The general spherically symmetric constant mean curvature foliations of the Schwarzschild solution

TL;DR

This paper constructs and analyzes spherical spacelike slices with constant mean curvature in Schwarzschild spacetime, revealing barriers, conditions for foliation, and behavior at critical points.

Contribution

It provides explicit expressions for positive lapse functions, demonstrates how to construct complete foliations with unbounded mean curvature, and analyzes behavior at the bifurcation sphere and critical points.

Findings

Foliations with mean curvature from zero to infinity are possible.

A barrier prevents mean curvature from becoming arbitrarily large.

Lapse function exponentially approaches zero at critical points.

Abstract

We consider a family of spherical three dimensional spacelike slices embedded in the Schwarzschild solution. The mean curvature is constant on each slice but can change from slice to slice. We give a simple expression for an everywhere positive lapse and thus we show how to construct foliations. There is a barrier preventing the mean curvature from becoming large, and we show how to avoid this so as to construct a foliation where the mean curvature runs all the way from zero to infinity. No foliation exists where the mean curvature goes from minus to plus infinity. There are slicings, however, where each slice passes through the bifurcation sphere $R = 2M$ and the lapse only vanishes at this one point, and is positive everywhere else, while the mean curvature does run from minus to plus infinity. Symmetric foliations of the extended Schwarzschild spacetime degenerate at a critical point, where we show that the lapse function exponentially approaches zero.