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Existence of maximal hypersurfaces in some spherically symmetric spacetimes
We prove that the maximal development of any spherically symmetric spacetime
with collisionless matter (obeying the Vlasov equation) or a massless scalar
field (obeying the massless wave equation) and possessing a constant mean
curvature 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
in any spacetime satisfying the timelike convergence condition in terms of the
volume and mean curvature of a fixed Cauchy surface and the maximal
distance between and . 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.Comment: 8 pages, REVTeX 3.