Isotropic Solutions of the Einstein-Liouville Equations

The gravitational field generated by a gas whose one-particle distribution function obeys the Liouville equation is examined under the following assumptions: First, the distribution is locally isotropic in momentum space with respect to some world-velocity field; second, if the particles have rest-mass zero, the gas is irrotational. It is shown that the model is then either stationary or a Robertson-Walker model. The time dependence of the radius in the Robertson-Walker models is given in terms of integrals containing the distribution function.