Topological Obstructions To Maximal Slices

A necessary condition for a globally hyperbolic spacetime ${\mathbb R}\times \Sigma$ to admit a maximal slice is that the Cauchy slice $\Sigma$ admit a metric with nonnegative scalar curvature, $R\ge 0$. In this paper, the two cases considered are the closed spatial manifold and the asymptotically flat spatial manifold. Although most results here will apply in four or more spacetime dimensions, this work will mainly consider 4-dimensional spacetimes. For $\Sigma$ closed or asymptotically flat, all topologies are allowed by the field equations. Since all $\Sigma$ occur as Cauchy slices of solutions to the Einstein equations and most $\Sigma$ do not admit metrics with $R\ge 0$, it follows that most globally hyperbolic spacetimes never admit a maximal slice, i.e. a slice with zero mean extrinsic curvature. In particular, asymptotically flat globally hyperbolic spacetimes which admit maximal slices are the exception rather than the rule. The reason for this is due to topological obstructions to constructing such slices. In the asymptotically flat case, this will be shown by smooth compactification of the manifold in order to use the results for spatially closed manifolds.