Null Geometry and the Einstein Equations

Abstract

This paper is based on a series of lectures given by the author at the Cargèse Summer School on Mathematical General Relativity and Global Properties of Solutions of Einstein’s Equations, held in Corsica, July 29—august 10, 2002. The general aim of those lectures was to illustrate with some current examples how the methods of global Lorentzian geometry and causal theory may be used to obtain results about the global behavior of solutions to the Einstein equations. This, of course, is a long standing program, dating back to the singularity theorems of Hawking and Penrose [24]. Here we consider some properties of asymptotically de Sitter solutions to the Einstein equations with (by our sign conventions) positive cosmological constant, ⋀> 0. We obtain, for example, some rather strong topological obstructions to the existence of such solutions, and, in another direction, present a uniqueness result for de Sitter space, associated with the occurrence of eternal observer horizons. As described later, these results have rather strong connections with Friedrich’s results [11, 13] on the nonlinear stability of asymptotically simple solutions to the Einstein equations with ⋀ > 0; see also Friedrich’s article elsewhere in this volume. The main theoretical tool from global Lorentzian geometry used to prove these results is the so-called null splitting theorem [16]. This theorem is discussed here, along with relevant background material.