Time delay as a probe of multiple photon spheres

Abstract

Black hole shadow images are primarily determined by the properties of photon spheres and can exhibit degeneracies across different spherically symmetric spacetime geometries. We show that time delay observables associated with higher-order images of transient sources provide a robust probe to break such degeneracies in spacetimes admitting multiple photon spheres. Adopting a theory-agnostic, parametrized, static, spherically symmetric framework that captures the generic features of double-peak effective potentials, we investigate photon geodesics and quantify them in terms of angular deflection, travel time, and the order of the image. We identify distinctive signatures of trajectories probing the region between the unstable photon spheres. In particular, we find that these trajectories are characterized by the nontrivial temporal behavior, including a minimum travel time, a minimum angular deflection, and a characteristic triplet structure of higher-order images with a specific arrival sequence. We further show that the influence of the depth of the potential well, between the two photon spheres, on the observed time delays provides a direct handle on otherwise inaccessible regions of the spacetime. Although we consider a parametrized framework, these identified signatures are primarily influenced by the potential structure and will therefore occur in any spacetime with a similar double-peak effective potential. Our results highlight that time-domain lensing observables encode information beyond static shadow images and offer a promising avenue for probing the structure of compact objects and the strong-field regime of gravity.