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arxiv: 2107.05218 · v3 · submitted 2021-07-12 · ✦ hep-th · gr-qc

Hawking-Page phase transition, Page curve and islands in black holes

Dao-Quan Sun This is my paper

Pith reviewed 2026-05-24 13:39 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords Hawking-Page phase transitionPage curveblack hole information paradoxVaidya metricBTZ black holeentanglement islandsentanglement entropyphase transition
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The pith

The Hawking-Page phase transition releases all black hole information at once via a quantum jump.

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

The paper investigates how the Hawking-Page phase transition interacts with the Page curve in evaporating BTZ black holes modeled by the Vaidya metric. It finds that the transition can happen before, during, or after the Page time, and when it precedes the Page time, the Page curve is still reproduced without requiring islands. The entanglement temperature, derived from the first law, switches from negative to positive at the Page time. The main claim is that the phase transition causes a sudden release of all information in a quantum jump, suggesting phase transitions as an alternative information escape route independent of Hawking radiation.

Core claim

In the Vaidya model of evaporating BTZ black holes, the Hawking-Page phase transition leads to the instantaneous release of all black hole information through a quantum jump, regardless of whether it occurs before, at, or after the Page time, and independent of the presence of islands. The entanglement temperature changes sign at the Page time, with negative values corresponding to the information deposition phase and positive to the release phase.

What carries the argument

The Vaidya metric for BTZ black holes allowing the Hawking-Page transition timing to be adjusted relative to the Page time while obeying the first law of entanglement entropy.

If this is right

  • The information release does not require an island when the transition is before the Page time.
  • The entanglement temperature diverges at the Page time.
  • First-order phase transitions may provide a universal mechanism for black hole information release.
  • Information deposition corresponds to negative entanglement temperature and release to positive.

Where Pith is reading between the lines

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

  • This suggests phase transitions could resolve the information paradox in other black hole spacetimes.
  • Future calculations could test if the quantum jump occurs in higher-dimensional models.
  • The sign change in temperature might link to thermodynamic interpretations of entanglement in other systems.

Load-bearing premise

The Vaidya metric and boundary conditions permit independent variation of the Hawking-Page transition timing from the Page time while still obeying the first law of entanglement entropy.

What would settle it

A calculation or simulation showing that the information is not released in a single jump during the Hawking-Page transition, or that the entanglement temperature does not change sign at the Page time.

Figures

Figures reproduced from arXiv: 2107.05218 by Dao-Quan Sun.

Figure 1
Figure 1. Figure 1: Penrose diagram of the BTZ black hole with auxiliary baths in thermal [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Page curve without a Hawking-Page phase transition. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: If the Hawking-Page phase transition happens before page time, that the Page [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: If the Hawking-Page phase transition happens after page time, that the Page [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: If a first order phase transition happens after page time, that the Page [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: If a first order phase transition happens before page time, that the Page [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
read the original abstract

We study the Page curve and the effect of the Hawking-Page phase transition in the Vaidya model of evaporating BTZ black holes, both with and without islands. The phase transition can occur before, at, or after the Page time. When it occurs before the Page time, the information release does not require an island yet still reproduces the Page curve. From the first law of entanglement, the entanglement temperature changes from negative to positive, diverging at the Page time. Thus, the information deposition phase corresponds to a negative temperature, and the release phase to a positive temperature. If the Hawking-Page phase transition takes place, all black hole information is released at once via a quantum jump. We speculate that first-order phase transitions universally affect black hole information, providing an alternative escape mechanism independent of Hawking radiation.

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

3 major / 2 minor

Summary. The manuscript examines the Page curve for evaporating BTZ black holes in the Vaidya model, both with and without islands, and analyzes the interplay with the Hawking-Page (HP) phase transition. It claims that the HP transition can occur before, at, or after the Page time; when occurring before the Page time, the Page curve is reproduced without islands. From the first law of entanglement entropy, the entanglement temperature changes sign (negative to positive) and diverges at the Page time. The central claim is that an HP transition causes all black hole information to be released at once via a quantum jump, offering an alternative information-release mechanism independent of Hawking radiation; the authors speculate this is universal for first-order phase transitions.

Significance. If the central claims hold with explicit derivations, the work would connect gravitational phase transitions to the black hole information problem in a novel way, suggesting first-order transitions as an independent escape route for information. The Vaidya BTZ setup with tunable transition timing relative to Page time, if shown to be consistent with the first law, would provide a concrete model for testing such ideas.

major comments (3)
  1. [Vaidya model setup and first-law application] The claim that the HP transition timing can be varied independently of the Page time (before/at/after) while still satisfying the first law dS_EE = dE/T_ent is load-bearing for the quantum-jump interpretation and the negative/positive temperature distinction. The Vaidya mass function and chosen boundary conditions must be shown explicitly to permit this decoupling; without a derivation or explicit parameter scan demonstrating consistency (e.g., in the section deriving the transition and Page times), the regime required for the central claim may not exist.
  2. [Hawking-Page transition and information release] The assertion that an HP transition produces an abrupt 'quantum jump' releasing all information at once requires a concrete calculation or plot showing the discontinuity in information content or entanglement entropy at the transition point. This is central to the claim that the transition provides an alternative to gradual Hawking radiation release; the manuscript should identify the specific equation or figure where the jump is quantified rather than asserted.
  3. [First law of entanglement entropy] The entanglement temperature divergence at the Page time and its sign change are used to interpret information deposition vs. release phases. The derivation from the first law must be shown in detail, including how T_ent is extracted from the Vaidya geometry and island or no-island configurations, to confirm it is not an artifact of the chosen normalization or boundary conditions.
minor comments (2)
  1. [Conclusion] The universality speculation in the abstract and conclusion lacks supporting calculations from other models or metrics; if retained, it should be clearly labeled as conjecture.
  2. [Notation] Notation for the entanglement temperature T_ent and its relation to the standard Hawking temperature should be clarified to avoid confusion with thermodynamic temperature.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. The comments highlight areas where explicit derivations and visualizations would strengthen the presentation of our central claims regarding the Vaidya BTZ model, the decoupling of Hawking-Page and Page times, and the entanglement temperature. We address each point below and will incorporate the requested details in a revised manuscript.

read point-by-point responses

  1. Referee: [Vaidya model setup and first-law application] The claim that the HP transition timing can be varied independently of the Page time (before/at/after) while still satisfying the first law dS_EE = dE/T_ent is load-bearing for the quantum-jump interpretation and the negative/positive temperature distinction. The Vaidya mass function and chosen boundary conditions must be shown explicitly to permit this decoupling; without a derivation or explicit parameter scan demonstrating consistency (e.g., in the section deriving the transition and Page times), the regime required for the central claim may not exist.

    Authors: The Vaidya mass function in our setup is parameterized by an evaporation rate that can be adjusted independently of the boundary conditions determining the Page time via the entanglement entropy computation. This allows the HP transition (governed by the free-energy comparison) to be tuned relative to the Page time while preserving the first-law relation. We agree that an explicit derivation and parameter scan were not sufficiently detailed; we will add this in a new subsection with numerical examples confirming consistency across the three regimes. revision: yes

  2. Referee: [Hawking-Page transition and information release] The assertion that an HP transition produces an abrupt 'quantum jump' releasing all information at once requires a concrete calculation or plot showing the discontinuity in information content or entanglement entropy at the transition point. This is central to the claim that the transition provides an alternative to gradual Hawking radiation release; the manuscript should identify the specific equation or figure where the jump is quantified rather than asserted.

    Authors: The jump arises from the first-order nature of the HP transition, which produces a discontinuous change in the on-shell action and thus in the entropy. When the transition precedes the Page time, this discontinuity accounts for the full information release without islands. We will add an explicit plot of the entanglement entropy versus time, marking the discontinuity at the transition, together with the corresponding equation for the jump magnitude derived from the free-energy difference. revision: yes

  3. Referee: [First law of entanglement entropy] The entanglement temperature divergence at the Page time and its sign change are used to interpret information deposition vs. release phases. The derivation from the first law must be shown in detail, including how T_ent is extracted from the Vaidya geometry and island or no-island configurations, to confirm it is not an artifact of the chosen normalization or boundary conditions.

    Authors: T_ent is obtained directly from the first-law relation applied to the computed S_EE and the energy E in the Vaidya geometry. The sign change and divergence follow from the Page curve shape (increasing then decreasing S_EE). We will expand the relevant section with the full step-by-step extraction for both island and no-island cases, including the explicit expressions for dE and dS_EE in the Vaidya metric to rule out normalization artifacts. revision: yes

Circularity Check

0 steps flagged

No significant circularity; model exploration is self-contained

full rationale

The paper applies the Vaidya metric to BTZ black holes and varies the Hawking-Page transition timing relative to the Page time as an explicit model choice. Conclusions about information release via quantum jump and the sign change in entanglement temperature follow directly from applying the first law of entanglement entropy to the resulting geometries, without any fitted parameters renamed as predictions, self-citations used as load-bearing premises, or definitions that presuppose the target result. The derivation remains independent of its inputs once the metric and boundary conditions are stated.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work rests on standard assumptions of black-hole thermodynamics and the first law of entanglement entropy; it introduces one new postulated mechanism (the quantum jump) without independent evidence.

axioms (1)
  • domain assumption The first law of entanglement entropy holds for the evaporating Vaidya BTZ geometry
    Invoked to define the entanglement temperature that changes sign at the Page time.
invented entities (1)
  • quantum jump releasing all information no independent evidence
    purpose: To account for instantaneous release of the remaining black-hole information at the Hawking-Page transition
    Postulated in the final speculation; no falsifiable prediction outside the model is given.

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Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

  1. [1]

    S. W. Hawking. Particle creation by black holes. Communications in Mathematical Physics, 43(3):199–220, Aug 1975

    work page 1975

  2. [2]

    The Large N limit of superconformal field theories and supergravity

    Juan Martin Maldacena. The Large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys. , 38:1113–1133, 1999. [Adv. Theor. Math. 17 Phys.2,231(1998)]

    work page 1999

  3. [3]

    Holographic derivation of entanglement entropy from the anti–de sitter space/conformal field theory correspondence

    Shinsei Ryu and Tadashi Takayanagi. Holographic derivation of entanglement entropy from the anti–de sitter space/conformal field theory correspondence. Phys. Rev. Lett., 96:181602, May 2006

    work page 2006

  4. [4]

    Don N. Page. Information in black hole radiation. Phys. Rev. Lett. , 71:3743–3746, 1993

    work page 1993

  5. [5]

    Don N. Page. Time Dependence of Hawking Radiation Entropy. JCAP, 09:028, 2013

    work page 2013

  6. [6]

    The Page curve of Hawking radiation from semiclassical geometry

    Ahmed Almheiri, Raghu Mahajan, Juan Maldacena, and Ying Zhao. The Page curve of Hawking radiation from semiclassical geometry. JHEP, 03:149, 2020

    work page 2020

  7. [7]

    Entanglement Wedge Reconstruction and the Information Para- dox

    Geoffrey Penington. Entanglement Wedge Reconstruction and the Information Para- dox. JHEP, 09:002, 2020

    work page 2020

  8. [8]

    The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole

    Ahmed Almheiri, Netta Engelhardt, Donald Marolf, and Henry Maxfield. The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole. JHEP, 12:063, 2019

    work page 2019

  9. [9]

    The entropy of Hawking radiation

    Ahmed Almheiri, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini. The entropy of Hawking radiation. Rev. Mod. Phys. , 93(3):035002, 2021

    work page 2021

  10. [10]

    Replica Wormholes and the Entropy of Hawking Radiation

    Ahmed Almheiri, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini. Replica Wormholes and the Entropy of Hawking Radiation. JHEP, 05:013, 2020

    work page 2020

  11. [11]

    Shenker, Douglas Stanford, and Zhenbin Yang

    Geoff Penington, Stephen H. Shenker, Douglas Stanford, and Zhenbin Yang. Replica wormholes and the black hole interior. JHEP, 03:205, 2022

    work page 2022

  12. [12]

    Thermodynamical property of entanglement entropy for excited states

    Jyotirmoy Bhattacharya, Masahiro Nozaki, Tadashi Takayanagi, and Tomonori Uga- jin. Thermodynamical property of entanglement entropy for excited states. Phys. Rev. Lett., 110:091602, Mar 2013. 18

    work page 2013

  13. [13]

    Blanco, Horacio Casini, Ling-Yan Hung, and Robert C

    David D. Blanco, Horacio Casini, Ling-Yan Hung, and Robert C. Myers. Relative Entropy and Holography. JHEP, 08:060, 2013

    work page 2013

  14. [14]

    Pando Zayas, and Diana Vaman

    Gabriel Wong, Israel Klich, Leopoldo A. Pando Zayas, and Diana Vaman. Entan- glement Temperature and Entanglement Entropy of Excited States. JHEP, 12:020, 2013

    work page 2013

  15. [15]

    Renormalization group flow of entanglement en- tropy to thermal entropy

    Ki-Seok Kim and Chanyong Park. Renormalization group flow of entanglement en- tropy to thermal entropy. Phys. Rev. D , 95:106007, May 2017

    work page 2017

  16. [16]

    Holographic entangle- ment entropy and generalized entanglement temperature

    Ashis Saha, Sunandan Gangopadhyay, and Jyoti Prasad Saha. Holographic entangle- ment entropy and generalized entanglement temperature. Phys. Rev. D , 100:106008, Nov 2019

    work page 2019

  17. [17]

    S. W. Hawking and Don N. Page. Thermodynamics of black holes in anti-de sitter space. Communications in Mathematical Physics , 87(4):577–588, Dec 1983

    work page 1983

  18. [18]

    The cosmological constant as a canonical variable

    Marc Henneaux and Claudio Teitelboim. The cosmological constant as a canonical variable. Physics Letters B , 143(4):415 – 420, 1984

    work page 1984

  19. [19]

    Mirjam Cvetic and Steven S. Gubser. Phases of r-charged black holes, spinning branes and strongly coupled gauge theories. Journal of High Energy Physics , 1999(04):024, 1999

    work page 1999

  20. [20]

    Anti-de Sitter space, thermal phase transition, and confinement in gauge theories

    Edward Witten. Anti-de Sitter space, thermal phase transition, and confinement in gauge theories. Adv. Theor. Math. Phys. , 2:505–532, 1998

    work page 1998

  21. [21]

    The cosmological constant as a thermodynamic black hole pa- rameter

    Claudio Teitelboim. The cosmological constant as a thermodynamic black hole pa- rameter. Physics Letters B , 158(4):293 – 297, 1985

    work page 1985

  22. [22]

    Enthalpy and the mechanics of ads black holes

    David Kastor, Sourya Ray, and Jennie Traschen. Enthalpy and the mechanics of ads black holes. Classical and Quantum Gravity , 26(19):195011, 2009

    work page 2009

  23. [23]

    Jorge V. Rocha. Evaporation of large black holes in AdS: Coupling to the evaporon. JHEP, 08:075, 2008. 19

    work page 2008

  24. [24]

    An Apologia for Firewalls

    Ahmed Almheiri, Donald Marolf, Joseph Polchinski, Douglas Stanford, and James Sully. An Apologia for Firewalls. JHEP, 09:018, 2013

    work page 2013

  25. [25]

    Evaporating Firewalls

    Mark Van Raamsdonk. Evaporating Firewalls. JHEP, 11:038, 2014

    work page 2014

  26. [26]

    Island, Page curve, and superradiance of rotating BTZ black holes

    Ming-Hui Yu, Cheng-Yuan Lu, Xian-Hui Ge, and Sang-Jin Sin. Island, Page curve, and superradiance of rotating BTZ black holes. Phys. Rev. D , 105(6):066009, 2022

    work page 2022

  27. [27]

    Page Curve and Island in EGB gravity

    Ankit Anand. Page Curve and Island in EGB gravity. 5 2022

    work page 2022

  28. [28]

    Mutual information, islands in black holes and the Page curve

    Ashis Saha, Sunandan Gangopadhyay, and Jyoti Prasad Saha. Mutual information, islands in black holes and the Page curve. Eur. Phys. J. C , 82(5):476, 2022

    work page 2022

  29. [29]

    Entanglement entropy of two dis- joint intervals in conformal field theory

    Pasquale Calabrese, John Cardy, and Erik Tonni. Entanglement entropy of two dis- joint intervals in conformal field theory. J. Stat. Mech. , 0911:P11001, 2009

    work page 2009

  30. [30]

    Casini and M

    H. Casini and M. Huerta. Entanglement entropy in free quantum field theory. J. Phys. A , 42:504007, 2009

    work page 2009

  31. [31]

    Black holes as mirrors: Quantum information in random subsystems

    Patrick Hayden and John Preskill. Black holes as mirrors: Quantum information in random subsystems. JHEP, 09:120, 2007

    work page 2007

  32. [32]

    Brian P. Dolan. The cosmological constant and the black hole equation of state. Class. Quant. Grav., 28:125020, 2011

    work page 2011

  33. [33]

    Entanglement entropy and quantum field theory

    Pasquale Calabrese and John Cardy. Entanglement entropy and quantum field theory. Journal of Statistical Mechanics: Theory and Experiment , 2004(06):P06002, jun 2004

    work page 2004

  34. [34]

    Hubeny, Henry Maxfield, Mukund Rangamani, and Erik Tonni

    Veronika E. Hubeny, Henry Maxfield, Mukund Rangamani, and Erik Tonni. Holo- graphic entanglement plateaux. JHEP, 08:092, 2013

    work page 2013

  35. [35]

    Hubeny and Mukund Rangamani

    Veronika E. Hubeny and Mukund Rangamani. Causal Holographic Information. JHEP, 06:114, 2012

    work page 2012

  36. [36]

    Insight into the Micro- scopic Structure of an AdS Black Hole from the Quantization

    Dao-Quan Sun, Jian-Bo Deng, Ping Li, and Xian-Ru Hu. Insight into the Micro- scopic Structure of an AdS Black Hole from the Quantization. Class. Quant. Grav. , 37(1):015008, 2020. 20

    work page 2020

  37. [37]

    P-v criticality of charged ads black holes

    David Kubizˇ n´ ak and Robert B Mann. P-v criticality of charged ads black holes. Journal of High Energy Physics , 2012(7):33, 2012. 21

    work page 2012