A Toy Model for Blandford-Znajek Mechanism

A toy model for the Blandford-Znajek mechanism is investigated: a Kerr black hole with a toroidal electric current residing in a thin disk around the black hole in the equatorial plane. The toroidal electric current generates a poloidal magnetic field threading the black hole and disk. The rotation of the black hole and disk relative to the magnetic field induces an electromotive force, which can power an astrophysical load at remote distance. The power of the black hole and disk is calculated. It is found that, for a wide range of parameters specifying the rotation of the black hole and the distribution of the electric current in the disk, the power of the disk dominates the power of the black hole. The torque provided by the black hole and disk is also calculated. The torque of the disk is comparable to the torque of the black hole. As the disk loses its angular momentum, the mass of the disk gradually drifts towards the black hole and gets accreted. Ultimately the power comes from the gravitational binding energy between the disk and the black hole, as in the standard theory of accretion disk, instead of the rotational energy of the black hole. This suggests that the Blandford-Znajek mechanism may not be effective in extracting energy from a rotating black hole with a disk. While this work is not a formal general proof of energetic non-effectiveness of the Blandford-Znajek mechanism, any serious suggestion to the contrary should be supported with a specific quantitative counter-example, demonstrating that there is a model for which the mechanism is effective. Subject headings: black hole physics – galaxies: jets – gamma-rays: bursts