Force-free currents and the Newman-Penrose tetrad of a Kerr black hole: Exact local solutions

In a previous article we derived a class of solutions to the force-free magnetosphere in a Kerr background. Here, the streaming surface, defined by constant values of the toriodal component of the electromagnetic vector potential $A$, was generated by constant values of $\theta$. The electromagnetic current vector flowed along the infalling principle null geodesic vector of the geometry. Subsequently, we generalized this to obtain an outgoing principle null geodesic vector as well. In this article, we derive solutions that are complimentary to the above-mentioned criteria. Namely, here the solution has a streaming surface generated by spheres of constant radial coordinate $r$, and the current vector is generated by linear combinations of $m$ and $m^{\star }$, the remaining bases vectors in the Newman-Penrose null tetrad.