INCOMPLETENESS THEOREMS FOR THE SPHERICALLY SYMMETRICAL SPACETIMES

The closed-universe recollapse conjecture is studied for the spherically symmetric spacetimes. It is proven that there exists an upper bound to the lengths of timelike curves in any spherically symmetric spacetime that possesses S1 x S2 Cauchy surfaces and that satisfies the non-negative-pressures and dominant-energy conditions. Further, an explicit bound is obtained that is determined by the initial data for the spacetime on any Cauchy surface. The conjecture is further studied for the spherically symmetric spacetimes possessing an extra spatial symmetry-the Kantowski-Sachs spacetimes. It is proven, for example, that there exists an upper bound to the lengths of time-like curves in any Kantowski-Sachs spacetime that possesses compact Cauchy surfaces and that satisfies the non-negative-sum-pressures condition.