Greg Galloway: Topology & General Relativity 4

Summer School »Between Geometry and Relativity«, Part 25: Greg Galloway: Topology & General Relativity 4

Abstract: An initial data set in spacetime consists of a spacelike hypersurface $V$, together with its its induced (Riemannian) metric h and its second fundamental form $K$. A solution to the Einstein equations influences the curvature of $V$ via the Einstein constraint equations, the geometric origin of which are the Gauss-Codazzi equations. After a brief introduction to Lorentzian manifolds and Lorentzian causality, we will study some topics of recent interest related to the geometry and topology of initial data sets. In particular, we will consider the topology of black holes in higher dimensional gravity, inspired by certain developments in string theory and issues related to black hole uniqueness. We shall also discuss recent work on the geometry and topology of the region of space exterior to all black holes, which is closely connected to the notion of topological censorship. Many 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.
Greg Galloway is professor at the Department of Mathematics, U of Miami.

1. Lead 00:00:00
2. Topology of the exterior region 00:00:10
3. Two theorems on MOTS 00:05:15
4. Remarks on the proof. The no horizon case 00:13:44
5. Cosmological spacetimes 00:24:43
6. A theorem on compact Cauchy surfaces 00:32:28