The transition to phenomenological behaviour of static solutions of the Einstein–Dirac system for an increasing number of fermions

Abstract

Static spherically symmetric solutions to the Einstein–Dirac system were constructed numerically for the first time in 1999 by Finster et al (1999 Phys. Rev. D59 104020) in the case of two fermions. In 2020 this result was generalized by Leith et al (2020 Phys. Rev. D101 106012) to a system consisting of an even number κ of fermions. They constructed solutions for . The purpose of the present investigation is to compare the properties of static solutions of the Einstein–Dirac system with static solutions of the Einstein–Vlasov system as the number of fermions increases, that is, for . Since the Einstein–Vlasov system is a fully classical physical model, whereas the Einstein–Dirac system is semiclassical and thus has a quantum signature, this framework provides an excellent opportunity to study the transition from quantum to classical behaviour. It turns out that even for a comparatively small number of particles, the features of the solutions are remarkably similar. For both systems, we find highly relativistic solutions having a multi-peak structure with strikingly similar characteristics. We also investigate the maximum compactness ratio of the solutions. The solutions of both systems share the fundamental properties regarding the maximum compactness ratio and obey the inequality derived in Andréasson (2008 J. Differ. Equ.245 2243–66). Furthermore, we investigate the sign of the pressure components of solutions of the Einstein–Dirac system. For small values of κ, there are regions where the radial pressure is negative. These regions disappear as κ increases. This supports the interpretation we make as a transition from quantum to classical behaviour as the number of fermions increases.