A toy model of black hole complementarity

TL;DR

This paper explores the algebra of simple operators in a holographic CFT, demonstrating how they approximate states and relate to bulk excitations, shedding light on black hole complementarity and bulk locality.

Contribution

It introduces a Reeh-Schlieder theorem for a specific algebra in holographic CFTs and provides explicit formulas for precursors, advancing understanding of bulk-boundary relations.

Findings

The algebra obeys a version of the Reeh-Schlieder theorem.

Local excitations outside the diamond relate to boundary precursors.

Implications for black hole complementarity and bulk locality.

Abstract

We consider the algebra of simple operators defined in a time band in a CFT with a holographic dual. When the band is smaller than the light crossing time of AdS, an entire causal diamond in the center of AdS is separated from the band by a horizon. We show that this algebra obeys a version of the Reeh-Schlieder theorem: the action of the algebra on the CFT vacuum can approximate any low energy state in the CFT arbitrarily well, but no operator within the algebra can exactly annihilate the vacuum. We show how to relate local excitations in the complement of the central diamond to simple operators in the band. Local excitations within the diamond are invisible to the algebra of simple operators in the band by causality, but can be related to complicated operators called "precursors". We use the Reeh-Schlieder theorem to write down a simple and explicit formula for these precursors on the boundary. We comment on the implications of our results for black hole complementarity and the emergence of bulk locality from the boundary.