  The Blandford and Znajek (BZ) split-monopole serves as an important
theoretical example of the mechanism that can drive the electromagnetic
extraction of energy from Kerr black holes. It is constructed as a perturbative
low spin solution of Force Free Electrodynamics (FFE). Recently, Armas $et~al.$
put this construction on a firmer footing by clearing up issues with apparent
divergent asymptotics. This was accomplished by resolving the behavior around
the outer light surface, a critical surface of the FFE equations. Building on
this, we revisit the BZ perturbative expansion, and extend the perturbative
approach to higher orders in the spin parameter of the Kerr black hole. We
employ matched-asymptotic-expansions and semi-analytic techniques to extend the
split-monopole solution to the sixth-order in perturbation theory. The
expansion necessarily includes novel logarithmic contributions in the spin
parameter. We show that these higher order terms result in non-analytic
contributions to the power and angular momentum output. In particular, we
compute for the first time the perturbative contributions to the energy
extraction at seventh- and eighth-order in the spin parameter. The resulting
formula for the energy extraction improves the agreement with numerical
simulations at finite spin. Moreover, we present a novel numerical procedure
for resolving the FFE equations across the outer light surface, resulting in
significantly faster convergence and greater accuracy, and extend this to
higher orders as well. Finally, we include a general discussion of light
surfaces as critical surfaces of the FFE equations.
