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REVIEW 3 major objections 5 minor 67 references

Effect of Non-Extensive Parameter on Page Curve

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Applying the quantum extremal island prescription to black holes with Barrow, Kaniadakis, Renyi, or Tsallis-Cirto entropy, this paper shows that the entropy of Hawking radiation saturates to the corresponding non-extensive generalization…

desk verdict All four non-extensive corrections just rescale the Schwarzschild mass, so the Page curves are reparameterizations and the paper's central saturation claim does not follow. read the letter →

arxiv 2502.00723 v1 pith:QNDZROWW submitted 2025-02-02 hep-th gr-qc

Ankit Anand , Dinesh Kumar , Aditya Singh This is my paper · ORCID
classification hep-thgr-qc
keywords non-extensiveentropyPagecurveislandprescriptionquantumextremalsurfaceblackholeinformationparadoxBarrowKaniadakisRenyi
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

The paper investigates whether the island prescription resolves the black-hole information paradox when black-hole entropy is one of four non-extensive generalizations: Barrow, Kaniadakis, Renyi, and Tsallis-Cirto entropy. For each model, the authors construct a corrected Schwarzschild-like metric from the temperature relation $1/T=dS/dM$, compute the entanglement entropy of Hawking radiation with and without an island, and compare the two. They find that without an island the radiation entropy grows linearly with time, while at late times a quantum extremal island forms near the horizon and the entropy saturates to the non-extensive version of Bekenstein-Hawking entropy. The upshot is that the island mechanism extends to deformed black-hole spacetimes and that the non-extensive parameter controls when the Page transition happens.

What carries the argument

The load-bearing object is the quantum extremal surface prescription applied to the generalized entropy functional $S_{\rm gen}$, with the area term coming from the island boundary and the matter term computed by the four-point conformal distance formula $S_{\rm matter}=(c/3)\log[d(a_+,a_-)d(b_+,b_-)d(a_+,b_+)d(a_-,b_-)/(d(a_+,b_-)d(a_-,b_+))]$ in Kruskal-like coordinates. The non-extensive correction enters through the inverse temperature $\beta=1/T=dS/dM$, which fixes the conformal factor $\Omega_{NE}$ and therefore the corrected metric and tortoise coordinate. Extremizing with respect to the island position $a$ and the time coordinate $t_a$ selects the late-time saddle $t_a=t_b$, which kills the time dependence and pins the island just inside the shifted horizon.

What would settle it

Compute the ADM or Komar mass of the corrected metrics (for example $f_B(r)=1-2\Delta(\Delta+2)\pi^{\Delta/2}GM(GM^2)^{\Delta/2}/r$) and compare it with the $M$ used in the non-extensive entropy; if the two masses differ, then the temperature used in the Page-curve comparison is not the Hawking temperature of that spacetime. A direct check of the first law $dM=T\,dS$ with $T=1/(dS/dM)$ on these metrics would settle the issue.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the generalized entropy functional $S_{\rm gen}=\mathrm{Area}(\partial I)/(4G_N)+S_{\rm matter}(R\cup I)$ admits a late-time saddle with an island for each of the four non-extensive entropy models, and at that saddle the entanglement entropy of the radiation becomes time-independent. The no-island configuration gives linear growth, $S\sim (c/6)(2\pi t/\beta)$, reproducing the usual paradox; the island configuration yields a constant that equals the corresponding non-extensive entropy (Barrow, Kaniadakis, Renyi, or Tsallis-Cirto) evaluated at the black-hole mass. The authors therefore claim that unitarity is restored in these corrected backgrounds, with the Page time shifted by the deformation parameter.

Load-bearing premise

The load-bearing premise is that the mass $M$ appearing in each non-extensive entropy formula is the same mass that appears in the corrected Schwarzschild metric, so the entropy–temperature relation $1/T=dS/dM$ belongs to the spacetime whose Page curve is being computed.

Editorial extensions

If this is right

  • Without an island, each corrected model shows the same linear growth of radiation entropy, so the information paradox would persist in these spacetimes.
  • At late times the island saddle exists, placing the island boundary near the shifted horizon and making the entanglement entropy constant.
  • The late-time constant equals the corresponding non-extensive entropy, so the island prescription restores unitarity for Barrow, Kaniadakis, Renyi, and Tsallis-Cirto black holes.
  • The Page time depends on the deformation parameter: Barrow and Tsallis-Cirto corrections increase it, Kaniadakis increases it as $\kappa^2$, and Renyi decreases it as $\lambda$.
  • Setting all deformation parameters to zero recovers the standard Schwarzschild Page curve and Bekenstein-Hawking entropy.

Reading between the lines

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

  • Because the entropy formulas and the metric both use the same symbol $M$ without an explicit ADM-mass check, a fully self-consistent back-reaction calculation could change the magnitude of the reported Page-time shifts; this is a consistency question the paper leaves open.
  • The same method could be applied to Sharma-Mittal entropy, which the paper mentions but does not analyze; the expected result is a one-parameter interpolation between the Tsallis and Renyi Page times.
  • The single-island approximation could be probed by including multiple islands; the authors note that this would smooth the sharp Page transition, and one could check whether the late-time plateau value changes.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper applies the quantum extremal surface / island prescription to four non-extensive entropy models (Barrow, Kaniadakis, Rényi, and Tsallis-Cirto). For each model it defines a temperature through 1/T = dS/dM, constructs a Schwarzschild-like metric whose Hawking temperature matches this T, computes the no-island and with-island entanglement entropy of Hawking radiation, and reads off a Page time. The abstract claims that at late times the entanglement entropy saturates to the non-extensive extension of the Bekenstein-Hawking entropy, and that the non-extensive parameters shift the Page time.

Significance. If correct, the paper would extend the island prescription to non-extensive entropy corrections and produce falsifiable, parameter-dependent Page times. The manuscript does contain a complete analytic treatment for four entropy models in the standard island-formula language, and it explicitly acknowledges the restriction to zero or one island; these are strengths. However, the central claim fails: the metric constructed from 1/T = dS/dM is Schwarzschild with a mass parameter M_eff that differs from the mass M appearing in the non-extensive entropy, and the late-time island entropy saturates to the Bekenstein-Hawking area entropy evaluated at the displaced horizon, not to S_non-ext(M). Several displayed formulas are also dimensionally inconsistent. The paper therefore does not establish the advertised result.

major comments (3)
  1. [§2 and Eqs. (3.1)–(3.2), (4.1)–(4.2), (5.1)–(5.2), (6.1)–(6.2)] The load-bearing construction is the equation 1/T = dS/dM combined with identifying T with the Hawking temperature of a Schwarzschild-like metric f(r) = 1 - 2G M_eff/r. For each model this yields M_eff as a nonlinear function of the shell mass M; for Barrow, Eq. (3.2) and the horizon radius give M_eff = 2^{Δ-1}(Δ+2)π^{Δ/2}G^{1+Δ/2}M^{1+Δ}. The ADM mass of the constructed spacetime is M_eff, not M, yet every entropy formula, island location, and Page time in Sections 3–6 uses M as if it were both the shell mass and the mass appearing in the first law. Equation (2.5) is therefore not the first law of the spacetime in which the island calculation is performed. This unsupported identification is the basis for the abstract's saturation claim.
  2. [§3, Eq. (3.9); §4, Eq. (4.10); §5, Eq. (5.9); §6, Eq. (6.10)] The late-time island entropy is the area contribution S_late = π r_h²/G_N = 4π G_N M_eff² with r_h = 2G M_eff, not the prescribed non-extensive entropy S_non-ext(M). For Barrow, for example, Eqs. (3.2) and (3.9) give S_late ≈ 4πG_N M²[1 + Δ(1 + log(4πG_N M²))], whereas S_B from Eq. (2.1) is 4πG_N M²[1 + (Δ/2) log(4πG_N M²)]. The difference is first order in Δ for generic M. Thus the entropy saturates to the standard Bekenstein-Hawking area entropy evaluated on the displaced horizon, reparameterized in terms of M, and the statement in the abstract that the entropy saturates to the non-extensive extension of the Bekenstein-Hawking entropy is not supported. The analogous failure occurs for the Kaniadakis, Rényi, and Tsallis-Cirto models.
  3. [§4, Eq. (4.7); §5, Eq. (5.7); §6, Eq. (6.7)] Several displayed entropy formulas are dimensionally inconsistent. In Eq. (4.7), π c t_b/3 has dimension of length (in units with G=1), while c log(b - 2GM cosh(4πGκM²))/(6π²b) has dimension 1/length; the large-time limit of Eq. (4.6) should have coefficient cπ/(3β) = c/[24GM cosh(4πGκM²)], not πc/3. Equation (5.7) has coefficient ct_b/6 instead of ct_b/(24GM) and a correction of order cλt_b with the wrong dimension; the correct correction is of order cλM t_b. Equation (6.7) also misses the 1/(24GM) prefactor and the logarithmic mass dependence in the δ correction. These errors propagate into the no-island entropy and hence into the Page-time formulas (4.11), (5.10), and (6.11).
minor comments (5)
  1. [§3, Eq. (3.2)] The displayed factor '2∆' should presumably be '2^Δ'; as printed, f_B(r) does not reduce to 1 - 2GM/r at Δ = 0.
  2. [§6, Eq. (6.3)] The Kretschmann scalar for the Tsallis-Cirto metric contains a factor δ² and vanishes at δ = 0, so it does not reproduce the Schwarzschild value K = 48G²M²/r^6. This should be corrected.
  3. [§5.1, discussion after Eq. (5.9)] The sentence 'comparing results (3.6) and (3.9) confirms...' appears to be a copy-paste from the Barrow section; it should refer to Eqs. (5.7) and (5.9).
  4. [§3–§6, concluding paragraphs] The statements that the island boundary 'contributes precisely to the Barrow entropy' (and the analogous statements for Kaniadakis, Rényi, and Tsallis-Cirto entropies) are asserted without derivation and are inconsistent with Major Comment 2; they need either proof or removal.
  5. [Throughout] There are numerous typographical errors, including 'Kandiakis', 'Kadiakis', 'Schwarchschild', 'Krestchmann', and 'Einstien's Equation', which would need correction in any revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the island calculation is a standard application of the quantum extremal surface prescription to explicit Schwarzschild-like metrics; the central claim's difficulty is an unjustified identification, not a circular reduction.

full rationale

The paper's derivation chain is self-contained: it defines non-extensive entropies S(M) (Barrow, Kaniadakis, Rényi, Tsallis-Cirto), fixes the temperature by 1/T = dS/dM, constructs static spherically symmetric metrics whose surface gravity reproduces that temperature, and then applies the standard quantum extremal surface/island prescription to compute the Page curve and Page time. No parameter is fitted to a target datum; the island extremization is performed explicitly, and the Page times are obtained by intersecting the no-island and island entropy curves. The only in-house citation is Ref. [59] by A. Anand and R. Campos Delgado, and it appears in a passing remark about Barrow-corrected black holes; it is not load-bearing for the derivation. The skeptical concern that the late-time island entropy is the area entropy 4π G_N M_eff^2 of the temperature-matched Schwarzschild metric rather than the prescribed non-extensive entropy S_non-ext(M) is a substantive correctness objection to the abstract's saturation claim, but it is not circularity: the paper does not define S_non-ext as the area entropy of the corrected metric, and its own equations (e.g., Eq. (3.2) versus Eq. (2.1)) can be used to exhibit a first-order mismatch. An unsupported or incorrect identification is a validity flaw, not a reduction of the output to the input by construction. Therefore, under the circularity criteria specified, the paper receives a score of 0.

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

All four corrected metrics reduce to Schwarzschild with a rescaled mass, so the only new ingredients are the deformation parameters and the assumption that the island prescription applies unchanged.

free parameters (4)
  • Barrow parameter Delta = not fitted (model parameter)
    Controls the fractal correction in S_B = (S_BH)^(1+Delta/2); results expanded to first order in Delta around 0.
  • Kaniadakis parameter kappa = not fitted (model parameter)
    Controls the deformation of S_K = sinh(kappa S_BH)/kappa; results expanded to order kappa^2.
  • Renyi parameter lambda = not fitted (model parameter)
    Controls S_R = log(1+lambda S_BH)/lambda; results expanded to first order in lambda.
  • Tsallis-Cirto parameter delta = not fitted (model parameter)
    Controls S_TC = S_BH^(delta+1); results expanded to first order in delta.
assumptions (4)
  • domain assumption Quantum extremal surface / island prescription validity for these corrected spacetimes
    Equation (1.1) is assumed to hold universally, including for metrics that are Schwarzschild with a rescaled mass.
  • domain assumption 2D s-wave approximation and CFT entanglement entropy formula (2.11)
    The 4D black hole is approximated by a 2D CFT; the entropy formula used in Eq. (2.11) is standard for 2D CFTs.
  • ad hoc to paper Temperature-entropy relation 1/T = dS/dM determines a consistent metric function
    The corrected temperature from non-extensive entropy is used to define f(r) with a matching horizon temperature; the paper does not justify that the resulting metric has ADM mass equal to the M in the entropy.
  • standard math Corrected metrics satisfy Einstein field equations
    All corrected metrics are vacuum Schwarzschild solutions with a rescaled mass parameter, so they satisfy R_mu_nu = 0; the paper verifies this and computes Kretschmann scalars.

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Cite this review

Pith. "Pith review of Effect of Non-Extensive Parameter on Page Curve." pith.science (2026). https://pith.science/paper/QNDZROWW

@misc{pith2026250200723,
  author       = {Pith},
  title        = {Pith review of: Effect of Non-Extensive Parameter on Page Curve},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QNDZROWW}},
  note         = {Machine review of arXiv:2502.00723}
}
read the original abstract

This work employs the quantum extremal surface framework to compute the Page curve for black holes corrected by non-extensive entropy. The entropy of Hawking radiation increases linearly with time, leading to the persistence of the information paradox for non-extensive entropy-corrected black holes. At late time, we extremize the generalized entropy functional; incorporating contributions from both matter and the quantum extremal island, we establish that the entanglement entropy of Hawking radiation saturates to the non-extensive extension of the Bekenstein-Hawking entropy. Finally, we study the dependence of non-extensive parameters on the Page time.

Figures

Figures reproduced from arXiv: 2502.00723 by the authors.

Figure 1
Figure 1. The Penrose diagram of Schwarchschild black hole. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Plot of Barrow metric function versus r for various values of Barrow parameters. So, in the Barrow-corrected spacetime, the singularity at r = r Barrow h is a coordinate singu￾larity, arising due to the choice of the coordinate and can be resolved through an appropriate transformation. In contrast, the singularity at r = 0 corresponds to a true curvature divergence and can be seen through the Eq. (3.3), as the Krets… view at source ↗
Figure 3
Figure 3. Plot of Sgen with and without island. 11 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Plot of the Kaniadakis metric function versus [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Plot of Sgen with and without island. by its eventual saturation to a constant value in Eq. (4.10). This behavior demonstrates the preservation of information during black hole evaporation and resolves the information paradox. 15 [PITH_FULL_IMAGE:figures/full_fig_p016…
Figure 6
Figure 6. Figure 6: Plot of the Renyi metric function versus [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Plot of Sgen with and without the island. 5.1 Page Time Again, the Page time can be determined by comparing the entropies for scenarios without an island (5.7) and with an island (5.9). This critical time corresponds to the intersection of the two entropy curves, marki…
Figure 8
Figure 8. Figure 8: Plot of Tsallis-Cirto metric function versus [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]
Figure 9
Figure 9. Figure 9: Plot of Sgen with and without the island. In conclusion, the comparison of results confirms that the island configuration emerges at 23 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]

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