Static solutions of the Einstein-Dirac system for an increasing number of particles behave as solutions of the Einstein- Vlasov system

Abstract

In this thesis we will study static solutions to the spherically symmetric Einstein- Dirac system. This system couples Einstein’s theory of general relativity to Dirac’s relativistic description of quantum mechanics. The goal was to study the transition from a quantum mechanical description to a classical description by comparing properties of the solutions to the Einstein-Dirac system to solutions of the Einstein- Vlasov system as the number of particles of the former system increases. In 1999 Finster et al. [10] found for the first-time static solutions to the Einstein-Dirac 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 [14]. In particular, they construct highly relativistic solutions. The structure of the solutions is strikingly similar to the structure of highly relativistic solutions of the Einstein-Vlasov system. In both cases multi-peak solutions are obtained, and moreover, the maximum compactness of the solutions is very similar. The compactness is measured by the quantity m/r, where m is the mass and r the areal radius, and in both cases the maximum value appears to be 4/9. Furthermore, in quantum mechanics the pressure may be negative whereas classically it is non-negative. We find that already for 16 particles the pressure is non-negative and thus behaves classically. In order to compare the solutions, I need to construct solutions numerically to the Einstein- Dirac system in the case of a large number of particles. This requires a delicate procedure with significant numerical precision when the number of particles in the system grows.