On the stability of a new relativistic kinetic equation

Abstract

Recently, Israel and Kandrup have formulated a new relativistic kinetic equation appropriate for a self-gravitating system. That equation allos for the combined influences of (i) collective mean fields, which define an ''average'' spacetime geometry, and (ii) ''fluctuations'' from mean field conditions, which are modeled in a Landau-type approximation. In the limit that the fluctuations are compleely ignored, one recovers the mean field theory discussed by such authors as Ipser and Thorne. It is shown here that this n ew kinetic equation admits the relativistic isothermal distribution as an exact stationary solution. Linearized perturbation equations are then derived for small spherically symmetric departures from the isothermal ''equilibrium.'' Attention next focuses upon an ''obit-averaged'' version of the kinetic equation which eliminates from explicit consideration the collective mean field effects. One finds, in analogy with the Newtonian theory considered by Ipser and Kandrup, that the isothermal solution to this averaged equation will be stable with respect to linearized spherically symmetric perturbtions if and only if the mean field Boltzmann entropy, which is extremized by the isothermal distribution, is in fact a local maximum. Finally, this result is shown to be a simple manifestation of a more general H-theorem inequality.