Light Rings around Five Dimensional Stationary Black Holes and Naked Singularities

The existence of light rings in a spacetime is closely related to the existence of black hole horizons and observables such as the ringdown and the shadow. Black holes, compared to nonvacuum ultracompact objects, have rather unique environments. To this aim, recently [P.V.P. Cunha and C.A.R. Herdeiro, Phys. Rev. Lett. \textbf{124}, 181101 (2020)] topological arguments, independent of the underlying gravity theory, were developed to prove the existence of unstable light rings outside the Killing horizon of {\it four} dimensional asymptotically flat, stationary, axisymmetric, nonextremal black holes. Here we extend these arguments to five-dimensional stationary black holes. Generically in five dimensions, there are two possible conserved angular momenta, hence the four-dimensional discussion does not extend verbatim to five dimensions; nevertheless, we prove that there is a light ring for each rotation sense for a stationary black hole. We give the static and the Myers-Perry rotating black holes as examples. We also show that when the horizon of the black hole disappears and the singularity becomes naked, only one of the light rings survives; a similar phenomenon also occurs in four dimensions which might allow testing the cosmic censorship hypothesis.