Photon surfaces in less symmetric spacetimes

We investigate photon surfaces and their stability in a less symmetric spacetime, a general static warped product with a warping function acting on a Riemannian submanifold of codimension two. We find a one-dimensional pseudopotential that gives photon surfaces as its extrema regardless of the spatial symmetry of the submanifold. The maxima and minima correspond to unstable and stable photon surfaces, respectively. It is analogous to the potential giving null circular orbits in a spherically symmetric spacetime. We also see that photon surfaces indeed exist for the spacetimes which are solutions to the Einstein equation. The parameter values for which the photon surfaces exist are specified. As we show finally, the pseudopotential arises due to the separability of the null geodesic equation, and the separability comes from the existence of a Killing tensor in the spacetime. The result leads to the conclusion that photon surfaces may exist even in a less symmetric spacetime if the spacetime admits a Killing tensor.