Rotating Clouds of Charged Vlasov matter in General Relativity

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 relies on the methods of Andr\ue9asson et al (2014 Commun. Math. Phys. 329, 787–808) 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