Comoving Einstein-Maxwell equation in the Newman-Penrose formalism

The Einstein-Maxwell equations are considered in a spherically symmetric coordinate system with the space-time filled with charged dust matter with pressure. The study is performed in the spinor formalism of Newman and Penrose by requiring matter and charge conservation. Symmetry conditions imply that F12 and J1, J2 are the only non-zero components of the electromagnetic field and current. The scheme is developed in the special case J1 = J2. If pressure is non-zero, the Bianchi and the Einstein Ricci equations together imply that the entire solution results to be time independent. Its explicit form is then determined that depends on three general functions two of which remain arbitrary. If, on the other hand, the dust matter has zero pressure the scheme is no more static. The solution can then be found by solving a generalized Kepler-like equation. The model is exemplified in some particular situations. The presence of a state equation is studied in general and explicitly solved in a special case. In the Robertson-Walker space-time case, the scheme results to be completely determined.