Singularities, geodesics and Green functions in the BTZ black hole

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 boundary two-point Green function. We explore how positions inside the horizon are encoded in the boundary theory. The main idea is to compare two different semi-classical approximations of the Green function and see how this indicates the bulk-boundary relation. Writing the Green function in the Fourier integration of the momentum space correlation function, we can approximate it by the steepest descent method and the Green function is dominated by saddle points. Alternatively, writing the Green function in the form of the Feynman paths integration, we can apply the geodesic approximation and the Green function is dominated by certain geodesics joining the two points. To relate the two approximations, we deduce a geodesic approximation from the saddle point approximation by using a key observation of Festucia and Liu, which is a frequency-geodesic identification, arising from comparing the WKB wave equation and the space-like geodesic equation. 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