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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 $A$, were generated by constant values of $θ$. The electromagnetic current vector flowed along the in-falling principle null geodesic vector of the geometry. Subsequently, we generalized this to obtain an out-going 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.