Fourth-order split monopole perturbation solutions to the Blandford-Znajek mechanism

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The Blandford-Znajek (BZ) mechanism describes a physical process for the energy extraction from a spinning black hole (BH), which is believed to power a great variety of astrophysical sources, such as active galactic nuclei (AGNs) and Gamma ray bursts (GRBs). The only known analytic solution to the BZ mechanism is a split monopole perturbation solution up to O(a²), where a is the spin parameter of a Kerr black hole. In this paper, we extend the monopole solution to higher order O(a⁴). We carefully investigate the structure of the BH magnetosphere, including the angular velocity of magnetic field lines Ω, the toroidal magnetic field Bϕ as well as the poloidal electric current I. In addition, the relevant energy extraction rate E and the stability of this high-order monopole perturbation solution are also examined.