Exact non-equilibrium solutions of the Einstein-Boltzmann equations

We give a review of exact results in non-equilibrium general relativistic kinetic theory, and a comprehensive analysis of the relaxation-time model of collisions. We find the conditions imposed by conservation of particle number and energy--momentum, and by the H-theorem. The exact truncated Boltzmann solution is shown to obey exact thermodynamic laws similar to the approximate laws of Israel and Stewart, and to be subject to the consistency conditions of shear-free flow and a restriction on the anisotropic stress. The Einstein--Boltzmann equations with the relaxation-time model of collisions are solved in FRW spacetime. The solution has vanishing bulk viscous and anisotropic pressures and zero energy and particle fluxes, but is nevertheless a non-equilibrium solution provided m=0.