Generalized Relativistic Magnetohydrodynamic Equations for Pair and
  Electron-Ion Plasmas
  We derived one-fluid equations based on a relativistic two-fluid
approximation of e$^\pm$ pair plasma and electron-ion plasma to reveal the
specific relativistic nature of their behavior. Assuming simple condition on
the relativistic one-fluid equations, we propose generalized relativistic
magnetohydrodynamic (RMHD) equations which satisfy causality. We show the
linear analyses of these equations regarding various plasma waves to show the
validity of the generalized RMHD equations derived here and to reveal the
distinct properties of the pair plasma and electron-ion plasma. The distinct
properties relate to (i) the inertia effect of electric charge, (ii) the
momentum of electric current, (iii) the relativistic Hall effect, (iv) the
thermal electromotive force, and (v) the thermalized energy exchange between
the two fluids. Using the generalized RMHD equations, we also clarify the
condition that we can use standard RMHD equations and that we need the distinct
RMHD equations of pair and electron-ion plasmas. The standard RMHD is available
only when the relative velocity of the two fluids is nonrelativistic, a
difference of the enthalpy densities of the two fluids is much smaller than the
total enthalpy density, and the above distinct properties of the
pair/electron-ion plasma are negligible. We discuss a general relativistic
version of the equations applicable to the pair and electron-ion plasmas in
black hole magnetospheres. We find the effective resistivity due to shear of
frame ragging around a rotating black hole.
