Probing Phase Structure of Black Holes with Lyapunov Exponents

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.

AdS black holes, which discovered a broad range of new phenomena . In the extended phase space, the analogy between RN-AdS black holes and the van der Waals fluid becomes more complete, in that the coexistence lines in the P -T diagram are both finite and terminate at critical points, and the P -V criticality matches with one another [20].
Since the nature of black hole thermodynamics has not yet been fully understood, it is of great interest to explore phase structure of black holes from various perspectives. For example, the Ruppeiner geometry can be exploited to probe the microstructure of black holes [43][44][45][46][47][48][49]. Motivated by the Ruppeiner geometry, RN-AdS black holes have been proposed to be built of some unknown micromolecules [50,51]. More interestingly, there have been attempts to associate phase transitions of black holes with some observational signatures, such as quasinormal modes [52][53][54][55][56], circular orbit radius of a test particle [57][58][59] and black hole shadow radius [60,61]. It showed that phase structure of black holes can be revealed by behavior of the aforementioned physical quantities, and the discontinuity in the physical quantities across phase transitions behaves similarly to an order parameter.
Lyapunov exponents characterize the rate of separation of adjacent trajectories, and positive/negative Lyapunov exponents correspond to divergent/convergent trajectories [62]. Lyapunov exponents can be used to study chaotic dynamics in general relativity, which is a nonlinear dynamical theory. The chaotic motion of particles in black hole spacetime has been extensively studied, such as static axisymmetric spacetimes [63,64], rotating charged black hole spacetimes [65,66], multi-black hole spacetimes [67], bumpy spacetimes [68], weakly magnetized Schwarzschild black holes [69], black holes with discs or rings [70], Schwarzschild-Melvin black holes [71], accelerating black holes [72], spacetimes with a quadrupole mass moment [73] and black holes with quantum gravity corrections [74,75]. Particularly, the motion of a particle near the black hole horizon was studied in [76,77], which found that the Lyapunov exponent obeys an universal upper bound proposed in the framework of gauge/gravity duality [78]. Nevertheless, counterexamples that violate the upper bound have been reported [79,80]. Moreover, partly motivated by gauge/gravity duality, chaotic dynamics of a ring string has been studied in Schwarzschild-AdS and charged AdS black holes [81][82][83][84][85][86][87]. Intriguingly, Lyapunov exponents of unstable null geodesics have been revealed to be closely related to the imaginary part of a class of quasinormal modes of perturbations in black hole spacetime [88,89].
In this paper, we aim to explore the relationship between phase structure of RN-AdS black holes and Lyapunov exponents of particles and ring strings moving in the black holes. The rest of this paper is organized as follows. Thermodynamics and phase transitions of RN-AdS black holes are briefly reviewed in Sec. II. Focusing on particles, we examine the relationship between Lyapunov exponents of unstable circular geodesics and phase structure of RN-AdS black holes in Sec. III.
The case of a ring string is discussed in Sec. IV. We summarize our results with a brief discussion in Sec. V. For simplicity, we set G = = k B = c = 1 in this paper.

II. PHASE STRUCTURE OF RN-ADS BLACK HOLES
In this section, we review thermodynamic properties and phase structure of RN-AdS black holes. The 4-dimensional static charged RN-AdS black hole solution is described by where the metric function f (r) is and l is the AdS radius. Here, the parameters M and Q can be interpreted as the black hole mass and charge, respectively. The RN-AdS black hole has an event horizon at r = r + , and the horizon radius r + satisfies f (r + ) = 0. In terms of r + , the Hawking temperature T and the mass M are given by [47] respectively. Moreover, the RN-AdS black hole obeys the first law of thermodynamics, where S = πr 2 + and Φ = Q/r + are the entropy and the potential of the black hole, respectively. By computing the Euclidean action in the semiclassical approximation, we obtain the free energy By dimensional analysis, we find that the physical quantities scale as powers of l, Q = Q/l,r + = r + /l,T = T l,F = F/l,M = M/l,r = r/l, where the tildes denote dimensionless quantities.
(3), we can expressr + as a function ofT . Ifr + (T ) is multivalued, there is more than one black hole solution for fixed values ofQ andT , corresponding to multiple phases in a canonical ensemble. The critical point is an inflection point determined by which gives the corresponding quantities evaluated at the critical point, To study phase transitions, we expressF with respect toT by pluggingr + (T ) into eqn. (5) and plot is only one black hole solution and no phase transition, which is shown in the right panel.

III. PHASE TRANSITIONS AND LYAPUNOV EXPONENTS OF PARTICLES
In this section, we investigate the relationship between Lyapunov exponents of massless and massive particles and phase transitions of RN-AdS black holes. In particular, we focus on unstable circular geodesics on the equatorial hyperplane with θ = π/2, which are described by the Lagrangian Here dots and primes denote derivatives with respect to the proper time and the areal radius r, respectively. Then the radial motion can be expressed aṡ where the constant E can be treated as the energy and the energy per unit mass for massless and massive particles, respectively. Here, we introduce the effective potential, where L is identified as the angular momentum of the particles, and δ 1 = 1 and 0 correspond to massless and massive particles, respectively. The radius of an unstable circular geodesic is determined by

A. Massless Particles
For massless particles, there is always (except for L = 0) an unstable circular geodesic outside the event horizon at which is independent of L. Furthermore, the Lyapunov exponent of the unstable null circular geodesic is given by [88] which depends only onQ andr + . We present the 3D plot of log 100 λ as a function ofQ andr + in FIG. 2, where black hole solutions with the event horizon atr + do not exist in the black region. It shows that λ diverges whenr + decrease to zero. In addition, λ approaches 1 asQ orr + increases to infinity. In fact, eqns. (13) and (14) indicate that, whenr + orQ approaches infinity,r o goes to infinity, and hence λ goes to 1.
Pluggingr + (T ) into eqn. (14), one can express the Lyapunov exponent λ in terms ofT . In which gives that the critical exponent of ∆λ is 1/2. Our result shows that the critical exponent of ∆λ is identical to that of the order parameter in the van der Waals fluid predicted by the mean field theory. It is worth emphasizing that the critical exponent of the circular orbit radius was also found to be 1/2 for charged AdS black holes [57].  time-like circular geodesics cease to exist.
The requirement V eff (r) = 0 for a circular orbit at r = r o yields which can be used to express r o in terms of L. The Lyapunov exponent of the time-like circular orbit is given by [88] which can be rewritten as a function of L,Q andr + by using eqn. (16). indicates that ∆λ can serve as an order parameter. Near the critical temperature, we find which confirms that the critical exponent of ∆λ is also 1/2.

IV. PHASE TRANSITIONS AND LYAPUNOV EXPONENTS OF STRINGS
In contrast to motion of particles in RN-AdS black holes, equations of motion governing strings are non-integrable, showing chaotic behavior of strings in RN-AdS black holes [90]. Therefore, numerical computations are often required to obtain Lyapunov exponents of motion of strings.
Following [83], we consider a ring string coaxially moving in the RN-AdS black hole spacetime, which is illustrated in FIG. 7. The equations of motion for a string is determined by the Polyakov action, where X µ are the target space coordinates, the indices {α, β} = 1 and 2 correspond to the (τ, σ) coordinates on the worldsheet of the string, respectively, γ αβ is the worldsheet metric, and G µν is the target space metric.
The ring string configuration considered in our paper is described by the following ansatz for the coordinates of the target space where n is the winding number of the string along the φ direction, and τ is the proper time. For the above ansatz with the conformal gauge γ αβ = η αβ , the Lagrangian for the ring string in RN-AdS black holes becomes where dots and primes denote derivatives with respect to r and τ , respectively. Using the Legendre transformation, one obtains the Hamiltonian where P t , P r and P θ are the canonical momenta, and the Hamiltonian satisfies the constraint H = 0. The canonical equations of motion are then given bẏ θ = πα P θ r 2 , P θ = − n 2 r 2 sin θ cos θ πα , which gives that P t = E is the conserved energy. As shown in [83], there are three different scenarios depending on the initial conditions of the string and the black hole parameters: (1) The string oscillates back and forth around the black hole.
(2) The string oscillates a finite number of times around the black hole before being captured by it.
(3) The string oscillates a finite number of times around the black hole before escaping to infinity.
To calculate the Lyapunov exponent of the motion of the string, we evolve two adjacent trajectories with an initial distance d 0 in the phase space spanned by r, θ, P r and P θ . When the distance between the trajectories d t i exceeds the upper threshold at t = t i , one initializes a rescaling of one trajectory back to having the initial distance d 0 . The maximum Lyapunov exponent is the average of the time-local Lyapunov exponent maximum Lyapunov exponent can be considered as an indication of deterministic chaos. Here, we adopt to Verner's "most efficient" Runge-Kutta 9(8) method [91], which can achieve high accuracy solving (tolerances like < 10 −12 ).
In the bottom row of FIG. 3, we plot the Lyapunov exponent λ against the temperatureT for ring strings of the scenarios (1) and (3)

V. CONCLUSIONS
In this paper, we calculated Lyapunov exponents of massless particles, massive particles and ring strings in RN-AdS black holes, and found that the behavior of the Lyapunov exponents can be employed to explore phase structure of black holes. In particular, when the black hole charge is less than the critical charge, the Lyapunov exponents as a function of the temperature demonstrate three branches, which correspond to three coexisting black hole phases. When the charge is greater than the critical charge, the Lyapunov exponents are singled-valued functions of the temperature, which coincides with one black hole phase. At the first-order phase transition, the discontinuity in the Lyapunov exponent ∆λ can act as an order parameter to characterize the black hole phase transition. Remarkably, ∆λ was shown to have a critical exponent of 1/2 at the critical point.
Our results support the conjectured relationship between Lyapunov exponents and phase transition for RN-AdS black holes, which could open a new window to study thermodynamics of black holes. It will be of great interest if our analysis can be generalized to more general black hole spacetimes beyond RN-AdS black holes. More importantly, it is highly desirable to investigate the relationship between Lyapunov exponents and black hole phase transitions in the extended phase space, in which the cosmological constant is identified as a thermodynamic pressure.

ACKNOWLEDGMENTS
We are grateful to Guangzhou Guo, Xin Jiang and Yiqian Chen for useful discussions and valuable comments. This work is supported in part by NSFC (Grant Nos. 11747171, 12105191,