Correlation functions in finite temperature CFT and black hole singularities

We compute thermal 2-point correlation functions in the black brane $AdS_5$ background dual to 4d CFTs 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 proper time from the event horizon to the black hole singularity.