A Class of Exact Solution to the Blandford-Znajek Process

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, we conclude that poloidal surfaces are independent of the radial coordinate for large values of $r$. Using this fact 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.