Macroscopic Einstein-Maxwell equations for a system of interacting   particles to second-order accuracy in the interaction constant
  In this paper the macroscopic Einstein and Maxwell equations for system, in which the electromagnetic interactions are dominating (for instance, the cosmological plasma before the moment of recombination), are derived. Ensemble averaging of the microscopic Einstein - Maxwell equations and the iouville equations for the random functions leads to a closed system of macroscopic Einstein - Maxwell equations and kinetic equations for one-particle distribution functions. The macroscopic Einstein equations for a relativistic plasma differ from the classical Einstein equations in that their left-hand sides contain additional terms due to particle interaction. The terms are traceless tensors with zero divergence. An explicit covariant expression for these terms is given in the form of momentum-space integrals of expressions depending on one-particles distribution functions of the interacting particles of the medium. The macroscopic Maxwell equations alsow differ from the classical macroscopic Maxwell equations in that their left-hand sides contain additional terms due to particle interaction as well the effects of general relativity. 
Alexei V. Zakharov