Causal production of the electromagnetic energy flux and role of the negative energies in the Blandford-Znajek process

The Blandford-Znajek process, the steady electromagnetic energy extraction from a rotating black hole (BH), is widely believed to work for driving relativistic jets in active galactic nuclei, gamma-ray bursts, and Galactic microquasars, although it is still under debate how the Poynting flux is causally produced and how the rotational energy of the BH is reduced. We generically discuss the Kerr BH magnetosphere filled with a collisionless plasma screening the electric field along the magnetic field, extending the arguments of Komissarov [S. S. Komissarov, Mon. Not. R. Astron. Soc., 50, 427 (2004)] and our previous [K. Toma and F. Takahara, Mon. Not. R. Astron. Soc., 442, 2855 (2014)] paper, and propose a new picture for resolving the issues. For the magnetic field lines threading the equatorial plane in the ergosphere, we find that the inflow of particles with negative energy as measured in the coordinate basis is generated near that plane as a feedback from the Poynting flux production, which appears to be a similar process to the mechanical Penrose process. For the field lines threading the event horizon, we first show that the concept of the steady inflow of negative electromagnetic energy is not physically essential, partly because the sign of the electromagnetic energy density depends on the coordinates. Then we build an analytical toy model of a time-dependent process in both the Boyer-Lindquist and Kerr-Schild coordinate systems, in which the force-free plasma injected continuously fills a vacuum, and suggest that the structure of the steady outward Poynting flux is causally constructed by the displacement current and the cross-field current at the ingoing boundary between the plasma and the vacuum. In the steady state, the Poynting flux is maintained without any electromagnetic source.