Causal connectability between quantum systems and the black hole interior in holographic duality

TL;DR

The paper investigates how bulk causal structure, horizons, and interiors of an eternal AdS black hole emerge from boundary CFTs in the thermofield double state. It constructs a boundary evolution operator $U(s)$ generated by a positive $G$ that couples both CFTs and uses half-sided modular translations to generate emergent boundary times, revealing horizons and interior regions via the notion of causal connectability. A central result is that sharp bulk horizons require the large-$N$ limit where boundary algebras become type III$_1$, enabling boundary descriptions of Rindler and Kruskal regions and their causal structure, including the BTZ black hole interior. The framework links boundary operator algebras, modular theory, and holographic reconstruction to provide a boundary account of bulk horizons, singularities, and UV-divergence issues, and outlines directions for extending these ideas to more general spacetimes and dynamical black holes.

Abstract

In holographic duality an eternal AdS black hole is described by two copies of the boundary CFT in the thermal field double state. This identification has many puzzles, including the boundary descriptions of the event horizons, the interiors of the black hole, and the singularities. Compounding these mysteries is the fact that, while there is no interaction between the CFTs, observers from them can fall into the black hole and interact. We address these issues in this paper. In particular, we (i) present a boundary formulation of a class of in-falling bulk observers; (ii) present an argument that a sharp bulk event horizon can only emerge in the infinite $N$ limit of the boundary theory; (iii) give an explicit construction in the boundary theory of an evolution operator for a bulk in-falling observer, making manifest the boundary emergence of the black hole horizons, the interiors, and the associated causal structure. A by-product is a concept called causal connectability, which is a criterion for any two quantum systems (which do not need to have a known gravity dual) to have an emergent sharp horizon structure.

Figures (11)

• Figure 1: The Penrose diagram of an eternal black hole. The dashed lines are event horizons and the wavy lines are the singularities. Two observers from $R$ and $L$ can meet and interact behind the horizon despite the fact that there is no interaction between the left and right CFTs.
• Figure 2: Rindler regions of Minkowski spacetime.
• Figure 3: The identifications \ref{['ejnn']} establish duality of CFT$_{R, L}$ with the exteriors of the black hole spacetime, but they do not directly say anything about the existence of the $F, P$ regions of a connected bulk.
• Figure 4: AdS Rindler regions of the bulk spacetime. The vertical lines denote the boundary and the dashed lines are Rindler horizons.
• Figure 5: Boundary subregions that are used for the construction of a one-parameter group of unitaries implementing bulk Poincaré $U-$ and $V-$translations, respectively. $t, x$ are boundary Minkowski coordinates.