Static solutions of the Einstein-Dirac system for an increasing number of particles behave as solutions of the Einstein- Vlasov system
Typ
Examensarbete för masterexamen
Master's Thesis
Master's Thesis
Program
Engineering mathematics and computational science (MPENM), MSc
Publicerad
2023
Författare
Blomqvist, Joakim
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
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.
Beskrivning
Ämne/nyckelord
General relativity, relativistic quantum mechanics, phenomenological matter model, field theoretic matter model.