Influence of observer’s inclination and spacetime structure on photon ring observables

Recent millimeter-wavelength observations of the near-horizon regions of $\mathrm{M}{87}^{*}$ and Sgr ${\mathrm{A}}^{*}$ by the Event Horizon Telescope have significantly advanced our understanding of gravity and astrophysics under extreme conditions. A key goal for future observations, with improved angular resolution and sensitivity, is the detection of the photon ring in these black holes. For Kerr black holes, photon rings exhibit universal scaling relations in terms of purely geometric observables: the demagnification exponent, time delay, and rotation parameter. Our study employs a nonperturbative, parametrized spinning spacetime to investigate how these observables vary with deviations from the no-hair theorem and the observer's inclination. Particular emphasis is placed on polar observers, which is highly relevant for $\mathrm{M}{87}^{*}$. Our findings reveal that these observables encode vital information about the strong-field spacetime structure, including the morphology of the ergosphere. By analyzing specific non-Kerr spacetimes, we demonstrate that potential measurements of these observables, including the shadow size, can directly constrain the spin and Kerr-deviation functions. For polar observers, time delay measurements offer an independent estimate of the shadow size, providing a complementary probe of the black hole's geometry. These results highlight the potential of using photon rings for precision tests of general relativity and for constraining strong-field deviations from Kerr spacetime.