This paper studies how to pull rotational energy out of a rotating black hole that has extra structure, or “hair,” in a modified theory of gravity called Horndeski gravity. The authors apply the magnetic Penrose process: a charged particle breaks into two near the black hole, one fragment with negative energy falls in, and the other escapes with more energy than the original. They map the regions where this can happen and calculate how efficient the extraction can be when the hole sits in a uniform magnetic field.

The work combines analytic formulas for charged-particle motion with numerical maps of the black hole’s ergosphere (the region outside the horizon where spacetime is swept around by the hole) and the negative-energy regions (where a falling fragment can carry away energy from the hole). The black hole model has three familiar parameters — mass, spin and a hair parameter h that measures deviation from the usual Kerr black hole. When h = 0 the solution reduces to the standard Kerr case.

At a high level, the Penrose idea relies on frame dragging in the ergosphere: one piece of a split particle can have negative energy relative to infinity, so the other piece can escape with a net energy gain. Adding a magnetic field relaxes the very fast relative-velocity requirement of the original, mechanical Penrose process. The authors note that the magnetic version can reach very large efficiencies, even above 100% for charged particles in modest magnetic fields, as prior literature has shown.

The main findings are about how the hair parameter h and the product q/m × B (the charge-to-mass ratio of the particle times the magnetic field) change the picture. Larger h makes both the ergosphere and the negative-energy regions smaller. For q/m × B ≥ 0 and for a given split radius r_x (the radius where the particle breaks): if r_x > 2 (in their units) the efficiency falls as h grows; if r_x < 2 the efficiency rises with h; if r_x = 2 the efficiency does not depend on h. When q/m × B < 0 the behavior is more complicated: apart from the special r_x = 2 case, how efficiency varies with h depends on the exact values of r_x and q/m × B and can either drop steadily or rise first and then fall.