Correlation functions in finite temperature CFT and black hole singularities

Abstract

We compute thermal 2-point correlation functions in the black brane AdS 5 background dual to 4d CFT's at finite temperature for operators of large scaling dimension. We find a formula that matches the expected structure of the OPE. It exhibits an exponentiation property, whose origin we explain. We also compute the first correction to the two-point function due to graviton emission, which encodes the time travel to the black hole singularity.

Thermal correlation functions in CFT and factorization

“…For strongly coupled theories with a gravity dual this structure naturally arises from the computation of the 2-point function in the black brane background, as recently shown for operators of large dimension in [3]. 3 It is interesting to note that, through the Eigenstate Thermalization Hypothesis, the thermal 2-point function is related to Heavy-Light-Light-Heavy 4-point correlators at T = 0 (e.g. [5,6,7,8]).…”

“…This was studied in general in [1] (see also [2]) for the case of the 2-point function, finding an elegant expansion in terms of Gegenbauer polynomials, whose coefficients incorporate both the OPE data as well as the non-zero vacuum expectation values in the thermal background. For strongly coupled theories with a gravity dual this structure naturally arises from the computation of the 2-point function in the black brane background, as recently shown for operators of large dimension in [3]. 3 It is interesting to note that, through the Eigenstate Thermalization Hypothesis, the thermal 2-point function is related to Heavy-Light-Light-Heavy 4-point correlators at T = 0 (e.g.…”

“…As a result, the value of the bulk cubic coupling enters only as an overall constant in the final result. Just as in [3], we will focus on correlators of operators with large dimension. In this case, the propagator can be computed in the WKB approximation, where it is represented as the exponential of the geodesic length of the trajectory for a particle moving between the boundary point where the operator is inserted up to the point of interaction.…”

“…In holographic theories, it has been shown that they can probe the interior of the black hole in the gravitational dual (e.g. [13,14,15,16,17,18,19,20,21,3]). In particular, they are expected to shed new light on the nature of spacetime singularities, although this may require going far beyond the leading strong coupling order.…”

The generalized holographic $c$-function for regular AdS black holes

“…There have already been some methods to probe the interior of a BH, especially those proposed in the frame of AdS/CFT correspondence [25]. One powerful and successful approach in probing the interior of BHs is the correlation function [26][27][28][29][30][31][32][33]. The function is divergent [29] at the singularity and is dominated [26] by the spacelike geodesics for the large mass bulk field.…”