Alfvénic superradiance for a monopole magnetosphere around a Kerr black hole

We explore superradiance for Alfvén waves (Alfvénic superradiance) in an axisymmetric rotating magnetosphere of a Kerr black hole within the force-free approximation. On the equatorial plane of the Kerr spacetime, the Alfvén wave equation is reduced to a one-dimensional Schr\"{o}dinger-type equation by separating variables of the wave function and introducing a tortoise coordinate mapping the inner and outer light surfaces to \(-\infty\) and \(+\infty\), respectively, and we investigate a wave scattering problem for Alfvén waves. An analysis of the asymptotic solutions of the wave equation and the conservation of the Wronskian give the superradiant condition for Alfvén waves, and it will be shown that the condition coincides with that for the Blandford-Znajek process. This means that when Alfvénic superradiance occurs, the Blandford-Znajek process also occurs in the force-free magnetosphere. Then, we evaluate the reflection rate of Alfvén waves numerically and confirm that Alfvénic superradiance is indeed possible in the Kerr spacetime. Moreover, we will discuss a resonant scattering of Alfvén waves, which is related to a "quasinormal mode" of the magnetosphere.