Constant mean curvature foliations in cosmological spacetimes

Foliations by constant mean curvature hypersurfaces provide a possibility of defining a preferred time coordinate in general relativity. In the following various conjectures are made about the existence of foliations of this kind in spacetimes satisfying the strong energy condition and possessing compact Cauchy hypersurfaces. Recent progress on proving these conjectures under supplementary assumptions is reviewed. The method of proof used is explained and the prospects for generalizing it discussed. The relations of these questions to cosmic censorship and the closed universe recollapse conjecture are pointed out.Comment: 11 pages. Contribution to the Journees Relativiste

Cosmic Censorship for Some Spatially Homogeneous Cosmological Models

The global properties of spatially homogeneous cosmological models with collisionless matter are studied. It is shown that as long as the mean curvature of the hypersurfaces of homogeneity remains finite no singularity can occur in finite proper time as measured by observers whose worldlines are orthogonal to these hypersurfaces. Strong cosmic censorship is then proved for the Bianchi I, Bianchi IX and Kantowski-Sachs symmetry classes.Comment: 14 pages, Plain TeX, MPA-AR-93-

Asymptotics of solutions of the Einstein equations with positive cosmological constant

A positive cosmological constant simplifies the asymptotics of forever expanding cosmological solutions of the Einstein equations. In this paper a general mathematical analysis on the level of formal power series is carried out for vacuum spacetimes of any dimension and perfect fluid spacetimes with linear equation of state in spacetime dimension four. For equations of state stiffer than radiation evidence for development of large gradients, analogous to spikes in Gowdy spacetimes, is found. It is shown that any vacuum solution satisfying minimal asymptotic conditions has a full asymptotic expansion given by the formal series. In four spacetime dimensions, and for spatially homogeneous spacetimes of any dimension, these minimal conditions can be derived for appropriate initial data. Using Fuchsian methods the existence of vacuum spacetimes with the given formal asymptotics depending on the maximal number of free functions is shown without symmetry assumptions.Comment: 23 page

Existence of constant mean curvature foliations in spacetimes with two-dimensional local symmetry

It is shown that in a class of maximal globally hyperbolic spacetimes admitting two local Killing vectors, the past (defined with respect to an appropriate time orientation) of any compact constant mean curvature hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces. Moreover, the mean curvature of the leaves of this foliation takes on arbitrarily negative values and so the initial singularity in these spacetimes is a crushing singularity. The simplest examples occur when the spatial topology is that of a torus, with the standard global Killing vectors, but more exotic topologies are also covered. In the course of the proof it is shown that in this class of spacetimes a kind of positive mass theorem holds. The symmetry singles out a compact surface passing through any given point of spacetime and the Hawking mass of any such surface is non-negative. If the Hawking mass of any one of these surfaces is zero then the entire spacetime is flat.Comment: 22 page

The structure of singularities in inhomogeneous cosmological models

Recent progress in understanding the structure of cosmological singularities is reviewed. The well-known picture due to Belinskii, Khalatnikov and Lifschitz (BKL) is summarized briefly and it is discussed what existing analytical and numerical results have to tell us about the validity of this picture. If the BKL description is correct then most cosmological singularities are complicated. However there are some cases where it predicts simple singularities. These cases should be particularly amenable to mathematical investigation and the results in this direction which have been achieved so far are described.Comment: 5 pages, to appear in proceedings of conference on mathematical cosmology, Potsdam, 199

The Einstein-Vlasov system

Rigorous results on solutions of the Einstein-Vlasov system are surveyed. After an introduction to this system of equations and the reasons for studying it, a general discussion of various classes of solutions is given. The emphasis is on presenting important conceptual ideas, while avoiding entering into technical details. Topics covered include spatially homogenous models, static solutions, spherically symmetric collapse and isotropic singularities.Comment: Lecture notes from Cargese worksho

The nature of spacetime singularities

Present knowledge about the nature of spacetime singularities in the context of classical general relativity is surveyed. The status of the BKL picture of cosmological singularities and its relevance to the cosmic censorship hypothesis are discussed. It is shown how insights on cosmic censorship also arise in connection with the idea of weak null singularities inside black holes. Other topics covered include matter singularities and critical collapse. Remarks are made on possible future directions in research on spacetime singularities.Comment: Submitted to 100 Years of Relativity - Space-Time Structure: Einstein and Beyond, A. Ashtekar (ed.