Lux in obscuro: Photon Orbits of Extremal Black Holes Revisited

Abstract: It has been shown in the literature that the event horizon of an extremal asymptotically flat Reissner-Nordstrom black hole is also a stable photon sphere. We further clarify this statement and give a general proof that this holds for a large class of static spherically symmetric black hole spacetimes with an extremal horizon. In contrast, in the Doran frame, an extremal asymptotically flat Kerr black hole has an unstable photon orbit on the equatorial plane of its horizon. In addition, we show that an extremal asymptotically flat Kerr-Newman black hole exhibits two equatorial photon orbits if a < M/2, one of which is on the extremal horizon in the Doran frame and is stable, whereas the second one outside the horizon is unstable. For a > M/2, there is only one equatorial photon orbit, located on the extremal horizon, and it is unstable. There can be no photon orbit on the horizon of a non-extremal Kerr-Newman black hole.