On the Topology of Vacuum Spacetimes

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 Σ n to an asymptotically Euclidean solution of the constraints on R n . For any Σ n 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 [16], which is restricted to constant mean curvature data.

A gluing construction for non-vacuum solutions of the Einstein-constraint equations

“…We do this because, as with the vacuum case, even when the constraint solutions we are gluing together have non-constant mean curvature, the analysis we rely on to carry out the gluing is based primarily on the CMC version of the conformal treatment of the constraints. (See [19]. )…”

Gluing Initial Data Sets for General Relativity

“…In the more general setting of constant mean curvature initial data sets a gluing construction was developed [2], in the context of the well known conformal method of Lichnerowicz, Choquet-Bruhat and York which reduces the constraints equations to a determined elliptic system. The construction of [2], and subsequently [3], allowed one to demonstrate how space-times can be joined by means of a geometric connected sum, or how a wormhole can be added between two points in a given space-time (on the level of the initial data). This was flexible enough to address a number of issues concerning the relation of the spatial topology to the geometry of solutions of the constraints and the constructibility of multi black hole solutions (see also [4]).…”

Topological censorship and chronology protection

“…This is not enough to show that they arise as isolated structures: A flat universe with the topology of a 3-torus satisfies the field equations, but its existence does not imply that one can find a vacuum geometry for which the universe looks everywhere like a 3-sphere except in an isolated region. Isolated systems are ordinarily modeled as asymptotically flat spacetimes; and a recent result by Isenberg et al [10] shows that all 3-manifolds do in fact occur as isolated structures, as spacelike hypersurfaces of asymptotically flat, vacuum spacetimes.…”