  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.
