Static solutions of the spherically symmetric Vlasov–Einstein system

Rein, Gerhard

doi:10.1017/s0305004100072303

Cited by 60 publications

(167 citation statements)

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“…The field either falls into the black hole or disperses to infinity. In the case of collisionless matter, this is not the case, as static solutions supported outside the horizon have been constructed by Rein [33]; thus, many interesting questions arise. Some of these are summarized below in Turning to the cases that are "successfully" handled in this paper, although our results indeed prove strong cosmic censorship in its C 2 formulation, they do not resolve all interesting questions about the maximal development.…”

Section: Open Questions and Conjecturesmentioning

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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Section: Open Questions and Conjecturesmentioning

confidence: 99%

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

Dafermos

Rendall

2016

Comm Pure Appl Math

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“…Existence of solutions to this system was first proven in the case of isotropic pressure in [6], and extended to anisotropic pressure in [7]. The main difficulty is to show that the solutions have finite ADM mass and compact support.…”

Section: Existence Of Spherically-symmetric Static Solutionsmentioning

confidence: 99%

“…The argument in these works to obtain a solution of compact support is to perturb a steady state of the Vlasov-Poisson system, which is known to have compact support. Two different types of solutions are constructed, those with a regular centre [6,7], and those with a Schwarzschild singularity in the centre [7].…”

Section: Existence Of Spherically-symmetric Static Solutionsmentioning

confidence: 99%

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On the existence, structure and stability of static and stationary solutions of the Einstein-Vlasov system

Andréasson

2014

AIP Conference Proceedings

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Abstract. The present status on the existence, structure and stability of static and stationary solutions of the Einstein-Vlasov system is reviewed. Under the assumptions that a spherically symmetric static object has isotropic pressure and non-increasing energy density outwards, Buchdahl showed 1959 the bound M/R<4/9, where M is the ADM mass and R the outer radius. Most static solutions of the Einstein-Vlasov system do not satisfy these assumptions. The bound M/R<4/9 nevertheless holds and it is sharp. An analogous bound in the charged case is also given. The important question of stability of spherically symmetric static solutions is presently open but numerical results are available and these are reviewed. A natural question is to go beyond spherical symmetry and consider axially symmetric solutions, and a recent result on the existence of axially symmetric stationary solutions is also discussed.Keywords: Einstein-Vlasov system, Buchdahl inequalities, axially symmetric spacetimes PACS: 04.20-q, 04.40 Dg STATIONARY ASYMPTOTICALLY-FLAT SOLUTIONSEquilibrium states in galactic dynamics can be described as static or stationary solutions of the Einstein-Vlasov system, or of the Vlasov-Poisson system in the Newtonian case. Here we consider the relativistic case and we refer to the review paper [1] for the Newtonian case. First, the spherically-symmetric case for which the structure is well understood is discussed. The question of the stability of the spherically-symmetric static solutions of the Einstein-Vlasov system is basically open, which is in sharp contrast to the situation for the Vlasov-Poisson system. A numerical study on stability is reviewed. At the end the results [2, 31] on axisymmetric static solutions are presented. The equations become much more complicated by going beyond spherical symmetry and only solutions close to spherically symmetric ones have been analyzed. This review is to a large extent an extract from the the section on static and stationary solutions in the review article [3]. Existence of spherically-symmetric static solutionsFor a general introduction to relativistic kinetic theory and the Einstein-Vlasov system I refer to [3] and the notation below follows the one given in [3].Let us assume that spacetime is static and spherically symmetric. Let the metric have the formwhere Plasma Physics and Relativistic Fluids AIP Conf.

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“…T µν = (ρ + p)u µ u ν + pg µν , (2.62) 63) and f is chosen so that u µ is future-pointing and u µ u µ = −1. With this specialization the quantities τ, τ i , τ ij entering in the field equation (2.46,47,48) become 66) where η i = h ij η j .…”

Section: Axial Symmetrymentioning

confidence: 99%