BTZ dynamics and chaos
Rohan R. Poojary








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        ^1Department of Theoretical Physics, 
        
        Tata Institute of Fundamental Research,
        
        Mumbai 400005, India.
        
        ^2Chennai Mathematical Institute,
        
        H1-Sipcot IT Park, Siruseri 603103, India.
        
        
        rpglaznos@gmail.com,ronp@theory.tifr.res.inWe find an effective action for gravitational interactions with scalars in AdS_3 to first order in G_N at the conformal boundary. This action can be understood as an action for the Brown-Henneaux modes and is given by the square-root product of right and left moving Schwarzian derivatives for conformal transformations of the boundary. We thus reproduce the result  λ_L=2π/β for OTOC computed first in  arXiv:1412.6087 for a Schwarzchild black hole in AdS_3. Applying the same procedure to rotating BTZ we find the Lyapunov index to be λ_L=2π/β_+>2π/β where β_+=β(1-μ_L), with μ_L=r_-/r_+ being the chemical potential for angular momentum. We thus comment on a possible generalization to a part of the proof given in arXiv:1503.01409 to accommodate this result.  
        
        
        
        
        
        BTZ dynamics  and chaos.  
            Rohan R. Poojary^1,2
            
        ==========================
        
        
        
        
        § INTRODUCTION
        
        A very intriguing phenomena of strongly coupled thermal systems is chaos. In a classical sense, phase space trajectories which differ in their initial values by a small amount tend to grow exponentially far apart at later times.  A good quantum mechanical analogue of this would be the time scale when
        
            C(t)=⟨[W(t),V(0)]^2⟩_β
          
        becomes equal to 2 ⟨ WW⟩_β⟨ VV⟩_β. Here, W & V are simple Hermitian operators with 𝒪(1) degrees of freedom. The exponential increase in the time of C(t)≈ e^λ_L t can be considered as the Lypunov index generically associated with chaotic systems <cit.>. The chaotic behaviour of thermal large N CFTs is related to the chaotic behaviour of black holes via the gauge-gravity duality, the latter   are conjectured to be the fastest "scramblers" of information <cit.>. Shenker and Stanford <cit.> first computed the out-of-time-ordered (otoc) term in (<ref>)
        
            ⟨ W(t)V(0)W(t)V(0) ⟩_β
        
        holographically using the eikonal approximation. In <cit.> the next order in probe approximation in G_N was computed for a 2→2 scattering for 2 minimally coupled scalars in AdS_d Schwarzchild interacting with each other only via gravity. The Lyapunov index thus obtained was λ_L=2π/β, β being the inverse temperature of the AdS_d Schwarzchild. This lead Maldacena, Shenker and Stanford <cit.> to propose a bound on the Lyapunov index of large N thermal QFTs to be λ_L ≤ 2π/β by using generic arguments of unitarity and analyticity of Whightman functions on the complex plane. It was assumed that holographic CFTs saturate this bound as evidenced by <cit.>.
        
        
        
        Further interest in chaotic systems was heightened by the study of the SYK <cit.>and SYK-like models initiated first in <cit.> and <cit.>; in <cit.> Maldacena and Stanford found that the otoc for the fermions has λ_L=2π/β, this computation was done in the strong coupling (zero temperature) limit  of the SYK model where the model is conformal. In order to compute the leading contribution to the 4pt. function they had to break the conformal invariance at zero temperature. The modes which are responsible for maximizing chaos where shown to be the modes related by diffeomorphism which now have an action due to breaking of conformal invariance. Their effective action was computed and found to be the Schwarzian derivative of reparametrizations of the thermal circle. Many interesting properties of the SYK model have since been uncovered <cit.> to quote a few. 
        
        
        
        
        
        There have been many variations to the original SYK problem which had relied upon averaging over a space of couplings. A unitary model proposed by Gurav<cit.> and Witten<cit.> showed a similar behaviour to the SYK model at large N. There have also since been higher-dimensional and super-symmetric avatars of this model, <cit.> study interesting properties.
        
        
        
        
        
        This lead to investigations to ascertain the bulk degrees of freedom which are responsible for similar chaotic behaviour. The dynamics of near extremal black holes was found to be captured by a 2d dilaton-gravity theory of Jackiw<cit.> and Teitelboim<cit.> in <cit.>. The   nAdS_2 dynamics of  Jackiw-Teitelboim (JT) action is essentially dictated by its asymptotic symmetries since the theory possesses zero propagating degrees of freedom. The effective action for these modes was captured by the Schwarzian action for the AdS_2 boundary diffeomorphisms <cit.>, similar ideas where pursued in <cit.>. Explicit computations on near extremal RN AdS_4 black holes <cit.> corroborated this understanding from a higher dimensional perspective.  
        
        
        
        The dynamics of the 2d gravity theory reproducing the Schwarzian effective action have since been studied 
        <cit.>. Apart from the JT action, the Polyakov action with a cosmological constant was also studied and found to describe the AdS_2 bulk dynamics dual to the soft modes of SYK <cit.>. This was done by analysing the action for the co-adjoint orbits of the Virasoro group which can be thought to describe the soft modes. Different aspects of AdS_2 gravity were also covered in  
        <cit.>. The effect of AdS_2 arising in rotating horizons was also studied in <cit.>, here a large N SYK like system was modelled to mimic the near horizon near extremal symmetries of Kerr-Neuman blackholes in AdS_4. For past works the reader may refer to <cit.> and references therein. 
        
        
        
        There have also been efforts to realize the SYK model completely by a providing a holographic description <cit.>. These have also been studied for the SYK tensor models too in some detail <cit.>. 
        
        
        
        It is also worth noting that a theory of open string governed by the Nambu-Goto action  probing an AdS Schwarzchild geometry also exhibits maximal chaos <cit.>. In such a system the scrambling time is governed by the string tension. A Schwarzian effective action has also been uncovered for such systems as being comprised of the reparametrizations of the world sheet <cit.>.  
        
        
        
        It would be an interesting question to ask if such modes can be found in thermal large N CFTs such that their effective actions govern the chaotic behaviour of the system, like in the SYK model studied in <cit.>. The present holographic understanding of this phenomenon allows one to visualize these modes close to extremality in the near horizon region for at least non-rotating geometries.  The near extremal geometries possess a near horizon AdS_2 throat, and bulk scattering of the form studied in <cit.> excite graviton modes which can be described by a JT theory confined to this throat region <cit.>.  It is worth noting that the holographic computations in <cit.> do not assume extremality. It would therefore be worthwhile to understand how these modes behave away from extremality and also in the entirety of a black hole in AdS. 
        
        
        
        To this end we address a simpler problem of that in AdS_3 which like in the dilaton-gravity theory in AdS_2 has only boundary degrees of freedom. In fact in AdS_3 these have been well studied and are called the Brown-Henneaux modes <cit.> which are in one-to-one correspondence with 2d infinite conformal symmetries of the boundary CFT_2. In section 2 using the gauge gravity prescription we first formally equate the computation of eikonal scattering in the bulk done in <cit.> to computing correlators in the boundary CFT upto linear order in G_N. This we do by computing the effective action for conformal transformations on the boundary obtained from the bulk on-shell path integral. We then (section 3) compute the effective action for the Brown-Henneaux modes about a rotating BTZ and find it to be the product of square-root Schwarzian derivatives, each for left and right moving conformal transformations of the boundary.
        
        
        
        In section 4 we proceed to compute the correction to the 4pt function of 2 boundary operators -  computed in the probe approximation; to linear order in G_N. We thus reproduce the answer of <cit.> for the non-rotating BTZ case of λ_L=2π/β. The similar procedure when used for the rotating BTZ yields λ_L=2π/β_+ where 
        β_±=β(1∓μ_L) with μ_L=r_-/r_+ being the chemical potential associated with angular momentum. We thus find that for the rotating BTZ λ_L=2π/β_+>2π/β.
        
        
        
        We end with section 5 with some conclusions and discuss the possible implications of the result. In particular we point out a possible modification of a part of the proof given in <cit.> so as to allow for a modified bound in the presence of a chemical potential for angular momentum.  
        
        
        
        
        § BULK COMPUTATION
        
        In this section we heuristically equate the eikonal approximate calculation of <cit.> to the one generally done while introducing the AdS/CFT correspondence i.e. equating the bulk on-shell (small G_N) path-integral to the generating function of boundary correlators:
        
            ϕ→ϕ_0g→η∫𝒟[g]𝒟[ϕ_i]  e^i(S_grav+S_matter)=Z_CFT[ϕ_0]=⟨ e^i∫_∂ϕ_0𝒪⟩_CFT
        
        where ϕ_0 is the boundary value of the scalar field in the bulk which sources a scalar operator 𝒪 in the boundary CFT.
        Like in <cit.>
        we will consider 2 minimally coupled scalar fields in the bulk with masses m_1& m_2[There is summation in i and the space time integrals are suppressed for brevity.], with no interaction terms between them
        
            S_matter=-∫√(-g)12[(∂ϕ_i)^2-m^2_iϕ_i^2],  S_grav=-1/16π G_N∫√(-g)(R-2Λ).
        
        Here we have not written down the boundary terms for the actions which make the variational problem well defined and render the on-shell action finite.
        
        
        
        We will concern ourselves with the computation of the bulk 4pt. function ⟨ϕ_1ϕ_1ϕ_2ϕ_2⟩, which is equal to the boundary 4pt. function
        
            ⟨𝒪_1𝒪_1𝒪_2𝒪_2⟩≈lim_r→∞r^-2(2d-Δ_1-Δ_2)⟨ϕ_1ϕ_1ϕ_2ϕ_2⟩
        
        where each of the bulk coordinates is taken to the boundary[Δ_i=d2+√(d^24+m_i^2l^2), where l is the AdS radius.]. Using the bulk path integral expression, the 4pt function is given by
        
            ⟨ϕ_1ϕ_1ϕ_2ϕ_2⟩=∫𝒟[g]𝒟[ϕ_i]  ϕ_1ϕ_1ϕ_2ϕ_2   e^i(S_grav+S_matter)
        
        Here for simplicity of notation we have not mentioned the space time dependence. Since we would be concerned with the limit in which classical gravity dominates we would be interested in the saddle point evaluation of the above path-integral. Further  one usually considers the probe approximation in which the scalars act as probes for a given metric which satisfies vacuum Einstein's equation with cosmological constant Λ. We will denote this solution as g̅_μν. In this limit since the matter action is quadratic the answer is readily computed from the expression
        
            ⟨ϕ_1ϕ_1⟩_g̅⟨ϕ_2ϕ_2⟩_g̅ =∫𝒟[ϕ_i]  ϕ_1ϕ_1ϕ_2ϕ_2   e^i(S_grav[g̅]+S_matter[ϕ̅_̅i̅]),
        
        here ϕ̅_̅i̅ solves the Klein-Gordon equation in the background g̅_μν[This is basically the expression for the propagator in the form of a path integral for a free theory. ]. Let us consider the effect of first order back reaction in orders of G_N by considering
        
            R_μν-1/2Rg_μν+Λ g_μν=4π G_N T_μν
        
        where g_μν=g̅_μν+h_μν and T_μν is determined entirely in terms of ϕ̅_̅i̅s. Therefore we can rewrite (<ref>) as
        
            ⟨ϕ_1ϕ_1ϕ_2ϕ_2⟩   =   ∫𝒟[h]𝒟[ϕ_i]  ϕ_1ϕ_1ϕ_2ϕ_2   e^i{ S_grav[g̅]+δ S_grav[h]+S_matter[g̅,ϕ̅_̅i̅]+δ S_matter[h]}   =   ∫𝒟[h]⟨ϕ_1ϕ_1⟩_g̅+h⟨ϕ_2ϕ_2⟩_g̅+h   e^iδ S_grav[h]   =   ∫𝒟[h]⟨ϕ_1ϕ_1⟩_g̅+h⟨ϕ_2ϕ_2⟩_g̅+h   exp[il16π G_N∫12h D^2h]
        
        where ⟨ϕ_1ϕ_1⟩_g̅+h denotes the 2pt. function evaluated in background metric g̅_μν+h_μν for an h_μν determined by (<ref>) and constrained by boundary conditions on the bulk metric. Note, that here we have only considered diagrams in which the graviton lines attach to different scalar legs, corrections to scalar propagator due to gravitons attaching to the same scalar propagator have been ignored. 
        
        
        
        In the eikonal approximation i.e. in the limit when the scalar field momenta are taken to be large and light-like, (<ref>) reduces to <cit.>
            ∫𝒟[h]   exp[il/16π G_N∫12hD^2h +h_μνT^μν]
        
        where we have assumed that the T_μν due to the matter fields are like shockwaves with light-like momenta. In this approximation the incoming and the out-going momenta are the same and we get the effect of infinite graviton ladder exchanges between the 2 scalar propagators. This precisely the origin of the e^iδ(s) factor in <cit.>. Here h_μν is the response to the shockwave generated by high momentum scalar propagators in T_μν.
        This approximation is justified in the shock wave analysis of <cit.> since the scattering is arranged to have a maximum contribution from the near horizon region. By the time any scalar perturbation reaches the horizon area it would be blue-shifted exponentially, thus giving the first leading contribution to the correction to the 4pt function. The eikonal approximation used in <cit.> effectively means that the contribution comes from the bifurcate horizon. One can then use bulk to boundary propagators to  compute the correlation function on the boundary. 
        
        
        
        Let us now try to rephrase the same computation in AdS_3. We  write the probe approximate 4pt function as
        
            ⟨𝒪_1𝒪_1⟩_g̅⟨𝒪_2 𝒪_2⟩_g̅=lim_r→∂∫𝒟[ϕ_i]  ϕ_1ϕ_1ϕ_2ϕ_2   e^iS_matter[g̅,ϕ̅_̅i̅]
        
        here we have haven't included S_grav[g̅] since it would a constant. The correction it would receive from the gravity path integral would be captured in 
        
            ⟨𝒪_1𝒪_1𝒪_2𝒪_2⟩=∫𝒟[g] ⟨𝒪_1𝒪_1⟩_g⟨𝒪_2 𝒪_2⟩_g   exp[il16π G_N∫√(-g)(R-2Λ)].
        
        We could now consider the above path integral being dominated by the gravity saddle by considering the small G_N limit. Therefore we would only be seeking contributions from geometries satisfying vacuum Einstein's equations. This way we should be able to reproduce the leading correction in the small G_N limit. 
        In the above metric path integral only boundary degrees of freedom contribute in AdS_3. These are in one-to one correspondence with the boundary conformal transformations[Barring the truly small diffeomorphisms which we ignore.]. Further ⟨𝒪_1𝒪_1⟩_g≈⟨𝒪_1𝒪_1⟩_g̅+h would simply correspond to the change in the 2pt. functions due to conformal transformations. Therefore atleast in AdS_3 we must be able to capture the effect of chaos via (<ref>).
        
        
        
        § ADS_3 STORY FROM THE BOUNDARY
        
        In this section we will derive the effective action for the soft modes by calculating the gravity bulk on-shell action about an arbitrary BTZ geometry.
        
        As justified in the previous section this amounts to finding the effective action for the diffeomorphisms in the bulk respecting Dirichlet boundary conditions. It is known that such configurations in the bulk are dual to states in the boundary CFT which are created by the action of 2 copies of commuting Virasoro algebra. We begin by writing the most general bulk configuration with the boundary metric being flat, we begin with a Lorenztian metric in the Fefferman-Graham gauge <cit.>.
        ds^2/l^2   =   dr^2/r^2-r^2dx^+dx^-/4+1/4(T_++dx^+2+T_–dx^-2)-1/4r^2T_++T_–dx^+dx^-,
         
        where T_++=T_++(x^+)  T_–=T_–(x^-) and x^±=t±ϕ. One can cast the BTZ <cit.> metric in AdS_3ds^2/l^2   =   r^2dr^2/(r^2-r_+^2)(r^2-r_-^2)-(r^2-r_+^2)(r^2-r_-^2)dt^2/r^2+r^2(dϕ-r_+r_-/r^2dt)^2      
        M   =   r_+^2+r_-^2,  J=2lr_+r_-.
        
        in the Fefferman-Graham gauge with T_±±=(r_+± r_-)^2. It is worthwhile to notice that the radial coordinate in (<ref>) sees the horizon at r_h=√(r_+^2-r_-^2)[Throughout the text r_± would refer to outer or inner horizons in the metric in (<ref>)and will be casual in the use of r as the radial coordinate in any metric.  ]. 
        
        
        The bulk action with relevant boundary(counter) terms is<cit.>
        16π G_N S_bulk   =   ∫ d^3x √(-g)(R-2Λ)+2∫_∂d^2x√(-h)(K+1/l),
         
        where Λ=-1/l^2 and l is the length of AdS_3. The 1/l term in the boundary action is used to make the on-shell action finite. Therefore the on-shell value of (<ref>) for arbitrary metrics given by (<ref>) is
        
        S^on-shell_bulk   =   l/64π G_N∫_∂(r_h^2+T_++T_–/r_h^2)   =   l/32π G_N∫_∂√(T_++T_–) 
        Since gravity in 3-dim is non-dynamical in the bulk, all solutions to the bulk action (<ref>) can be obtained from one another via diffeomorphisms. In fact the Fefferman-Graham theorem <cit.> allows one to express any solution in the form (<ref>) as a diffeomorphisms of the other and we thus only need to concern ourselves with diffeomorphisms that preserve this form to generate all solutions. We would therefore like to know the action (<ref>) associated with such a diffeomorphism, having started from a particular solution (for eg.) of the form of the BTZ metric (<ref>). To this end we would like to know how T_±± depend on these diffeomorphisms. 
        
        
        
        We notice that for infinitesimal diffeomorphisms which maintain the form of (<ref>), the change in T_±± is given by <cit.>δ T_±±   =   ξ^±_(0)T'_±±+2ξ_0^±'T_±±-2ξ_0^±”'
        
        where[The primes denote derviatives w.r.t. respective coordinate dependence.]ξ^μ∂_μ   =   ξ^r∂_r+ξ_(0)^+(x^+)∂_+ + ξ_(0)^-(x^-)∂_-+𝒪(1/r)ξ^r   =   -r/2(ξ_(0)^+'+ξ_(0)^-').
            
        We also note that change in a Schwarzian derivative {T(u),u} due to a diffeo u→ u+ ϵ(u) is:
        {T(u)+ϵ(u)T'(u),u}   =   {T,u}+ϵ(u)∂_u{T,u}+2ϵ'(u){T,u}+ϵ”'(u).
        
        One can find the full non-linear completion of (<ref>) <cit.> which takes a Poincáre AdS_3 (with T_±±=0) to (<ref>). Under such a diffeomorphism the stress tensor is proportional to the Schwarzian for boundary conformal transformations. Comparing (<ref>) and (<ref>) we can deduce that under x^±→ X^±(x^±)
        T_±±   =   -2{X^±,x^±} where    {X,x}   =   2X'X”'-3X”^2/2X'^2
        where for infinitesimal diffeomorphisms X^±≡ x^± + ξ_(0)^±. One notices that for X^±=x^±T_±±=0. This value can be shifted by defining
        
        T_±±   =   -2{X^±,x^±}+L_±X^±'^2
        
        where the choice of X^±'^2 makes sure that the linear in T_±± terms in (<ref>) remain the same. Here, L_± define the charge of the BTZ metric about which the change in the parameters T_±± is measured[T_±± are components of the Brown-York stress tensor for the bulk metric, which are also the CFT_2 stress tensor components.].
        
        
        
        Therefore the on-shell action in (<ref>) is
        
        S^on-shell_bulk   =   l/32π G_N∫_∂√((-2{X^+,x^+}+L_+X^+'^2)(-2{X^-,x^-}+L_-X^-'^2))
        where L_± decides which bulk configuration one mesures the change from. The above action is defined on the boundary of AdS_3, the integral in the AdS radial direction receives contribution from r=r_h and the boundary r=∞. The divergent contribution from the boundary at r=∞ is cancelled by the holographic boundary counter-terms, the only finite contribution comes from the horizon. Under infinitesimal diffeomorphisms X^±→ x^± + ϵ^±(x^±) the quadratic action takes the form
              S^on-shell_bulk=-l64π G_N(L_+L_-)^3/2∫_∂(L_-^2(ϵ^+”'(x^+)^2+L_+ϵ^+”(x^+)^2)+L_+^2(ϵ^-”'(x^-)^2+L_-ϵ^-”(x^-)^2))      
        where we have ignored boundary terms. Since we would be interested in computing OTOC's later we Euclideanize the above action
              S^on-shell_bulk,E=l64π G_N(L_+L_-)^3/2∫_∂(L_-^2(ϵ^+”'(x^+)^2-L_+ϵ^+”(x^+)^2)+L_+^2(ϵ^-”'(x^-)^2-L_-ϵ^-”(x^-)^2))      
        The quadratic action divides itself into left and right sector. 
        
        
        The action (<ref>) evidently has the symmetries of the Schwarzian derivative, the infinitesimal versions of which are manifested in (<ref>). We would correspondingly have got the above action by working in the bulk in a Euclidean setting to begin with. This would have invariably required us to have the angular momentum associated with the BTZ metric to be imaginary so as to have a real Euclidean metric.
        
        
        
        One can in principle derive the one-loop exact action for the boundary gravitons in AdS_3 as was done recently by <cit.> using the Chern-Simmons prescription. The authors obtain a theory of re-parametrizations which encodes loop contributions of boundary gravitons in a perturbation in 1/c[c being the central charge.]. 
        
        
         §.§ Propagators
        
        We are now in the Euclidean setting where  the time τ=-it is along the imaginary direction while the space-like coordinate ϕ is real. The x^± coordinates of Euclideanised BTZ metric can be regarded to have complex periodicities <cit.>
        x^± = x^±+iβ_±,     β_±=β∓ iΩ=2π/√(L_±) 
        We regard the integral in (<ref>) to be in one such periodic interval, therefore the above action splits into two 1-dim actions
        
        S_+[ϵ^+]=α_+∫ϵ^+(∂̅^(6)+L_+∂̅^(4))ϵ^+,   
        S_-[ϵ^-]=α_-∫ϵ^-(∂^(6)+L_-∂^(4))ϵ^- ,
          
        where α_± = -2π l/64π G_N L_±^3/2. For convenience we have defined
        z̅=-ix^+=z̅+β_+,   z=-ix^-=z+β_-
        
        and we analyse the propagator for ϵ^+, for which we evaluate the Green's function G_+ for the operator
        𝕆_+=∂̅^(4)(∂̅^(2)+(2πβ_+)^2),   𝕆_+G_+=δ(z̅).
        
        Here, we observe that G_+ would depend on z̅ by a function of the ratio z̅/β_+. The zero modes themselves would look like {1,e^2πz̅β_+,e^-2πz̅β_+}. We will compute the  G_+ first for the Schwarzchild case and try and generalize for the rotating BTZ case.
        
        
        
        For AdS_3 Schwarzchild, β_+=β_-=β∈ℝ. Assuming G_+ be a function of z̅/β we solve (<ref>) for real values of z̅/βi.e.G_+(τ/β) and then using Schwarz's theorem analytically continue it for arbitrary complex values of z̅/β<cit.>.
        
        
        
        
        
        
        
        This allows us to express both sides of the (<ref>) as a discrete sum, thus
        
        G_+=1/α_+'∑_n=-∞^∞e^2π i n z̅/β/(2πβ)^6 n^4 (1-n^2),∀  z̅/β∈ℝ,
        
        where the prime on the sum denotes n∉{-1,0,1}. Doing the relevant Matsubara summation and analytically continuing in the complex z̅/β plane yields
        
        (π^3 l4G_Nβ^3)G_+(z̅)   =   124(2πz̅β-π)^4 -(π^2+6)12(2πz̅β-π)^2+      +π(2πz̅β-π)sin(2πz̅β)+
        
        +a+bcos(2πz̅β).
        
         Here,
        
            z̅/β=
            z̅/β,    if Re[z̅/β]>0
               
            -z̅/β,    if Re[z̅/β]<0.
        
        The last 2 terms[Explicitly: a=(1+π^26+7π^4360)&b=9/2. ] in (<ref>) are comprised of the zero modes we neglected in the sum and would drop out of any computation which respects the bulk isometries. An identical expression would exist for G_- in terms of z/β.
        
        
         §.§ Rotating BTZ propagators
        
        One could extend the above method naively to rotating BTZ case. This would require solving the (<ref>) for the real values of z̅/β_+z̅/β_+∈ℝ(τ+μϕ)+i(μτ-ϕ)/β(1+μ^2)∈ℝϕ=μτ,
        
        where Ω=μβ. Same holds true for z/β_-∈ℝ. It is quite clear from the outset that one could define Euclidean coordinates {τ̃,ϕ̃}.
        
            τ̃=τ+μϕ/(1+μ^2),ϕ̃=ϕ-μτ/(1+μ^2).
        
        Therefore z̅/β_+=z̅̃̅/β, where z̅̃̅=τ̃-iϕ̃. Which is a conformal transformation on the boundary metric:
        
            ds^2=dτ^2+dϕ^2→(1+μ^2)(dτ̃^2+dϕ̃^2).
        
        We do not however perform such a transformation on the propagator, we merely use (<ref>) for making the coordinated dependence look simple. Therefore solving (<ref>) for real values of z̅̃̅/β and analytically continuing we get
        (π^3 l4G_Nβ^3_+)G_+(z̅)   =   124(2πz̅β_+-π)^4 -(π^2+6)12(2πz̅β_+-π)^2+      +π(2πz̅β_+-π)sin(2πz̅β_+)+
        
        +a+bcos(2πz̅β_+).
        
        where
        
            z̅/β_+=
            z̅/β_+,    if Re[z̅/β_+]>0
               
            -z̅/β_+,    if Re[z̅/β_+]<0.
        
        Note that the conformal transformation (<ref>) isn't one of the SL(2,ℝ) zero modes.
        
        
        § 4PT CORRELATOR
        
        In this section we will use the propagators obtained in the last section to compute the next to leading order in G_N corrections to the 4pt. function. We consider the first the leading contribution to the Euclidean 4pt. function of four scalars <cit.>⟨ V_1V_2W_3W_4⟩ = 1/sin^2h̅
        _1(πz̅_12/β_+)sin^2h_1(π z_12/β_-)sin^2h̅2(πz̅_34/β_+)sin^2h_2(π z_34/β_-)
        where z_12=z_1-z_2 and V_1≡ V(z̅_1,z_1). We would be interested in seeing how they would depend on the bulk on-shell metrics in the path integral (<ref>). As explained before, these would correspond to computing the change in (<ref>) due to conformal transformations parametrized by ϵ^+(z̅)&ϵ^-(z). Under z̅→z̅+ϵ^+(z̅)&z→ z+ϵ^-(z) we have
        1/sin^2h̅(πz̅_12/β_+)sin^2h(π z_12/β_-)   →   ℬ(ϵ^±_1,ϵ^±_2)1/sin^2h(πz̅_12/β_+)sin^2h̅(π z_12/β_-),ℬ(ϵ^±_1,ϵ^±_2)   =   h̅[(ϵ^+'_1+ϵ^+'_2)-(2π/β_+)(ϵ^+_1-ϵ^+_2)/tan(πz̅_12/β_+)]+ c.c
        It can be seen that ℬ above is invariant under the SL(2,ℝ) zero modes of  ϵ^+={1,e^± 2π i z̅/β_+}&ϵ^-={1,e^± 2π i z/β_-}.
        The correction to (<ref>) is obtained by Wick contracting the ϵ^±s with each other using the propagator (<ref>) and it's complex conjugate. We will first analyse this around AdS_3 Schwarzchild and then in a generic rotating BTZ background.
        
        
        
         §.§ AdS_3 Schwarzchild
        
        For the case of AdS_3 Schwarzchild we have β_±=β. The reality condition of (<ref>) implies ϕ=0, i.e. we compute the propagators G_± along the τ real line and then analytically continue. Here to proceed we first have to order the Euclidean times for the operators in question and then use the appropriate propagator value for Wick contraction. We then add arbitrary Lorenztian time arguments corresponding to each operator and then read off the answer. The Euclidean answer to the expression
        ⟨ V_1 V_2 W_3 W_4 ⟩_grav/⟨ V_1 V_2 ⟩⟨ W_3 W_4⟩=
        ⟨ℬ(ϵ^±_1,ϵ^±_2)ℬ(ϵ^±_3,ϵ^±_4)⟩ 
        looks like
        
        h_1h_2      {83(πβ)^6(z_13^4+z_24^4-z_14^4-z_23^4)(π z_12β)(π z_34β).      +83(πβ)^5[ -2π (z_13^3+z_24^3-z_14^3-z_23^3)(π z_12β)(π z_34β).      .+2(z_24^3+z_14^3-z_23^3-z_13^3)(π z_34β)-2(z_24^3-z_14^3+z_23^3-z_13^3)(π z_12β)]      +43(πβ)^4[-6(z_13^2+z_14^2+z_23^2+z_24^2)+(12-4π^2)z_12z_34(π z_12β)(π z_34β).      .-12π(z_13+z_24)(z_12(π z_12β)+z_34(π z_34β))]      +83(πβ)^3[6π(z_13+z_24)+2(π^2-3)(z_12(π z_12β)+z_34(π z_34β))]      .+163(πβ)^2(π^2-3)} +  c.c.|_h_1→h̅_1,h_2→h̅_2
        Introducing Lorentzian time coordinates for each of operator i.e.z→τ +i ϕ -it, we fix τ_1=β,τ_2=β/4,τ_3=β/2,τ_4=3β/4 and further fix t_1=t_2=t, t_3=t_4=0&ϕ_1=ϕ_2=0,ϕ_3=ϕ_4=ϕ. Having done this (<ref>) shows a polynomial growth in time.
        
        
        
        For the case of OTOC, contracting ϵ_i in (<ref>) we similarly get
              83(πβ)^6(z_13^4+z_24^4-z_14^4-z_23^4)(π z_12β)(π z_34β)      +83(πβ)^5[ -2π (z_13^3+z_24^3-z_14^3+z_23^3)(π z_12β)(π z_34/β).      .+2(z_24^3+z_14^3-z_23^3-z_13^3)(π z_34β)-2(z_24^3-z_14^3+z_23^3-z_13^3)(π z_12β)]      +43(πβ)^4[-6(z_13^2+z_14^2+z_23^2+z_24^2)+(12-4π^2)z_12z_34(π z_12β)(π z_34β).      .-12π(z_13^2+z_12z_34)(π z_12β)-12π (z_24^2+z_12z_34)(π z_34β)]      +83(πβ)^3[6π(z_14+z_23(π z_12β)(π z_34β))+2(π^2-3)(z_12(π z_12β)+z_34(π z_34β))],
              +83(πβ)^2[-2(π^2-3)+3π(sin(π(z_12+z_34)β)+sin(π(z_13+z_24)β)sin(π z_12β)sin(π z_34β))] + c.c|_h_1→h̅_1,h_2→h̅_2
        Similarly we introduce Lorentzian time coordinates for each of the operators i.e.z→τ +i ϕ -it, we fix τ_1=β,τ_3=β/4,τ_2=β/2,τ_4=3β/4 and further fix t_1=t_2=t, t_3=t_4=0&ϕ_1=ϕ_2=0,ϕ_3=ϕ_4=ϕ. Here one clearly sees the exponential behaviour of the correlator for both the boundary null coordinates t±ϕ as
        ⟨ V_1(t,0) W_3(0,ϕ) V_2(t,0) W_4(0,ϕ) ⟩_grav/⟨ V_1(t,0) V_2(t,0) ⟩⟨ W_3(0,ϕ) W_4(0,ϕ)⟩∼G_Nβ/ l[h_1h_2 cosh(2π(t-ϕ)β)+h̅_1h̅_2cosh(2π(t+ϕ)β)]
        
        Thus we see that the Schwarzian action (<ref>) associated with Brown-Henneaux modes (<ref>) are responsible for the maximal Lypunov index at least in non-rotating BTZ. Note that since we have been cavalier about causality in our propagators (<ref>) we would not reproduce the (t-|ϕ|) behaviour in the exponent like <cit.>[If one does consider this then there will be possibly step functions multiplying each of the terms above.].  
        
        
         §.§ Rotating BTZ
        
        Let's do the similar exercise for rotating BTZ where β_± are complex parameters. Here in order to use the propagator in (<ref>) we would have to use a shifted Euclidean time in (<ref>) τ̃=(τ+μϕ)/(1+μ^2) for ordering the different Euclidean times. i.e. for time ordered correlator we arrange τ̃_1>τ̃_2>τ̃_3>τ̃_4 while for out-of-time-ordered τ̃_1>τ̃_3>τ̃_2>τ̃_4. We would then fix the τ̃ on a circle of period β. 
        We will return later to the detail of how specifying the Euclidean time ordering in shifted Euclidean time does not effect the end result. 
        
        
        
        For the time-ordered case the exact Euclidean answer is given in the appendix (<ref>) and omitted here for the sake of brevity. As before we introduce Lorentzian time t̃  for the shifted coordinate τ̃. We will infer the Lorenztian equivalent of (<ref>) later. We fix the Euclidean times to  τ̃_1=β,τ̃_̃2̃=β/4,τ_3=β/2,τ̃_4=3β/4 and further fix t̃_1=t̃_2=t̃, t̃_3=t̃_4=0&ϕ̃_1=ϕ̃_2=0,ϕ̃_3=ϕ̃_4=ϕ̃. For fixed ϕ̃ this corresponds to setting τ_1>τ_2>τ_3+μϕ̃>τ_4+μϕ̃. 
        
        Having done this (<ref>) shows a polynomial growth in time.
        
        
        
        Similar expression for the  out-of-time-ordered  Euclidean answer is given in  (<ref>).
        Introducing Lorentzian times and fixing Euclidean times to τ̃_1=β,τ̃_3=β/4,τ̃_2=β/2,τ̃_4=3β/4 so as to compute OTOC; we then fix  t̃_1=t̃_2=t̃, t̃_3=t̃_4=0&ϕ̃_1=ϕ̃_2=0,ϕ̃_3=ϕ̃_4=ϕ̃. This implies taking τ_1>τ_3+μϕ̃>τ_2>τ_4+μϕ̃. It is worth noting that this  out-of-time-ordering is unaffected by the value of ϕ̃.  Here we find the exponentially growing term in t̃ as
        ⟨ V_1(t,0) W_3(0,ϕ) V_2(t,0) W_4(0,ϕ) ⟩_grav/⟨ V_1(t,0) V_2(t,0) ⟩⟨ W_3(0,ϕ) W_4(0,ϕ)⟩∼G_Nβ/l[h_1h_2cosh(2π(t̃-ϕ̃)β)+h̅_1h̅_2cosh(2π(t̃+ϕ̃)β)]
        
        Let's convert this back to {t,ϕ} by the Lorentzian version of (<ref>) i.e.
            x̃^±=t̃±ϕ̃=x^±/(1∓μ_L)ϕ=ϕ̃-μ_Lt̃,   t=t̃-μ_Lϕ̃
        
        where we define the Lorenztian angular velocity as μ_L=iμ=r_-/r_+ as it would has risen in a Lorenztian bulk geometry, thus yielding[Here δ x^±=t∓ϕ.]
            G_Nβ/l[h_1h_2cosh(2π(t-ϕ)β_+)+h̅_1h̅_2cosh(2π(t+ϕ)β_-)]
        
        Note, that the transformation (<ref>) is a conformal transformation of the boundary in the metric (<ref>). The proper boundary coordinates along the lines of section 2 of <cit.> are {t,ϕ} which is what one must use to measure correlators in the boundary. It is clear from the last expression the Lypunov index for the each of the left and right moving modes  is governed by β_±=β(1∓μ_L) in stead of β. Thus the Lypunov index for the 4pt OTOC would be λ_L=2π/β_+>2π/β as it governs the fastest growth.
        
        
        
        A some what similar conclusion was reached in <cit.> with mutual information computed between the left and right intervals of |.TFD⟩ corresponding to a rotating BTZ. This was computed both by computing mutual information via Rényi entropy in the 2D CFT with a chemical potential μ perturbed by a heavy operator, and from the bulk by employing the Ryu-Takayanagi prescription of minimal surfaces in a shock wave background. In <cit.> the symmetry between β_± is broken by the spatial arrangement of the heavy operator relative to the entangling interval in question. This arrangement is such that only one of the modes with a smaller temperature effects the entangling region for positive times. For a different spatial arrangement one may seem to find that the scrambling time computed is governed by the higher of the 2 temperatures. 
        
        
        
        
        In the above computations we had chosen to order the Euclidean times in the shifted coordinate frame of τ̃=(τ+μϕ)/(1+μ^2) instead of ordering their τ coordinates. The reason why we still end up computing the desired correlator in the desired frame is as follows: The behaviour of the 4pt function is encapsulated in the sum conformal blocks. This in the limit of large central charge for holographic CFTs can be seen to be captured by only the Virasoro conformal block of the identity operator <cit.>. The conformal blocks are given by hypergeometric functions F(a,b,c,η)[Here a,b,c do not encode bulk information and depend upon the conformal block in question.] that are known to have a branch point at η=1 where η is the conformal cross ratio[Similar comments would hold true for the anti-holomorphic sector as the blocks factorize.]. It was shown in <cit.> that the behaviour of η for large Lorentzian times depends on the ordering of the Euclidean times[Here η is analytically continued to have both Lorentzian and Euclidean co-ordinates.]. For  out-of-time-ordering of the Euclidean times, η circles the branch point η=1 in the complex plane; while this is not the case when the Euclidean times are  time-ordered. The authors of <cit.> then use the monodromy of F(a,b,c,η) to compute it's behaviour for small η thus obtaining the Lyapunov exponent for 2dim holographic CFTs. One can therefore use this behaviour of the cross ratio to discern if one is indeed computing  time-ordered or  out-of-time-ordered correlators.   
        
        
        
        We define the cross ratio η as
        η=ω_12ω_34/ω_13ω_24,ω=e^-2π i z/β_-,z=τ	+ iϕ
        and similarly for the barred coordinates which are complex conjugates. η can be analytically continued to include Lorentzian times by τ→τ -it. The tilde coordinates are related to un-tilde coordinates by (<ref>)i.e.z/β_-=z̃/β,z̅/β_+=z̃̅̃/β      ω=e^-2π i z̃/β,ω̅=e^2π i z̃̅̃/β  
        In all the above cases we choose ϕ̃_1,2=0,t̃_3,4=0&t̃_1,2=t̃,ϕ̃_3,4=ϕ̃, therefore for fixed values of Euclidean times τ̃_i, η(t̃,ϕ̃) is a complex function. It was demonstrated in <cit.> that for |t̃|>|ϕ̃|>0, the  out-of-time-ordered cross-ratio η_oto(t̃,ϕ̃) circles the branch point η=1 as one varies t̃; while no such behaviour is found in the case for the  time-ordered cross-ratio η_to(t̃,ϕ̃). In order to know how the shifted Euclidean time ordering effects the answer in the un-tilde frame we would like to see how η (and η̅) behaves as a function of {t,ϕ} for the respective orderings in τ̃_i. 
        
        
        
        The cross-ratio η for arbitrary τ̃_i looks like
        η({τ̃_i},t̃,ϕ̃)   =   (e^-2π i/βτ̃_12-1)(e^-2π i/βτ̃_43-1)/(e^-2π i/βτ̃_13e^2π i/β(t̃+ϕ̃)-1)(e^-2π i/βτ̃_42e^-2π/β(t̃+ϕ̃)-1)   =   (e^-2π i/βτ̃_12-1)(e^-2π i/βτ̃_43-1)/(e^-2π i/βτ̃_13e^2π/β_-(t+ϕ)-1)(e^-2π i/βτ̃_42e^-2π/β_-(t+ϕ)-1).
        
        where in the second line above we have performed the Lorentzian coordinate transformation (<ref>). A similar expression holds for η̅ with β_-→β_+ and t+ϕ→ t-ϕ. Here β±=β(1∓μ_L). We  plot the second expression in (<ref>) for different values of μ_L∈{0,1}, τ̃_i=β-(i-1)3β/4+(i-1)ϵβ/4 and β=2π with ϵ∈{0,1} for the  time-ordered case. For the  out-of-time-ordered case we exchange τ̃_2↔τ̃_3 as compared to the  time-ordered case.  We find the same behaviour as seen in <cit.> for different values of chemical potential μ_L and ϵ. Here ϵ varies the adjacent Euclidean time differences on the thermal circle linearly from β/4 to zero. Therefore even though we ordered shifted Euclidean times we end up computing the desired correlator in the un-tilde Lorentzian frame spanned by {t,ϕ}[This might seem counter-intuitive at first,  but there is no precise way in which one can change frames when both Euclidean and Lorentzian space-time coordinates are present with non-zero chemical potential.].
        
        
        
        
        
        
        
        
        
        
        One could further ask the question  whether a purely CFT computation of the form of <cit.> can recover(or corroborate) the results obtained for the rotating BTZ. This would require using the hypergeometric function F(a,b,c,η)  in the large central charge limit to incorporate not only temperature but also μ_L due to rotation. In <cit.> it was shown that effect of introducing heavy operators at time-like infinities in 2dim CFT corresponds to evaluating the global conformal blocks on a new 2dim background  metric obtained by mapping the flat Euclidean space to a torus by[We are thankful to Justin David for having pointed this out.]ω=e^-2π i z/β_-,ω̅=e^2π i z̅/β_+.
         
        where {ω,ω̅} are the Euclidean flat-space light-cone coordinates.
        We leave this as an exercise for the near future.
        
        
        
        
        
        
        
        § RESULTS AND DISCUSSIONS
        
        The bulk understanding of how the Lyapunov index is 2π/β has as of yet always relied upon near the extremal property of an AdS black hole exhibiting an AdS_2 throat. In some sense the back reaction of the scalars in the bulk gets the most contribution from this region. Any deviation in the AdS_2 geometry is captured in an action like the Jackiw-Teitielboim thus giving rise to a Schwarzian action.
        What we have demonstrated here - at least in AdS_3; is that the Schwarzian arises even when one is far away from extremality. Moreover we find this as an effective action at the boundary of AdS_3 rather than at some screen in the interior.  It would be interesting to investigate how such an action can be arrived at for black holes in AdS_d>3, this would indeed give some understanding of the soft modes in higher dimensional large N CFTs.
        
        
        
        In the probe approximation there is an inherent conformal symmetry in the bulk emanating from the asymptotic symmetries of AdS_3. This would correspond to the 2 copies of Virasoro algebra in the boundary CFT. Any arbitrary solution to the Einstein's equation with matter would  not have such a symmetry. Expanding perturbatively  about the probe approximation in orders of G_N (i.e. back reaction) breaks this symmetry spontaneously. The action (<ref>) therefore can be seen as the action cost associated with conformal transformations at the boundary of AdS_3  when one tries to go away from the probe approximation to linear order in G_N. 
        
        
        
        Extremality can be reached in the simplest possible manner by turning on charges for the AdS black hole, the top down understanding of <cit.> in such a setting was explained in <cit.> in AdS_4. Here the authors studied a probe uncharged scalar in the bulk thus having no dynamics for the gauge field. It would be interesting to analyse how the near horizon picture in <cit.> is reached for rotating geometries close to extremality in AdS_d>3. In <cit.> 5d rotating Kerr geometries were analysed close to their near extremal limit in the near horizon throat region. There the authors have discovered a generalized JT action consisting of a dilaton and an additional scalar. 
        
        
        
        
        
        
         In (<ref>) we take the view that β_± are complex to begin with. The complex value of β_±=β∓ i Ω is required to make sense of the Euclidean BTZ metric as a real quantity. Further the iϵ[Here Euclidean time is τ instead of ϵ.] prescription that we use requires us to first compute a Euclidean correlator and then analytically continue it to desired Lorentzian times. This is similar to the technique employed in <cit.> for computing the mutual entanglement from the 2D CFT, as computing the Rényi entropy involves analytically continuing the Euclidean 4pt correlator ⟨ψσσ̃ψ^†⟩[ψ is the heavy operator generating the shock-wave in the BTZ while σ is the twist operator. ] to obtain the Lorentzian answer.
        This requires making the left and right moving temperatures real: β_±=β∓Ω[In our analysis β_±∼1/√(L_±)∼1/√(M± J/l) are associated with x^± respectively.] when all Euclidean times have been put to zero. (<ref>) would therefore yield a growth in the scrambling time that would be greater than  the Lyapunov index of 2π/β due to the left moving (anti-holomorphic) modes for x^+i.e.2π/β-Ω.
        
        
        
        
        The presence of rotation in the bulk implies a CFT with a chemical potential corresponding to angular momentum. 1d SYK and gauged-SYK models have been studied in the presence of a chemical potential <cit.>. Here the Lypunov index computed was found to be bounded by 2π/β. This also bodes well with the intuition that holding other charges fixed makes the system less chaotic. However the chemical potential present in such cases were associated to an internal symmetry and not a space-time symmetry.
        
        
        
        The analysis of section (4.2) for the case of rotating BTZ seems to yield a result in contradiction with the mathematical proof in <cit.>. The proof in section (4.1) of <cit.> is basically based on the maximal modulus theorem for a bounded holomorphic function. Here we try to give a simple understanding as to how one may try to reconcile the result of section (4.2) of this paper with that of <cit.>. 
        
        
        
        For the case of 2d CFTs one could argue a change in the Lyapunov index using the fact that the function of the form of ⟨ V_1 V_2 W_3 W_4 ⟩/⟨ V_1 V_2 ⟩⟨ W_3 W_4⟩ of Virasoro primaries must be a scalar under conformal transformations. 
        ⟨ V_1 V_2 W_3 W_4 ⟩/⟨ V_1 V_2 ⟩⟨ W_3 W_4⟩= F(x_13,x_24,x_14),    where   x_ij=x_i-x_j
        
        For the specific case of 2dim CFT dual to the Schwarzchild-AdS_3 this function was shown <cit.> to have a behaviour of 
        
            ⟨ V_1 V_2 W_3 W_4 ⟩_oto/⟨ V_1 V_2 ⟩⟨ W_3 W_4⟩∼ 1 + # G_Ne^2π/β(t-|x|)=
            
            1 + # G_Ne^2π/β(t+x)    for  x<0
               
            
            1 + # G_Ne^2π/β(t-x)    for  x>0.
        
        The finite diffeomorphism which takes Schwarzchild-AdS_3 to a generic rotating BTZ black- hole corresponds to a conformal transformation on the boundary of the form (<ref>)δ x^±/β→δx̃^±/β =δ x^±/β_±.
        
        where β_±=(1∓μ_L) and δ x is the difference in any two of the coordinates in the correlator.
        This would imply studying a holographic CFT with a chemical potential corresponding to rotation turned on.
        The proof in section (4.1) of <cit.> relies crucially upon mapping the half strip of width β/2 in Euclidean time τ to a disk via a conformal map  w=1-sinh[2πβ(t+iτ)]/1+sinh[2πβ(t+iτ)]. 
        This map has a periodicity under τ→τ+β which is exhibited by ⟨ V_1 V_2 W_3 W_4 ⟩/⟨ V_1 V_2 ⟩⟨ W_3 W_4⟩. It was argued that this half strip is the minimum area of analyticity a function of the form ⟨ V_1 V_2 W_3 W_4 ⟩/⟨ V_1 V_2 ⟩⟨ W_3 W_4⟩ can have. Smaller the width of the strip, greater would be the bound on chaos. However conformal transformation of the form (<ref>) suggests that there exist 2dim holographic CFTs for which ⟨ V_1 V_2 W_3 W_4 ⟩/⟨ V_1 V_2 ⟩⟨ W_3 W_4⟩ has a thinner region of analyticity of width β_-/2=β(1-μ_L)/2 controlled by the chemical potential μ_L. This suggests that repeating the analysis of <cit.> for systems at finite temperature and chemical potential might in certain cases yield a smaller region of analyticity in the Euclidean time direction.
        
        
        
        
        
        
        
        
        
        
        
        One could very well have guessed such a result simply by observing that the effective action (<ref>) doesn't mix the left and right movers, each of which have different inverse temperatures i.e.β. The conformal blocks used in <cit.> to study chaos in 2dim holographic CFTs also exhibit such a holomorphic and anti-holomorphic factorization. From such a point of view it seems reasonable to expect that there would be 2 Lyapunov indices governed by the 2 temperatures; this point of view does not seem to be at odds with the statement of maximal chaos first proposed in <cit.>.
        Therefore the maximal growth would be governed by the smaller of the two i.e.min[β_+,β-], while the surface gravity of the bulk would be related to the average  β_++β_-/2. 
        However one can easily see that this would also lead to extremal BTZ metric exhibiting chaotic behaviour with the Lyapunov index being one of the 2 temperatures which is  non-vanishing. This would precisely be the temperature which would be greater than that given by the surface gravity for a generic rotating BTZ. 
        This raises an important question in higher dimensional Kerr-AdS black holes of the possibility of extremal chaos. 
        It would be very interesting to see how these considerations would have to be modified when analysing rotating geometries in AdS_d>3 as unlike AdS_3 the bulk degrees of freedom of the metric would also participate in the dynamics.     
        It is also worth noting that bulk explanations of the dynamics of chaotic behaviour using the 2dim Jackiw-Teitelboim model in the near horizon region of near extremal black holes <cit.> cannot be used to explain such extremal chaos. Therefore it would be interesting to find a similar JT like model(if possible) in the near horizon region of extremal and/or near extremal region to explain such a behaviour.  This would provide a useful tool in analysing such behaviour in higher dimensional Kerr-AdS black holes.
        
        
        
        
        
        The result of section (4.2) is also validated by the analysis of mutual information for late times computed in the BTZ geometry subjected to a shock wave <cit.>. Here the author found the Lyapunov index to be related to the smaller of the two temperatures i.e.λ_L=2π/β_- in the conventions of this paper. The mutual information in the |TFD⟩ state corresponding to an eternal BTZ subjected to a shock wave is computed in  <cit.> on the boundary by taking the limit of the Rényi entropy, and in the bulk by employing the Ryu-Takayanagi prescription of minimal area. However the spatial arrangement of the heavy operator in <cit.>w.r.t. the entangling region under consideration only sees the effect of one of the modes.       
        
        
        
        We have analysed the bulk on-shell action (<ref>) upto quadratic order in fluctuations about a generic BTZ metric. It is in this approximation that the left and the right sectors seem to decouple and the scrambling times for the left and the right sectors are governed by their respective temperatures.
        The full non-linear action (<ref>) if expanded beyond quadratic order in fluctuations would indeed have terms which mix the left and the right moving sectors. It would be interesting to study how the three temperatures β&β_± participate in  the process of reaching a thermal equilibrium at late times when the boundary CFT is excited. This would require a more detailed analysis for longer time periods than the scrambling times of the left and right sectors[The author is indebted to the referee for drawing attention to this. ]. 
        
        
        
        
        
        
        The techniques used in this paper seem to be too well suited for AdS_3. As mentioned before that generalizing this to higher dimensional AdS black holes would be interesting, it would nonetheless be easier to analyse the rotating BTZ along the lines of <cit.> by computing bulk eikonal scattering which seems to be an analysis suited for all dimensions[Barring the difficulty of computing bulk to boundary propagators for rotating black holes in AdS_d>3.].   
        
        
        
        To conclude, this work also suggests that if the Lypunov index associated with rotating AdS black holes in Einstein-Hilbert theory have maximal chaos, then for large N thermal CFTs with chemical potential associated with angular momentum the chaos bound is λ_L=2π/β(1-μ_L). It would be interesting to find a more thorough generalization of the proof in <cit.> for large-N CFTs with chemical potential associated with angular momentum.  
        
        
        § ACKNOWLEDGEMENTS
        
        The author is indebted to Gautam Mandal for fruitful discussions throughout the duration of this work. The author also benefited by earlier work with Gautam Mandal, Pranjal Nayak, Nemani Suryanarayana and Spenta Wadia. The author also acknowledges Adwait Gaikwad, Anurag Kaushal, R. Loganayagam, Pranjal Nayak, Ronak Soni, Shiraz Minwalla & M. V. Vishal for their numerous discussions during the completion of this work. The author also acknowledges Abhijit Gadde and Justin David for giving inputs on this work. The author also found the academic environment of the Indian Strings Meet 2018 held in IISER Thiruvanathapuram conducive towards the completion of this work.
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        § APPENDIX
        
        The time ordered Euclidean answer for (<ref>) corresponding to the rotating BTZ is
              ⟨ V_1 V_2 W_3 W_4 ⟩_grav/⟨ V_1 V_2 ⟩⟨ W_3 W_4⟩|_TO=      h_1h_2{83(πβ(μ-i))^6(z_14^4+z_23^4-z_13^4-z_24^4)(π z_12β(1+iμ))(π z_34β(1+iμ)).      +163(iπβ(μ-i))^5[-3π z_12z_34(z_13+z_24)(π z_12β(1+iμ))(π z_34β(1+iμ)).      .+(z_13^3+z_23^3-z_14^3-z_24^3)(π z_34β(1+iμ))+(-z_13^3+z_23^3-z_14^3+z_24^3)(π z_12β(1+iμ))]      +83(πβ(μ-i))^4[-3(z_13^2+z_14^2+z_23^2+z_24^2)+2(3-π^2)z_12z_34(π z_12β(1+iμ))(π z_34β(1+iμ)).      .+3π(-z_13^2+z_24^2-z_14^2+z_23^2)(π z_12β(1+iμ))+3π(+z_13^2-z_24^2-z_14^2+z_23^2)(π z_34β(1+iμ))]      +83(πβ(μ-i))^3[6iπ (z_13+z_24)+2i(π^2-3)(z_34(π z_34β (1+iμ))+z_12(π z_12β (1+iμ)))]      .+163(πβ(μ-i))^2(π^2-3)}+  c.c.|_h_1→h̅_1,h_2→h̅_2
        The above expression yields a polynomial expression in terms of the coordinates after analytically continuing to the Lorentzian times.
        Similarly the out of time ordered Euclidean answer for the rotating BTZ case is
              ⟨ V_1 V_2 W_3 W_4 ⟩_grav/⟨ V_1 V_2 ⟩⟨ W_3 W_4⟩|_OTO=      h_1h_2{83(πβ(μ-i))^6(z_14^4+z_23^4-z_13^4-z_24^4)(π z_12β(1+iμ))(π z_34β(1+iμ)).      +163(iπβ(μ-i))^5[π(z_13^3-z_14^3+z_23^3+z_24^3)(π z_12β(1+iμ))(π z_34β(1+iμ)).      .+(z_13^3+z_23^3-z_14^3-z_24^3)(π z_34β(1+iμ))+(-z_13^3+z_23^3-z_14^3+z_24^3)(π z_12β(1+iμ))]      +83(πβ(μ-i))^4[-3(z_13^2+z_14^2+z_23^2+z_24^2)+2(3-π^2)z_12z_34(π z_12β(1+iμ))(π z_34β(1+iμ)).      .-3π(z_13^2-z_24^2+z_14^2+z_23^2)(π z_12β(1+iμ))-3π(-z_13^2+z_24^2+z_14^2+z_23^2)(π z_34β(1+iμ))]      +83(πβ(μ-i))^3[6iπ z_14+6iπ z_23(π z_12β (1+iμ))(π z_34β (1+iμ)).      .+2(π^2-3)(iz_34(π z_34β (1+iμ))+iz_12(π z_12β (1+iμ)))]      .-83(πβ(μ-i))^2[2(3-π^2)+3π((π z_12β(1+iμ))+(π z_34β(1+iμ)))-3πsin(π(z_13+z_24)β(1+iμ))/sin(π z_12β(1+iμ))sin(π z_34β(1+iμ))]}      + c.c.|_h_1→h̅_1,h_2→h̅_2 
        Introducing Lorentzian times and fixing τ_1=β̃-μϕ_1,τ_3=β̃/4-μϕ_3,τ_2=β̃/2-μϕ_2,τ_4=3β̃/4-μϕ_4 and further fixing t_1=t_2=t, t_3=t_4=0&ϕ_1=ϕ_2=0,ϕ_3=ϕ_4=ϕ. Here we find the exponentially growing term in t as
        ⟨ V_1(t,0) W_3(0,ϕ) V_2(t,0) W_4(0,ϕ) ⟩_grav/⟨ V_1(t,0) V_2(t,0) ⟩⟨ W_3(0,ϕ) W_4(0,ϕ)⟩∼G_Nβ/l[h_1h_2cosh(2π(t-ϕ)β_+)+h̅_1h̅_2cosh(2π(t+ϕ)β_-)]
        
        JHEP.bst
