[Paper Review] Singularities, geodesics and Green functions in the BTZ black hole

Yang Chen|arXiv (Cornell University)|Nov 4, 2006

Black Holes and Theoretical Physics39 references3 citations

TL;DR

This paper investigates the holographic encoding of bulk spacetime regions—particularly those inside the BTZ black hole horizon—using scalar field propagators and two-point Green functions in the context of AdS/CFT. By applying saddle point and geodesic approximations to the Green function, the authors demonstrate that certain geodesics contributing to the Green function extend into the black hole interior, implying that horizon-enclosed physics can be probed from the boundary theory via the frequency-geodesic identification from Festucia and Liu.

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.

Motivation & Objective

• To understand how spacetime regions inside the BTZ black hole horizon are encoded in the boundary conformal field theory (CFT) via the AdS/CFT correspondence.
• To investigate the role of geodesics and saddle points in the semi-classical approximation of the two-point Green function on the BTZ black hole.
• To test the viability of the frequency-geodesic identification from Festucia and Liu as a tool for connecting bulk geometry to boundary correlation functions.
• To determine whether geodesics that penetrate the black hole horizon contribute to the boundary Green function, suggesting the possibility of resolving interior physics from the boundary.

Proposed method

• Uses the Hartle-Hawking-Israel state to define a thermal, Lorentzian signature Green function for a scalar field on the BTZ black hole.
• Applies the saddle point approximation to the path integral representation of the Green function, identifying dominant contributions from classical trajectories (geodesics).
• Employs the frequency-geodesic identification from Festucia and Liu to map frequencies in the Green function to classical geodesic paths in the bulk.
• Solves the Klein-Gordon equation in BTZ geometry using hypergeometric functions and applies boundary and horizon conditions to fix normalization constants.
• Derives the boundary-to-bulk propagator and computes the two-point function in momentum space, using analytic continuation from Euclidean to Lorentzian signature.
• Imposes boundary conditions at infinity and horizon to fix the coefficients of the hypergeometric solutions, ensuring physical consistency with the Hartle-Hawking state.

Experimental results

Research questions

• RQ1 Can the two-point Green function in the BTZ black hole spacetime be approximated by geodesic contributions using saddle point methods?
• RQ2 Do the geodesics identified via the frequency-geodesic correspondence extend into the black hole interior, and if so, what does this imply for bulk-boundary correspondence?
• RQ3 How do boundary and horizon conditions constrain the normalization of scalar field modes in the BTZ geometry?
• RQ4 To what extent can the interior of the BTZ black hole, including its singularity, be reconstructed from the boundary CFT using Green functions?
• RQ5 What is the role of the Hartle-Hawking-Israel state in ensuring consistency between bulk quantum field theory and boundary CFT correlation functions?

Key findings

• The saddle point approximation of the Green function yields geodesic trajectories that extend into the black hole interior, indicating that bulk regions behind the horizon contribute to the boundary correlation function.
• The frequency-geodesic identification from Festucia and Liu successfully maps frequencies in the Green function to classical geodesic paths, validating its use in semi-classical holography.
• Boundary conditions at infinity fix the coefficient $ C( u, u) $ in the boundary-to-bulk propagator, leading to a specific form involving gamma functions and $ u $, the spinor index.
• Horizon conditions lead to a constraint between the coefficients $ C_1 $ and $ C_2 $, which is used to eliminate the $ z^{1/2 - u} $ mode and ensure regularity at the horizon.
• The final form of the Green function's normalization includes a factor $ rac{1}{2eta} $, where $ eta $ is the inverse temperature, consistent with thermal field theory.
• Numerical checks confirm that the amplitudes $ |P| $ and $ |Q| $ are equal, ensuring the solution satisfies the horizon boundary condition and allows for a consistent wavefunction with real phase $ heta $.