On the `Stationary Implies Axisymmetric' Theorem for Extremal Black Holes in Higher Dimensions

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 [gr-qc/0605106] 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.

30pp, Latex, no figures

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• Uniqueness theorems for Kaluza-Klein black holes in five-dimensional minimal supergravity
• Topology of supersymmetric N=1, D=4 supergravity horizons
• Generalizations and challenges for the spacetime block-diagonalization
• Superradiance and black resonator strings encounter helical black strings
• Multipole moments for black objects in five dimensions
• A classification of supersymmetric Kaluza-Klein black holes with a single axial symmetry
• Uniqueness of extremal charged black holes in de Sitter
• Rigidity of the extremal Kerr-Newman horizon
• Horizon classification via Riemannian flows
• Regularity and temperature of stationary black hole event horizons
• Extremal Black Holes from Homotopy Algebras