G R - Q C - 9 40 90 11 Lifetimes of spherically symmetric closed universes

It is proven that any spherically symmetric spacetime that possesses a compact Cauchy surface and that satis es the dominant-energy and non-negative-pressures conditions must have a nite lifetime in the sense that all timelike curves in such a spacetime must have a length no greater than 10max (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 di erent method, for the spherically symmetric massless scalar eld spacetimes, thereby proving a conjecture o ered in that work. Prospects for generalizing these results beyond the spherically symmetric case are discussed.