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On the ‘Stationary Implies Axisymmetric’ Theorem for Extremal Black Holes in Higher Dimensions

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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 [24] 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.

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Authors and Affiliations

  1. School of Mathematics, Cardiff University, Cardiff, United Kingdom

    Stefan Hollands

  2. Cosmophysics Group, Institute of Particle and Nuclear Studies, KEK, Tsukuba, Japan

    Akihiro Ishibashi

  3. Perimeter Institute for Theoretical Physics, Waterloo, Canada

    Akihiro Ishibashi

Authors
  1. Stefan Hollands
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  2. Akihiro Ishibashi
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Correspondence to Stefan Hollands.

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Communicated by G. W. Gibbons

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Hollands, S., Ishibashi, A. On the ‘Stationary Implies Axisymmetric’ Theorem for Extremal Black Holes in Higher Dimensions. Commun. Math. Phys. 291, 443–471 (2009). https://doi.org/10.1007/s00220-009-0841-1

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  • DOI: https://doi.org/10.1007/s00220-009-0841-1

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