Ohm’s Law in the Fast Lane: General Relativistic Charge Dynamics

Abstract

Fully relativistic and causal equations for the flow of charge in curved spacetime are derived. It is believed that this is the first set of equations to be published that correctly describes the flow of charge, as well as the evolution of the electromagnetic field, in highly dynamical relativistic environments on timescales much shorter than the collapse time (GM/c3). The equations will therefore be important for correctly investigating problems such as the dynamical collapse of magnetized stellar cores to black holes and the production of jets. Both are potentially important problems in the study of gamma-ray burst engine models and in predicting the dynamical morphology of the collapse and the character of the gravitational waves generated. This system of equations, given the name of "charge dynamics," is analogous to those of hydrodynamics (which describe the flow of mass in spacetime rather than the flow of charge). The most important equation in the system is the relativistic generalized Ohm's law, which is used to compute time-dependent four-current. Unlike other equations for the current now in use, this one ensures that charge drift velocities remain less than the speed of light, takes into account the finite current rise time, is expressed in a covariant form, and is suitable for general relativistic computations in an arbitrary metric. It includes the standard known effects (Lorentz force, Hall effect, pressure effect, and resistivity) and reduces to known forms of Ohm's law in the appropriate limits. In addition, the plasma particles are allowed to have highly relativistic drift velocities, resulting in an implicit equation for the "current beaming factor" γq. It is proposed that, short of solving the multifluid plasma equations or the relativistic Boltzmann equation itself, these are the most general expressions for relativistic current flow in the one-fluid approximation, and they should be made part of the general set of equations that are solved in extreme black hole accretion and fully general numerical relativistic collapse simulations.