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

TL;DR

This paper investigates the conditions under which static, globally hyperbolic, non-singular spacetimes can have Ricci curvature that causes geodesic convergence, revealing that Ricci must decay at least quadratically in certain directions.

Contribution

It characterizes the Ricci fall-off rate necessary for non-singular static spacetimes with geodesic completeness and global hyperbolicity.

Findings

Ricci curvature must decay at least quadratically in some spacelike direction.

Established global properties of the static observer space.

Analyzed the behavior of these spacetimes under universal coverings.

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.