Pole-Skipping in Rotating BTZ Black Holes

Motivated by the connection between pole-skipping phenomena of two point functions and four point out-of-time-order correlators, we study the pole-skipping phenomena for rotating BTZ black holes. In particular, we investigate the effect of rotations on the pole-skipping point for various fields with spin $s = 1/2, 1, 2/3$, extending the previous research for $s=0, 2$. We derive an analytic full tower of the pole-skipping points of fermionic ($s=1/2$) and vector ($s=1$) fields by the exact holographic Green's functions. For the \textit{non-extremal} black hole, the leading pole-skipping frequency is $\omega_{\text{leading}}=2\pi i T_h {(s-1+\nu \Omega)}/{(1-\Omega^2)}$ where $T_h$ is the temperature, $\Omega$ the rotation, and $\nu:=(\Delta_+ - \Delta_-)/2$, the difference of conformal dimensions ($\Delta_{\pm}$). These are confirmed by another independent method: the near-horizon analysis. For the \textit{extremal} black hole, we find that the leading pole-skipping frequency can occur at $\omega_{\text{leading}}^{\text{extremal}}=-2\pi i T_R {(s+1)}$ only when $\nu = s+1$, where $T_R$ is the temperature of the right moving mode. It is non-trivial because it cannot be achieved by simply taking the extreme limit ($T_h\rightarrow 0\,, \Omega\rightarrow 1$) of the non-extremal black hole result.

1 Introduction Retarded Green's functions, denoted as G R , play a fundamental role in the field of physics by providing essential insights into the behavior and properties of various physical systems.Notably, the exploration of the AdS/CFT duality or holographic duality [1][2][3][4] has uncovered a remarkable and universal characteristic inherent in Green's functions, commonly referred to as pole-skipping [5,6].The Green's function of the dual field theory in holography is typically associated with the fluctuations in the bulk.Perturbation problems in the bulk often involve different fields, including scalar fields, Dirac fields, Maxwell fields, and gravitational fields.However, when studying the dual Green's functions in the complex momentum space (ω, k), particularly at pole-skipping points (ω * , k * ), the uniqueness of Green's functions becomes uncertain.For instance, near the pole-skipping point, the Green's function can be expressed as where the Green's function becomes non-unique and its behavior is dependent on the slope δω/δk as it approaches the pole-skipping point.From a bulk perspective, the appearance of pole-skipping arises due to the non-uniqueness of the bulk solution, especially due to the existence of "two" independent ingoing solutions at the black hole horizon.The phenomenon of pole-skipping was initially discovered in the context of holographic chaos [5,6], where the presence of pole-skipping points in the energy density two-point function is closely related to the Lyapunov exponent (and butterfly velocity) that govern the behavior of out-of-time-ordered correlation functions.This intriguing pole-skipping phenomenon arises as a prediction of hydrodynamic effective theories that describe maximally chaotic systems [7,8], and it has been observed to be a generic feature of holographic theories dual to planar black holes [6,9]. 1ince the discovery of the connection to quantum chaos, significant attention has been focused on investigating the mathematical foundations of the pole-skipping phenomena.The striking progress in this research area includes the establishment of the universality of pole-skipping points, as demonstrated in studies such as [9,.
There exists a remarkable universality for the pole-skipping points, particularly at their leading expression, which includes Matsubara frequencies: where s denotes the spin of the bulk fields and T h the black hole temperature.For instance, the scalar, Dirac, Maxwell (or vector), and gravitational spin-2 field have the pole-skipping points as where the case of the vector field, ω 1 , corresponds to the hydrodynamic pole.In the context of the gravitational sound mode (spin-2 field), the corresponding pole-skipping point ω 2 has been proposed that it is associated with the many-body quantum chaos [44][45][46][47][48].
More recent investigations have explored the relationship between chaos and energy dynamics in rotating black holes in the context of pole-skipping phenomena.A notable case is the rotating BTZ black hole (three-dimensional gravity theories), which is dual to a two-dimensional conformal field theory featuring a chemical potential for rotation.Detailed examinations of this scenario were conducted, for instance, in [27]. 2 Additionally, the interplay of chaos and pole-skipping in the field theory dual to a higher-dimensional rotating Kerr-AdS black hole was recently investigated in [34,39]. 3hen taking into account finite rotation, an intriguing consequence arises: the universal pole-skipping point can be extended to incorporate rotational effects.Previous studies, such as [27,34,39], have revealed that ω 2 can be generalized as follows: Here, Ω represents the finite rotation and ν is associated with the conformal dimension of the dual boundary operator.In addition, through the analysis of scalar field fluctuations, [29] revealed that the value of ω 0 can also be further extended to In this paper, we focus our investigation on the Dirac (ω 1/2 ) and vector (ω 1 ) fields, with the objective of extending (1.2) to derive the more general universal pole-skipping point.This generalization incorporates both the spin information, represented by s, and the rotation parameter Ω.
Additionally, as a result of considering finite rotation, we also explore the pole-skipping point in the extreme limit (T h → 0 , Ω → 1).As we will show in the main context, it is imperative to note that this limit can only be approached when the analytic Green's function is applicable.Furthermore, in the context of extremal black holes, we will reveal an additional universal relationship associated with the pole-skipping point.
The structure of this paper is as follows.In section 2, we provide a quick review of the black hole background of our interest, specifically the rotating BTZ black holes.Building upon the BTZ black holes introduced in section 2, in section 3, we focus on the fermionic fluctuation field and derive the analytic Green's function.In addition, we perform the systematic analysis of the pole-skipping points from such Green's function.Likewise, in section 4, we investigate the exact Green's function of massive vector fields, and examine the analytic structure of pole-skipping points in presence of rotations.Section 5 is devoted to conclusions.

The rotating BTZ black holes
Let us first briefly review the rotating BTZ black hole [51,52] whose metic reads in terms of Schwarzschild coordinates (t, r, φ) as where φ is periodic with 2π and J angular momentum given in (2.3).The emblackening factor f (r) can be found as the solution of Einstein gravity where the black hole mass M and angular momentum J are Using the emblackening factor (2.2), one can also find the Hawking temperature T h and other thermodynamic quantities such as the entropy density S: where the angular velocity (or angular momentum potential) Ω is where 0 ≤ Ω ≤ 1.Note that the thermodynamic quantities (2.3)-(2.5)satisfy the first law of thermodynamics dM = T h dS + Ω dJ .
(2.6) Furthermore, using the angular velocity Ω (2.5), one may also introduce temperatures of the left/right moving modes in a two dimensional CFT as Following the standard convention we choose T L → 0 for the extreme limit in this paper.
In following sections, we will study the fluctuation fields (fermionic field ψ and gauge field A) on the background geometry (2.1).In particular, we aim to solve the corresponding fluctuation equations of motion and obtain the Green's function analytically.For this purpose, it is convenient to introduce another coordinate (T, ρ, X) where the coordinate transformation from (2.1) to (2.8) is given Note that the AdS boundary in the Schwarzschild coordinates (2.1), r → ∞, becomes ρ → ∞ in a new coordinate (2.8).
3 Fermionic fields (s = 1/2) In this section, we study the pole-skipping point of fermionic field in the presence of rotations.For this purpose, we will follow closely the presentation of [53] in order to obtain the Green's function.Furthermore, we consider the pole-skipping point from the Green's function not only for the non-extremal case, but also for the extremal case.As we will show shortly, the non-extremal case is consistent with [23], while the extremal case has not been reported in literature yet.

Dirac equations of motion
In this section, we study the fermionic pole-skipping points in the presence of the rotation.
For this purpose, we start with a Dirac equation written in terms of a Dirac spinor field ψ with the fermion mass m f : where Γ M is the gamma matrix and D M the covariant derivative.In this paper, we use the indices M, N = (T, ρ, X) for the bulk spacetime and a, b = (T , ρ, X) for the tangent spacetime where they are associated by the veinbein e a M as In terms of the veinbein, Γ M and D M can be expressed as where the spin connection (ω ab ) M and Γ ab are determined by with the Christoffel symbols Γ N M Q .Given all the relations (3.2)-(3.4), in order for solving the Dirac equation (3.1), we need to specify two things: i) the inverse vielbein e M a ; ii) the gamma matrix Γ a .In what follows, given the metric g M N (2.8), we choose the diagonal inverse vielbein as together with the gamma matrix Dirac equations in the Fourier space.Furthermore, introducing the Dirac spinor ψ in the Fourier space we find the Dirac equations of motion (3.1) as In order to solve this equation analytically, it is convenient to use the ansatz where the coordinate ρ is replaced by z.Note that in this z-coordinate, the AdS boundary is located at z → 1.Within the ansatz (3.9), the Dirac equations (3.8) become which allow the analytic incoming solutions in terms of the hypergeometric function 2 F 1 as where ) Therefore, using the ansatz (3.9) together with (3.11)-(3.13),one can find the analytic fermionic field ψ ± (z).
3.2 Exact fermionic Green's function 3.2.1 Holographic dictionary for fermionic Green's function Next, using the analytic solution ψ ± (z) obtained above, we study the fermionic Green's function.Let us first review how to read such a fermionic boundary correlator.
Non half-integer case.According to the holographic dictionary, the Green's function are related to the AdS boundary behavior of fermionic fields ψ ± (z).For instance, one can find that the equation of motion (3.8) within a z-coordinate (3.9), z = tanh 2 ρ, produces the following asymptotic behavior near the AdS boundary (z → 1) where ∆ ± is the conformal dimension Here, A and D are independent free parameters in which the remaining coefficients, B and C, are determined by Using the obtained analytic solution (3.11),A and D can be found, for instance (3.27).One can also notice that the relation (3.16) may not be well defined for the case m f = ±1/2.We will discuss this point later.
Half-integer case.It would also be convenient to introduce a new parameter ν composed of the conformal dimension (3.15) In particular, using ν, we can rewrite the boundary expansion (3.14) as In other words, in (3.14), it is intrinsically assumed that ν (or equivalently the mass m f ) is not a half-integer: In principle, it is also possible to consider the "negative" half-integer.However, it is enough to consider the positive half-integer case for our purpose: see the description around (3.24).Therefore, for the case of half-integer ν, ν ± 1 2 ∈ Z + , such as one can find the different boundary expansion ) where Ā and D are still the independent free parameters together with the same relation (3.16) Similar to the non half-integer case, using the obtained analytic solution (3.11), one can also find the analytic expression for Ā and D as (3.36).
Fermionic Green's function in holography.According to the holographic principle, the fermionic Green's function, GR , is given by the ratio between two independent parameters D and A as where the former one is for the non half-integer case, while the later is for the half-integer case.Strictly speaking, the Green's function in (3.23) corresponds to the case of the standard quantization where A (or Ā) is interpreted as the source and D (or D) is the corresponding vev.
In principle, it is also possible to consider the other quantization, alternative quantization, by replacing A ↔ D or Ā ↔ D. However, in this paper, we only consider the case of standard quantization (3.23) for our own purpose: notice that the structure of the pole-skipping, GR ∼ 0/0, is independent of the type of a quantization.
Equivalently, considering a standard quantization implies that we only focus on the positive ν, for instance, one can notice that the role of A and D in (3.18) can be exchanged when the sign of ν in (3.17) is reversed.
Therefore, in this paper, we only choose ν ≥ 0 hereafter, i.e., Mathematically, it is also possible to consider ν = m f = 1/2.However, this special case may produce the ill-defined relation (3.16) (or (3.22)) and requires the separate calculations for the Green's function: for instance, see [53].Furthermore, it is also worth noting that this special case can also give rise to the subtlety in the boundary expansion (3.18).
In this study, we do not explore this special case for simplicity.Note that the special case can also appear even in the scalar field case [29] with ν = 0. Nevertheless, it is shown [29] that the pole-skipping cannot occur in this case.We suspect that this may also be the case even for other fields.We leave this subject as future work.Fermionic Green's function in (t, r, φ) coordinate.Following the coordinate transformation (2.9), it is shown [53] that one can restore the Green's function in the original coordinate (t, r, φ) of (2.1), G R , from the one in the (T, z, X) coordinates, GR in (3.23), as where, as we will show shortly, the prefactor (2πT L ) ν− 1 2 plays an important role to study the pole-skipping in the extreme limit.Also, comparing ψ ∼ e −iωt + ikφ with (3.7), one can find the relation between (ω, k) and (k T , k X ) as where we use (2.7) together with (2.9).In summary, once GR (k T , k X ) is evaluated via (3.23), one can obtain G R (ω, k) by the relation (3.25)- (3.26).Following this procedure, we present the analytic result of G R (ω, k) for both the non half-integer case and half integer case below.

Non half-integer ν case
Plugging analytic solutions (3.11) into (3.9),one can find the analytic expression for the spinor ψ ± .Furthermore, expanding ψ ± near the AdS boundary (z → 1), we can read the coefficients A and D as where (a, b) is given in (3.12).Then, evaluating their ratio, (3.23), GR can be simply found as Here, we also define where we used (3.12) together with (3.26) in order to express GR in terms of (ω, k).Note that the temperature dependence (T L , T R ) of Green's function is encoded in the refined parameters (a L , b R ) in (3.29).Therefore, using (3.25), we can find the expression of G R (ω, k) as Note that this Green's function is for the non-extremal black hole case.
For the case of an extreme limit (T L → 0), one can notice that a L → ∞ in (3.29).Therefore, in order to describe the extreme limit of G R (ω, k), first it is useful to consider the following asympotics of the gamma function Then, using (3.31), one can easily check that the ratio between Γ(a L + ν − 1 2 ) and Γ(a L ) in (3.30) can be expressed in a L → ∞ limit as where we also used (3.29) in the last equality.
Therefore, now we can find the Green's function in the extreme limit, where the overall prefactor in (3.30), (2πT L ) ν− 1 2 , is canceled out with the one from (3.32).

Half-integer ν case
Next we discuss the half-integer case.For this purpose, it is useful to consider the following property of the hypergeometric function, which can be used in the analytic spinor (3.11).It is shown [23] that when n ∈ Z + the hypergeometric function 2 F 1 (ã, b; ã + b − n; z) can be expressed as where (ã) j := Γ(ã + j)/Γ(ã) is the Pochhammer symbol and Ψ(ã) the digamma function.
Note that hypergeometric functions in (3.11) can be expressed by (3.34) for positive integer n 0 = ν − 1 2 because χ 1 and χ 2 can be written as Therefore, one can expand the analytic spinior ψ ± near the AdS boundary using (3.34) and find the coefficients in (3.21) for the half-integer case as Then the Green's function is obtained via (3.25) together with (3.23) as where we use (3.29) and ignore the contact terms, 2Ψ ν + 1 2 + 2Ψ(1), not related with pole-skipping.
In the extreme limit (T L → 0 or a L → ∞), we can also use (3.32) even for the halfinteger case.Furthermore, we also make use of the property of the digamma function i.e., the digamma function in (3.37), Ψ a L + ν − 1 2 , can be expressed as where we omit a log T L term irrelevant for the structure of the pole-skipping. 5Therefore, the Green's function in the extreme limit becomes (3.40)

Fermionic pole-skipping points with rotations
In this section, we investigate the pole-skipping of the fermionic Green's function: (3.30), (3.33) for the non half-integer case and (3.37), (3.40) for the half-integer case.

Non half-integer ν case
Pole-skipping in the non-extremal case.Let us start with the non half-integer case.
In particular, we first discuss the case of the non-extremal Green's function (3.30).One can notice that we have two types of poles and zeros from (3.30) as: (left poles): where n p L , n p R = 0, 1, 2, • • • and (left zeros): where n z L , n z R = 0, 1, 2, • • • .Then, the pole-skipping point at which the pole line intersects with the zero line can be obtained from the following combinations (left poles & right zeros): It may be instructive to show the leading pole-skipping point to discuss the effect of rotation.One can check that the first combination, (3.43), produces the leading poleskipping point (ω leading , k leading ), in particular using (2.7), we find where the correction of rotation Ω appears both in frequency and wave-vector.
Pole-skipping in the extreme limit.We also discuss the pole-skipping in the extreme limit (T L → 0) for the non half-integer case.Note that such a pole-skipping cannot be achieved simply by taking T L → 0 on (3.43) because (3.41)-(3.42)are ill-defined in the extreme limit.Instead, we need to consider (3.33) in order to discuss the pole-skipping in the extreme limit.From the structure of (3.33), one can find the two types of the 'would-be' poleskipping points, (ω (I) , k (I) ) and (ω (II) , k (II) ), as where right poles are from (3.41) and right zeros from (3.42).However, as in the scalar field case [29], (3.45) may not be considered as the poleskipping points.Recall that the pole-skipping phenomena indicates that the Green's function near the pole-skipping point cannot give a uniquely-determined finite value due to the slope δω/δk (1.1).One can check that (3.45) does not correspond to this case.For instance, the Green's function (3.33) near the 'would-be' pole-skipping points (3.45) can be expressed as Therefore, (3.45) cannot be taken as the pole-skipping points since the Green's function can be solely determined as ∞ or 0.
In other words, one cannot find the pole-skipping points in the extreme limit for the non-half integer case.As we will see shortly, this is not the case for the half-integer case (e.g, ν = 3/2).

Half-integer ν case and pole-skipping in extreme limit
Pole-skipping in the non-extremal case.For the case of the half-integer case, it is useful to consider the following property of gamma function Using (3.47), one can notice that the gamma functions in (3.37) can be expressed as which give the following zeros (left zeros): where One can notice two things for the half-integer case.First, for the half-integer case, the gamma functions in the Green's function only produce zeros unlike the non half-integer case (3.41)-(3.42).Second, the condition for zeros is restricted by the value of ν: for instance, n z L is up to ν − 3/2.On the other hand, poles of (3.37) are associated with the digamma functions therein.
For our case, one can find poles from the three digamma functions, respectively, as (left poles): where Before continuing with our analysis, let us pause and discuss the 2nd right poles.As can be seen, the 2nd right poles have the same structure with the 1st right poles except for the leading condition: b R + ν − 1 2 = 0.However, we also find that this leading ("pole") condition may not only be irrelevant for our pole-skipping analysis, but also affect on the range for the zeros: n z R .For instance, one can find the same condition from the right "zeros" when n z R = ν − 1/2 in (3.49).This implies that the Green's function (3.37) can be expressed as when b R + ν − 1 2 = 0, which is a finite constant.In other words, the condition satisfying b R + ν − 1 2 = 0 cannot be considered as zeros (or poles).Thus, we need to refine the range for the right zeros as In summary, for the half-integer case, we have the (left/right) zeros (3.49) and the (left/1st right) poles (3.50) with Then, based on zeros and poles obtained above, we can discuss the pole-skipping for the half-integer case.We find the pole-skipping points as (right zeros & left poles): which have the same structure with the non half-integer case (3.43): however, recall that the range of zeros is restricted for the half-integer case (3.52).Also, notice that the functional form of the leading pole-skipping point is the same with (3.44).
Pole-skipping in the extreme limit.Next, let us discuss the extreme limit of the pole-skipping for the half-integer case.In order for this, we need to consider (3.40) (3.54) First, we can find that the factor 2 only give the zeros as where (3.52).Note that unlike the non half-integer case (3.45), (ω − k) ν− 1 2 only produces zeros, (left zeros), for the half-integer case since (3.24). 6 On the other hand, we can find the poles from the rest, 2 log(ω where (3.52).Notice that the two digamma functions give this right poles for the same reason described around (3.51).Also note that one may also find the poles from log(ω − k), ω − k = 0, however this cannot be considered as the poles since the structure of (3.54) shows that ω − k = 0 corresponds to the zeros: when ω → k.
Finally, combining the left zeros, ω − k = 0, with the right poles (3.56), we find the pole-skipping point (ω 0 , k 0 ) as (3.58) 6 One may also find the zeros from the rest, 2 log(ω − k) , however we do not consider this case in this paper since such zeros cannot make the pole-skipping phenomena: recall that the factor 2 cannot produce the pole structure.
Furthermore, as did in the non half-integer case (3.46), we can expand the Green's function (3.40) near (3.58) and find which implies that (3.58) can be considered as the pole-skipping point when ν = 3/2.Therefore, we can find the pole-skipping point even in the extreme limit for the half-integer case when ν = 3/2 as: in which its leading is 4 Vector fields (s = 1) In this section, our focus is on the pole-skipping phenomena of massive vector fields and their response to rotations.We begin by employing the approach used in computing the analytic solution for vector fields [54,55], enabling us to derive the analytic Green's function and investigate the pole-skipping behavior.Although the pole-skipping points for vector fields have been previously studied, they mostly have been explored in the context of non-rotating black holes, for instance [19].To the best of our knowledge, the literature lacks both the explicit form of analytic Green's function of the massive vector field and an examination of pole-skipping thereof in the rotating BTZ black hole.

Maxwell equations of motion
The analytic solution of the vector field A for the massive Maxwell equation is given in [54,55].Based on these papers, we review the analytic solutions for the gauge field in rotating BTZ.The field A satisfies with the mass m A .In the coordinates (T, ρ, X) represented in (2.8), as in the fermionic field (3.7), the gauge field fluctuation can be written as In order to solve the equation (4.1) analytically, it is convenient to use the ansatz Using this ansatz, the equations of motion in (4.1) are decoupled as for i = 1, 2. Here, Within the zcoordinate introduced in (3.9), the decoupled equations above can be expressed as Furthermore, the remaining equation of motion for A ρ can be derived from (4.1) as where the relation between (A T , A X ) and (A 1 , A 2 ) in (4.3) are used with the definition of z.It should be noted that this relation arises when considering a finite value of m A .The analytic incoming solutions of the equations in (4.5) are where and These parameters are different with (3.12) in the fermionic field part.The constants e 1 and e 2 have a relation by Lorenz gauge(∇ ν A ν = 0) and the representation theory of one forms on SL(2,R) manifolds [54,55]. 7.2 Exact vector Green's function

Holographic dictionary for vector Green's function
Using the analytic solution A 1,2 obtained above, we study the vector Green's function.Let us first review how to read such a vector boundary correlator.
Non integer case.According to the holographic dictionary, the AdS boundary behavior of the gauge field A are also related to the Green's function.For the equations in (4.5) within the z-coordinate, one can find the following asymptotic behavior near the AdS boundary (z → 1) where ∆ ± is the conformal dimension Furthermore, one can also study the asymptotic behavior for the gauge fields in terms of (t, r, φ) coordinates as follows.The gauge fields (A t , A r , A φ ) are defined as of (t, r, φ) coordinates in the metric (2.1).Comparing with the definition of (A T , A ρ , A X ) in (4.2), below relations are computed.
where we use the coordinate transformation (2.9).According to the transformations ( , the near boundary asymptotic behavior for the gauge fields (A t , A r , A φ ) are obtained as where For the obtained incoming analytic solution (A 1 , A 2 ) in (4.7), the coefficients E and F can be found as (4.26).However, the behavior (4.15) is not correct for the following integer case.
Integer case.As same as (3.17), we define the parameter ν as by the conformal dimension (4.12).In particular, using ν, we can rewrite the boundary expansion (4.11) as It is assumed that ν is not an integer for the boundary expansion (4.11): In principle, it is also possible to consider the negative integer.However, it is enough to consider the positive integer case for our purpose: see the description around (4.24).Therefore, for the case of integer ν, such as one can find the different boundary behaviors By the transformations (4.3), (4.6) and (4.14), the boundary asymptotic behaviors of (A t , A r , A φ ) are computed from the behaviors of (A 1 , A 2 ) in (4.21) as (4.22)We also found the relations corresponding to (4.16).However, its expression is not so illuminating so we do not display it here.By using the incoming analytic solution in (4.7).one can find the analytic expression for Ē, F, H, and J as (4.35) for integer case.
Vector Green's function in holography.According to the holographic principle, the Green's functions are given by the ratio between two independent parameters as where the former one is for the non integer case, while the later is for the integer case.Strictly speaking, the Green's functions in (4.23) corresponds to the case of the standard quantization where E (or Ē), H (or H) are interpreted as the sources and F (or F), I (or Ī) are corresponding vevs.
As same as the Fermionic Green's function, it is also possible to consider the other quantization, alternative quantization, by replacing E ↔ F and H ↔ I.However, in this paper, we only consider the case of standard quantization (4.23) for our own purpose: notice that the structure of the pole-skipping, GR ∼ 0/0, is independent of the type of a quantization.
Equivalently, considering a standard quantization implies that we only focus on the positive ν, for instance, one can notice that the role of E and F (H and I) in (4.15) can be exchanged when the ∆ ± in (4.12) is defined reversely with the reversed sign of ν in (4.17).Therefore, in this paper, we only choose ν > 0 hereafter 8   Mathematically, it is also possible to consider ν = 1.However, this special case may produce ill-defined boundary expansions.In this case, the coefficients C and D of (4.21) are ill-defined because for ν = 1.As explained around (3.24), we do not consider the special case in this paper.
Furthermore, similar to section 3.2, following the coordinate transformation (2.9), one can restore the Green's function in the original coordinates (t, r, φ) of (2.1), G R , from the one in the coordinates (T, z, X), GR in (4.23), as The relation between (ω, k) and (k T , k X ) is equally given as (3.26) by comparing (4.2) with (4.13).

Non integer ν case
Plugging the analytic solutions A 1,2 (4.7) into (4.3),one can find the analytic expressions for A T and A X .Using the solution of A ρ (4.6) and transformation (4.14), the analytic solutions for (A t , A r , A φ ) can be computed.Furthermore, expanding (A t , A r , A φ ) near the AdS boundary (z → ∞), we can read the coefficients in (4.15) as where and (a, b) is given in (4.8).Then, evaluating their ratio, (4.23), the analytic Green's functions can be simply found as (4.28) Using the coordinate transformation (4.25) and definition of ν (4.17), we can find the expression of G R (ω, k) as Here, we define where we used together (4.8) with (3.26) in order to express G R in terms of (ω, k).Note that the temperature dependence (T L , T R ) of Green's function is encoded in the refined parameters (a L , b R ) in (4.30).Also, note that the Green's functions (4.29) are for the non-extremal case.
For the case of an extreme limit, T L → 0, one can notice that a L → ∞ in (4.30).Similar to section 3.2.2, in order to describe the extreme limit of G R (ω, k), it is useful to consider the asymptotic gamma function (3.31).Then, one can easily check that the ratio between Γ(a L + ν − 1) and Γ(a L ) in (4.29) can be expressed in a L → ∞ limit as where we also used (4.30) in the last equality.
Therefore, now we can find the Green's function in the extreme limit, where the overall prefactor in (4.29), is canceled out with the one from (4.31).

Integer ν case
Next we discuss the integer case.For this purpose, it is useful to consider the property of the hypergeometric function (3.34), similar to section 3.2.3.Note that the hypergeometric functions in (4.7) can be expressed by (3.34) for integer n 1,2 as because A 1 and A 2 can be written as Therefore, one can expand the analytic solutions for (A t , A r , A φ ) near the AdS boundary using (3.34) and find the coefficients in (4.22) for the integer case as where Then, the Green's functions are obtained via (4.25) together with (4.23) and (4.10) as where we use (4.17) and (4.30) and ignore the contact terms, Ψ(ν) + Ψ(1), not related with pole-skipping.
In the extreme limit (T L → 0 or a L → ∞), we can use (4.31) even for the integer case.Similar to section 3.2.3,we also use the property of the digamma function (3.38).Then, the digamma functions in (4.37) can be expressed as where we also omit log T L irrelevant for structure of the pole-skipping.Therefore, the Green's functions in the extreme limit become In this section, we investigate the pole-skipping of the vector Green's function G R tt and G R φφ : (4.29), (4.32) for the non integer case and (4.37), (4.39) for the half-integer case.Note that we focus on G R tt in this section because G R φφ is obtained by G R tt for all the cases.

Non integer ν case
Pole-skipping in the non-extremal case.We first discuss the case of the non-extremal Green's function G R tt in (4.29).One can notice that we have two types of poles and zeros from (4.29) as (left poles): where where n z L , n z R = 0, 1, 2, • • • .Then, the pole-skipping point at which the pole line intersects with the zero line can be obtained from the following combinations Therefore, (4.44) cannot be taken as the pole-skipping points since the Green's function G R tt,0 can be solely determined as ∞ or 0.

Integer ν case and pole-skipping in extreme limit
Pole-skipping in the non-extremal case.For the integer ν case, because of the property of gamma function (3.47), one can notice that the gamma functions in G R tt of (4.37) can be expressed as These gamma functions and overall factor (ω − k) in the Green's function G R tt give the following zeros (left zeros): As same as section 3.3.2, the leading condition (n p R = 0) may not only be irrelevant for our pole-skipping analysis but also affect the range for the zeros: n z R .In this case, one can find the same condition as a "zero" condition in the right zeros (see b R = −n z R = −ν from (4.47)).This implies that at (n z R = ν) from right zeros or (n p R = 0) from right poles, the Green's function (4.37) can be expressed as when b R + ν = 0, which is a finite constant.The condition b R + ν = 0 cannot be considered as zeros (or poles).Thus, the range for the right zeros is redefined as In summary, for the integer case, we have the (left/right) zeros (4.47) and the (left/right) One can check that the first combination for G R rr in (4.42) and (4.61), produces the leading pole-skipping point (ω leading , k leading ) as same as one for G R tt (4.43).
Pole-skipping in the extreme limit.The pole-skipping in the extreme limit (T L → 0) for the non integer case cannot be achieved simply by taking T L → 0 on (4.61) because a L in (4.30) are ill-defined in the extreme limit.Instead, we need to consider G R rr,0 of (4.32) in order to discuss the pole-skipping in the extreme limit.From the structure of G R rr,0 , one can find the two types of the 'would-be' pole-skipping points, (ω (I) , k where right poles are from (4.40) and (4.60).Note that the overall factor (ω − k) ν only can make zeros because of the positive ν.This is a difference with those for G R tt,0 in (4.44).However, as same as the subsection 3.3.1,(4.62) may not be considered as the poleskipping points.The vector Green's function G R rr,0 near the 'would-be' pole-skipping points (4.62) can be expressed as

. 43 )
Note that the remaining combination, (left poles & left zeros) and (right poles & right zeros), cannot produce the pole-skipping point.