arXiv subjects

Claes Uggla

Publications and source records attributed to Claes Uggla.

At least 55 records · Page 3Linked to original sources

arXiv2008

Bianchi type I models with two tilted fluids

In this paper we investigate expanding Bianchi type I models with two tilted fluids with linear equations of state. Individually the fluids have non-zero energy fluxes w.r.t. the symmetry surfaces, but these cancel each other because of the Codazzi constraint. Asymptotically toward the past the solutions approach Kasner states if the speeds of sound are less than that of light. If one of the fluids has a speed of sound that is less or equal to 1/3 of the speed of light (radiation) then the models isotropize toward the future, but if both fluids are stiffer than radiation then the final state is anisotropic with non-zero Hubble-normalized shear. The significance of these results is discussed in a broader context.

gr-qc↗

arXiv2007

The Nature of Generic Cosmological Singularities

The existence of a singularity by definition implies a preferred scale--the affine parameter distance from/to the singularity of a causal geodesic that is used to define it. However, this variable scale is also captured by the expansion along the geodesic, and this can be used to obtain a regularized state space picture by means of a conformal transformation that factors out the expansion. This leads to the conformal `Hubble-normalized' orthonormal frame approach which allows one to translate methods and results concerning spatially homogeneous models into the generic inhomogeneous context, which in turn enables one to derive the dynamical nature of generic cosmological singularities. Here we describe this approach and outline the derivation of the `cosmological billiard attractor,' which describes the generic dynamical asymptotic behavior towards a generic spacelike singularity. We also compare the `dynamical systems picture' resulting from this approach with other work on generic spacelike singularities: the metric approach of Belinskii, Lifschitz, and Khalatnikov, and the recent Iwasawa based Hamiltonian method used by Damour, Henneaux, and Nicolai; in particular we show that the cosmological billiards obtained by the latter and the cosmological billiard attractor form complementary `dual' descriptions of the generic asymptotic dynamics of generic spacelike singularities.

gr-qc↗

arXiv2007

The cosmological billiard attractor

This article is devoted to a study of the asymptotic dynamics of generic solutions of the Einstein vacuum equations toward a generic spacelike singularity. Starting from fundamental assumptions about the nature of generic spacelike singularities we derive in a step-by-step manner the cosmological billiard conjecture: we show that the generic asymptotic dynamics of solutions is represented by (randomized) sequences of heteroclinic orbits on the `billiard attractor'. Our analysis rests on two pillars: (i) a dynamical systems formulation based on the conformal Hubble-normalized orthonormal frame approach expressed in an Iwasawa frame; (ii) stochastic methods and the interplay between genericity and stochasticity. Our work generalizes and improves the level of rigor of previous work by Belinskii, Khalatnikov, and Lifshitz; furthermore, we establish that our approach and the Hamiltonian approach to `cosmological billiards', as elaborated by Damour, Hennaux, and Nicolai, can be viewed as yielding `dual' representations of the asymptotic dynamics.

gr-qc↗

arXiv2006

Self-gravitating stationary spherically symmetric systems in relativistic galactic dynamics

We study equilibrium states in relativistic galactic dynamics which are described by solutions of the Einstein-Vlasov system for collisionless matter. We recast the equations into a regular three-dimensional system of autonomous first order ordinary differential equations on a bounded state space. Based on a dynamical systems analysis we derive new theorems that guarantee that the steady state solutions have finite radii and masses.

gr-qc↗

arXiv2006

Asymptotic silence-breaking singularities

We discuss three complementary aspects of scalar curvature singularities: asymptotic causal properties, asymptotic Ricci and Weyl curvature, and asymptotic spatial properties. We divide scalar curvature singularities into two classes: so-called asymptotically silent singularities and non-generic singularities that break asymptotic silence. The emphasis in this paper is on the latter class which have not been previously discussed. We illustrate the above aspects and concepts by describing the singularities of a number of representative explicit perfect fluid solutions.

gr-qc↗

arXiv2005

Dynamics of the spatially homogeneous Bianchi type I Einstein-Vlasov equations

We investigate the dynamics of spatially homogeneous solutions of the Einstein-Vlasov equations with Bianchi type I symmetry by using dynamical systems methods. All models are forever expanding and isotropize toward the future; toward the past there exists a singularity. We identify and describe all possible past asymptotic states; in particular, on the past attractor set we establish the existence of a heteroclinic network, which is a new type of feature in general relativity. This illustrates among other things that Vlasov matter can lead to quite different dynamics of cosmological models as compared to perfect fluids.

gr-qc↗

arXiv2005

Conformal regularization of Einstein's field equations

To study asymptotic structures, we regularize Einstein's field equations by means of conformal transformations. The conformal factor is chosen so that it carries a dimensional scale that captures crucial asymptotic features. By choosing a conformal orthonormal frame we obtain a coupled system of differential equations for a set of dimensionless variables, associated with the conformal dimensionless metric, where the variables describe ratios with respect to the chosen asymptotic scale structure. As examples, we describe some explicit choices of conformal factors and coordinates appropriate for the situation of a timelike congruence approaching a singularity. One choice is shown to just slightly modify the so-called Hubble-normalized approach, and one leads to dimensionless first order symmetric hyperbolic equations. We also discuss differences and similarities with other conformal approaches in the literature, as regards, e.g., isotropic singularities.

gr-qc↗

arXiv2005

Asymptotic silence of generic cosmological singularities

In this letter we investigate the nature of generic cosmological singularities using the framework developed by Uggla et al. We do so by studying the past asymptotic dynamics of general vacuum G2 cosmologies, models that are expected to capture the singular behavior of generic cosmologies with no symmetries at all. In particular, our results indicate that asymptotic silence holds, i.e., that particle horizons along all timelines shrink to zero for generic solutions. Moreover, we provide evidence that spatial derivatives become dynamically insignificant along generic timelines, and that the evolution into the past along such timelines is governed by an asymptotic dynamical system which is associated with an invariant set -- the silent boundary. We also identify an attracting subset on the silent boundary that organizes the oscillatory dynamics of generic timelines in the singular regime. In addition, we discuss the dynamics associated with recurring spike formation.

gr-qc↗

arXiv2004

Theory of Newtonian self-gravitating stationary spherically symmetric systems

We investigate spherically symmetric equilibrium states of the Vlasov-Poisson system, relevant in galactic dynamics. We recast the equations into a regular three-dimensional system of autonomous first order ordinary differential equations on a region with compact closure.Based on a dynamical systems analysis we derive theorems that guarantee that the steady state solutions have finite mass and compact support.

math-ph↗

arXiv2003

Gowdy phenomenology in scale-invariant variables

The dynamics of Gowdy vacuum spacetimes is considered in terms of Hubble-normalized scale-invariant variables, using the timelike area temporal gauge. The resulting state space formulation provides for a simple mechanism for the formation of ``false'' and ``true spikes'' in the approach to the singularity, and a geometrical formulation for the local attractor.

gr-qc↗

arXiv2003

Asymptotic isotropization in inhomogeneous cosmology

In this paper we investigate asymptotic isotropization. We derive the asymptotic dynamics of spatially inhomogeneous cosmological models with a perfect fluid matter source and a positive cosmological constant near the de Sitter equilibrium state at late times, and near the flat FL equilibrium state at early times. Our results show that there exists an open set of solutions approaching the de Sitter state at late times, consistent with the cosmic no-hair conjecture. On the other hand, solutions that approach the flat FL state at early times are special and admit a so-called isotropic initial singularity. For both classes of models the asymptotic expansion of the line element contains an arbitrary spatial metric at leading order, indicating asymptotic spatial inhomogeneity. We show, however, that in the asymptotic regimes this spatial inhomogeneity is significant only at super-horizon scales.

gr-qc↗

arXiv2003

The past attractor in inhomogeneous cosmology

We present a general framework for analyzing spatially inhomogeneous cosmological dynamics. It employs Hubble-normalized scale-invariant variables which are defined within the orthonormal frame formalism, and leads to the formulation of Einstein's field equations with a perfect fluid matter source as an autonomous system of evolution equations and constraints. This framework incorporates spatially homogeneous dynamics in a natural way as a special case, thereby placing earlier work on spatially homogeneous cosmology in a broader context, and allows us to draw on experience gained in that field using dynamical systems methods. One of our goals is to provide a precise formulation of the approach to the spacelike initial singularity in cosmological models, described heuristically by Belinski\vı, Khalatnikov and Lifshitz. Specifically, we construct an invariant set which we conjecture forms the local past attractor for the evolution equations. We anticipate that this new formulation will provide the basis for proving rigorous theorems concerning the asymptotic behavior of spatially inhomogeneous cosmological models.

gr-qc↗

arXiv2002

Newtonian Stellar Models

We investigate static spherically symmetric perfect fluid models in Newtonian gravity for barotropic equations of state that are asymptotically polytropic at low and high pressures. This is done by casting the equations into a 3-dimensional regular dynamical system with bounded dependent variables. The low and high central pressure limits correspond to two 2-dimensional boundary subsets, described by homology invariant equations for exact polytropes. Thus the formulation naturally places work about polytropes in a more general context. The introduced framework yields a visual aid for obtaining qualitative information about the solution space and is also suitable for numerical investigations. Last, but not least, it makes a host of mathematical tools from dynamical systems theory available. This allows us to prove a number of theorems about the relationship between the equation of state and properties concerning total masses and radii.

astro-ph↗

arXiv2001

Dynamical systems approach to G2 cosmology

In this paper we present a new approach for studying the dynamics of spatially inhomogeneous cosmological models with one spatial degree of freedom. By introducing suitable scale-invariant dependent variables we write the evolution equations of the Einstein field equations as a system of autonomous partial differential equations in first-order symmetric hyperbolic format, whose explicit form depends on the choice of gauge. As a first application, we show that the asymptotic behaviour near the cosmological initial singularity can be given a simple geometrical description in terms of the local past attractor on the boundary of the scale-invariant dynamical state space. The analysis suggests the name ``asymptotic silence'' to describe the evolution of the gravitational field near the cosmological initial singularity.

gr-qc↗

arXiv2000

Dynamics of spatially homogeneous locally rotationally symmetric solutions of the Einstein-Vlasov equations

The dynamics of the Einstein-Vlasov equations for a class of cosmological models with four Killing vectors is discussed in the case of massive particles. It is shown that in all models analysed the solutions with massive particles are asymptotic to solutions with massless particles at early times. It is also shown that in Bianchi types I and II the solutions with massive particles are asymptotic to dust solutions at late times. That Bianchi type III models are also asymptotic to dust solutions at late times is consistent with our results but is not established by them.

gr-qc↗

arXiv1998

Spatially self-similar spherically symmetric perfect-fluid models

Einstein's field equations for spatially self-similar spherically symmetric perfect-fluid models are investigated. The field equations are rewritten as a first-order system of autonomous differential equations. Dimensionless variables are chosen in such a way that the number of equations in the coupled system is reduced as far as possible and so that the reduced phase space becomes compact and regular. The system is subsequently analysed qualitatively with the theory of dynamical systems.

gr-qc↗

arXiv1998

Timelike self-similar spherically symmetric perfect-fluid models

Einstein's field equations for timelike self-similar spherically symmetric perfect-fluid models are investigated. The field equations are rewritten as a first-order system of autonomous differential equations. Dimensionless variables are chosen in such a way that the number of equations in the coupled system is reduced as far as possible and so that the reduced phase space becomes compact and regular. The system is subsequently analysed qualitatively using the theory of dynamical systems.

gr-qc↗

arXiv1996

General Relativistic 1+3 Orthonormal Frame Approach Revisited

The equations of the 1+3 orthonormal frame approach are explicitly presented and discussed. Natural choices of local coordinates are mentioned. A dimensionless formulation is subsequently given. It is demonstrated how one can obtain a number of interesting problems by specializing the general equations. In particular, equation systems for ``silent'' dust cosmological models also containing magnetic Maxwell fields, locally rotationally symmetric spacetime geometries and spatially homogeneous cosmological models are presented. We show that while the 3-Cotton--York tensor is zero for Szekeres dust models, it is nonzero for a generic representative within the ``silent'' class.

gr-qc↗