On the connection of spacetime separability and spherical photon orbits

The notion of non-equatorial spherical photon orbits is among the very special properties of the Kerr spacetime of rotating black holes and is one that leaves a clear mark on the electromagnetic and gravitational wave signature of these objects. In principle, one could use observations like the shadow a black hole casts when is externally illuminated or the gravitational quasi-normal mode ringdown of merging black holes in order to identify the Kerr metric via its light-ring and spherical photon orbit structure, or perhaps look for the presence of more exotic non-Kerr ultracompact objects. This approach would require some understanding of how circular photon orbits behave in alternative (and far less special) non-Kerr spacetimes. In this letter we explore the connection between the existence of spherical photon orbits and the separability of a general stationary and axisymmetric spacetime. We show that a spacetime cannot be separable if it possesses an equatorial photon ring but at the same time does not admit (in any coordinate system) non-equatorial spherical photon orbits. As a result, the separability-circularity connection could serve as a non-Kerr diagnostic of black hole candidates.