Force-free electrodynamics near rotation axis of a Kerr black hole

Despite their potential importance for understanding astrophysical jets, physically realistic ex- act solutions for magnetospheres around Kerr black holes have not been found, even in the force-free approximation. Instead approximate analytical solutions such as the Blandford-Znajek (split-)monopole, as well as numerical solutions, have been constructed. In this paper we consider a new approach to the analysis and construction of such magnetospheres. We consider force-free elec- trodynamics close to the rotation axis of a magnetosphere surrounding a Kerr black hole assuming axisymmetry. This is the region where the force-free approximation should work the best, and where the jets are located. We perform a systematic study of the asymptotic region with (split-)monopole, paraboloidal and vertical asymptotic behaviors. Imposing asymptotics similar to a (split-)monopole, we find under certain assumptions that demanding regularity at the rotation axis and the event hori- zon restricts solutions of the stream equation so much that it is not possible for a solution to be continuously connected to the static (split-)monopole around the Schwarzschild black hole in the limit where the rotation goes to zero. On the one hand, this result provides independent evidence to the issues discovered with the asymptotics of the Blandford-Znajek (split-)monopole in Ref. [1]. On the other hand, we also point out possible caveats in our arguments that one could conceivably exploit to amend the perturbative construction of the Blandford-Znajek (split-)monopole.