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On the `Stationary Implies Axisymmetric' Theorem for Extremal Black Holes in Higher Dimensions

Stefan Hollands, Akihiro Ishibashi

TL;DR

The paper extends the rigidity and axisymmetry results for stationary higher-dimensional black holes to extremal (degenerate) horizons under analyticity and a diophantine condition on the horizon's angular velocities. It constructs a horizon Killing field K^a in a neighborhood of the horizon and, when the stationary vector is not aligned with the horizon, proves the existence of additional commuting rotational Killing fields, extending these symmetries to the exterior. The authors generalize the results to Einstein-matter theories with scalars and abelian gauge fields and discuss the role and limitations of the diophantine condition. Together, these results reinforce symmetry constraints on extremal black holes in higher dimensions and inform the broader context of black hole uniqueness and horizon thermodynamics.

Abstract

All known stationary black hole solutions in higher dimensions possess additional rotational symmetries in addition to the stationary Killing field. Also, for all known stationary solutions, the event horizon is a Killing horizon, and the surface gravity is constant. In the case of non-degenerate horizons (non-extremal black holes), a general theorem was previously established [gr-qc/0605106] proving that these statements are in fact generally true under the assumption that the spacetime is analytic, and that the metric satisfies Einstein's equation. Here, we extend the analysis to the case of degenerate (extremal) black holes. It is shown that the theorem still holds true if the vector of angular velocities of the horizon satisfies a certain "diophantine condition," which holds except for a set of measure zero.

On the `Stationary Implies Axisymmetric' Theorem for Extremal Black Holes in Higher Dimensions

TL;DR

The paper extends the rigidity and axisymmetry results for stationary higher-dimensional black holes to extremal (degenerate) horizons under analyticity and a diophantine condition on the horizon's angular velocities. It constructs a horizon Killing field K^a in a neighborhood of the horizon and, when the stationary vector is not aligned with the horizon, proves the existence of additional commuting rotational Killing fields, extending these symmetries to the exterior. The authors generalize the results to Einstein-matter theories with scalars and abelian gauge fields and discuss the role and limitations of the diophantine condition. Together, these results reinforce symmetry constraints on extremal black holes in higher dimensions and inform the broader context of black hole uniqueness and horizon thermodynamics.

Abstract

All known stationary black hole solutions in higher dimensions possess additional rotational symmetries in addition to the stationary Killing field. Also, for all known stationary solutions, the event horizon is a Killing horizon, and the surface gravity is constant. In the case of non-degenerate horizons (non-extremal black holes), a general theorem was previously established [gr-qc/0605106] proving that these statements are in fact generally true under the assumption that the spacetime is analytic, and that the metric satisfies Einstein's equation. Here, we extend the analysis to the case of degenerate (extremal) black holes. It is shown that the theorem still holds true if the vector of angular velocities of the horizon satisfies a certain "diophantine condition," which holds except for a set of measure zero.

Paper Structure

This paper contains 10 sections, 1 theorem, 108 equations.

Table of Contents

  1. Introduction
  2. Proof of the rigidity theorem in the vacuum case
  3. Theorem 1:
  4. Theorem 2:
  5. Matter fields
  6. Discussion
  7. Acknowledgements:
  8. Ricci tensor in Gaussian null coordinates
  9. Gravity coupled to matter fields
  10. Chern-Simons theories in $n=5$

Key Result

Lemma 1

Let $J$ be a smooth function on $\Sigma$ with the property that Let ${\underline \Omega}=(\Omega_1, \dots, \Omega_N) \in {\mathbb R}^N$ [see eq. sadef] satisfy the following "diophantine condition": There exits a number $q$ such thatNote that ${\underline \Omega} \cdot {\underline m} \neq 0$ if ${\underline m} \neq 0$, since the entries of ${\underline \Omega}$ holds for all but finitely many ${

Theorems & Definitions (1)

  • Lemma 1