Collision-free gases in static space-times

Collision-free gases in static space-times are analyzed by developing previous work in static spherically symmetric space-times and extending the analysis to include the cases of planar and hyperbolic symmetry. By assuming that the distribution function of the gas inherits the space-time symmetries, distribution solutions to the Einstein-Liouville equations, which are without expansion, rotation, shear, and heat flow, but which have an anisotropic stress are found. The conditions for the gas to behave like a perfect fluid are considered and the relation between equations of state and the distribution function are investigated. In particular, distribution functions that generate the γ-law equation of state are found. The solutions are extended to find invariant Einstein-Maxwell-Liouville solutions for a charged gas, subject to a consistency condition on the invariant electromagnetic potential. Finally, the general solution of Liouville's equation in the static space-times is obtained and a particular nonstatic solution is considered, which can be shown to lead to a self-gravitating gas with expansion, shear, and heat flow.