New conservation laws in general-relativistic magnetohydrodynamics

Abstract

The flow of magnetized plasma is governed by a large number of coupled equations (Maxwell's, Euler's, conservation of energy and of baryon number) so that the solution of a problem in general-relativistic magnetohydrodynamics is very complicated, even if symmetries are present. We present here a number of new conservation laws which make the solution process easier. We obtain the general criteria for a flux conservation law to exist. We apply them to obtain the relativistic versions of the conservation of magnetic flux and of Kelvin's circulation theorem for an unmagnetized fluid, as well as a new flux conservation law for a charged fluid. For stationary and axial symmetry we find conservation laws for each component of the Maxwell tensor; these are valid even if the plasma is nonperfect. For perfect plasma we find magnetic generalizations of the relativistic Bernoulli theorems for an unmagnetized fluid. We also find a new conservation law without previous analog. As an application of our results we show that extraction of rotational energy from a black hole by interaction with a magnetized plasma is not possible in the stationary state. This contradicts previous conclusions based on the approximation of geodesic flow. Finally, still for stationary and axial symmetry, we find the magnetic generalization of Kelvin's circulation theorem. With its help we reduce the problem of solving for the field of flow and for the magnetic field to the solution of two equations: baryon conservation and a Hamilton-Jacobi-type equation. A by-product of our derivations is an explicit formula for the strength of the magnetic field in terms of fluid variables.