Extended monopole solution of the Blandford-Znajek mechanism: Higher order terms for a Kerr parameter

The Blandford-Znajek mechanism, by which the rotational energy of a black hole is extracted through electromagnetic fields, is one of the promising candidates as an essential process of the central engine of active compact objects such as gamma-ray bursts. The only known analytical solution of this mechanism is the perturbative monopole solution for Kerr parameter a up to the second order terms. In order to apply the Blandford-Znajek mechanism to rapidly rotating black holes, we try to obtain the perturbation solution up to the fourth order. As a result, we find that the fourth order terms of the vector potential diverge at infinity, and this implies that the perturbation approach breaks down at a large distance from the black hole. Although there are some uncertainties about the solution due to the unknown boundary condition at infinity for the fourth order terms, we can derive the evaluation of the total energy flux extracted from the black hole up to fourth order of a without any ambiguity. Furthermore, from the comparison between the numerical solution, which is valid for 0<a<1, and the fourth order solution, we find that the fourth order solution reproduces the numerical result better than the second order solution. At the same time, since the fourth order solution does not match well with the numerical result at large a, we conclude that more higher order terms are required to reproduce the numerical result.