Existence of Maximal Hypersurfaces in Some Spherically Symmetric Spacetimes

摘要

The authors prove that the maximal development of any spherically symmetricspacetime with collisionless matter (obeying the Vlasov equation) or a massless scalar field (obeying the massless wave equation) and possessing a constant mean curvature S(sup 1) X S(sup 2) Cauchy surface also contains a maximal Cauchy surface. Combining this with previous results establishes that the spacetime can be foliated by constant mean curvature Cauchy surfaces with the mean curvature taking on all real values, thereby showing that these spacetimes satisfy the closed-universe recollapse conjecture. A key element of the proof, of interest in itself, is a bound for the volume of any Cauchy surface sigma in any spacetime satisfying the timelike convergence condition in terms of the volume and mean curvature of a fixed Cauchy surface sigma(sub 0) and the maximal distance between sigma and sigma(sub 0). In particular, this shows that any globally hyperbolic spacetime having a finite lifetime and obeying the timelike-convergence condition cannot attain an arbitrarily large spatial volume.