[Paper Review] Page Curve and the Information Paradox in Flat Space

Chethan Krishnan, Vaishnavi Patil+1 more|arXiv (Cornell University)|May 6, 2020

Black Holes and Theoretical Physics73 references54 citations

TL;DR

The paper adapts asymptotic causal diamonds and holographic screens in flat space to define quantum extremal surfaces and entanglement wedges, showing a Page-curve-like phase transition for evaporating Schwarzschild black holes and suggesting a flat-space analog of the information paradox resolution.

ABSTRACT

Asymptotic Causal Diamonds (ACDs) are a natural flat space analogue of AdS causal wedges, and it has been argued previously that they may be useful for understanding bulk locality in flat space holography. In this paper, we use ACD-inspired ideas to argue that there exist natural candidates for Quantum Extremal Surfaces (QES) and entanglement wedges in flat space, anchored to the conformal boundary. When there is a holographic screen at finite radius, we can also associate entanglement wedges and entropies to screen sub-regions, with the system naturally coupled to a sink. The screen and the boundary provide two complementary ways of formulating the information paradox. We explain how they are related and show that in both formulations, the flat space entanglement wedge undergoes a phase transition at the Page time in the background of an evaporating Schwarzschild black hole. Our results closely parallel recent observations in AdS, and reproduce the Page curve. That there is a variation of the argument that can be phrased directly in flat space without reliance on AdS, is a strong indication that entanglement wedge phase transitions may be key to the information paradox in flat space as well. Along the way, we give evidence that the entanglement entropy of an ACD is a well-defined, and likely instructive, quantity. We further note that the picture of the sink we present here may have an understanding in terms of sub-matrix deconfinement in a large-$N$ setting.

Motivation & Objective

• Motivate and develop a flat-space analogue of holographic entanglement concepts using Asymptotic Causal Diamonds (ACDs) and holographic screens.
• Define quantum extremal surfaces (QES) and entanglement wedges in asymptotically flat space anchored to the conformal boundary.
• Introduce two formulations for handling Hawking radiation and information flow with a finite-radius screen (suitable for a Page-curve analysis).
• Demonstrate a phase transition of the flat-space entanglement wedge at the Page time analogous to AdS results.
• Discuss renormalization and the entropy of subregions on the screen to connect to a boundary-like entanglement entropy.
• Provide evidence that the entropy and entanglement wedge structures obey strong subadditivity in this flat-space setup.

Proposed method

• Define and employ Asymptotic Causal Diamonds (ACDs) as flat-space counterparts to AdS boundary causal diamonds.
• Construct relative extremal/maximin surfaces anchored to holographic screen sub-regions and relate them to ACDs.
• Formulate a generalized entropy including bulk entanglement and a screen-attached sink to define flat-space QES and entanglement wedges.
• Present two formulations for black hole evaporation: interior+sink factorization and an external-sink coupling, both yielding a Page-curve-like result.
• Use a crude renormalization scheme (background subtraction with Minkowski space) to define finite entropies for the screen sub-regions and the ACD shadows.

Experimental results

Research questions

• RQ1 Can flat-space holography realize quantum extremal surfaces and entanglement wedges anchored to the conformal boundary via ACDs?
• RQ2 How does a finite-radius holographic screen influence entanglement structure and entropy in evaporating flat-space black holes?
• RQ3 Do flat-space entanglement wedges exhibit a Page-time phase transition similar to AdS setups?
• RQ4 What is a robust, renormalized notion of entanglement entropy for ACDs and screen sub-regions in flat space?
• RQ5 What are the two complementary formulations for Hawking evaporation in flat space, and how do they relate to the Page curve?

Key findings

• There exist natural candidates for QES and entanglement wedges in flat space anchored to the conformal boundary via ACDs.
• A holographic screen at finite radius allows entanglement wedges and entropies to be defined for screen sub-regions, with the system coupled to a sink.
• The flat-space entanglement wedge undergoes a phase transition at the Page time in the background of an evaporating Schwarzschild black hole, reproducing the Page curve in this setting.
• Two formulations (interior+sink factorization and external-sink coupling) yield consistent, Page-curve-like behavior for Hawking evaporation in flat space.
• The entropy of an ACD is shown to be a well-defined quantity within this framework, suggesting a meaningful flat-space entanglement entropy.