On geodesic propagators and black hole holography

TL;DR

This work investigates how interior features of AdS3 spacetimes, including regions behind black hole horizons, are encoded in boundary CFT observables using nonlocal propagators. By applying a geodesic (WKB) approximation to the bulk Feynman path integral in analytic spacetimes (BTZ and RP^2 geon) and comparing with CFT expectations and toy models, it shows that the boundary correlators reflect natural bulk vacuum states (Hartle-Hawking-like for BTZ and geon vacuum for the RP^2 geon). Crucially, geodesics that traverse the interior contribute to boundary correlations, illustrating how interior information can be stored outside the horizon in a causally consistent framework. The results clarify the role of analyticity, boundary conditions, and mixed past/future data in the bulk-boundary map and inform how nonlocal CFT observables encode bulk interior dynamics.

Abstract

One of the most challenging technical aspects of the dualities between string theory on anti-de Sitter spaces and conformal field theories is understanding how location in the interior of spacetime is represented in the field theory. It has recently been argued that the interior of the spacetime can be directly probed by using intrinsically non-local quantities in the field theory. In addition, Balasubramanian and Ross [hep-th/9906226] argued that when the spacetime described the formation of an AdS_3 black hole, the propagator in the field theory probed the whole spacetime, including the region behind the horizon. We use the same approach to study the propagator for the BTZ black hole and a black hole solution with a single exterior region, and show that it reproduces the propagator associated with the natural vacuum states on these spacetimes. We compare our result with a toy model of the CFT for the single-exterior black hole, finding remarkable agreement. The spacetimes studied in this work are analytic, which makes them quite special. We also discuss the interpretation of this propagator in more general spacetimes, shedding light on certain issues involving causality, black hole horizons, and products of local operators on the boundary.