We derive a relativistic magnetohydrodynamics (RMHD) theory for a dilute electron-ion gas governed by the Boltzmann-Vlasov equation, using the method of moments. This yields an extended MHD framework beyond standard astrophysical formulations, which typically include only the shear-stress component parallel to the magnetic field. We analyze our framework in the linear regime and show that it leads to the firehose instability when the background longitudinal pressure becomes large. In these extreme scenario, the transverse and semi-transverse shear-stress components become large and may play a role in accretion disk dynamics.
Kushwah et al. (Fri,) studied this question.