Collision-free gases in static space-times

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Abstract
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.