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

Abstract

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.

Stationary solutions of the axially symmetric Einstein–Vlasov system: present status and open problems

“…Since solutions of the Einstein-Vlasov system share many properties of solutions of field theoretical models such as the Einstein-Dirac system, see [12], we find it to be a question of fundamental importance to understand whether or not such a transition is possible. Perhaps there is a maximum value of the compactness parameter Γ which is strictly below one as in the spherically symmetric case.…”