Time delay as a probe of multiple photon spheres

TL;DR

This paper demonstrates that time delay measurements of higher-order photon images can break degeneracies in black hole shadow observations caused by multiple photon spheres, revealing detailed spacetime structure.

Contribution

It introduces a model-independent approach showing how time delays and image sequences can probe regions between multiple photon spheres in spherically symmetric spacetimes.

Findings

Time delays reveal the region between photon spheres.

Higher-order images exhibit a triplet structure with specific arrival sequences.

Time-domain lensing encodes information beyond static shadow images.

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 model-independent, parametrized, static, spherically symmetric framework that captures the generic features of double-peaked 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. 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.