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

Fully relativistic and causal equations for the flow of charge in curvedspacetime are derived. It is believed that this is the first set of equationsto be published that correctly describes the flow of charge, and evolution ofthe electromagnetic field, in highly dynamical relativistic environments ontime scales much shorter than the collapse time ($GM/c^3$). The equations willbe important for correctly investigating problems such as the dynamicalcollapse of magnetized stellar cores to black holes and the production of jetsand gravitational waves. This system of equations, given the name of `chargedynamics', is analogous to those of hydrodynamics (which describe the flow of{\em mass} in spacetime rather than the flow of charge). The most important onein the system is the relativistic generalized Ohm's law, which is used tocompute time-dependent four-current. Unlike previous equations for the current,this one ensures that charge drift velocities remain less than the speed oflight, takes into account the finite current rise time, is expressed in acovariant form, and is suitable for general relativistic computations in anarbitrary metric. It includes the standard known effects (Lorentz force, Halleffect, pressure effect, and resistivity) and reduces to known forms of Ohm'slaw in the appropriate limits. In addition, the plasma particles are allowed tohave highly relativistic drift velocities, resulting in an implicit equationfor the `current beaming factor' $\gamma_q$.Comment: 23 pages, 0 figures; accepted for publication in the Astrophysical Journa