Magnetohydrodynamics with Post-Newtonian Corrections

Using the fully nonlinear and exact perturbation formulation with magnetohydrodynamics (MHD) in the Minkowski background, we derive first-order, post-Newtonian (1PN) equations without imposing the slicing (temporal gauge) condition. The 1PN-MHD formulation is complementary to our recently presented fully relativistic MHD combined with 0PN gravity available only in the maximal slicing. We present the 1PN-MHD equations in two gauge conditions previously used in the literature and provide gauge transformation relations between different gauges. We derive the PN effects on MHD waves in a static homogeneous medium.