Current driving mechanism and role of the negative energies in the Blandford-Znajek process

Blandford-Znajek process, the steady electromagnetic energy extraction from rotating black holes, is widely believed to work for driving relativistic jets, although it is still under debate how the Poynting flux is causally produced. In the membrane paradigm, assuming the unipolar induction (that works for pulsar winds), causality appears broken and the current driving mechanism is unclear.

We find that analogies to the mechanical Penrose process are not applicable either. Some researcher argues that a steady non-MHD gap is required to drive the electric current making the toroidal magnetic field. If this is the case, such a gap may produce characteristic non-thermal particles, which could be relevant for the upcoming high-resolution radio observations of M87 jet. We build an analytical toy model of a time-dependent process towards the steady state both in the
Boyer-Lindquist and Kerr-Schild coordinate system, and propose a causal mechanism of Poynting flux production. Then we show that the non-MHD gap is not necessary in the steady state.