On the steady states of the spherically symmetric Einstein–Vlasov system

AIP Conference Proceedings

2014

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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.

Section: Buchdahl-type Inequalitiesmentioning

confidence: 82%

Section: Existence Of Spherically-symmetric Static Solutionsmentioning

confidence: 99%

Section: Existence Of Spherically-symmetric Static Solutionsmentioning

confidence: 99%

Section: Existence Of Spherically-symmetric Static Solutionsmentioning

confidence: 99%

Section: Buchdahl-type Inequalitiesmentioning

confidence: 99%

See 3 more Smart Citations

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

AIP Conference Proceedings

2014

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Black Hole Formation from a Complete Regular Past for Collisionless Matter

2012

Ann. Henri Poincaré

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Initial data for the spherically symmetric Einstein-Vlasov system is constructed whose past evolution is regular and whose future evolution contains a black hole. This is the first example of initial data with these properties for the Einstein-matter system with a "realistic" matter model. One consequence of the result is that there exists a class of initial data for which the ratio of the Hawking mass • m = • m(r) and the area radius r is arbitrarily small everywhere, such that a black hole forms in the evolution. This result is in a sense analogous to the result [15] for a scalar field. Another consequence is that there exist black hole initial data such that the solutions exist for all Schwarzschild time t ∈ (−∞, ∞).

Models for Self-Gravitating Photon Shells and Geons

Fajman

Thaller

2016

Ann. Henri Poincaré

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Abstract. We prove existence of spherically symmetric, static, selfgravitating photon shells as solutions to the massless Einstein-Vlasov system. The solutions are highly relativistic in the sense that the ratio 2m(r)/r is close to 8/9, where m(r) is the Hawking mass and r is the area radius. In 1955 Wheeler constructed, by numerical means, so-called idealized spherically symmetric geons, i.e., solutions of the Einstein-Maxwell equations for which the energy momentum tensor is spherically symmetric on a time average. The structure of these solutions is such that the electromagnetic field is confined to a thin shell for which the ratio 2m/r is close to 8/9, i.e., the solutions are highly relativistic photon shells. The solutions presented in this work provide an alternative model for photon shells or idealized spherically symmetric geons.