23 research outputs found
On the geometry of silent and anisotropic big bang singularities
This article is the second of two in which we develop a geometric framework
for analysing silent and anisotropic big bang singularities. In the present
article, we record geometric conclusions obtained by combining the geometric
framework with Einstein's equations. The main features of the results are the
following: The assumptions do not involve any symmetry requirements and are
weak enough to be consistent with most big bang singularities for which the
asymptotic geometry is understood. The framework gives a clear picture of the
asymptotic geometry. It also reproduces the Kasner map, conjectured in the
physics literature to constitute the essence of the asymptotic dynamics for
vacuum solutions to Einstein's equations. When combined with Einstein's
equations, the framework yields partial improvements of the assumptions
concerning, e.g., the expansion normalised Weingarten map (one of
the central objects of the framework, defined as the Weingarten map of the
leaves of the foliation divided by the mean curvature). For example, the
expansion normalised normal derivative of can, under suitable
assumptions concerning the eigenvalues of , be demonstrated to
decay exponentially and can be demonstrated to converge
exponentially, even though we initially only impose weighted bounds on these
quantities. Finally, the framework gives a unified perspective on the existing
results. Moreover, in -dimensions, the only parameters necessary to
interpret the results are the eigenvalues of and an additional
scalar function determined by the geometry induced on the leaves of the
foliation.
In the companion article, we obtain conclusions concerning the asymptotic
behaviour of solutions to linear systems of wave equations on the backgrounds
consistent with the framework.Comment: 65 pages, 6 figure
The future asymptotics of Bianchi VIII vacuum solutions
Bianchi VIII vacuum solutions to Einstein's equations are causally
geodesically complete to the future, given an appropriate time orientation, and
the objective of this article is to analyze the asymptotic behaviour of
solutions in this time direction. For the Bianchi class A spacetimes, there is
a formulation of the field equations that was presented in an article by
Wainwright and Hsu, and we analyze the asymptotic behaviour of solutions in
these variables. We also try to give the analytic results a geometric
interpretation by analyzing how a normalized version of the Riemannian metric
on the spatial hypersurfaces of homogeneity evolves.Comment: 34 pages, no figure
Formation of quiescent big bang singularities
Hawking's singularity theorem says that cosmological solutions arising from
initial data with positive mean curvature have a past singularity. However, the
nature of the singularity remains unclear. We therefore ask: If the initial
hypersurface has sufficiently large mean curvature, does the curvature
necessarily blow up towards the singularity?
In case the eigenvalues of the expansion-normalized Weingarten map are
everywhere distinct and satisfy a certain algebraic condition (which in 3+1
dimensions is equivalent to them being positive), we prove that this is the
case in the CMC Einstein-non-linear scalar field setting. More specifically, we
associate a set of geometric expansion-normalized quantities to any initial
data set with positive mean curvature. These quantities are expected to
converge, in the quiescent setting, in the direction of crushing big bang
singularities. Our main result says that if the mean curvature is large enough,
relative to an appropriate Sobolev norm of these geometric quantities, and if
the algebraic condition is satisfied, then a quiescent (as opposed to
oscillatory) big bang singularity with curvature blow-up forms. This provides a
stable regime of big bang formation without requiring proximity to any
particular class of background solutions.
An important recent result by Fournodavlos, Rodnianski and Speck demonstrates
stable big bang formation for all the spatially flat and spatially homogeneous
solutions to the Einstein-scalar field equations satisfying the algebraic
condition. Here we obtain analogous stability results for any solution inducing
data at the singularity, in the sense introduced by the third author, in
particular generalizing the aforementioned result. Moreover, we are able to
prove both future and past global non-linear stability of a large class of
spatially locally homogeneous solutions.Comment: 77 page
Future asymptotic expansions of Bianchi VIII vacuum metrics
Bianchi VIII vacuum solutions to Einstein's equations are causally
geodesically complete to the future, given an appropriate time orientation, and
the objective of this article is to analyze the asymptotic behaviour of
solutions in this time direction. For the Bianchi class A spacetimes, there is
a formulation of the field equations that was presented in an article by
Wainwright and Hsu, and in a previous article we analyzed the asymptotic
behaviour of solutions in these variables. One objective of this paper is to
give an asymptotic expansion for the metric. Furthermore, we relate this
expansion to the topology of the compactified spatial hypersurfaces of
homogeneity. The compactified spatial hypersurfaces have the topology of
Seifert fibred spaces and we prove that in the case of NUT Bianchi VIII
spacetimes, the length of a circle fibre converges to a positive constant but
that in the case of general Bianchi VIII solutions, the length tends to
infinity at a rate we determine.Comment: 50 pages, no figures. Erronous definition of Seifert fibred spaces
correcte
The Bianchi IX attractor
We consider the asymptotic behaviour of spatially homogeneous spacetimes of Bianchi type IX close to the singularity (we also consider some of the other Bianchi types, e. g. Bianchi VIII in the stiff fluid case). The matter content is assumed to be an orthogonal perfect fluid with linear equation of state and zero cosmological constant. In terms of the variables of Wainwright and Hsu, we have the following results. In the stiff fluid case, the solution converges to a point for all the Bianchi class A types. For the other matter models we consider, the Bianchi IX solutions generically converge to an attractor consisting of the closure of the vacuum type II orbits. Furthermore, we observe that for all the Bianchi class A spacetimes, except those of vacuum Taub type, a curvature invariant is unbounded in the incomplete directions of inextendible causal geodesics
Cosmic Censorship for Gowdy Spacetimes
Due to the complexity of Einstein’s equations, it is often natural to study a question of interest in the framework of a restricted class of solutions. One way to impose a restriction is to consider solutions satisfying a given symmetry condition. There are many possible choices, but the present article is concerned with one particular choice, which we shall refer to as Gowdy symmetry. We begin by explaining the origin and meaning of this symmetry type, which has been used as a simplifying assumption in various contexts, some of which we shall mention. Nevertheless, the subject of interest here is strong cosmic censorship. Consequently, after having described what the Gowdy class of spacetimes is, we describe, as seen from the perspective of a mathematician, what is meant by strong cosmic censorship. The existing results on cosmic censorship are based on a detailed analysis of the asymptotic behavior of solutions. This analysis is in part motivated by conjectures, such as the BKL conjecture, which we shall therefore briefly describe. However, the emphasis of the article is on the mathematical analysis of the asymptotics, due to its central importance in the proof and in the hope that it might be of relevance more generally. The article ends with a description of the results that have been obtained concerning strong cosmic censorship in the class of Gowdy spacetimes