Singularities, geodesics and Green functions in the BTZ black hole

TL;DR

This paper investigates how the BTZ black hole's interior, including singularities, can be understood through boundary Green functions by relating semi-classical approximations and geodesic analysis within the AdS/CFT framework.

Contribution

It introduces a method to relate Green function saddle points to geodesics, revealing that some geodesics penetrate the horizon, aiding in resolving singularities via boundary theory.

Findings

Some geodesics extend inside the horizon.

Green function saddle points correspond to bulk geodesics.

Potential to resolve singularities from boundary data.

Abstract

In the context of studying black hole singularities by the AdS/CFT correspondence, we study the BTZ black hole by a scalar field propagating on it and the corresponding two-point Green functions. We explore how positions inside the horizon are encoded in the boundary theory. The main idea is to relate two different semi-classical approximations of the Green function and see how this indicates the bulk-boundary correspondence. From a key observation of Festucia and Liu, which is a frequency-geodesic identification, we deduce a geodesic approximation from the saddle point approximation. As an application, we find saddles of the Green function and hence their corresponding geodesics. The conclusion is that some of these geodesics do go inside the horizon. This gives the possibility of resolving the singularity from the boundary theory.