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 which has been obtained by Andréasson in [Comm. Math. Phys., 288 (2009), pp. 715--730], and which bounds the mass-to-radius ratio by a constant and the charge-to-radius ratio, becomes sharp. However, in this limit the charge terms in the inequality are shown to tend to zero.