Page Curve and the Information Paradox in Flat Space

TL;DR

The paper develops a flat-space holographic framework using Asymptotic Causal Diamonds (ACDs) and holographic screens to define Quantum Extremal Surfaces (QES) and entanglement wedges anchored to the conformal boundary. It demonstrates that flat-space evaporating black holes exhibit a Page-time entanglement-wedge phase transition, producing the Page curve via two complementary formulations of Hawking evaporation and a renormalization scheme for ACD entropies. By connecting ACD data to screen-based regulators, the work generalizes AdS insights to flat space and suggests a non-AdS route to resolving the information paradox, including a possible sub-matrix deconfinement interpretation. The results reinforce the role of entanglement wedge transitions as central to flat-space holography and information recovery, while highlighting unique nonlocal features of the flat-space holographic dual.

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.

Figures (14)

• Figure 1: The ACD defined by vertices $P, Q$, its causal surface and its Quantum Extremal surface (QES) are shown. This data defines the entanglement/reconstruction wedge completely. We have also shown the holographic screen. The shadow of the ACD is its intersection with the conformal boundary.
• Figure 2: Comparing acceptable and unacceptable holographic screens.
• Figure 3: Points in non-trivial intersections of subsets lie in intersections of the parent sets as well. A version of this fact, when the sets are certain ACDs, is used in the proof of theorem 2.2. We present this picture to emphasize that the idea is trivial.
• Figure 4: A cartoon of various points and surfaces associated to an ACD and the holographic screen. We emphasize that this really is a cartoon: it mixes some features of conformal coordinates and some features of spacetime coordinates. Note in particular, that the circle is actually of infinite radius and therefore the waist should be a straight line inside the cut-off. The "cylinder" should really be a cigar ending in future and past timelike infinity (assuming that the background contains no horizons etc.).
• Figure 5: The olive shaded region shows a chart where the asymptotically flat $u, r$ coordinates are well-defined on this geometry.