A class of exact solutions to the force-free, axisymetric, stationary magnetosphere of a Kerr black hole

Abstract

We analyze the constraint equation giving allowed solutions describing fields and currents in a force-free magnetosphere around a rotating black hole. Utilizing the divergence properties of the energy and angular-momentum fluxes, for physically allowed solutions with nonzero energy and angular momentum extraction, we conclude that poloidal surfaces are independent of the radial coordinate for large values of r. Imposing this requirement and the Znajek regularity condition, we explicitly derive all possible exact solutions admitted by the constraint equation for r independent poloidal surfaces which are given in terms of the electromagnetic angular velocity function \({\Omega = 1/a \sin^2 \theta}\) , where a is the angular momentum per unit mass of the black hole. Further, we show that for the class of solutions we have developed there is no electromagnetic extraction of energy.