Spacetime Measurements with the Photon Ring

We explore the universal symmetries of the black hole photon ring in a wide range of non-Kerr spacetimes, including the Kerr-Newman, Kerr-Sen, Kerr-Bardeen, and Kerr-Hayward metrics. The demagnification exponent ($\gamma$) controls the size and flux scaling of higher-order images, which appear in the photon ring, the time delay ($\tau$) determines the timing of their appearance, and the rotation parameter ($\delta$) relates their relative orientations on the image plane. Our investigation reveals that these critical parameters respond distinctly to variations in black hole spin, generalized charge, and observer inclination, establishing them as complementary probes of spacetime geometry: $\gamma$ is predominantly influenced by charge and spin, $\tau$ is strongly affected by inclination, especially for near-extremal black holes, and $\delta$ is highly sensitive to spin. Notably, we find that the time delay provides an independent constraint on shadow size for polar observers, while the rotation parameter facilitates metric-independent spin measurements. Specifically, for Kerr black holes, the total variation in $\gamma$, $\tau$, and $\delta$ across all possible inclinations and spins is $\lesssim 20\%$, $\lesssim 10\%$, and $\lesssim 60\%$, respectively. By contrast, the Kerr shadow radius varies by only $\lesssim 8\%$. A future joint measurement of these critical parameters -- along with the black hole shadow size -- will enable precise spacetime characterization, including measurements of the spin, inclination, and generalized charge.