Highly relativistic solutions of the Einstein-Vlasov system and a comparison to solutions of the Einstein-Dirac system

ABSTRACT: I will first introduce the Einstein-Vlasov (EV) system and discuss how to construct spherically symmetric static solutions to this system. I will then review some results on highly compact solutions, including a result in the charged case. These results apply to any matter model satisfying a certain energy condition. In the specific case of the EV system I will also discuss massless solutions and make a comparison to Wheeler's idealized spherically symmetric geons. I will then focus on a recent work where we carry out a comparison between solutions of the Einstein-Dirac (ED) system and the EV system. In 1999 Finster et al. found static solutions to the ED system in the case of two fermions with opposite spins. Recently this study has been extended to a larger number of particles by Leith et al. In particular, they construct highly relativistic solutions. The structure of the solutions is strikingly similar to the structure of highly relativistic solutions of the EV system. Since solutions of the EV system are classical whereas solutions of the ED system have a quantum signature this study provides insights about the transition from quantum to classical behaviour for a small number of particles.