Non-equilibrium statistical mechanics in the general theory of relativity III. Collisional stellar dynamics

Abstract

This paper assembles and extends earlier results to formulate a coherent theory of relativistic stellar dynamics appropriate for comparatively small systems of stars in which relativistic effects can be important. The structure of the Newtonian theory is outlined, culminating in the “collisional Boltzmann” or Fokker-Planck equation appropriate for an unconfined system of point masses. The theory of relativistic Fokker-Planck equations is then developed for general Lorentz-covariant interactions such as electromagnetism or scalar fields. The basic physical ingredients of Newtonian stellar dynamics are identified, and it is indicated how they can be reformulated relativistically. These considerations are then used to construct a relativistic Fokker-Planck equation appropriate for the evolution of a collection of point mass stars. The analysis is then generalized to allow, both Newtonianly and relativistically, for the effects of direct physical collisions between stars of finite size. By way of conclusion and illustration, the theory is applied to the study of a prototypical dense galactic nucleus which could evolve to contain a massive black hole. The paper ends by enumerating a number of tractable unsolved problems deserving of further consideration.