154 research outputs found
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.
- …