Electric Charge on a Black Hole Surrounded by Force-Free Magnetosphere

In the Blandford-Znajek process of extracting rotational energy from a black hole, a strong magnetic field near the horizon plays an essential role. When the magnetic field is strong enough so that the inertia of the plasma around the black hole can be ignored, the force-free condition is a good approximation. We consider the structure of stationary axisymmetric force-free black hole magnetospheres using the "3+1" formalism. The structure can be characterized by three scalar functions - the stream function, the current potential and the angular velocity of the magnetic field lines. We discuss the possible analytic solutions to the equations governing those scalar functions. From one such solution that can represent the electromagnetic field configuration in the space with an accretion disk on the equatorial plane of the black hole, we estimate the amount of the electric charge induced on the stretched horizon of the black hole. This value is compared with the amount of electric charge that may be induced on a typical magnetar.