Radiation Entropy in asymptotically AdS Black Holes within f(Q) Gravity

Baocheng Zhang; Wei Xu; Yipeng Liu

REVIEW 3 major objections 4 minor 63 references

Radiation Entropy in asymptotically AdS Black Holes within f(Q) Gravity

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read In f(Q) gravity, the island rule must carry an f_Q-weighted area term, yielding a finite radiation entropy with a logarithmic area correction for collapsing AdS black holes.

desk verdict First island-rule computation in f(Q) gravity, but the central factor-of-2 in the generalized entropy is unsupported and the eternal-case extremization drops a divergent term. read the letter →

arxiv 2510.17528 v2 pith:6VYNXSFA submitted 2025-10-20 hep-th

Yipeng Liu , Wei Xu , Baocheng Zhang This is my paper · ORCID

classification hep-th MSC 83C5783D0581T4083C47 PACS 04.70.Dy04.60.-m04.50.Kd

keywords islandrulef(Q)gravityradiationentropyPagecurveAdSblackholesgeneralizedholeinformationparadoxlogarithmiccorrection

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that the standard island rule for computing Hawking radiation entropy has to be modified when the underlying gravitational theory is f(Q) gravity, an extension of general relativity based on non-metricity rather than curvature. Computing the Euclidean action for an asymptotically AdS black hole, the authors find that the area term in the generalized entropy carries an extra factor f_Q, so the island rule becomes S_R = Min_X Ext_X[ f_Q A(X)/(2G_N) + S_semiclassical ]. With this corrected rule, an eternal AdS black hole gives a time-independent radiation entropy that diverges as the artificial cutoff surface is pushed outward, which they read as a failure of the s-wave approximation. For a collapsing AdS black hole, by contrast, the radiation entropy saturates to the finite value f_Q A_H/(2G_N) - (c/6) ln(A_H/4G_N) + const, independent of the cutoff. The result connects modified gravity to black hole information flow and shows that the specific f(Q) model is imprinted in the final radiation entropy and Page time.

What carries the argument

The central object is the f_Q-modified area term in the generalized entropy. After deriving S_gen = 2 f_Q pi r_h^2/G_N from the Euclidean action, the paper replaces the Ryu-Takayanagi-type area term (the holographic entanglement entropy formula) in the island rule by f_Q A/(2G_N) and extremizes the sum of this geometric term with the semiclassical entanglement entropy of the radiation and island regions. The f_Q factor and the extremization together determine where the island sits: for a collapsing black hole, the island is pushed so close to the horizon that the would-be logarithmic dependence on the cutoff collapses into a constant, leaving a finite answer. The mechanism is essentially a m

What would settle it

Compute the holographic entanglement entropy of the same planar AdS black hole directly from the f(Q) bulk action via the gravitational replica trick. If the extremal-surface area coefficient that emerges is f_Q/(4G_N) rather than f_Q/(2G_N), the paper's island rule, radiation entropy, and Page time are all off by a factor of two. A cheaper consistency check: set f_Q = 1 and compare the island-rule radiation entropy with the standard general-relativity result; a factor-of-two discrepancy would mean the boundary-term identification is wrong.

Watch

Extended reading notes

Core claim

The paper's central claim is that in f(Q) gravity the generalized entropy of a black hole is no longer A/(4G_N) but S_gen = 2 f_Q pi r_h^2/G_N = f_Q A_h/(2G_N), so the island rule should be written as S_R = Min_X Ext_X[ f_Q A(X)/(2G_N) + S_semiclassical(Sigma_R union Sigma_I) ]. For a collapsing asymptotically AdS black hole, extremizing this generalized entropy yields S_R approximately f_Q A_H/(2G_N) - (c/6) ln(A_H/(4G_N)) + const, a finite, cutoff-independent late-time entropy with a logarithmic area correction that matches quantum-gravity expectations. For an eternal black hole, the same rule gives a time-independent entropy that grows without bound as the cutoff moves outward; the paper

Load-bearing premise

The load-bearing premise is that the f_Q/(2G_N) coefficient obtained from the Euclidean action for black hole entropy also governs the area term in the island rule; if the correct coefficient is f_Q/(4G_N), every island location, radiation entropy, and Page time in the paper changes by a factor of two.

Editorial extensions

If this is right

Earlier island-rule results in f(Q) gravity must be redone with f_Q A/(2G_N) in place of A/(4G_N); omitting f_Q changes the entropy and island location by a model-dependent factor.
The eternal AdS black hole case establishes a concrete obstruction: under the s-wave approximation, island computations in this background inevitably diverge with the cutoff, so finite results require dropping the s-wave treatment.
For a collapsing AdS black hole, the radiation entropy reaches a finite plateau with a logarithmic area term, reproducing the expected Page-curve behavior within f(Q) gravity.
Both the plateau value and the Page time explicitly depend on f_Q and on the AdS radius set by the non-metricity scalar, so information-paradox data could constrain the functional form of f(Q).
In the limit f_Q = 1, the f(Q) entropy is twice the general-relativity result, indicating that boundary terms which leave the field equations unchanged still affect the entanglement entropy.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the vacuum case, the same f_Q-corrected island rule could be applied to charged f(Q) black holes, where the solutions genuinely differ from general relativity; those settings would make the model dependence of the Page curve stronger and more directly testable than the uncharged case treated here.
The factor-of-two discrepancy in the f_Q tending to 1 limit indicates that island-rule calculations are sensitive to the variational formulation of gravity, not just to the equations of motion; this suggests the island prescription may need independent derivation in any modified-gravity theory, not only f(Q).
Because the late-time entropy is dominated by the area term with a small logarithmic correction, an accurate measurement of the area-law coefficient would select among f(Q) models, for instance f_Q = 1/2 if the geometric term preserves the standard form, offering a theoretical consistency constraint independent of cosmology.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This is the first application of the island rule to f(Q) gravity, and it gets the structural story from the GR template: eternal case diverges with cutoff, collapsing case gives a finite entropy with a logarithmic correction. That is novel, and the paper is clearly written, following the standard pipeline and honestly flagging some of its own limitations (e.g., s-wave breakdown).

The load-bearing premise, though, is the modified area coefficient f_Q A/(2G_N) in the island rule (Eq. 28). That coefficient is inferred from the Euclidean-action entropy, Eq. (25), which gives twice the GR entropy when f_Q=1. The paper itself notes that f(Q)=Q-2λ is dynamically equivalent to GR, so this factor of two is a genuine discrepancy, not a boundary-term afterthought. The known f(Q) black-hole entropy in the cited literature (e.g., Ref. [60]) is f_Q A/(4G_N). If that is the correct coefficient, every subsequent result — the radiation entropy, the Page time — changes by a factor of two. The paper has no independent check of its inference, and this is exactly where the whole calculation rests.

There is also an algebraic problem in the eternal case. In Eq. (39), the expansion coth y ≈ 1+2e^{-2y} leaves a constant term −cκ/(3a(r_I)) that is not small (it diverges at the horizon), and the claim that a'(r_I)−2κ vanishes to second order in (r_I−r_h) is wrong; it is first order. Dropping that term changes the extremum condition and island location. This is a fixable error, but as written that section is unreliable.

Secondary issues: the planar horizon treats area as 4πr² without specifying a compactification of the x,y coordinates, and the modified RT formula (Eq. 29) is asserted by analogy, not derived from a holographic functional. These are softer but still need attention.

Overall, this is not a desk reject. The mistakes are specific and checkable, and the topic is legitimate. A serious referee could sort out the factor of two and the dropped term. As it stands, the central claim is not supported, and the paper needs major revision before it can be taken as a reliable extension of the island program.

Referee Report

3 major / 4 minor

Summary. The paper studies the island rule and the Page curve for asymptotically AdS, planar-horizon black holes in f(Q) gravity. After deriving a Euclidean-action generalized entropy S_gen = 2 f_Q π r_h^2/G_N, the authors posit a modified Ryu-Takayanagi area term S = f_Q A/(2G_N) and hence a modified island rule (Eq. 28). For an eternal black hole the resulting radiation entropy is time-independent but diverges as the cutoff surface is moved outward; for a collapsing black hole the island entropy saturates and is claimed to take the form S_R = f_Q A_H/(2G_N) - (c/6) ln(A_H/4G_N) + const, with a Page time v_Page = 16π f_Q ℓ^2 r_h/(cG_N). The paper concludes that the f(Q) model is encoded in the radiation entropy and Page time.

Significance. If the central construction were sound, the paper would extend the quantum-extremal-surface/island formalism to a non-Riemannian gravitational theory and would make a falsifiable, model-dependent prediction for radiation entropy. The calculations are explicit, and the treatment of the cutoff dependence in the collapsing case is clear. The main difficulty is that the paper's central premise—the modified area coefficient f_Q/(2G_N)—is not derived; it is inferred from a thermodynamic entropy computation that itself has an acknowledged factor-of-two discrepancy with the standard f(Q) black-hole entropy. Because every quantitative result inherits this coefficient, the paper's prediction of f_Q dependence is only as reliable as this unvalidated input.

major comments (3)

[Sec. II C, Eqs. (25)-(28)] The generalized entropy (25) gives S_gen = 2 f_Q π r_h^2/G_N. In the limit f_Q=1, which the paper itself identifies as recovering general relativity, this equals A/(2G_N) rather than the Bekenstein-Hawking entropy A/(4G_N). The paper acknowledges this at the end of Sec. II C but only attributes it to boundary terms and offers no derivation. This factor of two is load-bearing: it enters the island rule (28) and then propagates into Eqs. (43), (56), (57), and (59). If, as in the established f(Q) black-hole entropy literature cited as Ref. [60], the correct area coefficient is f_Q/(4G_N), then all of the paper's quantitative results change by a factor of two. The discrepancy must be resolved with a concrete derivation, not a qualitative remark about boundary terms.
[Sec. III A, Eq. (29)] The modified RT formula S = f_Q A_M/(2G_N) is inferred from the thermodynamic entropy of a static black hole, not derived from a holographic entanglement-entropy functional or from a replica calculation in f(Q) gravity. The sentence 'one can infer that the RT formula itself will also be modified' is exactly the assumption on which the entire island rule (28) rests. Without a derivation or independent check, the f_Q dependence appearing in the final radiation entropy and Page time is not a prediction but a restatement of the input. A correct derivation, or at least a rigorous argument connecting the thermodynamic entropy to the entanglement entropy of extremal surfaces, is essential.
[Eqs. (36), (43) vs. Eq. (25)] There is an internal factor-of-two inconsistency in the area term. Eq. (25) gives the area contribution as 2π f_Q r_h^2/G_N, and the collapsing-case formulas (50), (56), and (57) are consistent with this. However, Eqs. (36) and (43) use 4π f_Q r_I^2/G_N (and 4π f_Q r_h^2/G_N in the final eternal result). Since Eq. (43) is the reported radiation entropy for the eternal black hole, the manuscript is internally inconsistent about the central coefficient. This needs to be fixed before any of the quantitative results can be assessed.

minor comments (4)

[Sec. II A] Typo: 'nation fQ' should read 'notation fQ'; in Sec. III A, 'cental charge' should be 'central charge'.
[Eq. (26)] The final expression on the right-hand side contains an undefined symbol r; it should presumably be r_h (the horizon radius). Please define all symbols in the Euclidean action computation.
[Eq. (54)] From the extremality equation (52), the condition is v_A - v_I = 2/(κ coth χ) = (2/κ) tanh χ, not (2/κ) coth χ as written. The large-χ final result is unchanged, but the displayed equation is misleading.
[Eq. (12)] The transverse coordinates x,y in the planar metric are not compactified or normalized, so factors of 4π in the horizon 'area' are ambiguous. Specify ∫dx dy or otherwise fix the area normalization; otherwise the numerical coefficients in the entropy are not well defined.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the f_Q dependence is propagated from the Euclidean-action entropy, not re-imported from the outputs.

full rationale

The central modified island rule derives from an explicit Euclidean-action calculation: Eq. (25) follows from Eqs. (23)-(26), not from the radiation entropy it is later used to compute. The RT-area replacement in Eqs. (28)-(29) is an inference from that independently obtained generalized entropy, and the f_Q factors in Eqs. (56)-(59) are the direct propagation of that coefficient through the extremization. No parameter is fitted to the predicted entropy, and the one self-citation (Ref. [41]) is not load-bearing. The factor-of-two mismatch with GR at f_Q=1, acknowledged in Sec. II C ('the radiation entropy in f(Q) gravity is twice that in general relativity' and attributed to boundary terms), is a significant correctness/derivation-risk flag, and the modified RT formula is asserted rather than derived; but these are assumptions or possible errors, not definitional equivalences. The analysis therefore does not reduce to its inputs by construction.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central calculation depends on three unevidenced inputs: the factor-of-2 generalized entropy (which conflicts with GR in the f_Q=1 limit), the inferred modification of the RT formula, and the spherical-area treatment of a planar horizon. The rest is standard island-rule technology.

free parameters (1)

f_Q = df/dQ evaluated at the horizon = unspecified (f_Q=1 for STEGR; f_Q=1/2 would match the GR area-law coefficient)
The entire modified island rule and radiation entropy scale with this model parameter. It is not determined by the paper; it is an input of the f(Q) model.

assumptions (4)

ad hoc to paper The generalized entropy of f(Q) black holes is S_gen = 2 f_Q π r_h²/G_N (Eq. 25).
Derived in the paper from the Euclidean action, but it gives a factor-of-2 discrepancy with GR in the f_Q=1 limit, so it functions as an unverified assumption.
ad hoc to paper The holographic RT formula in f(Q) gravity is S = f_Q A/(2G_N) (Eq. 29).
Inferred from the thermodynamic entropy, not derived from a holographic calculation; the modified island rule (Eq. 28) rests on this.
domain assumption The s-wave approximation and conformal-map entanglement entropy formulas (Eqs. 30-31) from GR remain valid for these f(Q) black holes.
The paper assumes the matter entanglement entropy is unchanged in f(Q) gravity aside from the background metric.
domain assumption The planar (flat-horizon) AdS black hole solution (Eq. 20) is a valid background and its transverse area can be identified with 4π r².
The solution is the GR planar AdS BH, but the paper uses spherical area 4π r² consistently, without compactification or justification.

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Pith. "Pith review of Radiation Entropy in asymptotically AdS Black Holes within f(Q) Gravity." pith.science (2026). https://pith.science/paper/6VYNXSFA

@misc{pith2026251017528,
  author       = {Pith},
  title        = {Pith review of: Radiation Entropy in asymptotically AdS Black Holes within f(Q) Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6VYNXSFA}},
  note         = {Machine review of arXiv:2510.17528}
}

read the original abstract

We employ the island rule to investigate the radiation entropy of asymptotically AdS black holes in the framework of f(Q) gravity. Through an analysis based on the Euclidean action, we find that the area term of the generalized entropy must be modified, which in turn leads to a modification of the island rule itself. Using the corrected rule to compute the radiation entropy for the eternal asymptotically AdS black hole reveals that, the result diverges as the cutoff surface is moved outward, indicating the breakdown of the s-wave approximation. For a collapsing asymptotically AdS black hole, the radiation entropy is dominated by the area term, with a logarithmic correction proportional to the area, which is consistent with the predictions of quantum gravity theories. Furthermore, both the radiation entropy and the Page time are ultimately influenced by the choice of the f(Q) model, implying that information regarding the underlying gravitational model is encoded in the final radiation entropy.

Figures

Figures reproduced from arXiv: 2510.17528 by the authors.

**Figure 2.** FIG. 2: The vacuum fluctuations near the island and the cutoff s [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3: A black hole formed by the collapse of an AdS vacuum, wi [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

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