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Gravito-magnetic instabilities of Relativistic Magnetohydrodynamics

We study gravito-magnetic instabilities of a static homogeneous medium with an aligned magnetic field in the two contexts of relativistic magnetohydrodynamics (MHD): first, MHD with post-Newtonian (PN) corrections, and second, special relativistic (SR) MHD with weak gravity. The analysis in the PN MHD is made without taking the temporal gauge condition, thus results are gauge-invariant. The PN corrections of the internal energy, pressure, sound velocity and the Alfvén velocity lower the critical (Jeans) wavelength. {All relativistic effects tend to destabilize the system.} Although the SR MHD with weak gravity is presented in the harmonic gauge, in the presence of gravity the stability analysis is strictly valid to Newtonian order. In the absence of gravity, the SR MHD is independent of the gauge condition. We present the plane wave velocities and the stability criteria in both cases.