Thermalization of Green functions and quasinormal modes

We develop a new method to study the thermalization of time dependent retarded Green function in conformal field theories holographically dual to thin shell AdS Vaidya space times. The method relies on using the information of all time derivatives of the Green function at the shell and then evolving it for later times. The time derivatives of the Green function at the shell is given in terms of a recursion formula. Using this method we obtain analytic results for short time thermalization of the Green function. We show that the late time behaviour of the Green function is determined by the first quasinormal mode. We then implement the method numerically. As applications of this method we study the thermalization of the retarded time dependent Green function corresponding to a minimally coupled scalar in the AdS3 and AdS5 thin Vaidya shells. We see that as expected the late time behaviour is determined by the first quasinormal mode. We apply the method to study the late time behaviour of the shear vector mode in AdS5 Vaidya shell. At small momentum the corresponding time dependent Green function is expected to relax to equilibrium by the shear hydrodynamic mode. Using this we obtain the universal ratio of the shear viscosity to entropy density from a time dependent process.


Introduction
The question of how a highly excited state in a quantum system relaxes to equilibrium or how a quantum system relaxes when one of the parameters describing its Hamiltonian is quenched is of phenomenological interest. The relaxation of the state produced initially by high energy nuclear collisions from a highly excited state to quark-gluon plasma is an important question in the RHIC experiments [1]. Similarly the question of quenching of a quantum system can be studied experimentally in cold atoms where the coupling of an interacting system can be tuned to almost any value and in short times scales [2][3][4][5]. There have been various approaches, both analytical and numerical, developed to study these questions for a variety of quantum systems. When the question of thermalization is asked in quantum field theories which admit a holographic dual, the gauge/gravity correspondence links the question of approach to equilibrium in the field theory to the formation of a black hole in the bulk. See [6][7][8][9][10] for early studies pursuing this idea. More recently motivated by success of the gauge/gravity duality to describe near equilibrium physics and hydrodynamic behaviour in strongly coupled field theories this question has received renewed attention which has resulted in more quantitative understanding [11][12][13][14][15][16][17]. In [13] the excited state in the quantum field theory was created by a translational invariant perturbation along the boundary of a minimally coupled scalar field . This perturbation lasted for a short duration of time. It was shown by solving the bulk equations that for a small amplitude of the perturbation, the metric outside the in-falling shell of matter is that of a black brane at the leading order. This result encouraged subsequent authors to model the collapse to a black hole by a homogenous in-falling shell of matter [18][19][20][21][22][23].
For definiteness we consider the thin shell model of collapse [20] given by the following metric for v ≥ 0.
Here z refers to the radial co-ordinate, the boundary is at z = 0. x = x 1 , · · · x d−1 are the spatial co-ordinates at the boundary. The metric for v < 0 can be seen to be that of AdS d+1 using the following co-ordinate transformation v = t − z. (1.2) While the metric for v > 0 reduces to that of the black brane in AdS d+1 under the co-ordinate transformation From these co-ordinate transformations, we see that v coincides with time t at the boundary. We have chosen to work with units in which the radius of AdS d+1 is unity. The radius of the horizon is also unity. The Penrose diagram of the collapse is given in figure 1.
In [20] the study of how probes such as two point functions, Wilson loops and the entanglement entropy [24] thermalize in the thin shell Vaidya collapse was initiated. This study was mainly confined to the saddle point approximation of the probes. The behaviour of the probes were characterized in terms of their minimal geometric volume. In [22] the analysis was extended to study the two point function of operators dual to a minimally coupled massive scalar beyond the geodesic approximation for the case of the AdS 3 Vaidya shell. The retarded Green function G R (t 2 , t 1 ; k) with time t 1 < 0, t 2 > 0, before and after the collapse of the shell was evaluated numerically. Translational invariance in the spatial direction of the collapsing shell (1.1) enabled the characterization of the Green function in the Fourier k space corresponding to the spatial directions. The analysis was numerical and it showed that the relaxation of the Green function is determined by the first quasinormal mode. In [23] the study of thermalizing Green functions was extended to fermions in the AdS 4 Vaidya shell. The analysis was again done numerically.
In this paper we develop a new method to evaluate the retarded two point function G R (t 2 , t 1 ; k) of an operator in the dual theory corresponding to the collapsing Vaidya AdS shell in the bulk. The method is general and can be implemented in arbitrary dimensions and for arbitrary types of fields in the bulk. The method relies on performing the matching of the wave functions corresponding to the dual fields before and after the shell term by term in the expansion of the radial co-ordinate z of (1.1). We will show that this enables the determination of all the time derivatives of G R just after the collapse, v = 0 + of the shell. It is then possible to evolve the Green function to an arbitrary future time t 2 . On implementing this method we see that to obtain information of more and more higher derivatives of the Green function one needs to perform the matching of the wave functions of the bulk fields to higher powers in the radial coordinate z. This implies that one needs the information of the wave functions of these fields closer to the horizon to obtain long time behaviour of the Green function. It also implies that short time behaviour of the two point function after the collapse can be determined analytically from the near boundary information of the wave functions. Using the fact that the Green function at long time is determined by the near horizon behaviour of the wave functions we show that the relaxation of the Green function to equilibrium is determined by the first quasi-normal mode of the dual bulk field corresponding to the operator of interest in the black hole background.
We implement the method numerically and re-visit the case of the minimally coupled massive scalar in the AdS 3 Vaidya shell. We reproduce the results of [22]. We then study the case of the minimally coupled massless scalar in AdS 5 Vaidya shell. The Green function corresponding to this scalar is the retarded two point function of the spin 2 part of the stress tensor. We show that this Green function relaxes by the first quasi-normal mode which was determined numerically in [31]. We then study the vector fluctuations of the metric and obtain the Green function of the vector part of the stress tensor. For small momentum k, it is known [32] that this mode admits a hydrodynamic quasi-normal mode which obeys the dispersion relation given by where η is the shear viscosity, s the entropy density and T the temperature of the fluid. We show the time dependent Green function corresponding to the vector fluctuations of the metric relaxes to equilibrium at small k by the hydrodynamic mode. Using this we determine the universal ratio of shear viscosity to entropy density from a time dependent process. This paper is organized as follows. In the next section we detail the new method developed in this paper to evaluate the retarded Green function in collapsing AdS Vaidya thin shell backgrounds. We see that the method results in a recursion formula for the derivatives of the Green function just after the collapse of the shell. In section 3 we apply the method to obtain the Green function of the operator dual to the minimally coupled scalar in AdS 3 Vaidya shell. In section 4 we show that the long time behaviour of the Green function is determined by the first quasi-normal mode. This is first done for the case of AdS 3 Vaidya for which wave functions before and after the collapse of the shell are known exactly. Then the argument is extended in general for any Green function. In section 5 we turn to the case of AdS 5 Vaidya. We first study the thermalization of the shear correlator of the stress tensor by solving the minimally coupled scalar in AdS 5 Vaidya. We then examine the vector perturbations of the metric to evaluate the two point function of the spin-1 part of the stress tensor and show that it relaxes at small momentum by the shear hydrodynamic mode. Section 6 contains the conclusions. Appendices A to D deal with technical details required for the analysis in the paper. Appendix E describes the Mathematica notebooks which enable the evaluation of the Green functions numerically using the recursive method developed in this paper.

Recursion method for time dependent Green functions
In this section we will outline the general method to obtain the retarded Green function in the thin shell Vaidya AdS geometry. The method is general, and can be applied to any field in the AdS d+1 Vaidya geometry given in (1.1). For definiteness let us focus on the minimally coupled scalar of mass m. The differential equation obeyed by φ is given by and k is the Fourier conjugate of direction x d−1 .
The solution φ AdS in the region v < 0 corresponding to before the formation of the black hole admits a closed form in terms of Bessel functions. The analytical solution will be explicitly discussed in the examples we will consider subsequently. The solution has the form where ∆ + > ∆ − are the two solutions of the equation The solution we choose satisfies the boundary condition with t 1 < 0. This is necessary to obtain the retarded Green function. In (2.3) note that due to time translational symmetry for v < 0, the wave function just depends on the combination v − t 1 . In the black hole region v > 0, the bulk equations of motion usually do not admit a closed form solution. But the solution in the frequency ω and momentum k domain can be constructed in terms of a Frobenius series around the boundary z = 0. Then the most general solution in the time domain v > 0 can be obtained by taking Fourier transform of the two independent solutions obtained by the Frobenius method with respect to the frequency, ω. We write this as (2.6) Note that here we have assumed that the roots of the indicial equation of (2.1), ∆ + , ∆ − do not differ from each other by an integer. The discussion can be carried out for the case when the roots differ by an integer but as we will see that we will only need the less singular solution which falls of as z ∆ + to construct the retarded Green function. C(ω, k) and D(ω, k) in (2.6) are functions which must be determined by continuity at v = 0. Note that the coefficients of the differential equation (2.1) are discontinuous, but the discontinuity is finite across v = 0, therefore the solution φ must be continuous at v = 0. Thus we have We impose continuity by equating each term of the power series in z about the boundary. Thus we expand both sides of (2.7) in powers of z and obtain the equation We have suppressed the dependence ofJ n on t 1 , k and the dependence of C(ω), A n , D(ω), B n on k to un-clutter the equations. Equating the coefficients of z n in the terms proportional to z ∆ + we obtaiñ From the examples considered in the paper, it is seen that A n (ω) is an n-th order polynomial in ω 1 . Therefore we write A n as A n = n j=0 a j n ω j . (2.10) Substituting this expansion in (2.9) we obtaiñ Let us define the j-th moment of C(ω) as, Substituting this in equation (2.11), we can rewrite it as, This equation can be inverted to obtain the moments, M j , which contain information about C(ω). Note that a j n are known from Frobenius series solution φ BH . Therefore we obtain M n = 1 a n n J n − n−1 j=0 a j n M j ; n > 0, (2.14) Knowledge of all the moments M j is sufficient to construct the retarded Green function. To see this consider the near boundary behaviour of the field φ. From equations (2.5), (2.6) and (2.8), the near boundary behaviour of the field for v > 0 is given by Together with the boundary condition (2.5), the AdS/CFT recipe for the retarded Green function [26] states that it is given by Here we have ignored overall proportionality constants in the Green function to simplify the discussion. Expanding the exponential as Taylor series, and using the definition of the moments of C(ω) we obtain Thus the knowledge of all the moments of C(ω) is sufficient to construct the Green function. Note that knowledge of D(ω) in (2.6) is not necessary. We call this the recursion method to obtain the Green function since each term in (2.18) is given recursively from the knowledge of the lower moments using (2.14). It allows for the construction of the the Green function as a power series in time for v > 0. The information of t 1 is present inJ n and all terms depend on momentum k. There is another way to view this construction of the retarded Green function. Note that the j-th moment of C(ω) is essentially the j-th derivative of the Green function, evaluated at v = 0 + , Thus this method determines the Green function for v > 0 from the knowledge of all its time derivatives at v = 0 + .
We now make some general properties of this method of determining the retarded Green function. Since the method relies on construction of φ BH using the Frobenius series we can apply it in general to all black hole backgrounds even if closed form solutions do not exist. The method can be applied even if the background is only known in terms of a power series in z about the boundary for example in finite temperature versions of RG flow solutions [27]. Note that for short time development of the Green function for v > 0, knowledge of only a few moments is needed. This implies from (2.18) and (2.14) we need the knowledge of wave functions before and after v = 0 close to the boundary. That is we need the knowledge of the wave functions to a few powers of z. Thus short time development of the Green function can be written down analytically by obtaining a few moments. However for the long time behaviour of the Green function we need to know a large number of moments. Again from (2.14), this implies we need the information of the wave function for large powers of z which in turn implies that we need the behaviour close to the horizon. This fits with the general intuition that long time behaviour is controlled by the behaviour near the horizon. It is this property which will enable us to prove that the long time behaviour of the Green function is determined by the quasinormal mode in section 4. Finally, it will turn out that even moments are real and odd moments are purely imaginary. This is because the coefficients a j n for odd j are imaginary. This is easily seen due to the fact that each power of ω comes with a factor of i. Then from (2.14) it is easy to see that even moments are real and odd moments are purely imaginary. This then ensures that the Green function given by (2.18) is real.

Thermalization in AdS Vaidya
In this section we will implement the method developed in section 2 for the case of thin shell Vaidya metric in AdS 3 . We study the thermalization of the retarded Green function of the operator corresponding to the minimally coupled scalar of mass m. We will show that using our method we reproduce the results of [22]. We also determine a few low moments analytically to obtain the short time behaviour of the Green function. A simplification that occurs for the case of the AdS 3 is that the solutions of the minimally coupled scalar in the BTZ black hole are known in closed form in terms of hypergeometric functions [28]. These solutions will provide the initial starting point in our argument to demonstrate that the long time behaviour of the Green function is determined by the lowest quasi-normal mode.
The thin-shell AdS-Vaidya metric, in 3 dimensions is given by The co-ordinate φ parametrizes the spatial direction of the field theory and we assume that it is not compact. The scalar field equation in this metric is given by

Scalar wave functions in AdS 3 Vaidya shell
We solve the minimally coupled massive scalar equation given in (3.2). We discuss the solutions before and after the collapse of the shell below.
Solution for v < 0 The solution for the scalar field for v < 0 is given by [22] Φ Note that this solution is written down in mixed Fourier space (t, k). Here ∆ ± are solutions to the quadratic equation ∆(∆ − 2) = m 2 and are given by This solution satisfies the required boundary condition discussed in (2.5) which is required to evaluate the retarded Green function, To show this we first take the the limit This solution certainly diverges in the limit v → t 1 . All what one needs to show is that the expression in (3.7) is a representation of the delta function times z ∆ − . To demonstrate this we perform the integral over v as follows Therefore we conclude that the solution in (3.3) satisfies the required boundary condition given in (3.6).
The strategy to obtain a closed form solution for v > 0 is as follows. Note that under the transformation the metric given in (3.1) for v > 0 reduces to that of the BTZ black hole in Poincaré coordinates. This is given by (3.10) Now the solutions to the minimally coupled scalar in the BTZ black hole is known [28]. The two independent solutions are given by Note that these two independent solutions reduce to the ingoing and outgoing Fourier modes at the horizon. We can now obtain the solution in the coordinates (z, v, φ) by performing the substitution given in (3.9). This leads to the following independent solutions for the Vaidya metric in the region v > 0, Now the above solutions admit an expansion around the horizon z = 1, however we have seen in section 2, to obtain the Green function we need an expansion around the boundary z = 0. This can be achieved by using the transformation properties of the hypergeometric functions. We have performed the required transformation in appendix A. This results in the following independent solutions, It can be explicitly verified that the above solutions satisfy the equations of motion given in (3.2) for v > 0. The partial Fourier transform of this solution with respect to ω is given by The above solution admits an expansion around the boundary z = 0. Therefore we have written the solution in the black hole region in the required form given in (2.6).
Note that in the case of the BTZ black hole, the Frobenius expansion around the boundary can be written in closed form.

Matching at v = 0 and Green function
We follow the general procedure discussed in section 2 to construct the time dependent Green function. To do this we first obtain the moments of the function C 2 (ω) by matching the wave function The moments of C 2 (ω) are determined by comparing powers of z in the terms proportional to z ∆ + . We will demonstrate this procedure explicitly and obtain moments up to the second order. We expand the LHS of (3.3) and the z ∆ + coefficient of (3.13) to quadratic order in z to obtain the following equation Comparing the terms we can read out the moments to quadratic order. These are given by As discussed in section 2, see equation (2.18), the retarded Green function is given by 2 Thus it is clear that the short time expansions of the Green function can be easily obtained.
It is interesting to take the k = 0 limit of the moments and construct the short time expansion of the Green function. In this limit the Bessel function reduces to a rational function and the first few moments are given by Note that it is clear that the Green function does not vanish identically for k = 0 for arbitrary t 2 , v, since these moments do not vanish.

Recursive numerical construction of the Green function
It is easy to set up an algorithm in Mathematica to evaluate the moments recursively as discussed in section 2. This algorithm is used to evaluate 56 moments of the function C 2 (ω). From (2.18) we can construct the Green function to O(v 56 ) 3 To improve accuracy we then approximate the Green function using the (28|28)th Padé approximant. The results for the Green function are given in the 3 figures which we will discuss. Figure 2, shows the thermalizing Green function (solid blue curve), the vacuum Green function (dot-dashed red curve), and the thermal Green function (green dashed curve), as a function of future time v. The thermalizing Green function starts close to the vacuum Green function, for small time, however it deviates away from the vacuum Green function within one horizon time. At large time, the thermalizing Green function approaches the thermal one. At v = 0, the thermal Green function starts at a different value than the thermalizing and the vacuum Green functions. The thermal Green function is plotted from (B.2) derived in the appendix B. Figures  2 and 3, are plotted for specific values, ν = 2 3 , k = π 2 and t 1 = 1.7. Figure 3 is the logarithmic plot of the absolute value of the thermalizing Green function (solid blue curve), and the imaginary part of the lowest quasinormal mode (dot-dashed red curve), e −∆ + v . It is seen that for large time, the decay of the thermalizing Green function is given by this lowest quasinormal mode. Here, large time means time of the order of a few horizon radius, as can be seen from the plot. It is important to note that thermalization, i.e. decay of thermalizing Green function follows the lowest quasinormal mode, is achieved within a few (∼ O(1)) horizon radius. The dips in the plot are the points where the thermalizing Green function crosses the time axis in figure 2, and indicates the oscillations of this Green function. Figures 2 and 3 reproduce those found in [20] by the direct numerical integration of the differential equation (3.2).   large values of n. Form the recursive relations for the moments in (2.14) we see that this can be done if we implement the matching of the wave functions to order z n . This in turn implies that we need the knowledge of the less dominant wave function in (2.6) Φ + defined by closer to the horizon. This is because the coefficients of z n for large values of n will be determined by the singular behaviour at the horizon. The behaviour of this wave function near the horizon can be determined by the general properties of solutions of wave equations in AdS black holes. Using this information we will show that the long time behaviour of the Green function is determined by the first quasinormal mode. We will first demonstrate this for the case of the minimally coupled scalar in AdS 3 Vaidya. This is easy to do explicitly since, the wave function Φ BT Z(+) is known in closed form. We will then show in general that the long time behaviour of the Green function is determined by the first quasinormal mode of the wave functions in the corresponding black hole background.

Green function in AdS 3 Vaidya
From the preceding discussion we need to examine the behaviour of the function Φ + (ω, k, z) close to the horizon. For the case of the minimally coupled scalar in the BTZ black hole this function is known in closed form (3.14) and is given by (4.2) The near horizon limit of this wave function can be easily determined from the properties of the hypergeometric function given in (A.1). We see that the singular behaviour near z = 1 is given by .

(4.3)
This equation will serve as the starting point of obtaining the long term behaviour of the Green function. It is important to note that the mode Φ + vanishes near the horizon when the frequency is given by These are the quasinormal modes of the minimally coupled scalar in the BTZ black hole [29]. Also note that these modes are located in the lower half ω-plane. We now impose the matching condition at v = 0. This will lead to the following equation .
We expect this approximation near the horizon to estimate the behaviour of the moments suitable to obtain long time behaviour of the Green function. We can now solve for C 2 (ω) by substituting and multiplying both sides of the equation in (4.5) by e −iω y and formally integrating over y from −∞ to ∞ 4 . This leads to the following expression for C 2 (ω) (4.8) Finally the Green function is given by Note that the factor multiplyingĨ(ω, k) in (4.9) is a function of ω such that for ω → −i∞ it behaves as , (4.10) ∼ e +iω ln 2 (ω) ν+ 1 2 .
I(ω, k) is essentially a Fourier transform of the Bessel function. We assume that this is analytic in the lower half ω-plane and grows at the most exponentially in ω as ω → −i∞, given by lim Here M > 0 is a fixed constant. With these assumptions and the behaviour in (4.10) we see that integrand in (4.9) goes to zero in the limit ω → −i∞ for sufficiently large but fixed v > 0. This is because of the exponentially dying term e −iωv in the integrand of (4.9). Therefore the integral can be performed by completing the contour in the lower half ω-plane. Then the integral localizes to a sum over the poles of the Gamma functions. The poles of the Gamma functions are at the quasinormal modes given by (4.4). The result of the integral then reduces to It is now clear from (4.12) that the long time behaviour of the Green function is determined by the lowest quasinormal mode as we have seen in our explicit numerical evaluation of the Green function. The decay of the Green function is controlled by the imaginary part of the lowest quasinormal mode and the period of oscillations is determined by the real part of the lowest quasinormal mode. Note also from (4.12) we see that since the expression involves a sum over all the quasinormal modes, the rough time scale over which the lowest quasinormal modes takes over is of the order of a few horizon times. This is also clearly seen in the numerical evaluation of the Green function. It is important to note that the starting point of the analysis was the behaviour of Φ BT Z(+) near the horizon given in (4.3), which vanished at frequencies determined by the quasinormal mode spectrum in the lower half ω-plane.

Green function in AdS d+1 Vaidya
Using the intuition gained by the explicit solutions of the minimally coupled scalar in the BTZ black hole we now generalize the discussion. We show that the long time behaviour of the retarded Green function in arbitrary AdS d+1 Vaidya background is determined by the lowest quasinormal mode of the corresponding bulk field. Consider the differential equation given in (2.1) for the minimally coupled scalar in the AdS d+1 Vaidya for v > 0. Substituting we obtain (4.14) This equation has two regular singular points, z = 0 and z = 1, corresponding to the boundary and the horizon of the black hole. One can set up a Frobenius solution around either z = 0 or z = 1. Let the two independent solutions around z = 0 be Φ + ω and Φ − ω . From the indicial equation for the expansion at z = 0 we know that these solutions behave as follows 5 Therefore Φ − ω is the dominant singular mode at the boundary. Similarly one can set up an expansion around z = 1, let the two independent solutions around z = 1 be Φ in ω and Φ out ω . It is easy to see from the indicial equation around z = 1, that their behaviour near z = 1 is given by The reason we have labeled these solutions as in and out is because they correspond to the ingoing and outgoing solutions when these wave functions are transformed to the t, z coordinates. To see this note that the coordinate transformation near z = 1 can be obtained by integrating (1.3). This is given by Substituting this coordinate transformation we obtain the corresponding wave functions in the t, z coordinates

Now we have 2 sets of linearly independent solutions {Φ
Therefore by the uniqueness theorem of second order ordinary linear differential equations we should be able to express one set in terms of the other as linear combinations. Let us write Let us now use the definition of quasinormal modes to obtain some information of the coefficients M 12 and M 22 . We have seen that Φ in ω , Φ out ω correspond to the ingoing and outgoing modes in the t, z coordinates. Now by definition quasinormal modes are those values of frequencies for which these modes obey Dirichlet boundary conditions at the horizon. This implies that the more dominant mode proportional to Φ − ω should vanish at these frequencies. Thus we have the equation where ω out n are the quasinormal frequencies which lie in the upper half ω-plane for the outgoing modes and ω in n are the quasinormal frequencies which lie in the lower half ω-plane for the ingoing modes [29,30]. Let us invert the equation (4.19), we .

(4.21)
Note that the inverse exists that is detM = 0. This is because Φ + ω and Φ − ω can be written as linear combinations of Φ in ω and Φ out ω . We can now easily read out the near horizon behaviour of the solution Φ + ω from the above equation. The singular behaviour of Φ + ω is given by Here we have used the near horizon behaviour given in (4.16). Now M 22 (ω, k) vanishes at ω = ω in n , with zeros in the lower half plane. This is the general form of the equation (4.3) seen explicitly for the case of the BTZ black hole for black holes in AdS d+1 . Note that we arrived at this result from the general definition of quasinormal modes. From this point onwards we can follow the rest of the argument in section 4.1 to arrive at the conclusion that the long time behaviour is determined by the leading quasinormal mode. In general the gaps between quasinormal modes are of the order of horizon scales. Therefore we expect the leading behaviour to set in order of a few horizon times. Though here we have used the minimally coupled scalar to demonstrate our argument for simplicity, the analysis can be carried out for other fields. The steps involved will result in similar equations. We will see this explicitly for the vector fluctuations of the metric in the subsequent section.
This concludes our general argument of why the long time behaviour is determined by the leading quasinormal mode. Thus any retarded correlator which is used to probe the onset of thermalization caused due to injection of energy at an instance of time in the field theory will decay with the time scale set by the first quasinormal mode. In [33] it was conjectured that thermalization time scales in the field theory are determined by quasinormal modes. We have shown that the time dependent Green functions considered in this paper provides an explicit realization of this statement. From our general argument we expect this to be true for correlators whose lowest quasinormal modes are determined by hydrodynamics if the probing momentum scales are sufficiently small. In the next section we will verify this expectation for stress tensor correlators in N = 4 Yang-Mills whose lowest quasinormal mode is sensitive to the shear viscosity.

Thermalization in AdS Vaidya
In this section we study the thermalization of the retarded two point functions in the thin shell AdS 5 Vaidya geometry. We first consider the equation satisfied by the spin 2 metric fluctuation h x 1 x 2 . This fluctuation, perpendicular to the momentum k which is along the x 3 direction, forms the shear spin 2 mode. Evaluating the retarded Green function holographically for this mode provides information of the retarded two point function of the stress tensor T x 1 x 2 T x 1 x 2 , for the strongly coupled N = 4 Yang-Mills. The lowest quasinormal mode of shear fluctuations of the metric has been studied earlier in [31] numerically. We see our results are consistent with this earlier calculation. The more interesting metric fluctuation to consider is h x 1 x 3 . It is known that this mode admits a hydrodynamic quasinormal mode in the AdS 5 black hole for small momentum [32]. This mode appears as a pole in the thermal two point function of the stress tensor T x 1 x 3 T x 1 x 3 . We will implement the recursion method numerically and evaluate the thermalizing Green function for various values of small momentum k. From the long time behaviour of the Green function we show that this Green function relaxes by the hydrodynamic quasinormal mode. This enables us to read out the universal shear viscosity to entropy density ratio from a time dependent process.
The thin shell AdS-Vaidya metric in 5 dimensions is given by, here, x is a 3 vector, with components, x i , where i = 1, 2, 3. The x i 's parametrize the spatial directions in the dual field theory. The transformation which reduces the v > 0 part of the above metric to the AdS 5 black hole metric in Poincaré co-ordinates is given by In these coordinates we obtain the planar black hole metric in AdS 5 , for v > 0, Here again the radius of AdS as well as the black hole has been set to unity. The temperature of the black hole is then given by To study two point functions of the stress energy tensor of the boundary field theory we consider small perturbations to the background metric, g (0) µν , of (5.1), µν + h µν . The differential equations satisfied by the metric perturbations are obtained by linearizing the Einstein's equations which is given by Here R µν is the Ricci tensor and the value of cosmological constant, Λ, has been set to Λ = −6, in units of AdS radius. The linearized Einstein's equations are given by where R (1) µν is the linearized Ricci curvature.

Spin 2 metric perturbations
First, we consider the case when only the scalar mode of metric perturbation, h x 1 x 2 , is turned on. We substitute the ansatz h x 1 x 2 (t, x 3 , z) = e −iωt+ikx 3h x 1 x 2 (z) in (5.7). Here, the momentum is chosen to be along the x 3 direction. After the redefinition Φ(z) = z 2h x 1 x 2 (z) we obtain the following equation from (5.7) , where the prime means derivative with respect to z. Note that as expected these are the equations of the minimally coupled massless scalar in the AdS 5 background for v < 0 and the AdS 5 black hole background for v > 0.
Solution for v < 0 In appendix C we have obtained the solution which satisfies the boundary condition (2.5) for a minimally coupled massive scalar in AdS 5 . On setting the parameter m 2 = 3 or ν = 2 in (C.1), the equation reduces to (5.8). Therefore we can read out the solution in mixed Fourier space which satisfies the required boundary condition from (C.18). This is given by

Solution for v > 0
Unlike the case of the BTZ black hole, closed form solutions to minimally coupled scalar in the AdS 5 black holes are not available. However the recursive method we developed in section 2 just relies on information of the Frobenius expansion of the solution near the boundary. The roots of the indicial equation of (5.9) are given by ∆ ± = 4, 0. As discussed earlier to obtain the retarded Green function we need to obtain the Frobenius solution with the less dominant mode, that is with ∆ + = 4. The recursion relations for the coefficients a j n s defined in (2.10), necessary for the construction of the solution in the region v > 0, are given by the following equations with a 0 0 = 1, We can now extract the moments of the Green function by substituting the values of a j n determined from (5.11),(5.12) into (2.14). The values of the coefficients J n in (2.14) can be obtained from the recursion relation (D.4) with ν = 2. Finally the retarded Green function is constructed using (2.18). This procedure is clearly algorithmic and can be easily implemented numerically. Using Mathematica we implemented this procedure of obtaining the Green function. We have evaluated up to 200 moments C 2 (ω) and the Green function is constructed to O(v 200 ) in future time, v. We then approximated the Green function by the (100|100) Padé Approximant for better accuracy. For the values of k studied, 200 moments were sufficient to obtain convergent results. Figure 5a shows the thermalizing Green function (solid blue curve) for t 1 = 15 and a particular value of k = 10. As expected, the thermalizing Green function starts close to the vacuum Green function (red dot-dashed curve) at small time, and deviates from it within one horizon time. From the logarithmic plot, figure 5b the slope as shown by the dot-dashed red line of the thermalizing Green function (solid blue curve) is measured at large time, i.e. at time of the order of a few horizon radius. This slope gives the negative imaginary part of the lowest quasinormal mode (−Im ω), as shown in section 4. From the gaps between consecutive zeroes of the thermalizing Green function, real part of ω can be calculated using, Re ω = π/gap. The values of real and imaginary part of ω, for several values of k, are listed in table 1. These values agree with figures 5,6 of [31] where the numerical values of the real and imaginary part of the quasinormal modes for the minimally coupled scalar with ∆ = 4 in AdS 5 was obtained for various values of k. To compare our results to that of figures 5,6 of [31], note that we need to perform the following scalings k ours = 2q theirs , ω ours = 2ω theirs . As a simple check note that k ours = 10 corresponds to q theirs = 5, looking at their figure 6 we note that Im ω theirs = .8 which corresponds to Im ω ours = 1.6 which agrees with our result in table 1. Similarly note that for small values of q theirs , the value of Re ω theirs is above the 45 • line. This is also the case from our results in table 1.

Vector metric perturbations and shear viscosity
It is known that vector metric perturbations h x 1 x 3 , h tx 1 in the AdS 5 black hole with momentum is along the x 3 direction admit a hydrodynamic mode at small momentum in addition to the usual gapped quasinormal frequencies [32]. The hydrodynamic quasinormal mode corresponds to the hydrodynamic pole in the thermal correlator T tx 1 T tx 1 . This was used to read out the ratio of shear viscosity to entropy density in [32]. Using the methods developed in this paper we can evaluate the time dependent thermalizing retarded two point function T tx 1 T tx 1 in the AdS 5 thin shell Vaidya background. From the general analysis of section 4 we expect that the time dependent Green function in the AdS 5 thin shell Vaidya background should relax to equilibrium by the hydrodynamic quasi-normal mode. Therefore from the decay it should be possible to read out the ratio of shear viscosity to entropy density from a dynamical Green function. In this section we perform this analysis using the methods developed in this paper and obtain the universal ratio of shear viscosity to entropy density 6 .
We first turn on the following metric fluctuations with momentum along the x 3 direction, After the formation of the AdS 5 black hole for v > 0, the equations are given by Note that a j n is defined in (2.10) determine the Frobenius expansion around the boundary for the equation (5.18). The thermalizing Green function T tx 1 T tx 1 can be read out by by applying the matching condition, (2.7), on p v . This is because H v can be obtained by integrating p v . This just introduces a power of z which adjusts to give the right ∆ + root for the fluctuation H v . Now H v is related to the vector fluctuation h vx 1 which reduces to h tx 1 at the boundary v equals the boundary time t. Therefore reading out the time dependence of p v is sufficient to extract the time dependence of the the two point function T tx 1 T tx 1 .

Green function and quasinormal modes
Using the equations (5.20),(5.21),(2.14) and (D.4), with ν = 1, the moments M i are obtained. Then the retarded two point function of (v,x 1 ) component of the stress energy tensor of the boundary theory, is evaluated using (2.18). We used Mathematica to evaluate up to 3500 moments C 2 (ω), the results converged at this order of moments. The Green function is constructed to O(v 3500 ) in future time, v and then we approximated it by its (1750|1750) Padé approximant. From these moments the Green function is evaluated for various values of the momentum k. Figure 6a shows the thermalizing Green function (solid blue curve), which starts close to the vacuum Green function (dot-dashed red curve), and deviates away from it within one horizon time.
The hydrodynamic pole of the thermal two point function T tx 1 T tx 1 or the quasinormal mode of the vector perturbations of the metric obeys the dispersion relation η, s and T are the shear viscosity, entropy density and temperature, respectively. The universal value of η/s for the vector mode of metric perturbation is known to be (4π) −1 . The temperature of the AdS 5 black hole is T = π −1 , in which the radius of AdS and that of the horizon are normalized to unity. Therefore the dispersion relation for the hydrodynamic quasinormal frequencies is given by As discussed in section 4 the long time behaviour of the thermalizing Green function is dictated by the lowest quasinormal mode. Therefore it is possible to extract the value of the −Imω from the slope of the logarithmic plot of the thermalizing Green function for k << 1 and show that it obeys the dispersion relation given in (5.23).
In figure 6b, the logarithmic plot of the absolute value of the thermalizing Green function (solid blue curve) is plotted along with the line measuring the slope of its large time behaviour (dot-dashed red line). The slope of this straight line gives the value of −Im ω. The plot is for fixed values of t 1 = 15 and k = 10. Figure 7, shows the red dots for values of −Im ω for several values of k obtained from Mathematica. For small k, these values lie very close to the expected parabolic behaviour (solid blue curve), predicted by the hydrodynamic dispersion relation (5.23). For larger k, these values lie inside the parabola. This behaviour agrees with figure 14 of [31] which studied this quasinormal mode for large values of k in the AdS 5 black hole.

Conclusions
We have developed a recursive method to obtain the time dependent Green functions in the thin shell AdS Vaidya background. Using the intuition developed from this method we showed that the long time behaviour of the Green function is determined by the lowest quasinormal mode of the corresponding black hole. Thus our analysis provides an explicit realization of the general conjecture made in [33] that time scales in thermalization are determined by the quasinormal modes. We applied the method to study Green functions in thin shell Vaidya geometries in AdS 3 and AdS 5 . Using this method we obtained the universal ratio of shear viscosity to entropy density by studying the relaxation of the time dependent Green function of the vector metric perturbation in the AdS 5 Vaidya shell.
The methods developed in this paper to study time dependent Green functions can be generalized for fermion Green function which was obtained by direct numerical integration in [23]. It will be interesting to generalize this method to Vaidya shell geometries which have a finite width found in [13]. These geometries are a more accurate description of the thermalization process. The dependence of the width of the shell on the time dependence of the Green function will be interesting to extract. It is interesting to study thermalization of Green function of higher spin fields to study the spin dependence in thermalization. It is known that a non local probe like entanglement entropy is the slowest to relax to equilibrium [20]. It is will be interesting to see if higher spin fields relax faster or slower compared to scalars and entanglement entropy. In this context the exact solutions to wave functions of higher spin fields found in the BTZ background [34,35] will prove to be useful to obtain analytical results. Finally it will be useful to use these lessons learned in holography to understand time dependent Green function in conformal field theories during thermalization along the lines developed in [36].
with the graphics used in the paper. We thank the participants of the Bangalore area string meeting organized by the ICTS, TIFR, the string groups of IMSc, Chennai and IIT Kanpur for opportunities to present the results at various stages of the project and the resulting discussions. Finally J.R.D. thanks Dieter Lust and the string group at Max Planck Institute for Physics, Munich for hospitality during the initial phase of this project.

A Scalar wave functions in BTZ
In this appendix we derive the scalar wave functions in the BTZ geometry as an expansion around the boundary by transforming the known wave functions which are written as expansions around the horizon. Consider the wave functions given in (3.12) which admit a natural expansion near the horizon. We can use the following transformation property of the Hypergeometric function to write it as an expansion around the boundary, .
The coefficients B i , i = 1, . . Thus, the only two independent near boundary solutions are,
(C. 16) The following properties of the Bessel functions, λ (x), (C.17) are used in last two equations, to obtain the retarded Green function which is given by . Though we started out by assuming ν is not an integer and the two independent solutions are given by (C.2), the discussion can be generalized to the case when ν is an integer, leading to the same final result given in (C.18). A simple way to see this is that in the final result the order of the Bessel function is fractional.

D Recursion relation for retarded Green functions in AdS
In this appendix for completeness we obtain the recursion relation for the mixed Fourier space retarded Green function in AdS d+1 at v = 0. This is useful to implement the recursive algorithim to obtain the thermalizing Green function in thin shell Vaidya geometries. The solution for a scalar field in mixed Fourier space in AdS d+1 , has the following dependence on the Bessel function, Φ AdS (v = 0, k, z; t 1 ) = z ∆ + R(z). (D.1) Here we have assumed t 1 < 0.

E Details of Mathematica files
The recursion method of section 2 is used to numerically construct the Green function using Mathematica for the three cases considered in this paper. The files for each case is attached.
1. ads3-green.nb: The Green function for scalar field in AdS 3 Vaidya is numerically constructed to order 56 in moments and figures 2 and 3 are obtained.
2. shear-green.nb: The Green function for shear metric perturbations h x 1 x 2 = 0 in AdS 5 Vaidya is numerically constructed to order 200 in moments and figure 5 is obtained.
3. vector-green.nb: The Green function for metric perturbations h vx 1 , h x 1 x 3 = 0 in AdS 5 Vaidya is numerically constructed to order 3500 in moments and figure  6 is obtained.