Unified understanding of Alfvénic superradiance and the Blandford-Znajek process in a force-free black hole magnetosphere

We investigate the extraction of energy from a Kerr black hole via Alfv\'en waves (i.e., Alfv\'enic superradiance) in a force-free magnetosphere, in which the plasma inertia effects are ignored. We analyze the Poynting flux generated by Alfv\'en waves that propagate toward the event horizon across the inner light surface, the causal boundary for the waves. We find the relationship between the energy flux associated with Alfv\'en waves and that of the Blandford-Znajek (BZ) process. That is, both mechanisms can be described within a unified formulation of the Poynting flux, where the BZ process can be regarded as the long wavelength limit of the Alfv\'enic superradiance, and depending on the wave's frequency, Alfv\'en waves enhance or suppress the Poynting flux in the BZ process. This unified framework for the BZ process and the Alfv\'enic superradiance would offer a valuable perspective for understanding the energy sources of high-energy astrophysical phenomena, such as relativistic jets.