Gaussian curvature and Lyapunov exponent as probes of black hole phase transitions (2509.05103v1)

Abstract: We study the Gaussian curvature of unstable null orbits. The Gaussian curvature exhibits multivaluedness near the phase transition point of a first-order phase transition. Numerical investigations of Reissner-Nordstrom Anti-de Sitter (RN-AdS), Hayward-AdS, and Hayward-Letelier-AdS black holes demonstrate that this geometric multivalued region coincides precisely with the spinodal region calculated by black hole thermodynamics. Using the known relation $K=-\lambda^2$ linking orbital geometry to chaotic dynamics, we show that this geometric feature also satisfies the critical exponents predicted by mean-field theory, consistent with those derived from Lyapunov exponents. Our work demonstrates that Gaussian curvature can serve as an alternative effective tool to study the phase structure of black holes.