Closed spherically symmetric massless scalar field spacetimes have finite lifetimes

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 $6{\max }_{\mathrm{\Sigma }}(2m)$, where $m$ is the mass associated with the spheres of symmetry and $\mathrm{\Sigma }$ 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.