Existence of maximal hypersurfaces in some spherically symmetric spacetimes

TL;DR

This paper proves the existence of maximal hypersurfaces in certain spherically symmetric spacetimes with matter fields, and demonstrates that these spacetimes can be foliated by constant mean curvature surfaces, supporting the closed-universe recollapse conjecture.

Contribution

It establishes the existence of maximal hypersurfaces and a foliation by constant mean curvature surfaces in spherically symmetric spacetimes with matter, extending previous results.

Findings

Spacetimes with collisionless matter or scalar fields admit maximal Cauchy surfaces.

Such spacetimes can be foliated by constant mean curvature surfaces with all real mean curvatures.

A volume bound for Cauchy surfaces is derived under the timelike convergence condition.

Abstract

We prove that the maximal development of any spherically symmetric spacetime with collisionless matter (obeying the Vlasov equation) or a massless scalar field (obeying the massless wave equation) and possessing a constant mean curvature $S^1 \times S^2$ Cauchy surface also contains a maximal Cauchy surface. Combining this with previous results establishes that the spacetime can be foliated by constant mean curvature Cauchy surfaces with the mean curvature taking on all real values, thereby showing that these spacetimes satisfy the closed-universe recollapse conjecture. A key element of the proof, of interest in itself, is a bound for the volume of any Cauchy surface $\Sigma$ in any spacetime satisfying the timelike convergence condition in terms of the volume and mean curvature of a fixed Cauchy surface $\Sigma_0$ and the maximal distance between $\Sigma$ and $\Sigma_0$. In particular, this shows that any globally hyperbolic spacetime having a finite lifetime and obeying the timelike-convergence condition cannot attain an arbitrarily large spatial volume.