On the topology of vacuum spacetimes

TL;DR

This paper demonstrates that in higher dimensions, vacuum spacetimes can have arbitrary spatial topologies, expanding the understanding of possible universe shapes in general relativity.

Contribution

It introduces a new gluing method to construct vacuum solutions with arbitrary topology, generalizing previous constant mean curvature approaches.

Findings

Existence of vacuum spacetimes with arbitrary topology in higher dimensions.

Construction of solutions on manifolds without positive scalar curvature.

Extension of previous gluing techniques to non-constant mean curvature cases.

Abstract

We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n+1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary compact manifold to an asymptotically Euclidean solution of the constraints on R^n. For any compact manifold which does not admit a metric of positive scalar curvature, this provides for the existence of asymptotically flat vacuum spacetimes with no maximal slices. Our main theorem is a special case of a more general gluing construction for nondegenerate solutions of the vacuum constraint equations which have some restrictions on the mean curvature, but for which the mean curvature is not necessarily constant. This generalizes the construction [IMP] (gr-qc/0109045), which is restricted to constant mean curvature data.