[Paper Review] No Holography for Eternal AdS Black Holes

S. Avery, Borun D. Chowdhury|arXiv (Cornell University)|Dec 11, 2013

Black Holes and Theoretical Physics46 references14 citations

TL;DR

This paper argues that the eternal AdS black hole is not dual to two conformal field theories in a thermofield double state, as commonly believed. It shows that while the CFTs describe the exterior regions, they lack the necessary interaction to entangle degrees of freedom across the horizon, invalidating the 'entanglement=geometry' and 'ER=EPR' proposals. The correct dual geometry caps off before the global horizon, preventing observers from one side from meeting those from the other.

ABSTRACT

It is generally believed that the eternal AdS black hole is dual to two conformal field theories with compact spatial sections that are together in a thermofield double state. We argue that this proposal is incorrect, and by extension so are the "entanglement=geometry" proposal of Van Raamsdonk and "ER=EPR" proposal of Maldacena and Susskind. We show that in the bulk there is an interaction needed between the two halves of the Hilbert space for connectivity across the horizon; however, there is no such interaction between the CFTs. This rules out the possibility of the dual to the CFTs being the eternal AdS black hole. We argue the correct dual "geometries" resemble the exterior of the black hole outside the stretched horizon but cap off before the global horizon. This disallows the possibility of a shared future (and past) wedge where Alice falling from one side can meet Bob falling from the other. We expect that in the UV complete theory the aforementioned caps will be fuzzballs.

Motivation & Objective

• To challenge the widely accepted duality between eternal AdS black holes and two CFTs in a thermofield double state.
• To demonstrate that the absence of interaction between CFTs prevents the emergence of a shared spacetime wedge across the horizon.
• To argue that the proposed entanglement-based geometric constructions (entanglement=geometry, ER=EPR) fail in this context.
• To show that the correct dual description must cap off before the global horizon, disallowing connectivity between the two asymptotic regions.
• To clarify that Bell measurements—required to verify entanglement—cannot be performed by boundary observers due to decoupled CFTs.

Proposed method

• Analyzes the thermofield double state as a purification of a thermal density matrix, emphasizing that the state's meaning depends on the Hamiltonian and operators used.
• Compares two Hamiltonians: one with no interaction (H_L + H_R) and one with an interaction term (H_L + H_R + H_int), showing that only the latter allows for horizon-crossing physics.
• Uses the concept of Bell measurements to show that observers (like Alice) falling into the black hole must interact with both inside and outside modes to verify entanglement.
• Applies the method of images to CFT correlators on a cylinder, demonstrating its failure to reproduce correct finite-temperature correlation functions with quasi-normal modes.
• Contrasts infinite-line CFT correlators (with quasi-normal modes) with finite-cylinder CFTs (without such modes), showing that the method of images gives incorrect results in the latter case.
• Uses free field theory on a compact spatial circle to derive the correct two-point function, which lacks poles away from the real axis, contrasting with the image method's incorrect analytic structure.

Experimental results

Research questions

• RQ1 Can the eternal AdS black hole be dual to two decoupled CFTs in a thermofield double state, given the absence of interactions between them?
• RQ2 Does the lack of an interaction term between CFTs prevent the emergence of a shared spacetime region across the horizon?
• RQ3 Can boundary observers perform Bell measurements to verify entanglement between the two CFTs, as required for the ER=EPR and entanglement=geometry proposals?
• RQ4 Why does the method of images fail to reproduce the correct finite-temperature CFT correlator on a compact spatial section?
• RQ5 What is the correct dual description of the eternal AdS black hole if it cannot be described by two entangled CFTs?

Key findings

• The eternal AdS black hole is not dual to two CFTs in a thermofield double state because the CFTs are decoupled and cannot perform Bell measurements to verify entanglement.
• The absence of an interaction term between the CFTs means that no observer on either boundary can detect the entanglement necessary for a shared spacetime geometry.
• The correct dual geometry must cap off before the global horizon, disallowing any causal connection between observers from opposite asymptotic regions.
• The method of images produces a correlator with quasi-normal modes, which is incorrect for a finite-temperature CFT on a compact spatial section, invalidating its use in this context.
• The two-point function on a finite cylinder (torus) does not have poles away from the real axis, unlike the infinite-line limit, showing that the image method fails when applied to compact CFTs.
• The standard CFT correlator derived from summing over discrete modes on a compact space correctly captures the finite phase space, while the image method incorrectly assumes infinite extent.