Stability of null orbits on photon spheres and photon surfaces

Stability of a photon sphere, or stability of circular null geodesics on the sphere, plays a key role in its applications to astrophysics. For instance, an unstable photon sphere is responsible for determining the size of a black hole shadow, while a stable photon sphere is inferred to cause the instability of spacetime due to the trapping of gravitational waves on the radius. A photon surface is a geometrical structure first introduced by Claudel, Virbhadra, and Ellis as the generalization of a photon sphere. The surface does not require any symmetry of spacetime and has its second fundamental form of pure trace. In this paper, we define the stability of null geodesics on a photon surface. It represents whether null geodesics perturbed from the photon surface are attracted to or repelled from the photon surface. Then, we define a strictly (un)stable photon surface as a photon surface on which all null geodesics are (un)stable. We find that the stability is determined by Riemann curvature. Furthermore, it is characterized by the normal derivative of the second fundamental form. As a consequence, for example, a strictly unstable photon surface requires nonvanishing Weyl curvature on it if the null energy condition is satisfied.