On non-existence of static vacuum black holes with degenerate components of the event horizon

We present a simple proof of the non-existence of degenerate components of the event horizon in static, vacuum, regular, four-dimensional black hole spacetimes. We discuss the generalization to higher dimensions and the inclusion of a cosmological constant. The classical proof of uniqueness of static vacuum black holes [3] assumes that all components of the event horizon are non-degenerate. The argument has been extended1 to include degenerate components [5] by studying the orbit-space geometry near the event horizon, and applying an appropriate Partially supported by a Polish Research Committee grant 2 P03B 073 24. E-mail [email protected], URL www.phys.univ-tours.fr/∼piotr † E–mail: [email protected] ‡ E–mail: [email protected] A static configuration with all components degenerate is easy to exclude using Komar integrals and the positive energy theorem, see [5, Section 4] for precise statements. However, one also wants to exclude solutions with both degenerate and non-degenerate components.