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Applying the island formula to an acoustic black hole yields a Page curve for phonon radiation, with late-time entropy set by the acoustic horizon.

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2026-08-03 17:27 UTC pith:XTI6OTX3

load-bearing objection A clean but unreflective application of the island formula to an acoustic black hole; the Page curve is real only if you grant the formula's transfer to phonons, which the paper assumes. the 4 major comments →

arxiv 2512.09460 v2 pith:XTI6OTX3 submitted 2025-12-10 hep-th gr-qc

Island of acoustic black hole in Schwarzschild spacetime

Yu-Ye Cheng , Jia-Rui Sun This is my paper
classification hep-th gr-qc MSC 83C5781T2081P40 PACS 04.70.-s04.62.+v03.65.Ud
keywords acoustic black holeisland formulaPage curveanalogue gravityentanglement entropyphonon Hawking radiationSchwarzschild spacetimeextremal horizon
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies an analogue black hole formed by sound waves in a superfluid around a Schwarzschild black hole, with an effective metric that contains both optical and acoustic horizons. It applies the island formula—a prescription that adds a gravitational area term to matter entanglement entropy—to the phonons emitted as analogue Hawking radiation. In the non-extremal case, the phonon entanglement entropy first grows linearly and then saturates near twice the acoustic horizon area, reproducing a Page curve. In the extremal case, the no-island entropy diverges because the surface gravity vanishes, making the Page time ill-defined. The paper argues this verifies the unitarity of the analogue gravity system and that the island region lies outside the optical event horizon.

Core claim

The central claim is that the information paradox is resolved for this acoustic black hole: the generalized entanglement entropy of phonon Hawking radiation, computed with the island formula, obeys S_gen = min[(c/3)κ_+ t, S_with-island_gen]. At late times the entropy saturates to a value close to 2πr_+²/G_N, where r_+ is the outer acoustic horizon, matching its area entropy. In the extremal limit (coincident acoustic horizons), the no-island entropy diverges as κ_+ → 0, so the Page time is not well defined. The authors interpret these results as evidence that unitarity is preserved in the analogue gravity system and that the island sits just outside the optical horizon.

What carries the argument

The central object is the island formula, S_R = min ext[Area(∂I)/(4G_N) + S_matter(I∪R)], which augments matter entanglement entropy with a gravitational area term for the boundary of an 'island' region. In this paper it is applied to a phonon effective field theory on the acoustic metric ds² = √3 c_s²[-F(r)dt² + dr²/F(r) + r²dΩ²] with F(r) = (1 - r_h/r)(1 - ξ r_h/r (1 - r_h/r)). The island's position is found by extremizing this generalized entropy, placing it just outside the outer acoustic horizon r_+.

Load-bearing premise

The entire Page-curve result rests on the assumption that the island formula, including its gravitational area term and the background Newton constant, can be transferred from real gravity to the phonon effective field theory on an acoustic metric; the paper provides no independent derivation of this transfer.

What would settle it

Compute the phonon von Neumann entropy for a spherically symmetric Gross-Pitaevskii acoustic black hole without using the island formula—either by direct quantum simulation or by measuring second-order correlations in a Bose-Einstein condensate—and check whether the entropy saturates near 2πr_+²/G_N. If the entropy keeps growing linearly in time, the island-based Page curve is falsified. Alternatively, deriving the area term from the fluid action and showing it does not equal Area/(4G_N) with the background Schwarzschild G_N would undermine the quantitative prediction.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The phonon radiation from this acoustic black hole is unitarily evolving, so the analogue system does not exhibit an information-loss paradox.
  • The island region resides outside the optical event horizon, making the quantum extremal region in principle observable in an acoustic setup.
  • In the extremal limit (ξ = 4), the Page time diverges because the surface gravity vanishes, predicting no well-defined unitary saturation for zero-temperature acoustic black holes.
  • The island formula appears to be a universal mechanism for restoring unitarity in any effective theory with a horizon, not only gravitational systems.
  • The late-time entanglement entropy is controlled by the area of the outer acoustic horizon, giving a concrete analogue prediction for an entropy bound.

Where Pith is reading between the lines

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

  • If the island prescription applies to acoustic metrics, the same method could be used to analyse other emergent spacetimes (e.g., optical or gravity-wave analogues) without invoking the Einstein equations, suggesting a broader class of quantum extremal surfaces in condensed matter.
  • The paper leaves open the origin of the Newton constant in the area term; deriving it from the fluid's microscopic parameters via fluid/gravity duality would directly test the universality claim.
  • The extremal divergence hints that vanishing surface gravity generically spoils the definition of a Page time for any horizon-based entropy, which may be relevant to extremal black hole remnants in real gravity.
  • A direct measurement of phonon entanglement entropy in a Bose-Einstein condensate analogue black hole could probe the predicted saturation value and Page time, offering a laboratory test of the island mechanism.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper applies the island formula to the acoustic black hole obtained from a relativistic Gross-Pitaevskii superfluid on a Schwarzschild background. It considers the phonon Hawking radiation from the outer acoustic horizon and computes its entanglement entropy in both the non-extremal and extremal acoustic geometries. In the non-extremal case the no-island entropy grows linearly in time, while an island configuration gives a late-time entropy of order 2πr_+^2/G_N, producing a Page curve and a Page time. In the extremal case the no-island entropy diverges as the surface gravity tends to zero, so the Page time is ill-defined, although an island phase gives a finite entropy. The paper concludes that unitarity of the analogue gravity system is verified and that the island lies outside the optical event horizon.

Significance. If the island formula is legitimately transferable to the phonon effective field theory, the paper would extend the island/Page-curve program to analogue gravity systems and identify an experimentally accessible setting where the island is outside the event horizon. The non-extremal calculation is a competent application of the standard QES template: the island positions are solved, the late-time saturation is exhibited, and the extremal island-existence constraint is treated explicitly (Sec. IV B, Eq. (42)). The paper is also honest in distinguishing the near- and far-horizon limits. However, the central significance is conditional: the area term in the island formula is imported from background gravity without a microscopic derivation for a nongravitational phonon system, and the extremal limit is handled by taking κ_+→0 despite the paper's own statement that the extremal geometry is not a continuous limit. These issues are load-bearing for the advertised conclusions.

major comments (4)
  1. [Secs. I, III B; Eqs. (1), (24), (27)] The late-time entropy Sgen≈2πr_+²/G_N is obtained by inserting the gravitational area term Area(∂I)/(4G_N) into the island formula for a phonon system. The acoustic metric (10) is an effective metric for phase fluctuations of the GP scalar; it is not a solution of Einstein gravity, and the fluid action (2) does not fix the coefficient 1/G_N. No replica or microscopic argument is given that the phonon path integral produces a replica-wormhole saddle whose action equals the area term with G_N taken from the background Schwarzschild spacetime. Reference [64] is cited in the introduction but not engaged, even though it appears to address exactly this question. Without such a derivation or a clear justification, the Page curve and the 'verification of unitarity' are consequences of the assumed formula rather than of the acoustic model. The authors should either supply the missing justificatio
  2. [Sec. III A, Eq. (20)] The coefficient in Eq. (20) appears incorrect. From Eq. (12), κ_+ = (r_+−r_h)(r_+−r_−)/(2r_+³). Substituting Eq. (11) gives κ_+ = 2 y(ξ+y−2)/(r_h(ξ+y)^3) with y=√(ξ(ξ−4)), so Sgen≃(c/3)κ_+ t_b = (2c/3) y(ξ+y−2) t_b/[r_h(ξ+y)^3]. For ξ=5, this is about 0.0206 c t_b/r_h. Equation (20), however, gives (c/12)(4−ξ+y)ξ r_h t_b ≈ 0.515 c r_h t_b for ξ=5, which is wrong both numerically and dimensionally; the correct dimensionless combination appears to be (4−ξ+y)/(12 ξ r_h), not (4−ξ+y) ξ r_h/12. This error affects the slope of the no-island growth and should be corrected before publication.
  3. [Sec. IV, Eqs. (37), (44)] The paper states at the beginning of Sec. IV that the extremal geometry is not obtained by taking the continuous κ_+→0 limit, but the no-island entropy and the far-region island entropy are then computed precisely by taking κ_+→0 of the non-extremal Kruskal expressions (Eqs. (37) and (44)). The Kruskal coordinates (15) with U,V built from e^{±κ_+ r_*} become singular as κ_+→0, so this limit is not a well-defined prescription for the extremal spacetime. Consequently, the claimed divergence of the no-island entropy and the ill-defined Page time in the extremal case are not established by the computation as written. The authors should either define properly the extremal Kruskal coordinates and redo the calculation, or explicitly justify why the κ_+→0 limit of the non-extremal expression is reliable despite the degeneracy.
  4. [Secs. V, VI; Eq. (50)] The conclusion that the model 'verifies the unitarity of the analogue gravity system' is stronger than what the calculation shows. Equation (50) is the minimum of the standard linear-growth entropy and the island-formula result; the transition at the Page time is built into the island prescription, not derived from the fluid dynamics. What the paper demonstrates is that, if one assumes Eq. (1) for the acoustic system, the entanglement entropy takes the Page form. The final claims in Sec. VI should be weakened accordingly, e.g., to 'conditional on the island formula, the acoustic system is consistent with unitary evolution.'
minor comments (5)
  1. [Throughout] There are several typos: 'the follows' should be 'the following', 'sown' should be 'shown', 'temperture' should be 'temperature', and 'Hawking-Unurh' should be 'Hawking-Unruh'.
  2. [Eq. (21)] The same symbol κ is used for the dimensionless constant in Eq. (21) and for the surface gravity κ_+ everywhere else. The warning that they should not be confused is easy to miss; using e.g. k for the dimensionless constant would improve readability.
  3. [Eq. (21)] The quantity 'Area' in Eq. (21) is not defined before use; from Eq. (24) it is effectively 4πb². Please state this explicitly.
  4. [Introduction, Ref. [64]] Reference [64] is cited in the list of related works but is never discussed in the body. Since that paper is directly about analogue gravity and the island prescription, it should be engaged, especially in view of the main conceptual issue raised above.
  5. [Fig. 5] Figure 5 would be easier to read with labeled axes and explicit scales for the Page time and the late-time entropy.

Circularity Check

0 steps flagged

No significant circularity: the island formula is an explicit external input, and the saturation value follows from extremizing the generalized entropy, not from re-labeling an input.

full rationale

The paper's central chain is: acoustic metric (Eq. (10)) -> s-wave/matter entanglement entropies (Eqs. (18)-(21), (28)) -> generalized entropy S_gen = Area(∂I)/(4G_N) + S_matter (Eq. (1) applied in Eqs. (24), (29), (39), (45)) -> extremization over island position a and time t_a -> S_gen ≈ 2πr_+²/G_N (Eqs. (27), (36), (43), (49)) -> min with the no-island branch (Eq. (50)). The late-time saturation is not the island formula itself: the island location is solved from ∂S_gen/∂a = 0 (Eqs. (25)-(26), (34), (47)-(48)), and the matter contributions are calculated, not fitted. No parameter is tuned to force the final entropy; c, κ_c, G_N, r_+, and the acoustic-metric parameters are independent inputs. The only serious caveat is the physical applicability of the island formula, with its (4G_N)^{-1} area term, to a nongravitational phonon system; the paper cites [8-11] for the formula and [64] for analogue gravity, but does not justify the transfer. While this is a real correctness/scope risk, it is not a circularity: the result would not follow if the transfer is invalid, but the paper's equations do not reduce by construction to its inputs. The self-citations ([33], [40], [55], [56]) are used for an elementary extremum condition, a special-case comparison, and prior fluid/gravity duality; none carries the central claim. Hence score 0.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The calculation is built on the island formula, which is a substantive assumption when applied to a phonon system with no dynamical gravity. The fluid tuning parameter ξ, the near-horizon constant κ, and the central charge c are all left unspecified, so the quantitative entropy and Page time cannot be connected to an experimental fluid without additional model-building. No new particles or forces are introduced; the island is an application of the known prescription.

free parameters (3)
  • ξ (flow tuning parameter) = ξ ≥ 4; ξ=4 extremal
    Introduced ad hoc in the flow velocity v^r ∼ sqrt(2Mξ/r) to create two acoustic horizons; controls Hawking temperature, Page time, and horizon radii (Eqs. (10)–(12), (51)).
  • κ (near-horizon dimensionless constant)
    Introduced in Eq. (21) for the near-horizon matter entropy S_phonon = -2κc Area/L²; not specified or linked to fluid parameters.
  • c (central charge of s-wave phonon CFT)
    Appears in all entropy formulas and the Page time; assumed O(1) but never computed from the GP superfluid parameters.
axioms (4)
  • domain assumption The island formula (1) with the gravitational area term is valid for the phonon sector of the acoustic metric.
    Eq. (1) is imported from gravitational path-integral/AdS/CFT and applied to a condensed-matter analogue without derivation; the unitarity conclusion rests on it.
  • domain assumption 2D CFT entanglement entropy formulas (Eqs. (18), (21), (28)) describe the 4D phonon field after s-wave reduction.
    Standard in island literature but assumes the phonon modes reduce to a 2D CFT with central charge c; the reduction details are not given.
  • ad hoc to paper The extremal acoustic geometry is described by the κ_+→0 limit of the non-extremal Kruskal expressions.
    Sec. II end and Sec. IV.A use this limit even though the paper notes the extremal geometry is not the continuous limit of the non-extremal one.
  • domain assumption The background Schwarzschild metric and the relativistic GP acoustic metric (Eq. (10)) faithfully model the superfluid around the black hole.
    Taken from [57]; no experimental parameters for density, sound speed, or flow profile are specified.

pith-pipeline@v1.3.0-alltime-deepseek · 14503 in / 28377 out tokens · 271379 ms · 2026-08-03T17:27:07.825703+00:00 · methodology

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read the original abstract

We study an analogue information paradox in acoustic black holes which are emerged from the superfluid surrounding a Schwarzschild black hole. The resulting acoustic black hole contains both acoustic horizons and optical horizon, with analogue Hawking radiation, i.e. phonons, emitted from the outer acoustic horizon. By using the island formula, we calculate the entanglement entropy of analogue Hawking radiation of the acoustic black hole in both non-extremal and extremal cases. In the non-extremal case, the entanglement entropy of phonons follow the Page curve due to the emergence of islands, and it is approximately proportional to the area of the acoustic horizon at late time. While in the extremal case, the entanglement entropy of phonons diverges, leading to an ill-defined Page time. Our study verifies the unitarity of the analogue gravity system, and provides further insight into the connection between the entanglement entropy and the causal structure of spacetime.

Figures

Figures reproduced from arXiv: 2512.09460 by Jia-Rui Sun, Yu-Ye Cheng.

Figure 1
Figure 1. Figure 1: FIG. 1: The part of Penrose diagram for a non-extremal Schwarzschild acoustic black hole without islands. [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The part of Penrose diagram for a non-extremal Schwarzschild acoustic black hole with island. The [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The Penrose diagram of the Schwarzschild acoustic black hole in the extremal case with and without [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The diagram of function [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The Page curve of the entanglement entropy in the non-extremal case. The entanglement entropy is [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗

discussion (0)

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Forward citations

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Reference graph

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