  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.
