Collision-free gases in spatially homogeneous space-times

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Abstract
The kinematical and dynamical properties of one‐component collision‐free gases in spatially homogeneous, locally rotationally symmetric (LRS) space‐times are analyzed. Following Ray and Zimmerman [Nuovo Cimento B 4 2, 183 (1977)], it is assumed that the distribution function f of the gas inherits the symmetry of space‐time, in order to construct solutions of Liouville’s equation. The redundancy of their further assumption that f be based on Killing vector constants of the motion is shown. The Ray and Zimmerman results for Kantowski–Sachs space‐time are extended to all spatially homogeneous LRS space‐times. It is shown that in all these space‐times the kinematic average four‐velocity ui can be tilted relative to the homogeneous hypersurfaces. This differs from the perfect fluid case, in which only one space‐time admits tilted ui, as shown by King and Ellis [Commun. Math. Phys. 3 1, 209 (1973)]. As a consequence, it is shown that all space‐times admit nonzero acceleration and heat flow, while a subclass admits nonzero vorticity. The stress πij is proportional to the shear σij by virtue of the invariance of the distribution function. The evolution of tilt and the existence of perfect fluid solutions is also discussed.