Cosmological solutions of the Vlasov-Einstein system with spherical, plane, and hyperbolic symmetry

Rein, Gerhard

doi:10.1017/s0305004100074569

Cited by 93 publications

(123 citation statements)

References 12 publications

(40 reference statements)

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“…Finally this definition corresponds (besides the replacement {β, π, Q, P } → {β [0] , π [0] , Q [0] , P [0] }) to setting to zero the symmetry walls e −2(β a −β b ) (with a > b) in the full momentum constraints. Along the solution (21) of the evolution equations (20), the only time dependent term in…”

Section: Definition Of the Asymptotic Constraintsmentioning

confidence: 99%

“…modulo equations (20). (30) From this relation, we conclude that, when the Hamiltonian H [0] = 0 constraint is satisfied, the momentum constraints are conserved when the asymptotic evolution system (20) is satisfied.…”

Section: Definition Of the Asymptotic Constraintsmentioning

confidence: 99%

“…and is conserved modulo the asymptotic equations of motions (20). We impose that the asymptotic Gauss constraint (37) hold.…”

Section: Constraintsmentioning

confidence: 99%

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Describing general cosmological singularities in Iwasawa variables

2008

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Belinskii, Khalatnikov, and Lifshitz (BKL) conjectured that the description of the asymptotic behavior of a generic solution of Einstein equations near a spacelike singularity could be drastically simplified by considering that the time derivatives of the metric asymptotically dominate (except at a sequence of instants, in the 'chaotic case') over the spatial derivatives. We present a precise formulation of the BKL conjecture (in the chaotic case) that consists of basically three elements: (i) we parametrize the spatial metric g ij by means of Iwasawa variables (β a , N a i ); (ii) we define, at each spatial point, a (chaotic) asymptotic evolution system made of ordinary differential equations for the Iwasawa variables; and (iii) we characterize the exact Einstein solutions β, N whose asymptotic behavior is described by a solution β [0] , N [0] of the previous evolution system by means of a 'generalized Fuchsian system' for the differenced variablesβ = β − β [0] ,N = N − N [0] , and by requiring thatβ andN tend to zero on the singularity. We also show that, in spite of the apparently chaotic infinite succession of 'Kasner epochs' near the singularity, there exists a well-defined asymptotic geometrical structure on the singularity : it is described by a partially framed flag. Our treatment encompasses Einstein-matter systems (comprising scalar and p-forms), and also shows how the use of Iwasawa variables can simplify the usual ('asymptotically velocity term dominated') description of non-chaotic systems.

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Section: Definition Of the Asymptotic Constraintsmentioning

confidence: 99%

Section: Definition Of the Asymptotic Constraintsmentioning

confidence: 99%

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Describing general cosmological singularities in Iwasawa variables

2008

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“…Future inextendibility under additional small-data assumptions has been shown in [35] in the case of hyperbolic symmetry. Past inextendibility for small data under various assumptions on k and ƒ has been shown in [34,46].…”

Section: Previous Resultsmentioning

confidence: 98%

“…The study of the system under spherical symmetry was initiated by Rein and the second author [36] and by Rein [34] in the surface-symmetric "cosmological" case. There are now a host of results under these symmetry assumptions, in particular, several cases in which strong cosmic censorship has been shown.…”

Section: Introductionmentioning

confidence: 99%

Strong Cosmic Censorship for Surface‐Symmetric Cosmological Spacetimes with Collisionless Matter

Dafermos

Rendall

2016

Comm Pure Appl Math

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This paper addresses strong cosmic censorship for spacetimes with self-gravitating collisionless matter, evolving from surface-symmetric compact initial data. The global dynamics exhibit qualitatively different features according to the sign of the curvature k of the symmetric surfaces and the cosmological constant ƒ. With a suitable formulation, the question of strong cosmic censorship is settled in the affirmative if ƒ D 0 or k Ä 0, ƒ > 0. In the case ƒ > 0, k D 1, we give a detailed geometric characterization of possible "boundary" components of spacetime; the remaining obstruction to showing strong cosmic censorship in this case has to do with the possible formation of extremal Schwarzschild-de Sitter-type black holes. In the special case that the initial symmetric surfaces are all expanding, strong cosmic censorship is shown in the past for all k; ƒ. Finally, our results also lead to a geometric characterization of the future boundary of black hole interiors for the collapse of asymptotically flat data: in particular, in the case of small perturbations of Schwarzschild data, it is shown that these solutions do not exhibit Cauchy horizons emanating from i C with strictly positive limiting area radius.

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The Vlasov Equation and Related Systems

Cercignani

Kremer

2002

The Relativistic Boltzmann Equation: Theory and Applications

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Cosmological solutions of the Vlasov-Einstein system with spherical, plane, and hyperbolic symmetry

Cited by 93 publications

References 12 publications

Describing general cosmological singularities in Iwasawa variables

Describing general cosmological singularities in Iwasawa variables

Strong Cosmic Censorship for Surface‐Symmetric Cosmological Spacetimes with Collisionless Matter

The Vlasov Equation and Related Systems

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