Singularities from the Topology and Differentiable Structure of Asymptotically Flat Spacetimes

Abstract

We prove that certain asymptotically flat initial data sets with nontrivial topology and/or differentiable structure collapse to form singularities. The class of such initial data sets is characterized by a new smooth invariant, the maximal Yamabe invariant, defined through smooth compactification of the asymptotically flat manifold. Our singularity theorem applies to spacetimes admitting a Cauchy surface of nonpositive maximal Yamabe invariant with initial data that satisfies the dominant energy condition. This class of spacetimes includes simply connected spacetimes with a single asymptotic region, a class not covered by prior singularity theorems for topological structures. The maximal Yamabe invariant can be related to other invariants including, in 4 dimensions, the A-genus and the Seiberg-Witten invariants. In particular, 5-dimensional spacetimes with asymptotically flat Cauchy surfaces with non-trivial Seiberg-Witten invariants are singular. This singularity is due to the differentiable structure of the manifold.