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To study phenomena of plasmas around rotating black holes, we have derived aset of 3+1 formalism of generalized general relativistic magnetohydrodynamic(GRMHD) equations. Especially, we investigated general relativistic phenomenawith respect to the Ohm's law. We confirmed the electromotive force due to thegravitation, centrifugal force, and frame-dragging effect in plasmas near theblack holes. These effects are significant only in the local small-scalephenomena compared to the scale of astrophysical objects. We discuss thepossibility of magnetic reconnection, which is triggered by one of theseeffects in a small-scale region and influences the plasmas globally. We clarifythe conditions of applicability of the generalized GRMHD, standard resistiveGRMHD, and ideal GRMHD for plasmas in black hole magnetospheres.

Generalized General Relativistic MHD Equations and Distinctive Plasma Dynamics around Rotating Black Holes