# Retarded Green's function The retarded Green's function is a special [[0103 Two-point functions|two-point function]]. In this note we discuss the (thermal) retarded Green's function in holography. The Green's function encapsulates the relation between the source and the response. In holography, the thermal state is described by a black hole (see [[0574 Thermofield double]] for more details). Therefore, to compute the Green's function, one needs to perturb the system and study the linearised equations of motion. Once the perturbed solution is obtained, the Green's function can then be read off from it using the [[0001 AdS-CFT|AdS/CFT dictionary]]. More specifically, the [[0011 Fefferman-Graham expansion|FG expansion]] tells us how to do it in the case of metric perturbations, and other types of perturbations are analogous. There is however a question of whether the linearised equation of motion actually have a good boundary-value problem. In [[2002#Son, Starinets]], it was proposed that, since we are computing the retarded Green's function, the solution should satisfy ingoing boundary condition at the future horizon. This was later proved (see references below) to be the correct procedure. ## Refs - proposal in [[2002#Son, Starinets]] - proofs - [[HerzogSon2002]] - [[2008#Skenderis, van Rees (May)]] - [[2009#Iqbal, Liu]] - zero temperature - see [[0472 CMT for extremal BH]] ## Small frequency - see [[2009#Faulkner, Liu, McGreevy, Vegh]] for a good discussion in holography - the exact zero-frequency retarded Green's function $G_R(\omega=0, \vec{k})$ is governed by UV physics - the small $\omega$ expansion has two parts - analytic part governed by UV physics - non-analytical part proportional to the retarded Green's function of $\mathcal{O}_{k}$ in the IR CFT ## Explicit computations - [[2023#Bhatta, Chakrabortty, Mandal, Maurya]]: scalar operator, hyperbolic CFT ## Related - [[0103 Two-point functions]] - [[0179 Pole skipping]]