\Sigma, h)$ be a Riemannian manifold of dimension $s$, with $N$ mutually commuting Killing fields $\psi_i, i=1, \dots, N$. Let $f_{ij}$ be the Gram matrix, and let $x$ be a point such that ${\rm rank}[f(x)] = r$. Then it follows that $N-r \le [(s-r)/2]$.

Theorems & Definitions (13)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Lemma 6
  • ...and 3 more