Existence of maximal hypersurfaces in some spherically symmetric spacetimes

Abstract

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 $S^1\times S^2$ 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 ${\Sigma }_0$ and the maximal distance between $\Sigma$ and ${\Sigma }_0$. 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.

8 pages, REVTeX 3.0

References in corpus (4)

• Crushing singularities in spacetimes with spherical, plane and hyperbolic symmetry
• Optical scalars in spherical spacetimes
• A regularity theorem for solutions of the spherically symmetric Vlasov-Einstein system
• Lifetimes of spherically symmetric closed universes

Cited by in corpus (21)

• The Einstein-Vlasov System/Kinetic Theory
• The Einstein-Vlasov system/Kinetic theory
• Theorems on existence and global dynamics for the Einstein equations
• On the Einstein-Vlasov system with hyperbolic symmetry
• Constant mean curvature foliations in cosmological spacetimes
• The nature of spacetime singularities
• Strong cosmic censorship for surface-symmetric cosmological spacetimes with collisionless matter
• Superconducting loop quantum gravity and the cosmological constant
• The Einstein-Vlasov sytem/Kinetic theory
• Global Prescribed Mean Curvature foliations in cosmological spacetimes with matter, Part I
• Local Prescribed Mean Curvature foliations in cosmological spacetimes
• Local and global existence theorems for the Einstein equations
• Global Prescribed Mean Curvature foliations in cosmological spacetimes with matter, Part II
• The global existence problem in general relativity
• Maximal slicings in spherical symmetry: local existence and construction
• The Einstein-Lambda flow on product manifolds
• An introduction to the Einstein-Vlasov system
• Constant mean curvature slicings of Kantowski-Sachs spacetimes
• Existence and non-existence results for global constant mean curvature foliations
• Solutions of the Einstein equations with matter
• Recollapsing spacetimes with $\Lambda <0$