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arxiv: 1908.10996 · v2 · pith:OQOMQCOCnew · submitted 2019-08-29 · ✦ hep-th · gr-qc

The Page curve of Hawking radiation from semiclassical geometry

Ahmed Almheiri , Raghu Mahajan , Juan Maldacena , Ying Zhao This is my paper

Pith reviewed 2026-05-18 19:14 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords Page curveHawking radiationquantum extremal surfacesentanglement wedgeblack hole information paradoxholographic dualityisland rule
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The pith

The entropy of Hawking radiation follows the Page curve because its quantum extremal surface includes an island in the black hole interior.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In a gravity theory coupled to matter that possesses a higher-dimensional holographic dual, the problem of locating quantum extremal surfaces for the radiation reduces directly to locating classical RT/HRT surfaces in the dual geometry. This reduction shows that the radiation's entanglement wedge extends into the black hole interior, so that the entropy first rises and then falls in the manner of the Page curve. A reader would care because the result supplies a concrete semiclassical account of how the radiation can carry information out of the black hole while remaining consistent with unitarity. The construction relies on the higher-dimensional geometry to realize an ER=EPR-type connection between the radiation and the interior.

Core claim

When the matter sector admits a higher-dimensional holographic dual, quantum extremal surfaces that compute the entropy of Hawking radiation become equivalent to RT/HRT surfaces in that dual. The resulting entropy follows the Page curve, and the black hole interior is thereby included in the radiation's entanglement wedge. The paper proposes this equivalence as a general rule for determining the entanglement wedge of any quantum system coupled to gravity.

What carries the argument

The exact reduction of the quantum extremal surface problem to the classical RT/HRT surface problem in the higher-dimensional holographic dual of the matter sector.

If this is right

  • The radiation entropy rises linearly until the Page time and then decreases.
  • The black hole interior becomes part of the entanglement wedge of the outgoing radiation.
  • Entropy computations for other systems entangled with gravity proceed by searching for islands.
  • The higher-dimensional geometry supplies a concrete realization of the ER=EPR connection.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same island mechanism may appear in calculations of other information-theoretic quantities such as mutual information or reflected entropy.
  • The rule could be tested in lower-dimensional models that admit an explicit higher-dimensional lift.
  • Similar island contributions might resolve apparent non-unitarity in other semiclassical gravitational processes.

Load-bearing premise

The matter sector possesses a higher-dimensional holographic dual in which the quantum extremal surface problem reduces exactly to the classical RT/HRT surface problem.

What would settle it

An explicit calculation of the radiation entropy in a concrete model lacking such a higher-dimensional dual that yields an entropy curve different from the Page curve.

read the original abstract

We consider a gravity theory coupled to matter, where the matter has a higher-dimensional holographic dual. In such a theory, finding quantum extremal surfaces becomes equivalent to finding the RT/HRT surfaces in the higher-dimensional theory. Using this we compute the entropy of Hawking radiation and argue that it follows the Page curve, as suggested by recent computations of the entropy and entanglement wedges for old black holes. The higher-dimensional geometry connects the radiation to the black hole interior in the spirit of ER=EPR. The black hole interior then becomes part of the entanglement wedge of the radiation. Inspired by this, we propose a new rule for computing the entropy of quantum systems entangled with gravitational systems which involves searching for "islands" in determining the entanglement wedge.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper considers gravity coupled to matter fields possessing a higher-dimensional holographic dual. In this setup, the quantum extremal surface (QES) problem for the entropy of Hawking radiation reduces exactly to the classical Ryu-Takayanagi/Hubeny-Rangamani-Takayanagi (RT/HRT) problem in the higher-dimensional geometry. The authors show that the resulting entropy follows the Page curve because the radiation entanglement wedge includes an 'island' in the black hole interior, connected via ER=EPR-like geometry. They propose a new rule for entropy computation in systems entangled with gravity that involves searching for such islands.

Significance. If the central reduction holds, the work supplies a semiclassical geometric derivation of the Page curve for Hawking radiation, directly linking it to established holographic tools without additional parameters. It strengthens the case for islands in the entanglement wedge and offers a concrete proposal for extending the QES prescription. The approach is internally consistent within the stated holographic setup and provides falsifiable geometric predictions for the location of the island.

major comments (1)
  1. [§2] §2: The central claim that the QES problem reduces exactly to the classical RT/HRT problem relies on the matter sector having a higher-dimensional holographic dual. The manuscript should explicitly address the regime of validity (e.g., large-N limit, negligible backreaction from the radiation) to confirm that no additional quantum corrections arise that would alter the island location or the Page curve.
minor comments (2)
  1. [Abstract] Abstract: the reference to 'recent computations of the entropy and entanglement wedges for old black holes' would benefit from a specific citation to clarify the connection to prior work.
  2. [Conclusion] The proposed new rule for entropy computation (final paragraph) is stated qualitatively; a more precise mathematical formulation, even if heuristic, would aid readers in applying it beyond the specific setup.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation of minor revision. We address the single major comment below, agreeing that an explicit discussion of the regime of validity will improve the manuscript.

read point-by-point responses

  1. Referee: [§2] §2: The central claim that the QES problem reduces exactly to the classical RT/HRT problem relies on the matter sector having a higher-dimensional holographic dual. The manuscript should explicitly address the regime of validity (e.g., large-N limit, negligible backreaction from the radiation) to confirm that no additional quantum corrections arise that would alter the island location or the Page curve.

    Authors: We agree that the regime of validity should be stated more explicitly. The setup assumes that the matter sector admits a higher-dimensional holographic dual, which is valid in the large-N limit of the dual field theory. In this limit the bulk geometry is classical, the RT/HRT formula receives only subleading 1/N corrections, and the semiclassical treatment of Hawking radiation treats the radiation as a small perturbation whose backreaction does not shift the leading-order location of the quantum extremal surface. We will add a short paragraph at the end of §2 that spells out these conditions and confirms that they suffice to preserve the island and the Page curve at the order we compute. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies established holographic tools to new setup

full rationale

The paper explicitly assumes a gravity theory coupled to matter possessing a higher-dimensional holographic dual, under which quantum extremal surfaces reduce to classical RT/HRT surfaces. It then applies this equivalence together with the ER=EPR connection to show that the radiation entanglement wedge includes an interior island, yielding the Page curve for Hawking radiation entropy. This is an application of prior independent results (RT/HRT formula and ER=EPR) to a constructed geometry rather than a self-definitional loop, parameter fit, or renaming that forces the target outcome by construction. The central claim retains independent content from the higher-dimensional geometry and does not reduce to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the holographic duality for the matter sector and the identification of quantum extremal surfaces with classical minimal surfaces; no new free parameters are introduced, but the setup assumes the validity of the semiclassical approximation and the replica trick.

axioms (2)
  • domain assumption Matter fields admit a higher-dimensional holographic dual in which the quantum extremal surface problem reduces to the RT/HRT problem.
    Invoked in the opening paragraph and section 2 to equate the two calculations.
  • domain assumption The semiclassical geometry remains valid throughout the evaporation process.
    Underlying assumption for using classical RT surfaces.
invented entities (1)
  • Island no independent evidence
    purpose: Region inside the black hole that becomes part of the radiation entanglement wedge.
    Postulated to make the entropy follow the Page curve; independent evidence would be a direct microscopic calculation showing the same entropy.

pith-pipeline@v0.9.0 · 5654 in / 1538 out tokens · 31454 ms · 2026-05-18T19:14:46.253124+00:00 · methodology

discussion (0)

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Forward citations

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