Generalized Ohm's law for relativistic plasmas

We generalize the relativistic expression of Ohm's law by studying a multifluid system of charged species using the 1 + 3 covariant formulation of general relativistic electrodynamics. This is done by providing a fully relativistic, fully non-linear propagation equation for the spatial component of the electric 4-current. Our analysis proceeds along the lines of the non-relativistic studies and extends previous relativistic work on cold plasmas. Exploiting the compactness and transparency of the covariant formalism, we provide a direct comparison with the standard Newtonian versions of Ohm's law and identify the relativistic corrections in an unambiguous way. The generalized expression of Ohm's law is initially given relative to an arbitrary observer and for a multicomponent relativistic charged medium. Then, the law is written with respect to the Eckart frame and for a hot two-fluid plasma with zero total charge. Finally, we apply our analysis to a cold proton–electron plasma and recover the well-known magnetohydrodynamic expressions. In every step, we discuss the approximations made and identify familiar effects, like the Biermann battery and the Hall effect.