Hawking's Local Rigidity Theorem Without Analyticity

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Abstract

We prove the existence of a Hawking Killing vector-field in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth vacuum Einstein manifold. The result extends a previous result of Friedrich, Rácz and Wald, see [FRW, Prop. B. 1], which was limited to the domain of dependence of the bifurcate horizon. So far, the existence of a Killing vector-field in a full neighborhood has been proved only under the restrictive assumption of analyticity of the space-time. Using this result we provide the first unconditional proof that a stationary black-hole solution must possess an additional, rotational Killing field in an open neighborhood of the event horizon. This work is accompanied by a second paper, where we prove a uniqueness result for smooth stationary black-hole solutions which are close (in a very precise, geometric sense) to the Kerr family of solutions, for arbitrary 0 < a < m.

Original languageEnglish (US)
Pages (from-to)845-869
Number of pages25
JournalGeometric and Functional Analysis
Volume20
Issue number4
DOIs
StatePublished - 2010

All Science Journal Classification (ASJC) codes

  • Analysis
  • Geometry and Topology

Keywords

  • Einstein vacuum equations
  • Killing vector-field
  • non-expanding bifurcatehorizon
  • unique continuation

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