Rigidity aspects of a cosmological singularity theorem

TL;DR

This paper refines a singularity theorem in General Relativity, linking the geometry of Cauchy surfaces to the global structure of spacetime, with implications for cosmological models.

Contribution

It improves existing singularity theorems by incorporating rigidity aspects and symmetry conditions, providing new topological and geometric constraints.

Findings

If the Cauchy surface is 2-convex, the spacetime is either incomplete or the surface has specific topologies.

Presence of a U(1) symmetry allows relaxation of convexity conditions.

Stronger results are obtained for non-orientable, non-prime, or certain Haken manifolds without covering.

Abstract

Improving a singularity theorem in General Relativity by Galloway and Ling we show the following (cf.\ Theorem 1): If a globally hyperbolic spacetime $M$ satisfying the null energy condition contains a closed, spacelike Cauchy surface $(V,g,K)$ (with metric $g$ and extrinsic curvature $K$) which is 2-convex (meaning that the sum of the lowest two eigenvalues of $K$ is non-negative), then either $M$ is past null geodesically incomplete, or $V$ is a spherical space, or $V$ or some finite cover is a surface bundle over the circle, with totally geodesic fibers. Moreover, (cf.\ Theorem 2) if $(V,g,K)$ admits a $U(1)$ isometry group with corresponding Killing vector $\xi$, we can relax the convexity requirement in terms of a decomposition of $K$ with respect to the directions parallel and orthogonal to $\xi$. Finally, (cf. Propositions 1-3) in the special cases that $V$ is either non-orientable, or non-prime, or an orientable Haken manifold with vanishing second homology, we obtain stronger statements in both Theorems without passing to covers.