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

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 $O(a^2)$, where $a$ is the spin parameter of a Kerr black hole. In this paper, we extend the monopole solution to higher order $\sim O(a^4)$. We carefully investigate the structure of the BH magnetosphere, including the angular velocity of magnetic field lines $\mathrm{\Omega }$, the toroidal magnetic field $B^{\varphi }$ as well as the poloidal electric current $I$. In addition, the relevant energy extraction rate $\dot{E}$ and the stability of this high-order monopole perturbation solution are also examined.