Modification of photon trapping orbits as a diagnostic of non-Kerr spacetimes

Photon circular orbits, an extreme case of light deflection, are among the hallmarks of black holes and are known to play a central role in a variety of phenomena related to these extreme objects. The very existence of such orbits when motion is not confined in the equatorial plane, i.e., spherical orbits, is indeed a special property of the separable Kerr metric and may not occur, for instance, in the spacetime of other more speculative ultracompact objects. In this paper we consider a general stationary-axisymmetric spacetime and examine under what circumstances spherical or more general, variable-radius, ``spheroidal'' nonequatorial photon orbits may exist with the ultimate goal of using the modifications---or even loss---of photon trapping orbits as a telltale sign of non-Kerr physics. In addressing this issue, we first derive a general necessary condition for the existence of spherical/spheroidal orbits and then go on to study photon trapping orbits in a variety of known non-Kerr metrics (Johannsen, Johannsen-Psaltis, and Hartle-Thorne). The first of these is an example of a separable spacetime which supports Kerr-like spherical photon orbits. A more detailed analysis reveals a deeper connection between the presence of spherical orbits and the separability of a metric (that is, the existence of a third integral of motion). Specifically, a spacetime that does not admit spherical orbits in any coordinates is necessarily nonseparable. The other two spacetimes considered here exhibit a clear non-Kerr behavior by having spherical photon orbits replaced by spheroidal ones. More importantly, subject to the degree of deviation from Kerr, equatorial photon rings give place to nonequatorial ones with an accompanying loss of low-inclination spheroidal orbits. The implications of these results for the electromagnetic and gravitational wave signature of non-Kerr objects are briefly discussed.