Black hole jet power from impedance matching

Black hole jet power depends on the angular velocity of magnetic field lines, $\Omega_F$. Force-free black hole magnetospheres typically have $\Omega_F/\Omega_H \approx 0.5$, where $\Omega_H$ is the angular velocity of the horizon. We give a streamlined proof of this result using an extension of the classical black hole membrane paradigm. The proof is based on an impedance-matching argument between membranes at the horizon and infinity. Then we consider a general relativistic magnetohydrodynamic simulation of an accreting, spinning black hole and jet. We find that the theory correctly describes the simulation in the jet region. However, the field lines threading the horizon near the equator have much smaller $\Omega_F/\Omega_H$ because the force-free approximation breaks down in the accretion flow.