On the Chaos Bound in Rotating Black Holes

We study out-of-time-order correlators (OTOCs) of rotating BTZ black holes using two different approaches: the elastic eikonal gravity approximation, and the Chern-Simons formulations of 3-dimensional gravity. Within both methods the OTOC is given as a sum of two contributions, corresponding to left and right moving modes. The contributions have different Lyapunov exponents, $\lambda_L^{\pm}=\frac{2\pi}{\beta}\frac{1}{1\mp \ell \Omega}$, where $\Omega$ is the angular velocity and $\ell$ is the AdS radius. Since $\lambda_L^{-} \leq \frac{2\pi}{\beta} \leq \lambda_L^{+}$, there is an apparent contradiction with the chaos bound. We discuss how the result can be made consistent with the chaos bound if one views $\beta_{\pm}=\beta(1\mp \ell \Omega)$ as the effective temperature of the left and right moving modes.


Introduction
In recent years, the gauge-gravity duality [1][2][3] has provided remarkable insights into the nature of quantum chaos 1 . An important lesson from the holographic approach to quantum chaos [6][7][8][9] is the fact that the onset of chaos can be efficiently diagnosed with the so-called out-of-time-order correlators (OTOCs) which, for chaotic systems, are expected to vanish at later times for almost any local operators V and W . In holographic systems at finite temperature, the OTOC has a very simple form where the multiplicative factor, ε ∆ V ∆ W , contains information about the operators V and W , while the exponential corresponds to an universal piece, and it is characterized by only three parameters: the Lyapunov exponent λ L , the scrambling time t * , and the butterfly velocity v B . All these quantities are determined from the geometry close to the black hole horizon. For black holes, one generically expects λ L = 2π β , where β is the system's inverse temperature. Surprisingly, the above value provides an upper bound (the chaos bound) for the Lyapunov exponent in generic large-N systems, namely [10] This highlights a special property of holographic systems which are dual to black holes -they saturate the chaos bound. This finding generated a lot of excitement in the community, leading to the speculation that the saturation of the chaos bound could be a sufficient condition for the existence of Einstein gravity dual [10]. However, despite being a necessary condition, the saturation of the chaos does not guarantee the existence of dual description purely in terms of Einstein gravity [11][12][13]. The observation that the SYK model saturates the chaos bound put this system into evidence as a prototypical example of a simple model for holography. Several aspects of this system can be capture by a 2-dimensional Einstein-dilation theory, and this has generated a lot of research within the framework of NAdS 2 /NCFT 1 and related areas. See, for instance, [14][15][16][17][18][19][20][21][22][23][24][25][26].
One expects the saturation of the chaos bound to be a property of quite generic black holes [10]. However, some recent studies suggest that rotating black holes have two Lyapunov exponents, one of which does not saturate the chaos bound, while the other violates it [27,28]. Both these works consider the 3-dimensional rotating BTZ black hole. In [27], the author finds an effective action for the boundary degrees of freedom and studies OTOCs for this effective theory. He finds that the OTOC is controlled by two modes (left and right moving modes), with Lyapunov exponents λ ± L = 2π 1∓ Ω , where Ω is the chemical potential for angular momentum. In [28], the author studies the disruption of the two-sided mutual information both in the CFT and in the bulk. In his particular configuration, the onset of chaos is controlled by λ − L , which indicates a non-saturation of the chaos bound. Moreover, the aforementioned works seem to be in contradiction with [29], in which the author studies chaos for the rotating BTZ black holes using the geodesic approximation and finds a saturation of the chaos bound.
In this paper, we revisit the calculation of OTOCs for rotating BTZ black holes using two different methods. We first use the elastic eikonal gravity approximation [9], in which the OTOCs have a very vivid holographic representation in terms of a high energy shock wave collision near the black hole's bifurcation surface. We then study chaos using the Chern-Simons formulation of 3-dimensional gravity. We show that the dynamics of the boundary degrees of freedom is governed by two copies of a Schwarzian-like action and derive the OTOCs from the analytic continuation of the euclidean 4-point. We compare both approaches with each other and with previous results in the literature, and discuss an apparent violation of the chaos bound in this system. The paper is organized as follows. In Section 2 we review the rotating BTZ black hole geometry and discuss a few aspects of the corresponding CFT dual description. In Section 3 we compute OTOCs using the gravity eikonal approximation. In Section 4 we review the Chern-Simons formulation of 3-dimensional gravity and we derive an effective action for the boundary degrees of freedom. We then evaluate OTOCs from the on-shell action via analytic continuation. We discuss our results in Section 5 and relegate some technical details to the Appendix A.

The rotating BTZ black hole
In this section we briefly review the rotating BTZ geometry [30,31]. This is a solution of Einstein gravity in 2+1 dimensions with a negative cosmological constant Λ = −1/ 2 . In terms of Schwarzschild coordinates (t, r, ϕ), the metric reads where ϕ is periodic with period 2π and the blackening factor has the form Here r + and r − are the radii of the outer and inner horizons, respectively. The black hole's mass M , angular momentum J, and temperature T are determined from r ± Since we will be interested in the near-horizon geometry, it is convenient to work with co-rotating coordinates (t, r, φ), with the new angular variable defined as where is the angular velocity of the outer horizon. In these coordinates the metric takes the form In the near-horizon limit, N φ (r) vanishes and the metric becomes diagonal. Figure 1 shows the Penrose diagram for the maximally extented BTZ geometry. Here we follow [32] and denote the left and right exterior regions as 1 +− and 1 ++ , while the future and past interiors are denoted as 2 ++ and 2 +− , respectively. In Figure 1 we only shows the region of interest. The same structure repeats itself in the vertical direction. interest. The same pattern repeats itself indefinitely above and below [31]. The regions 1 ++ and 1 +− denote the right and left exterior regions, while the regions 2 +− and 2 ++ denote the past and future interiors. The coordinates (Ũ ,Ṽ ) are defined as in [33]: For later purposes, we introduce Kruskal coordinates for the region 1 ++ where the tortoise coordinate r * is [29] In these coordinates the metric takes the form (2.10)

Embedding coordinates and bulk-to-boundary propagators
The AdS 3 space is defined as the hyperboloid −T 2 1 −T 2 2 +X 2 1 +X 2 2 = − 2 embedded in a space with metric ds 2 = −dT 2 1 − dT 2 2 + dX 2 1 + dX 2 2 . The rotating BTZ geometry can be thought of as piece of pure AdS 3 , with identifications in the angular coordinates, namely ϕ ∼ ϕ + 2π. This fact allows us to compute bulk-to-boundary propagators for this geometry.
We start by defining embedding coordinates for the region 1 ++ . In terms of co-rotating coordinates, the embedding coordinates take the form (2.14) In terms of Kruskal coordinates, we have By replacing the above formulas in the metric ds 2 = −dT 2 1 − dT 2 2 + dX 2 1 + dX 2 2 , we can recover the formulas (2.6) and (2.10).
Having defined the embedding coordinates, we can now compute the geodesic distance d between two points p = (T 1 , T 2 , X 1 , X 2 ) and p = (T 1 , T 2 , X 1 , X 2 ) as We will use the above formulas to compute bulk-to-boundary propagators. For our calculation, it will be convenient to write the boundary point in terms of co-rotating coordinates p = (t, r, φ ) and the bulk point in terms of Kruskal coordinates p = (U, V, φ). The geodesic distance between such points can be written as where ∆φ = φ − φ and r ∞ is an UV cutoff defining the radial position of the boundary point. From the above expression we can obtain the bulk-to-boundary propagator as proportional to r ∆ ∞ cosh d −∆ , i.e., 21) where c is a constant that depends on the operator O. Since we are only going to compute the bulk-to-boundary propagators when the bulk point is either at U = 0 or at V = 0, we can simplify the above expression even further and write Finally, to describe the black hole, we need to take into account the periodicity of the angular variable. This is done by shifting the angular coordinate by integer multiples of 2π and adding an infinite sum in front of the propagator as 2 [34] Φ (2.23) where ∆φ n = ∆φ + 2πn.

The dual CFT description
The rotating BTZ black hole is dual to a CFT with a chemical potential for angular momentum. The maximally extended rotating black hole is dual to thermofield double (TFD) state made of two such CFTs where each CFT has Hamiltonian H and angular momentum J, with eigenvalues E n and J n , respectively. Here Z(β, Ω) is the grand canonical partition function. We can also decompose the system into left and right moving modes, and write the TFD state as [35]

Eikonal approximation
In this section we compute the out-of-time-order correlator (OTOC) using the holographic prescription given be Shenker and Stanford (S&S) in [9]. Here, to avoid clutter, the subscripts φ 1 , φ 2 , · · · are used to denote the four angles. We consider all the operators acting on the right side of the geometry 3 . For simplicity, we also consider the operators V and W to be single-trace operators, because in this case the corresponding bulk-to-boundary propagators have a simple form. If the t i s are all real, the above OTOC diverges. A simple way to regularize this divergence is to consider imaginary times 4 The basic idea of the S&S approach is to view the OTOC as a scattering amplitude where the 'in' and 'out' states are given by The above states have a simple description in the bulk in terms of two-particle states. The V -operator creates a bulk particle, which we call the V -particle, while the particle created by the W -operator is called the W -particle. For large enough t these particles will be highly boosted with respect to the t = 0 slice of the geometry. The W -particle will have a very large momentum in the V -direction, while the V -particle will have a very large momentum in the U -direction. As explained in [9], the state V φ 3 (t 3 )W φ 4 (t 4 )|TFD can be thought of as an 'in' state, in which the particle's wave functions is represented at some early time (before the collision), while the state W φ 2 (t 2 ) † V φ 1 (t 1 ) † |TFD can be thought of as an 'out' state, with the particle's wave functions represented at some late time (after the collision). See figure 2.
For convenience, we choose to represent the V -particle in the U = 0 slice of the geometry, and we expand the corresponding state in the basis |p U , φ of well defined momentum and angular position. Likewise, we represent the W -particle in the V = 0 slice of the geometry, in the basis |p V , φ . The basis vectors have the following normalization 5 |TFD represented in a bulk spatial slice that touches the right boundary at time t 3 . Right: the 'out' state W φ2 (t 2 ) † V φ1 (t 1 ) † |TFD represented in a bulk slice that touches the right boundary at time t 2 .
with a similar formula for The 'in' state can then be written as while the 'out' state reads The wave functions ψ i appearing in the above formulas are Fourier transforms of bulk-toboundary propagators where Φ V and Φ W are bulk fields dual to the operators V and W , which are represented via the 'extrapolate' dictionary. In the eikonal approximation, the 'in' and 'out' are related by a phase shift where δ(s, φ 1 − φ 2 ) is the so-called Eikonal phase and s = ( The state |inelastic represents an inelastic contribution that is orthogonal to all two-particle 'in' states. The physics behind the eikonal approximation is the following: since the particles collide at very high energy, the interaction is dominated by the graviton exchange, and the sum of the corresponding ladder diagrams can be described by a simple phase shift [36,37]. The OTOC (3.1) can then be written as (3.5). The information of φ i and t i with i = 1, 2, 3, 4 in (3.1) is denoted by the subscripts of ψ i .

Wave functions
For the rotating BTZ black hole the bulk-to-boundary propagators in (3.8)and (3.9) are given by For the bulk-to-boundary propagators in (3.10)and (3.11) we only need to change t → t * with To avoid clutter, we use the same notation ∆φ n for φ 1 and φ 3 (∆φ n for φ 2 and φ 4 ). The wave functions can then be written as (3.18)

The eikonal phase
The operators V and W introduce particles in the bulk, which we call the V -particle and the W -particle, respectively. In the elastic eikonal gravity approximation, the phase shift is basically given by the on-shell action of these particles, namely where S classical is simply the sum of the on-shell actions for the V -particles and W -particles. When the W is very late, and the V operator is very early, the corresponding bulk particles follow an almost null trajectory very close the black hole horizon. The W -particle moves along U = 0, while the V -particle moves along V = 0. The particles collide very close to the bifurcation surface, at U = V = 0, and the phase shift is only sensitive to this region of the geometry, where the metric assumes the approximate form In the following, we denote the horizon values by A 0 ≡ A(0) and r + ≡ r(0). In this particular configuration, the stress-energy of these particles takes the form where φ and φ denote the position of the sources. The back-reaction of these particles in the geometry takes a simple shock wave form. For instance, the backreaction of the W -particles reads while the back-reaction of the V -particle reads In the above formulas, f (φ) is the shock wave transverse profile. For a rotating BTZ black The most general solution of the above equation has the form where c 1 and c 2 are constants 7 . The on-shell classical action for the W -particles and V -particles can be written as [9] S classical = 1 2ˆd Note that, while h U U refers to the W −particle, the stress-tensor T U U = g U V g U V T V V refers to the V −particle. Substituting the expression for the stress-energy tensors and the corresponding back-reactions, one finds

Evaluating the integrals
Inserting the above wave functions in the formula for the OTOC (3.13), we find The OTOC can be divided into two contributions where F 0 corresponds to the term where n 1 = n 2 = n 3 = n 4 = 0, and F n 1 ,n 2 ,n 3 ,n 4 corresponds to the remainder terms. The dominant contribution comes from F 0 .
The only ingredient missing to calculate F 0 is the phase shift δ(s, φ − φ ), which we know to have the following form (see (3.27)) To evaluate the above integral it is convenient to introduce the new variables (3.32) By specifying the times as in (3.2), the integral becomes where ij = i(e κ i − e κ j ) and C is a constant given by (3.34) Without the phase shift, the above integral just gives V V W W . The integral can be evaluated in the limit ∆ W >> ∆ V >> 1 and the result reads 8 (3.25) and using that we can write the OTOC as where the constants C 1 and C 2 are proportional to ∆ V ∆ W G N . These two solutions correspond to left and right moving modes. Both modes have a maximal Lyapunov exponent, One of these butterfly velocities is larger than the speed of light when Ω > 0. 8 Here we evaluate the integrals using the same tricks that were used in [9]. By writing the integrals in q and φ as´dqdφ e −F (q,φ ) , we can check that the result is dominated by the region where q ≈ 2∆W and φ ≈ φ2.
The integral in p can be done analytically, and the integral in φ can be done by a saddle point approximation.
In terms of the Schwarzschild angular coordinate, In this case the two butterfly velocities are equal to the speed of light, but the two modes have different Lyapunov exponents which satisfy the property λ − L ≤ λ L ≤ λ + L . Naively, this seems to indicate that one of the Lyapunov exponents is less than maximum, while the other one violates the chaos bound [10]. We come back to this issue in the discussion section.

Chaos in the Chern-Simons formulation of AdS 3 gravity
In the previous section, we calculated the Lyapunov exponent and the butterfly velocity using the gravity eikonal approximation. In this section, we use the SL(2, C) Chern-Simons formulation of Euclidean AdS 3 gravity to derive the on-shell action of the boundary graviton. This is analogous to the Schwarzian on-shell action of the 2D Jackiw-Teitelboim model [18,[39][40][41][42] as well as the dimensional reduction from higher dimensional black holes [43][44][45][46][47]. Moreover, by using a metric-like formulation, [27] obtained a similar on-shell action, which is equivalent to our result up to quadratic level. Since we will evaluate the leading Lyapunov exponent in large k cs ∼ c from the quadratic on-shell action, the leading Lyapunov exponent agrees with [27]. Also, the on-shell action from the Chern-Simons gravity and Lyapunov exponent thereof can easily be generalized for the case of higher spin gravity [48].

Review of Chern-Simons formulation of AdS 3 gravity
In this section, we review the SL(2, C) Chern-Simons formulation of Euclidean AdS 3 (EAdS 3 ) gravity. In particular, we follow [49] in the choice of the boundary term and the corresponding boundary condition. The case of Lorentzian AdS 3 is similar, except that one has to choose carefully the "Kruskal-like" gauges for the eternal black hole [50].
The Euclidean solution can be obtained by analytic continuation of (2.1) to imaginary values of t and The Euclidean coordinates (t E , ϕ) satisfy the periodic condition [34,51] (t E , ϕ) ∼ (t E + β, ϕ + w) , Note that w and r E are taken to be real in the Euclidean black hole, but they become purely imaginary in the Lorentzian BTZ black hole. We define holomorphic and anti-holomorphic coordinates (z,z) as The modular parameters τ andτ are defined as In terms of the (z,z) and (τ,τ ), the above periodic condition becomes (z,z) ∼ (z + τ,z +τ ) . In three dimensions, EAdS 3 gravity can be described by the Chern-Simons action with complex Lie algebra SL(2, C) [52][53][54][55] where k cs = l 4G is Chern-Simons level and CS(A) is defined by Here, A is the Chern-Simons connection, andĀ is its conjugate, defined bȳ We use the coordinates (r, z,z) where z andz are defined by For EAdS 3 Chern-Simons gravity, the manifold in our consideration is a solid torus, and the modular parameter τ of the boundary torus gives the periodicity of z coordinate It is useful to fix the gauge as [49,[54][55][56][57] A =b −1 (d + a z dz + azdz)b (4.12) (4.14) Here, we define the sl(2) generators L 0 , L ±1 as The asymptotic AdS 3 solution with a flat boundary metric in the Chern-Simons gravity was found to be [49,56,57] A where A AdS is the AdS 3 exact solution Hence, the asymptotic AdS 3 condition implies that a = L 1 dz + · · · ,ā = L −1 dz + · · · (4.18) The variation of the action (4.7) without the additional boundary term is given by Hence, choosing the boundary condition Az =Ā z = 0 together with the gauge symmetry [56,57], we can fix the gauge as follows Now, we have to take the variation of the modular parameter τ because it can also be varied under the variation of the action. For this, we will summarize the (τ,τ ) formalism discussed in [49]. It is useful to introduce the new coordinates (w,w), defined by which has a fixed periodicity w ∼ w + 2π ∼ w + 2πi . At a cost of fixing periodicity, the modular parameter appears in the boundary volume element as well as in the boundary metric where dz 2 = i 2 dz ∧ dz. The key idea of [49] is to keep the (boundary) volume element of (w,w) fixed under the variation. In (w,w) coordinates, the differential form is not varied under the variation of the bulk action in (4.7). Instead, in returning to (z,z) coordinates, the variation of a w , aw gives the variation of the modular parameter τ,τ because it appears in the transformation of (a w , aw) into (a z , az) For the variational principle, the authors of [49] choose the boundary term given by 27) and the variation of the total action I tot ≡ I CS + I b is found to be Hence, we can impose the boundary conditions With these boundary conditions, the on-shell action of I tot is found to be Note that though the authors of [49] mainly analyzed a constant solution a andā, they pointed out that their method could be applied to non-constant solutions. Furthermore, the sl(2, C) Chern-Simons gravity is simple because an additional chemical potential is not necessary, as opposed to the case of higher spin gravity. In the next section, we will analyze the non-constant solutions and their on-shell action.

On-shell action of asymptotic AdS solutions
As in the previous section, the boundary condition in (4.29) allows us to fix the connections to be Then, the on-shell action in (4.30) can be written as We consider non-constant connections, which are smoothly connected to a fixed constant solution, such as the BTZ black hole, i.e., where h = h(z) is a holomorphic residual gauge transformation parameter that can be smoothly connected to identity, and the constant solution a BTZ is given by with L 0 being a constant. Such a smooth residual gauge transformation does not change the holonomy (up to a similarity transformation) The smoothness of a holonomy along the contractible cycle of solid torus requires the holonomy to belong to the center of the gauge group [58][59][60]. For the BTZ black hole, the Euclidean time circle corresponds to the contractible cycle, and the smoothness condition implies that w = τ a z +τ az = u −1 (2πiL 0 )u (4.36) for some matrix u. Hence, by taking the determinant of the both sides we obtain In the variation of the action, we consider a fixed constant modular parameter τ andτ , and, accordingly, the smoothness condition of the holonomy also fixes the constant solution L 0 andL 0 . This means that we are considering the BTZ black hole with τ,τ and its smooth fluctuations. Now, we evaluate the non-constant connection a z (z), which is connected to the constant connection a BTZ via a smooth residual gauge transformation by h(z). In principle, using the Gauss decomposition of h(z), one can find the residual gauge symmetry parameter [61]. However, within this method, it is not clear how to distinguish the large gauge transformation from the smooth one (see Appendix A) Alternatively, one may consider an infinitesimal gauge transformation, which guarantees the smoothness of gauge transformation. For example, let us consider the following (infinitesimal) gauge transformation . one can express λ 0 (z) and λ −1 (z) in terms of λ 1 (z) by demanding the transformation to keep the gauge condition in (4.31). The residual gauge transformation parametrized by λ 1 (z) leads to the transformation of the constant L 0 : This is a special case of the asymptotic symmetry of AdS 3 [56,57], or equivalently, the residual gauge symmetry of Chern-Simons gravity, which leads to Virasoro algebra. Note that from the anomaly one can determine the central charge: For the conformal transformation, one can integrate the infinitesimal transformation into a finite one where f (z) is the finite residual gauge transformation (conformal transformation) parameter. Also, noting that we have one can write L(z) as With a similar analysis forāz, the on-shell action in (4.32) is found to be (4.46) which is analogous to the finite temperature Schwarzian action derived in [17].
In the metric-like formulation of 3-dimensional gravity, one implements Dirichlet boundary conditions with the Gibbons-Hawking term, which leads to interaction between the holomorphic and anti-holomorphic soft modes. This interaction, however, is suppressed at large-N (or at large central charges). On the other hand, in the Chern-Simons formalism, we did not include the Gibbons-Hawking term, and we have the decoupled on-shell action.

Four point function in Euclidean correlators
To evaluate the out-of-time-ordered correlator via analytic continuation, we start with the Euclidean four point function on the boundary. For simplicity, we consider cases where the four point function can be viewed as two point function of bi-local operators Φ(z 1 ,z 1 ; z 2 ,z 2 ) of which the leading contribution is one-point function (classical solution) of each bi-local operator where G i (z 12 ,z 12 ) (i = 1, 2) is defined by We now list some cases where such a bi-local operator can be considered.
Heavy and light scalar operators with sparse spectrum: One can consider the boundaryto-boundary four point function of heavy and light matter scalar fields. Then, the leading contribution will be factorized into a product of boundary-to-boundary two point functions of light and heavy operators, respectively. And, one may take the bi-local operators as follows where we take the r coordinate of each end symmetrically. For constant connections a andā, this becomes a boundary-to-boundary propagator (4.51) For sl(N ) Chern-Simons gravity, the conformal dimension h is negative In fact, a similar object was observed in the Vasiliev equation [67]. In the SL(N ) version of the Vasiliev system, the equation of motion for the matter master field C ∈ SL(N ) is given by dC + AC − CĀ = 0 . For the given background connections A andĀ, we have C(r 1 , z 1 ,z 1 ; z 2 ,z 2 ) = lim wherec 0 is a constant matrix. After taking r 2 → ∞ limit, it becomes "the highest weight state" i.e., lim It was shown [67,68] that the trace of the master field C corresponds to the physical scalar field in the higher spin gravity. In particular, for the constant connection a andā, it becomes a bulk-to-boundary propagator [67]. Note that the coordinates (z 2 ,z 2 ) was chosen as the initial condition of the equation of motion, and it corresponds to the position of operator inserted on the boundary. The conformal dimension h of the master field is the same as that of Wilson line This is not surprising in the higher spin AdS/CFT. In the hs[λ] higher spin gravity which contains the infinite tower of higher spin field together with scalar field, the conformal dimension of the scalar field is given by where λ ∈ [0, 1]. In the semi-classical limit where we take analytic continuation λ → −N , one can truncate the infinite tower of higher spin field, and the gauge sector can be described by SL(N ) Chern-Simons gravity. In this limit, the conformal dimension of the scalar field becomes (4.57). Like Wilson line, one can also take the boundary-to-boundary propagator from the master field C as our bi-local operator: Φ Master (z 1 ,z 1 ; z 2 ,z 2 ) = lim r 1 →∞ e 2hr 1 tr [C(r 1 , z 1 ,z 1 ; z 2 ,z 2 )] (4.59) In the BTZ background the modular parameter τ,τ becomes purely imaginary, and the periodicity of φ requires the one point function of bi-local operator (which will be the boundary-to-boundary propagator) to be Note that for the case of Wilson line this corresponds to the summation of Wilson lines winding the non-contractible cycle of solid torus. For convenience, we define and the boundary-to-boundary propagator can be written as For the non-constant background a andā, our bi-local operator can be understood as a gravitationally dressed operator, and we can expand our bi-local operator Φ(z 1 ,z 1 ; z 2 ,z 2 ) around that in the constant background G(z 1 , z 2 ;z 1 ,z 2 ) where is an infinitesimal expansion parameter. Recall that our non-constant solution is generated from the constant BTZ solution by an infinitesimal residual gauge transformation, which corresponds to a conformal transformation on the boundary. Hence, the expansion of the dressed operator can be understood as an infinitesimal conformal transformation of the bi-local operator on the boundary. Namely, under an infinitesimal conformal transformation the bi-local operator can be written as where δ n G m (z 1 , z 2 ;z 1 ,z 2 ) is found to be (4.66) Here, we used the center of bi-local coordinates and relative coordinates defined by and a similar relation for the anti-holomorphic coordinates. As in [69], one can also understand it as expansion of bi-local field to construct a bi-local conformal partial wave function. For this, one can consider correlation function of the bi-local field and −n , which is conjugate to This is nothing but the conformal Ward identity. It is more clear to see the soft mode expansion of the gravitational dressed Wilson line (as well as the master field). The bi-local operator from the Wilson line with non-constant connection a = h −1 a BTZ h + h −1 ∂h can be written as follows For infinitesimal residual gauge parameter h(z), we expand h(z) around identity h(z) = I + λ(z) + · · · (4.70) Recall that the infinitesimal residual gauge transformation is parametrized by λ 1 (z). Then, from the mode expansion of λ 1 (z) we can also expand the dressed Wilson line with respect to n 's For the BTZ background, we also consider the expansion of the image of the dressed bi-local operator Φ dressed m (z 1 ,z 1 ; z 2 ,z 2 ) due to the periodicity of φ ∼ φ + 2π, and we denote it byf n,m In evaluating OTOCs, a particular configuration of (z 1 , z 2 , z 3 , z 4 ) simplify the four point function [10,17,48,70,71]. Namely, we will consider the configuration (z 1 ,z 1 ; z 2 ,z 2 ) = (χ − τ 4 ,χ −τ 4 ; χ + τ 4 ,χ +τ 4 ) or (χ, σ,χ,σ) = (χ, − τ 4 ,χ, −τ 4 ). For example, G m becomes Note that the second term is odd in m, which will be cancelled when we sum them up over the integers m. Hence, the expansion of the dressed bi-local field becomes Note that a special choice of the configuration allow us to factor out G BTZ (χ, − τ 4 ; χ, −τ 4 ). In general, this is not true, in particular, the same choice of coordinates does not simplify δ n Φ dressed m for the case of higher spin gravity [48]. Nevertheless, each term in the summation over m has the same exponential growth in time. Hence, the summation would, at most, change the overall factor of the exponential growth, and therefore, could change the scrambling time.
Now, using the expansion of the dressed bi-local operator, we will evaluate the two point function of bi-local operators, which corresponds to the four point function of the local field. For this, we need to evaluate the correlation function of soft graviton modes, n 's. This can be evaluated from the on-shell action (4.46) of the soft graviton mode around the constant background. Since the on-shell action is non-linear, we evaluate the quadratic action by taking an infinitesimal fluctuation around the identity Expanding the on-shell action, the quadratic action takes the form Note that the on-shell action vanishes for n = 0, ±1, which corresponds to the isometry of constant solutions. 9 From the quadratic on-shell action, one can read off the two point function of the boundary graviton fluctuation n −n = κ n 2 (n 2 − 1) , where κ andκ are defined by (4.80) We will evaluate the following configuration of the OTOC regularized by the thermal density matrix ρ ≡ e −βH : For this, we consider the Euclidean four point function at Recalling the expansion of the dressed bi-local operator in (4.76) at the special configuration, one can further simplify that of (z 3 ,z 3 ; z 4 ,z 4 ) Finally, the contribution of the boundary graviton fluctuation to the Euclidean four point function can be evaluated as follows 10 where only the even terms give a contribution to the four point function due to the cos nπ 2 factor. Then, one can rewrite the summation as a contour integral where the contour C is a collection of circles centered at n = 2Z with small radius.

Boyer-Lindquist co-rotating frame
In this section, we will shortly discuss the Boyer-Lindquist co-rotating frame, which is necessary for a consistent analytic continuation from Euclidean time to Lorentzian time in the four point functions.
In [72] In our TFD formulation, the operators are defined on the space (t,φ), wheret denotes the complex time coordinates. Then, the periodicity condition becomes

Discussion
In this paper we have studied the onset of chaos in rotating BTZ black holes. We have computed OTOCs using two different methods: the elastic eikonal gravity approximation [9], and a new method that is based on the Chern-Simons formulation of 3-dimensional gravity.
In the first case the OTOC can be obtained from a high energy shock wave collision in the bulk, while in the second method one explicitly derives an effective Schwarzian-like action for the boundary degrees of freedom, and computes the OTOC from analytic continuation of the Euclidean 4-point function. Both methods give the same result, which, in terms of Schwarzschild coordinates, takes the form with Ω = r − r + being the chemical potential for angular momentum. The two terms correspond to left and right moving modes, which have the same butterfly velocity 11 , v B /c = 1, but different Lyapunov exponents This result is consistent with [27], where the author used the metric formulation of 3dimensional Euclidean gravity to derive an effective action for the boundary degrees of freedom of the rotating BTZ black hole. A somewhat similar result was also found in [28], where the author studied the disruption of the mutual information in TFD states. However, due the special configuration considered in [28], their result is only sensitive to the mode with the lowest Lyapunov exponent (λ L = 2π β − ). Naively, since λ − L ≤ 2π β ≤ λ + L , the above result seems to indicate that one of the Lyapunov exponents is less than maximal, while the one violates the chaos bound. However, in the decomposition described in section 2.2, the parameters β −1 ± are precisely the effective temperatures of the left and right moving modes. Therefore, the chaos bound is different for each mode, and our result shows that each mode saturates its own chaos bound. This interpretation is very natural if one views the BTZ black hole as a part of a D1-D5 brane system. From this perspective, β(1 ∓ Ω) are the inverse temperatures of the left and right moving excitations on the effective string [73].

Co-rotating frame
The black hole temperature, β −1 , is defined in co-rotating coordinates (t, φ), in terms of which the result takes the form OTOC(t, φ 12 ) ≈ 1 + C 1 e In this frame, both modes have the same temperature, which results in the same Lyapunov exponent, namely λ L = 2π β . This is consistent with [29], in which the authors (using the co-rotating frame) studied chaos in the rotating BTZ black hole using the geodesic approximation. Since the authors of [29] only consider homogeneous shock waves, their result has no angular dependence, and it is controlled by a single mode, with maximal Lyapunov exponent. The change of frame breaks the symmetry between the butterfly velocities, which now become v ± B c = 1 ± Ω . The rotation affects the butterfly velocities, and one of them becomes superluminal when Ω > 0. Since v B defines an effective light cone for the butterfly effect [8], one expects this quantity to be bounded by the speed of light. In fact, in asymptotically AdS geometries satisfying the Null Energy Condition (NEC) one can show that [74] v B ≤ c . (5.5) This bound is known to be violated when the boundary theory displays non-local effects [75], which are related to a non-asymptotically AdS geometry in the dual gravitational description.
Since the rotating BTZ geometry is asymptotically AdS and satisfies NEC, the violation of the bound (5.5) suggests some sort of curious property of the co-rotating frame.

Boyer-Lindquist co-rotating frame
In the Boyer-Lindquist coordinates, the OTOC takes the form OTOC(t,φ 12 ) = 1 + B 1 exp 2π β t −φ In this frame both modes have the same, maximal, Lyapunov exponent, but they have different butterfly velocities. Just like in the case of co-rotating coordinates, one of the butterfly velocities can become superluminal, i.e., v B > c. But here this only happens when Ω > √ 2−1, while in the co-rotating coordinates this happens for any value of Ω.

Extremal limit
The extremal limit occurs when r − → r + . In the co-rotating frame (or in the Boyer-Lindquist co-rotating frame), both modes have zero Lyapunov exponent. That is consistent with the idea that the co-rotating observer only sees the zero-temperature black hole. Interestingly, this is not the case in the non-rotating frame. In this case, λ − L = 0, but λ + L = r + / 2 . That means that one of the modes survives in the extremal limit, with effective temperature given by T = r + 2π 2 . A similar phenomenon was also observed in [76].