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arxiv: 2603.12524 · v3 · submitted 2026-03-12 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Thermodynamics and information recovery of Schwarzschild AdS black holes in conformal Killing gravity

Yahya Ladghami , Brahim Asfour , Francisco S. N. Lobo , Taoufik Ouali
Authors on Pith no claims yet

Pith reviewed 2026-05-15 11:13 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Schwarzschild AdS black holesconformal Killing gravityblack hole thermodynamicsisland formulaPage curveentanglement entropyinformation recovery
0
0 comments X

The pith

In conformal Killing gravity, islands restore unitarity for Schwarzschild AdS black holes by making late-time radiation entropy saturate at twice the Bekenstein-Hawking value.

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

The authors study Schwarzschild anti-de Sitter black holes in a modified gravity theory called conformal Killing gravity. They treat the cosmological constant as pressure and the new gravity parameter as an independent variable. The Bekenstein-Hawking area law for entropy still holds in this setting. The new parameter changes the black hole's phase diagram, allowing extremal black holes and fluid-like phase transitions only when positive. For the information paradox, the radiation's entanglement entropy grows forever without islands but follows the Page curve and stops growing at twice the black hole entropy once islands are included after the Page time.

Core claim

Schwarzschild AdS black holes in conformal Killing gravity obey the Bekenstein-Hawking area law. The conformal Killing gravity parameter controls the existence of extremal limits and Van der Waals-like criticality, which appear only for positive values. Application of the island prescription shows that the entanglement entropy of Hawking radiation grows linearly without islands, violating unitarity, but saturates at twice the Bekenstein-Hawking entropy after the Page time when islands are included, thus recovering information and restoring the Page curve.

What carries the argument

The island formula, which computes the entanglement entropy by including contributions from island regions in the black hole interior after the Page time.

Load-bearing premise

The island formula and the area law for entropy apply unchanged when the conformal Killing gravity parameter is added as a new thermodynamic variable.

What would settle it

A direct computation of the late-time entanglement entropy that fails to reach a plateau at exactly twice the Bekenstein-Hawking entropy would show that the Page curve is not restored in this model.

Figures

Figures reproduced from arXiv: 2603.12524 by Brahim Asfour, Francisco S. N. Lobo, Taoufik Ouali, Yahya Ladghami.

Figure 1
Figure 1. Figure 1: FIG. 1. Thermal evolution of Schwarzschild AdS black holes [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Gibbs free energy as a function of the Hawking tem [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Heat capacity as a function of the event horizon radius for different values of the thermodynamic pressure. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Thermal evolution of Schwarzschild AdS black holes in negative conformal Killing gravity for different values of the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Penrose diagram of Schwarzschild AdS black holes in [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Gibbs free energy as a function of the Hawking tem [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Heat capacity as a function of the event horizon [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Penrose diagram of Schwarzschild AdS black holes in [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Page time as a function of thermodynamic parameters [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Page time as a function of the event horizon radius [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
read the original abstract

We study Schwarzschild AdS black holes in conformal Killing gravity, focusing on their thermodynamics and information recovery via the island formula. Treating the cosmological constant as pressure and the conformal Killing gravity parameter as an independent variable, we find that the Bekenstein-Hawking area law holds, while the conformal Killing gravity parameter dramatically affects phase structure. For a positive conformal Killing gravity parameter, black holes admit an extremal limit and exhibit Van der Waals-like criticality with first and second order phase transitions; for a negative conformal Killing gravity parameter, no extremal limit or criticality occurs. Using the island prescription, we show that without islands, the entanglement entropy of Hawking radiation grows unboundedly, violating unitarity, while including islands after Page time restores the Page curve, with late-time entropy saturating at twice the Bekenstein-Hawking value. Page time can be expressed in terms of thermodynamic quantities, displaying critical behavior for positive conformal Killing gravity parameter, whereas in negative conformal Killing gravity small black holes recover information rapidly and large ones more slowly, with pressure reducing Page time. Our results reveal a direct link between black hole thermodynamics, quantum information recovery, and modified gravity.

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 studies Schwarzschild-AdS black holes in conformal Killing gravity, treating the cosmological constant as pressure and the conformal Killing parameter as an independent thermodynamic variable. It claims that the Bekenstein-Hawking area law remains valid, that positive values of the parameter produce extremal limits and Van der Waals-like criticality with first- and second-order phase transitions, while negative values do not. Using the island prescription, the paper asserts that Hawking radiation entanglement entropy grows without bound in the absence of islands but saturates at twice the Bekenstein-Hawking value once islands are included after the Page time, with the Page time itself expressible in thermodynamic quantities and exhibiting critical behavior for positive parameter values.

Significance. If the unmodified extension of the area law and island formula to this modified-gravity setting is justified, the results would connect an additional gravitational parameter to both black-hole phase structure and the restoration of unitarity in the Page curve, offering a concrete example of how modified gravity can alter information-recovery timescales while preserving the late-time saturation value.

major comments (3)
  1. [§3] §3 (thermodynamic quantities): The assertion that the Bekenstein-Hawking area law holds is stated without an explicit Wald-entropy or Euclidean on-shell action calculation that includes the conformal Killing term in the action; the manuscript must show whether this term contributes to the entropy functional, because any nonzero contribution would invalidate both the identification of thermodynamic quantities with entanglement entropy and the claimed late-time saturation value of 2 S_BH.
  2. [§4.2] §4.2 (island prescription): The generalized entropy is taken to be S_gen = A/4G + S_matter with no modification from the conformal Killing parameter; given that the gravitational action is altered, a derivation (via Noether charge, replica trick, or direct variation) is required to confirm that the new term drops out of the entropy while still entering the first law and phase structure.
  3. [§4.3] §4.3 (Page time and saturation): The claim that islands restore the Page curve with late-time entropy exactly 2 S_BH rests on the standard island formula; the manuscript should provide an explicit check that the island location and entropy extremization remain consistent with the modified field equations once the conformal Killing parameter is treated as independent.
minor comments (2)
  1. The abstract and introduction should explicitly state the sign and range of the conformal Killing parameter for which the extremal limit and Van der Waals criticality are reported.
  2. [Figure 3] Figure captions for the phase diagrams would benefit from indicating the location of the critical point and the first-order transition line for clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each of the major comments below and will revise the manuscript to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [§3] §3 (thermodynamic quantities): The assertion that the Bekenstein-Hawking area law holds is stated without an explicit Wald-entropy or Euclidean on-shell action calculation that includes the conformal Killing term in the action; the manuscript must show whether this term contributes to the entropy functional, because any nonzero contribution would invalidate both the identification of thermodynamic quantities with entanglement entropy and the claimed late-time saturation value of 2 S_BH.

    Authors: We agree that an explicit calculation is required to rigorously establish the entropy. In the revised manuscript, we will include a detailed Wald entropy computation for the conformal Killing gravity action. Our preliminary analysis indicates that the conformal Killing term, being a total derivative or boundary term in the relevant variation, does not contribute to the entropy at the horizon, preserving the area law. This will be shown explicitly, thereby justifying the thermodynamic identifications and the late-time saturation value. revision: yes

  2. Referee: [§4.2] §4.2 (island prescription): The generalized entropy is taken to be S_gen = A/4G + S_matter with no modification from the conformal Killing parameter; given that the gravitational action is altered, a derivation (via Noether charge, replica trick, or direct variation) is required to confirm that the new term drops out of the entropy while still entering the first law and phase structure.

    Authors: We acknowledge the need for a derivation. We will add a subsection deriving the generalized entropy using the replica trick adapted to the modified action. The calculation shows that the conformal Killing parameter modifies the equations of motion and thus the thermodynamics (first law and phase transitions), but the entropy functional remains the standard area term plus matter entropy because the additional term in the action does not affect the Noether charge associated with the Killing horizon. This resolves the apparent inconsistency. revision: yes

  3. Referee: [§4.3] §4.3 (Page time and saturation): The claim that islands restore the Page curve with late-time entropy exactly 2 S_BH rests on the standard island formula; the manuscript should provide an explicit check that the island location and entropy extremization remain consistent with the modified field equations once the conformal Killing parameter is treated as independent.

    Authors: We will provide the requested explicit check in the revision. By solving the extremization condition for the island location using the modified metric and field equations that incorporate the conformal Killing parameter, we confirm that the island position shifts accordingly but the extremal value of the generalized entropy still saturates at 2 S_BH for late times. The Page time, expressed in thermodynamic variables, will be re-derived consistently with these equations, and we will demonstrate its critical behavior remains intact. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extension of island formula and area law is assumed rather than self-referential

full rationale

The paper introduces the conformal Killing gravity parameter as an independent thermodynamic variable, states that the Bekenstein-Hawking area law holds, and applies the standard island prescription to recover the Page curve with late-time saturation at twice the Bekenstein-Hawking entropy. No equations reduce a claimed prediction or result to a fitted input by construction, nor does any load-bearing step rely on a self-citation chain whose content is unverified within the paper. Thermodynamic quantities are expressed in terms of the new parameter without the entropy functional itself being redefined in a tautological manner. The derivation is therefore self-contained against external benchmarks even if the extension of the area law to this modified gravity theory lacks an explicit Wald or replica-trick derivation in the manuscript.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim depends on extending the Bekenstein-Hawking law and island formula to this modified gravity without additional corrections, plus treating the conformal Killing parameter as a free independent variable.

free parameters (1)
  • conformal Killing gravity parameter
    Treated as an independent variable that controls extremal limits and phase transitions; no fitting procedure described but its sign determines qualitative behavior.
axioms (2)
  • domain assumption Bekenstein-Hawking area law remains valid in conformal Killing gravity
    Stated to hold while the parameter affects phase structure; invoked for entropy calculations.
  • domain assumption Island formula applies directly to Hawking radiation entanglement entropy in this model
    Used to restore the Page curve and unitarity after Page time.

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Reference graph

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