Dirac solitons in general relativity and conformal gravity

Abstract

Static, spherically-symmetric particle-like solutions to the coupled Einstein-Dirac and Einstein-Dirac-Maxwell equations have been studied by Finster, Smoller and Yau (FSY). In their work, FSY left the fermion mass as a parameter set to ±1. This thesis generalises these equations to include the Higgs field, letting the fermion mass become a function through coupling, μ. We discuss the dynamics associated with the Higgs field and find that there exist qualitatively similar solutions to those found by FSY, with well behaved, non-divergent metric components and electrostatic potential, close to the origin, going over to the point-particle solutions for large r; the Schwarzschild or Reissner-Nordström metric, and the Coulomb potential. We then go on to discuss an alternative gravity theory, conformal gravity, (CG), and look for solutions of the CG equations of motion coupled to the Dirac, Higgs and Maxwell equations. We obtain asymptotically nonvanishing, yet fully normalisable Dirac spinor components, resembling those of FSY, and, in the case where charge is included, non-divergent electrostatic potential close to the origin, matching onto the Coulomb potential for large r.