  We investigate the temporal evolution of an axisymmetric magnetosphere around a rapidly rotating, stellar-mass black hole, applying a two-dimensional particle-in-cell simulation scheme. Adopting a homogeneous pair production, and assuming that the mass accretion rate is much less than the Eddington limit, we find that the black hole's rotational energy is preferentially extracted from the middle latitudes, and that this outward energy flux exhibits an enhancement that lasts approximately 160 dynamical time scales. It is demonstrated that the magnetohydrodynamic approximations cannot be justified in such a magnetically-dominated magnetosphere, because the Ohm's law completely breaks down, and because the charge-separated electron-positron plasmas are highly non-neutral. An implication is given regarding the collimation of relativistic jets. 
Two-dimensional Particle-in-Cell simulations of axisymmetric black hole   magnetospheres