Existence of Static Solutions of the Einstein--Vlasov--Maxwell System and the Thin Shell Limit

Abstract

In this article the static Einstein-Vlasov-Maxwell system is considered in spherical symmetry. This system describes an ensemble of charged particles interacting by general relativistic gravity and Coulomb forces. First, a proof for local existence of solutions around the center of symmetry is given. Then, by virtue of a perturbation argument, global existence is established for small particle charges. The method of proof yields solutions with matter quantities of bounded support -among other classes, shells of charged Vlasov matter. As a further result, the limit of infinitesimally thin shells as solution of the Einstein-Vlasov-Maxwell system is proven to exist for arbitrary values of the particle charge parameter. In this limit the inequality (1.2) obtained by Andréasson in [3] becomes sharp. However, in this limit the charge terms in the inequality are shown to tend to zero. The Einstein-Vlasov-Maxwell systemIn this section we state the EVM-system in a spherically symmetric, static setting and introduce the relevant objects. The intention is mostly to fix notation, for a detailed derivation of the equations, see e.g. [24].Let M be a four dimensional manifold equipped with the Schwarzschild coordinates t ∈ R, r ∈ [0, ∞), ϑ ∈ [0, π], ϕ ∈ [0, 2π) and the Lorentzian metric g of signature

Static Spherically Symmetric Einstein-Vlasov Bifurcations of the Schwarzschild Spacetime

“…Similar arguments were adapted by Andréasson-Fajman-Thaller in [2] to prove the existence of static spherically symmetric solutions of the Einstein-Vlasov-Maxwell system with non-vanishing cosmological constant. Recently, Thaller has constructed in [37] static and axially symmetric shells of the Einstein-Vlasov-Maxwell system.…”

A numerical stability analysis for the Einstein–Vlasov system

“…We should mention that steady state configurations have also been studied for the situation where the particles do not only carry mass, but also charge, which results in the Einstein-Vlasov-Maxwell system, cf [3,4,36] and the references there. The stability of such charged steady states is an open problem.…”

Rotating clouds of charged Vlasov matter in general relativity

“…It is however conjectured that these solutions need to be highly relativistic. For charged particles the same variety of different static, spherically symmetric solutions can be constructed, at least for sufficiently small particle charges [5,37]. In contrast to the Vlasov-Poisson system, where rigorous existence of axisymmetric solutions which are not necessarily close to spherically symmetric are known [19], there is only little analytical understanding of the stationary solutions of the Einstein-Vlasov system beyond spherical symmetry.…”