Ricci fall-off in static and stationary, 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 or stationary, geodesically complete, and globally hyperbolic. The answer is that, in at least one spacelike direction, the Ricci curvature must fall off (in a generalized manner of speaking) at a rate inversely quadratic in a naturally-occurring Riemannian metric on the space of stationary observers. Along the way, we establish some global results on the stationary observer space, regarding its completeness and its behaviour with respect to universal covering spaces; we also define a new geometric invariant for stationary spacetimes, related to causal behaviour (among other things).