A Variational Approach to Resistive General Relativistic Two-Temperature Plasmas

Abstract

We develop an action principle to construct the field equations for dissipative/resistive general relativistic two-temperature plasmas, including a neutrally charged component. The total action is a combination of four pieces: an action for a multifluid/plasma system with dissipation/resistivity and entrainment; the Maxwell action for the electromagnetic field; the Coulomb action with a minimal coupling of the four-potential to the charged fluxes; and the Einstein–Hilbert action for gravity (with the metric being minimally coupled to the other action pieces). We use a pull-back formalism from spacetime to abstract matter spaces to build unconstrained variations for the neutral, positively, and negatively charged fluid species and for three associated entropy flows. The full suite of field equations is recast in the so-called “3+1” form (suitable for numerical simulations), where spacetime is broken up into a foliation of spacelike hypersurfaces and a prescribed “flow-of-time”. A previously constructed phenomenological model for the resistivity is updated to include the modified heat flow and the presence of a neutrally charged species. We impose baryon number and charge conservation as well as the Second Law of Thermodynamics in order to constrain the number of free parameters in the resistivity. Finally, we take the Newtonian limit of the “3+1” form of the field equations, which can be compared to existing non-relativistic formulations. Applications include main sequence stars, neutron star interiors, accretion disks, and the early universe.