An Enlargeability Obstruction for Spacetimes with both Big Bang and Big Crunch

TL;DR

This paper extends the enlargeability obstruction to initial data sets in general relativity, showing that certain topological conditions prevent the existence of spacetimes with both big bang and big crunch singularities.

Contribution

It generalizes previous scalar curvature obstructions to include Gromov-Lawson's enlargeability, impacting the understanding of spacetime singularities in relativity.

Findings

Obstructions prevent certain initial data configurations.

Results apply to many manifolds, including dimension 3.

Excludes specific globally hyperbolic spacetimes with singularities.

Abstract

Given a spacelike hypersurface $M$ of a time-oriented Lorentzian manifold $(\overline{M}, \overline{g})$, the pair $(g, k)$ consisting of the induced Riemannian metric $g$ and the second fundamental form $k$ is known as initial data set. In this article, we study the space of all initial data sets $(g, k)$ on a fixed closed manifold $M$ that are subject to a strict version of the dominant energy condition. Whereas the pairs of the form $(g, \tau g)$ and $(g, -\tau g)$, for a sufficiently large $\tau > 0$, belong to the same path-component of this space when $M$ admits a positive scalar curvature metric, it was observed in a previous work \cite{arXiv:1906.00099} that this is not the case when the existence of a positive scalar curvature metric on $M$ is obstructed by $\alpha(M) \neq 0$. In the present article we extend this non-connectedness result to Gromov-Lawson's enlargeability obstruction, which covers many examples, also in dimension $3$. In the context of relativity theory, this result may be interpreted as excluding the existence of certain globally hyperbolic spacetimes with both a big bang and a big crunch singularity