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