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 $6 \max_\Sigma(2m)$, 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. 
Gregory A. Burnett