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, Smoller and Yau \cite{FSY1} in the case of two fermions. In 2020 this result was generalized by Leith, Hooley, Horne and Dritschel \cite{LHHD} to a system consisting of an even number $\kappa$ of fermions. They constructed solutions for $2\leq\kappa\leq 90$. 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 $2\leq\kappa \leq 180$. 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 $\sup 2m/r$ of the solutions. The solutions of both systems share the fundamental properties regarding the maximum compactness ratio and obey the inequality derived in \cite{A2}. Furthermore, we investigate the sign of the pressure components of solutions of the Einstein-Dirac system. For small values of $\kappa$, there are regions where the radial pressure is negative. These regions disappear as $\kappa$ increases. This supports the interpretation we make as a transition from quantum to classical behaviour as the number of fermions increases.