Exact non-equilibrium solutions of the Einstein-Boltzmann equations

Abstract

In this thesis we use the exact solution of the Boltzmann equation, with a relaxation-time model of collisions, to find solutions of the Einstein-Boltzmann system of equations. A covariant harmonic decomposition of the distribution function is used to obtain exact results. The conditions imposed by the conservation of particle number and energy-momentum, and by the H-theorem are determined. The properties of exact truncated Boltzmann solutions with first and second order anisotropies are investigated. Exact entropy results are obtained for the solution with first order anisotropy, and the solution with second order anisotropy is shown to obey exact thermodynamics laws. The Einstein-Boltzmann equations with relaxation-time model of collisions are solved in FRW and Bianchi I spacetime. In FRW spacetime, a general anisotropic solution and an isotropic solution are obtained. The non-equilibrium anisotropic solution with arbitrary isotropic relaxation function has vanishing particle flux and an equilibrium energy-momentum tensor. Specific forms of the relaxation function permit tilted solutions and solutions with non-zero bulk viscosity. Exact entropy results are derived for the isotropic solution showing that the H-theorem is satisfied. The non-equilibrium isotropic solution has vanishing non-equilibrium pressures and fluxes. The FRW and Bianchi I solutions are used to demonstrate the generation of anisotropy in FRW cosmologies. A relaxation length model of collisions is introduced. This model is used to obtain solutions of the Einstein-Boltzmann equations in static spherically symmetric spacetime. In this static model, anisotropic pressure comes from the bulk viscosity.