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On the uniqueness of smooth, stationary black holes in vacuum

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Abstract

A fundamental conjecture in general relativity asserts that the domain of outer communication of a regular, stationary, four dimensional, vacuum black hole solution is isometrically diffeomorphic to the domain of outer communication of a Kerr black hole. So far the conjecture has been resolved, by combining results of Hawking [17], Carter [4] and Robinson [28], under the additional hypothesis of non-degenerate horizons and real analyticity of the space-time. We develop a new strategy to bypass analyticity based on a tensorial characterization of the Kerr solutions, due to Mars [24], and new geometric Carleman estimates. We prove, under a technical assumption (an identity relating the Ernst potential and the Killing scalar) on the bifurcate sphere of the event horizon, that the domain of outer communication of a smooth, regular, stationary Einstein vacuum spacetime of dimension 4 is locally isometric to the domain of outer communication of a Kerr spacetime.

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 480 Lincoln Drive, 53706, USA

    Alexandru D. Ionescu

  2. Department of Mathematics, Princeton University, Princeton, NJ, 705 Fine Hall, 08544-1019, USA

    Sergiu Klainerman

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  1. Alexandru D. Ionescu
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  2. Sergiu Klainerman
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Correspondence to Sergiu Klainerman.

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Ionescu, A., Klainerman, S. On the uniqueness of smooth, stationary black holes in vacuum. Invent. math. 175, 35–102 (2009). https://doi.org/10.1007/s00222-008-0146-6

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  • DOI: https://doi.org/10.1007/s00222-008-0146-6

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