Ricci Fall-off in Static, Globally Hyperbolic, Non-singular Spacetimes

Abstract

What restrictions are there on a spacetime for which the Ricci curvature is such as to produce convergence of geodesics (such as the preconditions for the Singularity Theorems) but for which there are no singularities? We answer this question for a restricted class of spacetimes: static, geodesically complete, and globally hyperbolic. The answer is that, in at least one spacelike direction, the Ricci curvature must fall off at a rate inversely quadratic in a naturally-occurring Riemannian metric on the space of static observers. Along the way, we establish some global results on the static observer space, regarding its completeness and its behavior with respect to universal covering spaces.