Closed spherically symmetric massless scalar field spacetimes have finite lifetimes

TL;DR

This paper proves that certain closed spherically symmetric spacetimes with massless scalar fields, perfect fluids, or null dust have finite lifetimes, supporting the idea that such universes cannot expand indefinitely.

Contribution

It establishes a universal bound on the lifetime of these spacetimes, providing a rigorous proof for the recollapse conjecture in this class.

Findings

Timelike curves have length less than 6 times the maximum mass function.

The result applies to spacetimes with scalar fields, perfect fluids, or null dust.

A conjecture is made that similar bounds hold more generally.

Abstract

The closed-universe recollapse conjecture is studied for a class of closed spherically symmetric spacetimes which includes those having as a matter source: (1) a massless scalar field; (2) a perfect fluid obeying the equation of state $\rho = P$; and (3) null dust. It is proven that all timelike curves in any such spacetime must have length less than $6 \max_\Sigma(2m)$, where $m$ is the mass associated with the spheres of symmetry and $\Sigma$ is any Cauchy surface for the spacetime. The simplicity of this result leads us to conjecture that a similar bound can be established for the more general spherically symmetric spacetimes.