Influence of Observer Inclination and Spacetime Structure on Photon Ring Observables

Recent observations of the near-horizon regions of BHs, particularly the images captured by the Event Horizon Telescope (EHT) collaboration, have greatly advanced our understanding of gravity in extreme conditions. These images reveal a bright, ring-like structure surrounding the central dark area of supermassive BHs, created by the images of unstable photon orbits. As observational capabilities improve, future studies are expected to resolve higher-order rings, providing new opportunities to test gravity through observables such as the Lyapunov exponent, time delay, and azimuthal shift. These observables offer valuable insights into the structure of spacetime, BH properties, and the inclination of the observer. In this study, we employ a non-perturbative and non-parametric framework to examine how these observables change with deviations from the no-hair theorem and varying inclinations. We focus particularly on polar observers, which are highly relevant for the supermassive compact object at the centre of the galaxy M87. Our analysis explores how each of these observables can reveal information about the structure of spacetime and the morphology and existence of the ergosphere and event horizon. Furthermore, we illustrate this characterization for several specific alternative spacetimes, investigating how these current and potential future measurements, including those of the shadow size, can provide direct insights into the spin parameter values for each of these spacetimes.