Photon Rings: Universal Penrose Limits and Fast Scrambling

We investigate the dynamics of unstable null geodesics through the universal Penrose limit of axisymmetric black holes. Utilizing the Killing-Yano tensor, we show how the resulting plane wave geometry directly encodes the instabilities of photon rings, allowing quasinormal modes and Lyapunov exponents to be extracted from a single effective Hamiltonian. We then connect these geometric properties to a fast scrambling instability conjecture in binary black hole mergers. By mapping a quasi-circular binary onto an effective Kerr metric, we demonstrate that the merger's final state naturally maximizes a photon ring's Lyapunov exponent. This suggests a connection between post-merger black hole remnants and the dynamics of unstable null orbits.