On the ‘Stationary implies axisymmetric’ theorem for extremal black holes in higher dimensions. (English) Zbl 1190.83056
The purpose of the paper is to establish a version of the rigidity theorem for the case of degenerate (extremal) black holes. This case corresponds to a vanishing Hawking temperature and is of particular importance for the investigation of the quantum properties of black holes in string theory. Stationary, asymptotically flat, analytic black hole solutions to the vacuum or electrovacuum Einstein equations with a non-degenerate (non-extremal) event horizon for general spacetime dimension \(n > 4\), fulfil two statements: The event horizon is in fact a Killing horizon and if it is rotating, then the space time must also be axisymmetric. The paper extends this result 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.
Reviewer: Johannes Viktor Feitzinger (Bochum)
MSC:
| 83C57 | Black holes |
| 53C80 | Applications of global differential geometry to the sciences |
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 83E05 | Geometrodynamics and the holographic principle |
| 83E30 | String and superstring theories in gravitational theory |
| 83C15 | Exact solutions to problems in general relativity and gravitational theory |
| 83C22 | Einstein-Maxwell equations |
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