Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local bulk operators in AdS/CFT: A holographic description of the black hole interior

Alex Hamilton, Daniel Kabat, Gilad Lifschytz, David A. Lowe

TL;DR

The paper advances the AdS/CFT dictionary by constructing local bulk operators as boundary operators with compact support on a complexified boundary, thereby sharpening bulk locality in a holographic setting. It provides two complementary constructions in AdS: a Poincaré-mode-sum approach and a de Sitter/retarded-green-function method, yielding disk-shaped smearing regions that reproduce bulk correlators, including near-horizon and singular regions. Extending to AdS3 in Rindler coordinates and BTZ, the authors show how interior physics is encoded in boundary data, using antipodal maps and analytic continuation to access regions behind horizons and in collapse geometries. They also discuss finite-N implications, arguing for a finite number of commuting bulk degrees of freedom per boundary area and highlighting how non-normalizable boundary operators encode bulk singularities, with implications for holographic entropy counting and the black hole interior. The results provide a concrete, computable framework for probing bulk locality, horizon interiors, and singularities from a dual CFT perspective, with clear pathways for extending to more general backgrounds and finite-N corrections.

Abstract

To gain insight into how bulk locality emerges from the holographic conformal field theory, we reformulate the bulk to boundary map in as local a way as possible. In previous work, we carried out this program for Lorentzian AdS, and showed the support on the boundary could always be reduced to a compact region spacelike separated from the bulk point. In the present work the idea is extended to a complexified boundary, where spatial coordinates are continued to imaginary values. This continuation enables us to represent a local bulk operator as a CFT operator with support on a finite disc on the complexified boundary. We treat general AdS in Poincare coordinates and AdS3 in Rindler coordinates. We represent bulk operators inside the horizon of a BTZ black hole and we verify that the correct bulk two point functions are reproduced, including the divergence when one point hits the BTZ singularity. We comment on the holographic description of black holes formed by collapse and discuss locality and holographic entropy counting at finite N.

Local bulk operators in AdS/CFT: A holographic description of the black hole interior

TL;DR

The paper advances the AdS/CFT dictionary by constructing local bulk operators as boundary operators with compact support on a complexified boundary, thereby sharpening bulk locality in a holographic setting. It provides two complementary constructions in AdS: a Poincaré-mode-sum approach and a de Sitter/retarded-green-function method, yielding disk-shaped smearing regions that reproduce bulk correlators, including near-horizon and singular regions. Extending to AdS3 in Rindler coordinates and BTZ, the authors show how interior physics is encoded in boundary data, using antipodal maps and analytic continuation to access regions behind horizons and in collapse geometries. They also discuss finite-N implications, arguing for a finite number of commuting bulk degrees of freedom per boundary area and highlighting how non-normalizable boundary operators encode bulk singularities, with implications for holographic entropy counting and the black hole interior. The results provide a concrete, computable framework for probing bulk locality, horizon interiors, and singularities from a dual CFT perspective, with clear pathways for extending to more general backgrounds and finite-N corrections.

Abstract

To gain insight into how bulk locality emerges from the holographic conformal field theory, we reformulate the bulk to boundary map in as local a way as possible. In previous work, we carried out this program for Lorentzian AdS, and showed the support on the boundary could always be reduced to a compact region spacelike separated from the bulk point. In the present work the idea is extended to a complexified boundary, where spatial coordinates are continued to imaginary values. This continuation enables us to represent a local bulk operator as a CFT operator with support on a finite disc on the complexified boundary. We treat general AdS in Poincare coordinates and AdS3 in Rindler coordinates. We represent bulk operators inside the horizon of a BTZ black hole and we verify that the correct bulk two point functions are reproduced, including the divergence when one point hits the BTZ singularity. We comment on the holographic description of black holes formed by collapse and discuss locality and holographic entropy counting at finite N.

Paper Structure

This paper contains 16 sections, 90 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Poincaré coordinates
  3. Preliminaries
  4. Poincaré mode sum
  5. de Sitter continuation
  6. Recovering bulk correlators
  7. AdS${}_3$ in Rindler coordinates
  8. Preliminaries
  9. Rindler smearing functions
  10. Reproducing bulk correlators
  11. BTZ black hole
  12. Collapse geometries
  13. Comments on finite $N$
  14. Conclusions
  15. Rindler mode sum
  16. ...and 1 more sections

Figures (4)

  • Figure 1: The Penrose diagram for de Sitter space. Flat FRW coordinates cover the lower triangle. Horizontal slices are spheres. Each point on the diagram represents an $S^{d-1}$ which shrinks to a point at the north and south poles (the right and left edges of the diagram).
  • Figure 2: A slice of constant $\phi$ in AdS${}_3$, drawn as an AdS${}_2$ Penrose diagram. The four Rindler wedges are separated by horizons at $r = r_+$.
  • Figure 3: de Sitter space in static coordinates.
  • Figure 4: An AdS black hole formed by collapse. The left edge of the diagram is the origin of AdS, the right edge is the AdS boundary. The dashed line is the black hole horizon while the solid diagonal line represents the infalling shell.