MSRI Evans Talk: On the topology of black holes and beyond.
Greg Galloway (University of Miami)
There is a widely held belief in physics that a true astrophysical black hole, formed from the gravitational collapse of some stellar object, can be described by a certain exact solution to the Einstein equations discovered by Kerr in the 60's. This belief is based largely on a series of powerful results which shows that the Kerr solution is the unique solution to the vacuum (source-free) Einstein equations with certain prescribed properties. A basic step in the proof is Hawking's theorem on the topology of black holes which asserts that, under physically natural conditions, the surface of a black hole (cross-section of the event horizon) must be topologically a 2-sphere.
Various developments in string theory have generated a great deal of interest in gravity in higher dimensions and, in particular, in higher dimensional black holes. The remarkable discovery of Emparan and Reall of a 4+1 dimensional vacuum black hole solution to the Einstein equations with nonspherical horizon topology raised the question as to what horizon topologies are allowable in higher dimensions.
In this talk we review Hawking's theorem on the topology of black holes in 3+1 dimensions and present a generalization of it to higher dimensions. The latter is a geometric result which places restrictions on the topology of black holes in higher dimensions. We shall also discuss recent work on the topology of space exterior to a black hole. This is closely connected to the Principle of Topological Censorship, which roughly asserts that the topology of the region outside of all black holes (and white holes) should be simple. All of the results to be discussed rely on the recently developed theory of marginally outer trapped surfaces, which are natural spacetime analogues of minimal surfaces in Riemannian geometry.
This talk is based primarily on joint work with Rick Schoen and with Michael Eichmair and Dan Pollack