Electromotive force in the Blandford–Znajek process

One of the mechanisms widely considered for driving relativistic jets in active galactic nuclei, Galactic microquasars and gamma-ray bursts is the electromagnetic extraction of the rotational energy of a central black hole, i.e. the Blandford–Znajek process, although the origin of the electromotive force in this process is still under debate. We study this process as the steady unipolar induction in the Kerr black hole magnetosphere filled with a collisionless plasma screening the electric field (the D field) along the magnetic field (the B field), i.e. D · B = 0. We extend the formulations and arguments made by Komissarov, and generally show that the origin of the electromotive force is ascribed to the ergosphere. It is explicitly shown that open magnetic field lines penetrating the ergosphere have a region where the D field is stronger than the B field in the ergosphere, and it keeps driving the poloidal currents and generating the electromotive force and the outward Poynting flux. The range of the possible value of the so-called angular velocity of the magnetic field line Ω_F is deduced for the field lines threading the equatorial plane in the ergosphere. We briefly discuss the relation between our conclusion and the ideal magnetohydrodynamic condition.