August 1, 1995
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.
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October 22, 2001
It is shown that a spacetime with collisionless matter evolving from data on a compact Cauchy surface with hyperbolic symmetry can be globally covered by compact hypersurfaces on which the mean curvature is constant and by compact hypersurfaces on which the area radius is constant. Results for the related cases of spherical and plane symmetry are reviewed and extended. The prospects of using the global time coordinates obtained in this way to investigate the global geometry o...
July 31, 2006
Some future global properties of cosmological solutions for the Einstein-Vlasov-Maxwell system with surface symmetry are presented. Global existence is proved, the homogeneous spacetimes are future complete for causal trajectories, and the same is true for inhomogeneous plane-symmetric solutions with small initial data. In the latter case some decay properties are also obtained at late times. Similar but slightly weaker results hold for hyperbolic symmetry.
April 26, 1994
The closed-universe recollapse conjecture is studied for a class of closed spherically symmetric spacetimes which includes those having as a matter source: (1) a massless scalar field; (2) a perfect fluid obeying the equation of state $\rho = P$; and (3) null dust. It is proven that all timelike curves in any such spacetime must have length less than $6 \max_\Sigma(2m)$, where $m$ is the mass associated with the spheres of symmetry and $\Sigma$ is any Cauchy surface for the s...
September 6, 1994
It is proven that any spherically symmetric spacetime that possesses a compact Cauchy surface $\Sigma$ and that satisfies the dominant-energy and non-negative-pressures conditions must have a finite lifetime in the sense that all timelike curves in such a spacetime must have a length no greater than $10 \max_\Sigma(2m)$, 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 ...
April 5, 2021
As is well known, constant mean curvature (CMC) spacelike hypersurfaces play an important role in solving the Einstein equations, both in solving the contraints and the evolution equations. In this paper we review the CMC existence result obtained by the authors in [10] and consider some new existence results motivated by a conjecture of Dilts and Holst [8]. We also address some issues concerning the conformal structure of cosmological spacetimes.
October 18, 2001
This work investigates some global questions about cosmological spacetimes with two dimensional spherical, plane and hyperbolic symmetry containing matter. The result is, that these spacetimes admit a global foliation by prescribed mean curvature surfaces, which extends at least towards a crushing singularity. The time function of the foliation is geometrically defined and unique up to the choice of an initial Cauchy surface. This work generalizes a similar analysis on cons...
August 18, 2003
This paper treats the global existence question for a collection of general relativistic collisionless particles, all having the same mass. The spacetimes considered are globally hyperbolic, with Cauchy surface a 3-torus. Furthermore, the spacetimes considered are isometrically invariant under a two-dimensional group action, the orbits of which are spacelike 2-tori. It is known from previous work that the area of the group orbits serves as a global time coordinate. In the pre...
April 22, 2006
This paper gives a new proof that maximal, globally hyperbolic, flat spacetimes of dimension $n\geq 3$ with compact Cauchy hypersurfaces are globally foliated by Cauchy hypersurfaces of constant mean curvature, and that such spacetimes admit a globally defined constant mean curvature time function precisely when they are causally incomplete. The proof, which is based on using the level sets of the cosmological time function as barriers, is conceptually simple and will provide...
April 6, 2009
We prove several global existence theorems for spacetimes with toroidal or hyperbolic symmetry with respect to a geometrically defined time. More specifically, we prove that generically, the maximal Cauchy development of $T^2$-symmetric initial data with positive cosmological constant $\Lambda >0$, in the vacuum or with Vlasov matter, may be covered by a global areal foliation with the area of the symmetry orbits tending to zero in the contracting direction. We then prove the...
February 23, 2019
In this paper, we review results on the existence (and nonexistence) of constant mean curvature spacelike hypersurfaces in the cosmological setting, and discuss the connection to the spacetime splittng problem. It is a pleasure to dedicate this paper to Robert Bartnik, who has made fundamental contributions to this area.