A numerical investigation of the steady states of the spherically symmetric Einstein-Vlasov-Maxwell system

We construct, by numerical means, static solutions of the spherically symmetric Einstein-Vlasov-Maxwell system and investigate various features of the solutions. This extends a previous investigation \cite{AR1} of the chargeless case. We study the possible shapes of the energy density profile as a function of the area radius when the electric charge of an individual particle is varied as a parameter. We find profiles which are multi-peaked, where the peaks are separated either by vacuum or a thin atmosphere, and we find that for a sufficiently large charge parameter there are no physically meaningful solutions. Furthermore, we investigate if the inequality \sqrt{M}\leq \frac{\sqrt{R}}{3}+\sqrt{\frac{R}{9}+\frac{Q^2}{3R}}, derived in \cite{An2}, is sharp within the class of solutions to the Einstein-Vlasov-Maxwell system. Here M is the ADM mass, Q the charge, and R the area radius of the boundary of the static object. We find two classes of solutions with this property, while there is only one in the chargeless case. In particular we find numerical evidence for the existence of arbitrarily thin shell solutions to the Einstein-Vlasov-Maxwell system. Finally, we consider one parameter families of steady states, and we find spirals in the mass-radius diagram for all examples of the microscopic equation of state which we consider.