Analytic properties of force-free jets in the Kerr spacetime- I

Blandford-Znajek (BZ) mechanism describes a process extracting rotation energy from a spinning black hole (BH) via magnetic field lines penetrating the event horizon of central BH. In this paper, we present a perturbation approach to study force-free jets launched by the BZ mechanism, and its two immediate applications: (1) we present a high-order split monopole perturbation solution to the BZ mechanism, which accurately pins down the energy extraction rate $\dot E$ and well describes the structure of BH magnetosphere for all range of BH spins ($0\leq a\leq 1$); (2) the approach yields an exact constraint for the monopole field configuration in the Kerr spacetime, $I = \Omega (1-A_\phi^2)$, where $A_\phi$ is the $\phi-$component of the vector potential of electromagnetic field, $\Omega$ is the angular velocity of magnetic field lines and $I$ is the poloidal electric current. The constraint is of particular importance to benchmark the accuracy of numerical simulations.