Probing Phase Structure of Black Holes with Lyapunov Exponents

Abstract: We conjecture that there exists a relationship between Lyapunov exponents and black hole phase transitions. To support our conjecture, Lyapunov exponents of the motion of particles and ring strings are calculated for Reissner-Nordstr\"{o}m-AdS black holes. When a phase transition occurs, the Lyapunov exponents become multivalued, and branches of the Lyapunov exponents coincide with black hole phases. Moreover, the discontinuous change in the Lyapunov exponents can be treated as an order parameter, and has a critical exponent of $1/2$ near the critical point. Our findings reveal that Lyapunov exponents can be an efficient tool to study phase structure of black holes.