Perturbing Monopolar Force-Free Magnetospheres to Slowly Rotating Black Holes

We study the effects of frame-dragging on the structure of force-free magnetospheres around rotating black holes. For slowly rotating black holes, analytic explorations often focus on transforming an exact force-free solution applicable to a static black hole into a solution approximately applicable to a slowly-rotating black hole via perturbations in black hole spin. We show that the single perturbed monopolar solution most commonly arrived at using such techniques is in fact a separatrix between two broad classes of solutions: those with poloidal magnetic fields that bend upwards towards the azimuthal axis and those that bend downwards towards the equatorial plane. This is because frame-dragging affects the toroidal magnetic field distributions, changing the force balance between poloidal magnetic field lines.