Generally covariant relativistic anisotropic magnetohydrodynamics

We derive generally covariant hydrodynamical equations for a plasma with an anisotropic pressure in the external electromagnetic field. The equations are formulated in terms of the variables defined in the local plasma rest frame, in which the electric field vanishes. Generally covariant generalization for the equation of state is derived, which reduces to the Chew-Goldberger-Low [Proc. R. Soc. London, Ser. A 236, 112 (1956)] form when the plasma temperature is nonrelativistic in the plasma rest frame. Various ultrarelativistic limits are analyzed. The obtained equations are applied to the simplest monopole geometry of the relativistic stellar wind and to the analysis of the linear waves in the limit of geometrical optics.