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-Nordström-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.

Lyapunov exponents and phase transitions of Born-Infeld AdS black holes

“…r +c (s) → 0. As β → ∞, the BI AdS black hole reduces to RN AdS type, and there is only one critical point as r +c = 1/ √ 6, q c = 1/6 and T c = 1/π 2/3 [31]. Another interesting point is that as β decreases, the gap between q c (s) and q c (l) becomes narrow.…”

“…By dimensional analysis, the powers of l can be used as the physical quantities scale [31] as q = q/l, r+ = r + /l, T = T l, F = F/l, M = M/l, r = r/l, (2.14) here all tildes are dimensionless quantities. For the convenience of writing, we will directly represent these quantities without displaying the tilde.…”

“…They can be used to measure the divergence and convergence rate of trajectories near the black hole. Lyapunov exponents in background metric have been explored [28], which could describe the stability of geodesics around RN AdS black holes [31]. It is of our interest to describe timelike and null geodesics of particles around BI AdS black holes for the existence of nonlinear electrodynamical effect.…”

“…Thus in the depiction of figure 3, we consider E n as a nonzero constant and reset V r = V r + E 2 n similar to refs. [28,31]. To learn the features of timelike geodesics, we also set L = 20l.…”

“…By building upon the aforementioned connections between quasinormal modes (QNMs) and black hole phase transitions, it is possible to establish a direct relation between Lyapunov exponents and such transitions [31]. In this work, the occurrence of a phase transition can be identified by a discontinuous jump in the value of the Lyapunov exponent λ and its difference ∆λ can be characterized as an order parameter.…”