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The Blandford and Znajek (BZ) split-monopole serves as an importanttheoretical example of the mechanism that can drive the electromagneticextraction of energy from Kerr black holes. It is constructed as a perturbativelow spin solution of Force Free Electrodynamics (FFE). Recently, Armas $et~al.$put this construction on a firmer footing by clearing up issues with apparentdivergent asymptotics. This was accomplished by resolving the behavior aroundthe outer light surface, a critical surface of the FFE equations. Building onthis, we revisit the BZ perturbative expansion, and extend the perturbativeapproach to higher orders in the spin parameter of the Kerr black hole. Weemploy matched-asymptotic-expansions and semi-analytic techniques to extend thesplit-monopole solution to the sixth-order in perturbation theory. Theexpansion necessarily includes novel logarithmic contributions in the spinparameter. We show that these higher order terms result in non-analyticcontributions to the power and angular momentum output. In particular, wecompute for the first time the perturbative contributions to the energyextraction at seventh- and eighth-order in the spin parameter. The resultingformula for the energy extraction improves the agreement with numericalsimulations at finite spin. Moreover, we present a novel numerical procedurefor resolving the FFE equations across the outer light surface, resulting insignificantly faster convergence and greater accuracy, and extend this tohigher orders as well. Finally, we include a general discussion of lightsurfaces as critical surfaces of the FFE equations.

Blandford--Znajek monopole expansion revisited: novel non-analytic contributions to the power emission