Bulk and Boundary Dynamics in BTZ Black Holes

TL;DR

This work extends Lorentzian AdS/CFT to BTZ black holes, showing that bulk scalar dynamics and their boundary counterparts reproduce thermal boundary correlators at the Hawking temperature $T_H$. By analyzing the bulk-boundary Green's function in BTZ coordinates, the paper demonstrates that the boundary two-point function is periodic in imaginary time, consistent with a thermal state seen by boundary operators. It also provides a detailed modal analysis for non-extremal and extremal BTZ spacetimes, classifying bulk solutions into normalizable and non-normalizable sectors and linking them to boundary Hilbert space states, with explicit solutions via hypergeometric and Whittaker functions. The discussion of asymptotic symmetries (Brown-Henneaux), global isometries, and the handling of vacuum states clarifies how bulk and boundary descriptions relate under quotient identifications. Together, these results advance the holographic understanding of black hole physics, information flow, and evaporation in the BTZ context, with implications for a Hamiltonian formulation of the bulk/boundary correspondence.

Abstract

Recently, the AdS/CFT conjecture of Maldacena has been investigated in Lorentzian signature by Balasubramanian et. al. We extend this investigation to Lorentzian BTZ black hole spacetimes, and study the bulk and boundary behaviour of massive scalar fields both in the non-extremal and extremal case. Using the bulk-boundary correspondence, we also evaluate the two-point correlator of operators coupling to the scalar field at the boundary of the spacetime, and find that it satisfies thermal periodic boundary conditions relevant to the Hawking temperature of the BTZ black hole.