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A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric

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Abstract

A key result in the proof of black hole uniqueness in 4-dimensions is that a stationary black hole that is “rotating”—i.e., is such that the stationary Killing field is not everywhere normal to the horizon—must be axisymmetric. The proof of this result in 4-dimensions relies on the fact that the orbits of the stationary Killing field on the horizon have the property that they must return to the same null geodesic generator of the horizon after a certain period, P. This latter property follows, in turn, from the fact that the cross-sections of the horizon are two-dimensional spheres. However, in spacetimes of dimension greater than 4, it is no longer true that the orbits of the stationary Killing field on the horizon must return to the same null geodesic generator. In this paper, we prove that, nevertheless, a higher dimensional stationary black hole that is rotating must be axisymmetric. No assumptions are made concerning the topology of the horizon cross-sections other than that they are compact. However, we assume that the horizon is non-degenerate and, as in the 4-dimensional proof, that the spacetime is analytic.

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Authors and Affiliations

  1. Institut für Theoretische Physik, Universität Göttingen, D-37077, Göttingen, Germany

    Stefan Hollands

  2. Enrico Fermi Institute and Department of Physics, The University of Chicago, Chicago, IL, 60637, USA

    Akihiro Ishibashi & Robert M. Wald

Authors
  1. Stefan Hollands
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  2. Akihiro Ishibashi
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  3. Robert M. Wald
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Correspondence to Akihiro Ishibashi.

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Communicated by G.W. Gibbons

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Hollands, S., Ishibashi, A. & Wald, R.M. A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric. Commun. Math. Phys. 271, 699–722 (2007). https://doi.org/10.1007/s00220-007-0216-4

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  • DOI: https://doi.org/10.1007/s00220-007-0216-4

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