Jet Formation in the magnetospheres of supermassive black holes: analytic solutions describing energy loss through Blandford-Znajek processes

In this paper, we provide exact solutions for the extraction of energy from a rotating black hole via both the electromagnetic Poynting flux and matter currents. By appropriate choice of a radially independent poloidal function $\mathrm{\Lambda }(\theta )$, we find solutions where the dominant outward energy flux is along the polar axis, consistent with a jet-like collimated outflow, but also with a weaker flux of energy along the equatorial plane. Unlike all the previously obtained solutions (Blandford & Znajek (1977), Menon & Dermer (2005)), the magnetosphere is free of magnetic monopoles everywhere.