We present the equations of relativistic hydrodynamics coupled to dynamical
electromagnetic fields, including the effects of polarization, electric fields,
and the derivative expansion. We enumerate the transport coefficients at
leading order in derivatives, including electrical conductivities, viscosities,
and thermodynamic coefficients. We find the constraints on transport
coefficients due to the positivity of entropy production, and derive the
corresponding Kubo formulas. For the neutral state in a magnetic field, small
fluctuations include Alfven waves, magnetosonic waves, and the dissipative
modes. For the state with a non-zero dynamical charge density in a magnetic
field, plasma oscillations gap out all propagating modes, except for
Alfven-like waves with a quadratic dispersion relation. We relate the transport
coefficients in the "conventional" magnetohydrodynamics (formulated using
Maxwell's equations in matter) to those in the "dual" version of
magnetohydrodynamics (formulated using the conserved magnetic flux).Comment: 42 page
