A Generalization of Hawking’s Black Hole Topology Theorem to Higher Dimensions
- 299 Downloads
Hawking’s theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking’s results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology S2 × S1. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation).
Unable to display preview. Download preview PDF.
- 4.Cai, M. Volume minimizing hypersurfaces in manifolds of nonnegative scalar curvature. In: Minimal Surfaces, Geometric Analysis, and Symplectic Geometry, Advanced Studies in Pure Mathematics, eds. Fukaya, K., Nishikawa, S., Spruck, J., 34, 1–7 (2002)Google Scholar
- 12.Hawking, S.W. The event horizon. In ‘Black Holes, Les Houches lectures’ (1972), edited by C. DeWitt, B. S. DeWitt Amsterdam: North Holland, 1972Google Scholar
- 14.Helfgott, C., OZ, Y., Yanay, Y. On the topology of black hole event horizons in higher dimensions. JHEPO2 (2006) 024Google Scholar