Vacuum magnetospheres around Kerr black holes with a thin disk

We construct a model for stationary and axisymmetric black hole magnetospheres by solving the vacuum Maxwell equations in Kerr backgrounds. The poloidal magnetic field is generated by a toroidal electric current in a thin disk on the equatorial plane with an inner edge, and the poloidal electric field is induced due to the black hole spin. We assume a current distribution whose direction reverses at a radius on the disk. The magnetospheric structure is divided into the inner region and the outer one due to the current reversal. In the inner region, some magnetic field lines connect the black hole and the disk, and others surround the inner edge of the disk. In the outer region, the field lines connect the disk and infinity. We investigate the black hole spin dependence of the magnetospheric structure. In the magnetosphere around the Kerr black hole, the electric fields are generated by the black hole spin. In our model, the charge distribution is induced on the disk and the event horizon. Except near the black hole, the electric field strength is proportional to the black hole spin, while the direction of the electric field remains almost the same regardless of the spin. On the other hand, the electromagnetic field near the event horizon strongly depends on the spin. In the maximally rotating case, the magnetic field lines are expelled from the event horizon and the electric equipotential surface coincides with the horizon.