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.

Les Annales de l'institut Fourier sont membres du Centre Mersenne pour l'édition scienti que ouverte www.centre-mersenne.org Ann. Inst. Fourier, Grenoble Article à paraître Mis en ligne le 7 juin 2021. THE CONFORMAL EINSTEIN FIELD EQUATIONS WITH MASSLESS VLASOV MATTER by Jérémie JOUDIOUX, Maximilian THALLER & Juan A. VALIENTE KROON (*)Abstract. -We prove the stability of de Sitter space-time as a solution to the Einstein-Vlasov system with massless particles. The semi-global stability of Minkowski space-time is also addressed. The proof relies on conformal techniques, namely Friedrich's conformal Einstein field equations. We exploit the conformal invariance of the massless Vlasov equation on the cotangent bundle and adapt Kato's local existence theorem for symmetric hyperbolic systems to prove a long enough time of existence for solutions of the evolution system implied by the Vlasov equation and the conformal Einstein field equations.Résumé. -Nous prouvons la stabilité de l'espace-temps de de Sitter, solution du système d'Einstein-Vlasov avec des particules sans masse. Nous considérons également la stabilité semi-globale de l'espace-temps de Minkowski pour le même système. La preuve de la stabilité repose sur l'usage de techniques conformes, et plus précisément les équations de champs conformes d'Einstein introduites par Friedrich. Nous exploitons l'invariance conforme de l'équation de Vlasov sans masse et adaptons le résultat d'existence locale en temps suffisamment long de Kato au système d'Einstein-Vlasov.

We prove the global asymptotic stability of the Minkowski space for the massless Einstein–Vlasov system in wave coordinates. In contrast with previous work on the subject, no compact support assumptions on the initial data of the Vlasov field in space or the momentum variables are required. In fact, the initial decay in v is optimal. The present proof is based on vector field and weighted vector field techniques for Vlasov fields, as developed in previous work of Fajman, Joudioux, and Smulevici, and heavily relies on several structural properties of the massless Vlasov equation, similar to the null and weak null conditions. To deal with the weak decay rate of the metric, we propagate well-chosen hierarchized weighted energy norms which reflect the strong decay properties satisfied by the particle density far from the light cone. A particular analytical difficulty arises at the top order, when we do not have access to improved pointwise decay estimates for certain metric components. This difficulty is resolved using a novel hierarchy in the massless Einstein–Vlasov system, which exploits the propagation of different growth rates for the energy norms of different metric components.

We construct spherically symmetric, static solutions to the Einstein-Vlasov system with non-vanishing cosmological constant Λ. The results are divided as follows. For small Λ > 0 we show existence of globally regular solutions which coincide with the Schwarzschild-deSitter solution in the exterior of the matter sources. For Λ < 0 we show via an energy estimate the existence of globally regular solutions which coincide with the Schwarzschild-Anti-deSitter solution in the exterior vacuum region. We also construct solutions with a Schwarzschild singularity at the center regardless of the sign of Λ. For all solutions considered, the energy density and the pressure components have bounded support. Finally, we point out a straightforward method to obtain a large class of globally non-vacuum spacetimes with topologies R × S 3 and R × S 2 × R which arise from our solutions using the periodicity of the Schwarzschild-deSitter solution. A subclass of these solutions contains black holes of different masses.(2.1) ds 2 = −e 2µ(r) dt 2 + e 2λ(r) dr 2 + r 2 dϑ 2 + r 2 sin 2 (ϑ)dϕ 2 .

The existence of stationary solutions of the Einstein-Vlasov-Maxwell system which are axially symmetric but not spherically symmetric is proven by means of the implicit function theorem on Banach spaces. The proof generalises the methods of [3] where a similar result is obtained for uncharged particles. Among the solutions constructed in this article there are rotating and non-rotating ones. Static solutions exhibit an electric but no magnetic field. In the case of rotating solutions, in addition to the electric field, a purely poloidal magnetic field is induced by the particle current. The existence of toroidal components of the magnetic field turns out to be not possible in this setting.

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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

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