Existence of constant mean curvature foliations in spacetimes with two-dimensional local symmetry

TL;DR

This paper proves that in certain symmetric spacetimes, the past of any compact constant mean curvature hypersurface can be foliated by similar hypersurfaces, revealing properties of singularities and a positive mass theorem.

Contribution

It establishes the existence of constant mean curvature foliations in spacetimes with two-dimensional local symmetry, including exotic topologies, and proves a positive mass theorem in this context.

Findings

The past of any compact CMC hypersurface can be foliated by compact CMC hypersurfaces.

The mean curvature of these hypersurfaces can become arbitrarily negative.

A positive mass theorem holds, with zero Hawking mass implying flat spacetime.

Abstract

It is shown that in a class of maximal globally hyperbolic spacetimes admitting two local Killing vectors, the past (defined with respect to an appropriate time orientation) of any compact constant mean curvature hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces. Moreover, the mean curvature of the leaves of this foliation takes on arbitrarily negative values and so the initial singularity in these spacetimes is a crushing singularity. The simplest examples occur when the spatial topology is that of a torus, with the standard global Killing vectors, but more exotic topologies are also covered. In the course of the proof it is shown that in this class of spacetimes a kind of positive mass theorem holds. The symmetry singles out a compact surface passing through any given point of spacetime and the Hawking mass of any such surface is non-negative. If the Hawking mass of any one of these surfaces is zero then the entire spacetime is flat.