A New Condition Implying the Existence of a Constant Mean Curvature Foliation

Summary

Abstract

It is shown that if a non-flat spacetime (M, g) whose future c-boundary is a single point satisfies R_abV^aV^b ≥ 0 for all timelike vectors V^a, equality holding only if R_ab = 0, then sufficiently close to the future c-boundary the spacetime can be uniquely foliated by constant mean curvature compact hypersurfaces. The uniqueness proof uses a variational method developed by Brill and Flaherty to establish the uniqueness of maximal hypersurfaces.

In 1976 Dieter Brill and Frank Flaherty (1976) published an extremely important paper, “Isolated Maximal Hypersurfaces in Spacetime”, establishing that maximal hypersurfaces are unique in closed universes with attractive gravity everywhere. That is, there is only one such hypersurface, if it exists at all. In an earlier paper, Brill had established that in three-torus universes, only suitably identified flat space possessed a maximal hypersurface, so the existence of a maximal hypersurface is not guaranteed. These results by Brill are important because maximal hypersurfaces are very convenient spacelike hypersurfaces upon which to impose initial data; on such hypersurfaces the constraint equations are enormously simplified. Furthermore, in asymptotically flat space, foliations of spacetime by maximal hypersurfaces often exist, and the simplifications of the constraint equations on such a foliation make it easy to numerically solve the full four-dimensional vacuum Einstein equations for physically interesting situations.