Potential equations for plasmas round a rotating black hole

The generalized Helmholtz equations of relativistic multifluid plasmas can be integrated for axisymmetric equilibria in close analogy to the magnetic flux conservation law in ideal magnetohydrodynamics (J. D. Bekenstein and E. Oron, Phys. Rev. D 18, 1809 (1978)). The results are, for each fluid component, two flux functions and a potential equation for the poloidal stream function. Ampere’s equation for the four-potential A_\nu is reduced to two coupled equations for the time-like and the toroidal component. So we have altogether four potential equations for a two-component plasma; they can be derived from a variational principle.