Singularities from the Topology and Differentiable Structure of Asymptotically Flat Spacetimes

TL;DR

This paper introduces a new invariant called the maximal Yamabe invariant to analyze the formation of singularities in asymptotically flat spacetimes with complex topology or differentiable structures, extending classical singularity theorems.

Contribution

It establishes a novel singularity theorem based on the maximal Yamabe invariant, linking topology, differentiable structure, and singularity formation in general relativity.

Findings

Spacetimes with nonpositive maximal Yamabe invariant develop singularities.

The maximal Yamabe invariant relates to the A-genus and Seiberg-Witten invariants.

Certain 5-dimensional spacetimes with non-trivial Seiberg-Witten invariants are singular.

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.