Black hole interior Petz map reconstruction and Papadodimas-Raju proposal

TL;DR

This paper demonstrates that the Papadodimas-Raju mirror operator construction for black hole interiors can be understood as a specific case of Petz map reconstruction in AdS/CFT, linking bulk interior modes to boundary operators.

Contribution

It shows that the Papadodimas-Raju proposal is an instance of Petz map reconstruction, providing a new perspective on bulk interior operator recovery in holography.

Findings

Petz map reproduces exterior black hole operators as HKLL maps.

Interior modes correspond to mirror operators in the boundary theory.

The approach applies to evaporating black holes and their radiation bath.

Abstract

We study the reconstruction of the bulk operators in AdS/CFT when the geometry contains a black hole. The black hole exterior can be mapped to the CFT via a very simple Petz map which coincides with the HKLL map reconstruction of the black hole exterior. For the interior modes of the bulk theory, using the definition of the Petz recovery channel in modular theory, we can find the mapping from the black hole interior to the dual boundary theory. In the case of the evaporating black hole, it is expected that the interior modes map to some operators which have support only on the bath system, the cavity that absorbs the Hawking radiation. The most important observation that we have here is that in the case that we have a typical black hole microstate in the bulk, the CFT dual of the interior modes that we can find using the Petz recovery channel are exactly the operators which so-called "mirror operator " in the Papadodimas-Raju proposal. Therefore, we can interpret Papadodimas-Raju proposal as an example of the Petz map reconstruction. It may help us answer some open questions about their procedure.