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The asymptotic expansion of the spacetime metric at the event horizon

Klaus Kroencke, Oliver Petersen

Abstract

Hawking's local rigidity theorem, proven in the smooth setting by Alexakis-Ionescu-Klainerman, says that the event horizon of any stationary non-extremal black hole is a non-degenerate Killing horizon. In this paper, we prove that the full asymptotic expansion of any smooth vacuum metric at a non-degenerate Killing horizon is determined by the geometry of the horizon. This gives a new perspective on the black hole uniqueness conjecture. In spacetime dimension $4$, we also prove an existence theorem: Given any non-degenerate horizon geometry, Einstein's vacuum equations can be solved to infinite order at the horizon in a unique way (up to isometry). The latter is a gauge invariant version of Moncrief's classical existence result, without any restriction on the topology of the horizon. In the real analytic setting, the asymptotic expansion is shown to converge and we get well-posedness of this characteristic Cauchy problem.

The asymptotic expansion of the spacetime metric at the event horizon

Abstract

Hawking's local rigidity theorem, proven in the smooth setting by Alexakis-Ionescu-Klainerman, says that the event horizon of any stationary non-extremal black hole is a non-degenerate Killing horizon. In this paper, we prove that the full asymptotic expansion of any smooth vacuum metric at a non-degenerate Killing horizon is determined by the geometry of the horizon. This gives a new perspective on the black hole uniqueness conjecture. In spacetime dimension , we also prove an existence theorem: Given any non-degenerate horizon geometry, Einstein's vacuum equations can be solved to infinite order at the horizon in a unique way (up to isometry). The latter is a gauge invariant version of Moncrief's classical existence result, without any restriction on the topology of the horizon. In the real analytic setting, the asymptotic expansion is shown to converge and we get well-posedness of this characteristic Cauchy problem.

Paper Structure

This paper contains 18 sections, 28 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Induced non-degenerate Killing horizon data
  3. Main result
  4. Motivating the data
  5. Degrees of freedom
  6. Relation to earlier results
  7. Uniqueness of the expansion
  8. A geometric gauge
  9. The expansion
  10. The first derivative
  11. The higher derivatives
  12. Notation and first identities
  13. The curvature components
  14. Expressions for the $V$-components
  15. Computation of the commutator
  16. ...and 3 more sections

Key Result

Theorem 1.2

The asymptotic expansion of a vacuum spacetime metric at a non-degenerate Killing horizon $\mathcal{H}$ is completely determined by a Riemannian metric $\sigma$ and a Killing vector field $V$ of constant length (w.r.t. $\sigma$) on the horizon.

Theorems & Definitions (67)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4: AIK2010
  • Theorem 1.5
  • Corollary 1.6
  • Example 1.7
  • Remark 1.8
  • Remark 1.9
  • Conjecture 1.10: The black hole uniqueness conjecture
  • ...and 57 more