Lux in obscuro II: photon orbits of extremal AdS black holes revisited

A large class of spherically symmetric static extremal black hole spacetimes possesses a stable null photon sphere on their horizons. For the extremal Kerr–Newman family, the photon sphere only really coincides with the horizon in the sense clarified by Doran. The condition under which a photon orbit is stable on an asymptotically flat extremal Kerr–Newman black hole horizon has recently been clarified; it is found that a sufficiently large angular momentum destabilizes the photon orbit, whereas an electrical charge tends to stabilize it. We investigated the effect of a negative cosmological constant on this observation, and found the same behavior in the case of extremal asymptotically Kerr–Newman–AdS black holes in $(3+1)$-dimensions. In $(2+1)$-dimensions, in the presence of an electrical charge, the angular momentum never becomes large enough to destabilize the photon orbit. We comment on the instabilities of black hole spacetimes with a stable photon orbit. (paper)