Lifetimes of spherically symmetric closed universes

TL;DR

This paper proves that spherically symmetric closed universes with certain energy conditions have a finite lifetime, providing a computable bound based on initial data and resolving a version of the recollapse conjecture.

Contribution

It establishes a universal, computable bound on the lifetime of spherically symmetric closed universes satisfying specific energy conditions, confirming a conjecture for scalar field spacetimes.

Findings

Finite lifetime bound proportional to maximum mass on initial surface

Bound is computable from initial data and has a simple form

Results extend to scalar field spacetimes, confirming previous conjectures

Abstract

It is proven that any spherically symmetric spacetime that possesses a compact Cauchy surface $\Sigma$ and that satisfies the dominant-energy and non-negative-pressures conditions must have a finite lifetime in the sense that all timelike curves in such a spacetime must have a length no greater than $10 \max_\Sigma(2m)$, where $m$ is the mass associated with the spheres of symmetry. This result gives a complete resolution, in the spherically symmetric case, of one version of the closed-universe recollapse conjecture (though it is likely that a slightly better bound can be established). This bound has the desirable properties of being computable from the (spherically symmetric) initial data for the spacetime and having a very simple form. In fact, its form is the same as was established, using a different method, for the spherically symmetric massless scalar field spacetimes, thereby proving a conjecture offered in that work. Prospects for generalizing these results beyond the spherically symmetric case are discussed.