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The classification of static vacuum space-times containing an asymptotically flat spacelike hypersurface with compact interior

Piotr T. Chruściel

Abstract

We prove non-existence of static, vacuum, appropriately regular, asymptotically flat black hole space-times with degenerate (not necessarily connected) components of the event horizon. This finishes the classification of static, vacuum, asymptotically flat domains of outer communication in an appropriate class of space-times, showing that the domains of outer communication of the Schwarzschild black holes exhaust the space of appropriately regular black hole exteriors.

The classification of static vacuum space-times containing an asymptotically flat spacelike hypersurface with compact interior

Abstract

We prove non-existence of static, vacuum, appropriately regular, asymptotically flat black hole space-times with degenerate (not necessarily connected) components of the event horizon. This finishes the classification of static, vacuum, asymptotically flat domains of outer communication in an appropriate class of space-times, showing that the domains of outer communication of the Schwarzschild black holes exhaust the space of appropriately regular black hole exteriors.

Paper Structure

This paper contains 8 sections, 20 theorems, 99 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Boundary conditions at Killing horizons
  4. A proof based on the Komar identity
  5. Considerations global in space
  6. Considerations global in space--time
  7. Concluding remarks
  8. Addendum, June 2010

Key Result

Theorem 1.1

$^{\hbox{\footnotesize $\bullet$1.2}}$ Let $(M,g)$ be a static solution of the vacuum Einstein equations with defining Killing vector $X$. Suppose that $M$ contains a connected space-like hypersurface $\Sigma$ the closure $\bar{\Sigma}$ of which is the union of a finite number of asymptotically flat Then $\Sigma$ is diffeomorphic to $\Bbb R^3$ minus a ball, so that it is simply connected, it has o

Figures (1)

  • Figure 1: A family of three embedded curves in the plane consisting of two circles, together with a curve of infinite length that spirals towards the circles.

Theorems & Definitions (21)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 3.1: Boyer Boyer
  • Proposition 3.2
  • Proposition 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 4.1: Vishveshwara--Carter Lemma VishveshwaraCarterJMP
  • Proposition 4.2
  • ...and 11 more