You have charged dust (pressure = 0, charge density/mass density = given constant). I suppose the total energy-momentum tensor of that system (including the rest energy and the EM field) cannot be expressed simply in terms of the arbitrary 4-velocity of the dust like for example the case of ideal fluid. That's why, lets specialize to the case of charged dust that is homogeneous and isotropic, basically charged dust that expands in FRW universe. What is the total energy momentum tensor of that system either as an abstract geometric formula involving the 4-velocity of the dust or by components in the comoving coordinates? I can't find a paper that discusses this case.