Slowly rotating black holes in nonlinear electrodynamics

We show how (at least in principle) one can construct electrically and magnetically charged slowly rotating black hole solutions coupled to non-linear electrodynamics (NLE). Our generalized Lense-Thirring ansatz is, apart from the static metric function \(f\) and the electrostatic potential \(\phi\) inherited from the corresponding spherical solution, characterized by two new functions \(h\) (in the metric) and \(\omega\) (in the vector potential) encoding the effect of rotation. In the linear Maxwell case, the rotating solutions are completely characterized by static solution, featuring \(h=(f-1)/r^2\) and \(\omega=1\). We show that when the first is imposed, the ansatz is inconsistent with any restricted (see below) NLE but the Maxwell electrodynamics. In particular, this implies that the (standard) Newman-Janis algorithm cannot be used to generate rotating solutions for any restricted non-trivial NLE. We present a few explicit examples of slowly rotating solutions in particular models of NLE, as well as briefly discuss the NLE charged Taub-NUT spacetimes.