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

Heinzle, Jakob; Uggla, Claes

doi:10.1088/0264-9381/23/10/016

Cited by 31 publications

(70 citation statements)

References 20 publications

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“…As regards possible sources, this naturally suggests a study of the influence of matter on singularities. As a simple example we refer to [46], which gives an indication that, e.g., Vlasov matter behaves differently in some respects than perfect fluids. Of particular interest is the question of structural stability of generic spacelike singularities, especially since generic singularities seem to uncover the essential properties of matter.…”

Section: Discussionmentioning

confidence: 99%

The cosmological billiard attractor

Heinzle¹,

Uggla²,

Röhr³

2009

Advances in Theoretical and Mathematical Physics

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231

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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 Hubblenormalized 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.e-print archive: http://lanl.arXiv.org/abs/gr-qc/0702141 J. MARK HEINZLE, CLAES UGGLA, AND NIKLAS RÖHR

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Section: Discussionmentioning

confidence: 99%

The cosmological billiard attractor

Heinzle¹,

Uggla²,

Röhr³

2009

Advances in Theoretical and Mathematical Physics

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231

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“…Another issue is how and if matter influences the connection between generic spacelike singularities and weak null singularities in asymptotically flat spacetimes (there are indications that e.g. Vlasov matter behaves differently than perfect fluids [35]); for special examples, see [25].…”

Section: Discussionmentioning

confidence: 99%

The Nature of Generic Cosmological Singularities

Uggla

2008

The Eleventh Marcel Grossmann Meeting

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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.

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“…For a general introduction to collisionless matter and the Vlasov-Einstein system we refer to [1,17]; the Bianchi type I case is discussed in detail in [6]. For a thorough discussion on the general relativistic theory of elasticity we refer to [2,4,9,10,20]; we will confine ourselves to deriving the energy-momentum tensor of elastic bodies.…”

Section: Anisotropic Matter Modelsmentioning

confidence: 99%

“…In contrast, a systematic analysis of the influence of the matter model on the (asymptotic) dynamics of solutions is not available at present. It is known that the dynamics of solutions (and the asymptotics towards the initial singularity, in particular) strongly depends on the matter source that is considered; the relatively simple behaviour of the vacuum and the perfect fluid case, see, e.g., [22], is replaced by a considerably more intricate behaviour in the case of anisotropic matter models [3,6,11,15,19]. The purpose of this paper is to analyze systematically and in detail in which way anisotropies of the matter source determine the dynamics of cosmological solutions.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of Bianchi Type I Solutions of the Einstein Equations with Anisotropic Matter

Calogero

Heinzle

2009

Ann. Henri Poincaré

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We analyze the global dynamics of Bianchi type I solutions of the Einstein equations with anisotropic matter. The matter model is not specified explicitly but only through a set of mild and physically motivated assumptions; thereby our analysis covers matter models as different from each other as, e.g., collisionless matter, elastic matter and magnetic fields. The main result we prove is the existence of an 'anisotropy classification' for the asymptotic behaviour of Bianchi type I cosmologies. The type of asymptotic behaviour of generic solutions is determined by one single parameter that describes certain properties of the anisotropic matter model under extreme conditions. The anisotropy classification comprises the following types. The convergent type A + : Each solution converges to a Kasner solution as the singularity is approached and each Kasner solution is a possible past asymptotic state. The convergent types B + and C + : Each solution converges to a Kasner solution as the singularity is approached; however, the set of Kasner solutions that are possible past asymptotic states is restricted. The oscillatory type D + : Each solution oscillates between different Kasner solutions as the singularity is approached. Furthermore, we investigate non-generic asymptotic behaviour and the future asymptotic behaviour of solutions.

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

Cited by 31 publications

References 20 publications

The cosmological billiard attractor

The cosmological billiard attractor

The Nature of Generic Cosmological Singularities

Dynamics of Bianchi Type I Solutions of the Einstein Equations with Anisotropic Matter

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