Generalized Ohm's law for relativistic plasmas

TL;DR

This paper derives a fully relativistic, nonlinear form of Ohm's law for plasmas using a covariant multi-fluid approach, extending previous non-relativistic and cold plasma models to include relativistic corrections and effects.

Contribution

It provides the first fully relativistic, nonlinear propagation equation for the electric current in multi-fluid plasmas, unifying and extending prior models with a covariant formalism.

Findings

Recover the classical MHD Ohm's law in the non-relativistic limit.

Identify relativistic corrections to Ohm's law, including the Biermann battery and Hall effects.

Present a covariant framework applicable to hot and cold relativistic plasmas.

Abstract

We generalise the relativistic expression of Ohm's law by studying a multi-fluid system of charged species using the 1+3 covariant formulation of general relativistic electrodynamics. This is done by providing a fully relativistic, fully nonlinear 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 generalised expression of Ohm's law is initially given relative to an arbitrary observer and for a multi-component 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.