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

To gain insight into how bulk locality emerges from the holographic conformal field theory (CFT), we reformulate the bulk-to-boundary map in as local a way as possible. In previous work, we carried out this program for Lorentzian anti-de Sitter (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 Poincaré coordinates and ${\mathrm{AdS}}_3$ in Rindler coordinates. We represent bulk operators inside the horizon of a Banados-Teitelboim-Zanelli (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 $N$.

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 $\varphi$ in ${\mathrm{AdS}}_3$, drawn as an ${\mathrm{AdS}}_2$ 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.