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Evolution of interior entropy of a loop quantum-corrected black hole pierced by a cosmic string

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For a large-mass loop quantum-corrected black hole pierced by an infinitely straight cosmic string, the paper derives the evolution relation between interior entropy and Bekenstein–Hawking entropy and shows that the total variation obeys…

desk verdict A competent extension of the interior-entropy recipe to an LQG black hole with a cosmic string; the headline α-dependent relation rests on an unproven area-law assumption that conflicts with the paper's own first law. read the letter →

arxiv 2501.00422 v2 pith:4GVW22GD submitted 2024-12-31 gr-qc

Wei Zhang This is my paper · ORCID
classification gr-qc MSC 83C5783C45 PACS 04.70.Dy04.60.Pp11.27.+d
keywords interiorentropyBekenstein-Hawkingloopquantum-correctedblackholecosmicstringHawkingradiationsecondlawofthermodynamicsinformationvolume
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 asks whether a cosmic string threading a black hole changes how the hole's interior entropy grows as the hole evaporates. For a large-mass loop quantum-corrected black hole pierced by an infinitely straight cosmic string, it derives the evolution relation $$dS_{\Sigma'} = -\frac{\$pi^{2}$ M r_+ \sqrt{$27M^{2}$-16\$\alpha$}}{180\,\$\sigma$(1-4\mu)(r_+^3-\$\alpha$ M)}\,dS_{\mathrm{BH}},$$ where $\mu$ is the string tension, $\alpha$ is the loop-quantum correction parameter, and $\sigma$ is the Stefan–Boltzmann constant. The paper shows the total variation $S_1=S_{\Sigma'}+S_{\mathrm{BH}}$ increases with advanced time, so the second law of thermodynamics is preserved during evaporation. The result matters because interior entropy is a candidate for encoding the information that the shrinking horizon appears to lose, and showing that a topology-defect string does not spoil that encoding extends the idea to a less idealized black hole.

What carries the argument

The central object is the interior volume $V_{\Sigma'}$ of the black hole, computed from the advanced Eddington–Finkelstein metric (12) by taking the maximal three-dimensional spacelike hypersurface inside the horizon, located at $r_v=3M/2$. The cosmic string enters as the angular factor $(1-4\mu)$ from the deficit angle $\delta=8\pi\mu$. Interior entropy is taken to be $S_{\Sigma'}=\pi^2 T_H^3 V_{\Sigma'}/45$, with the Hawking temperature $T_H$ obtained from surface gravity, and the evolution relation is assembled by combining the volume growth rate with the Stefan–Boltzmann law $dM/dv=-\sigma T_H^4 A$, the mass–radius relation (22), and the area-law horizon entropy change $dS_{\mathrm{BH}}=2\pi(1-4\mu)r_+dr_+$.

What would settle it

Compute the horizon entropy from the first law $dM=T_H\,dS$ using the paper's own temperature (16) and mass–radius relation (22); if the resulting $dS$ is not equal to $2\pi(1-4\mu)r_+dr_+$ unless $\alpha=0$, then the central evolution relation (26) fails its own consistency check.

Watch

Extended reading notes

Core claim

The paper's central claim is that for a large-mass loop quantum-corrected black hole with an infinitely straight cosmic string, the interior entropy and the Bekenstein–Hawking entropy are tied by the differential relation (26) during Hawking radiation. The relation has a negative coefficient, so as the horizon loses entropy ($dS_{\mathrm{BH}}<0$) the interior entropy increases, and the sum $S_1=S_{\Sigma'}+S_{\mathrm{BH}}$ satisfies $\dot S_1>0$, meaning the second law holds. In the double limit $\alpha\to0$ and $\mu\to0$, the relation reduces to the known Schwarzschild result $dS_{\Sigma'}=-\sqrt{3}\,\pi^2/(240\sigma)\,dS_{\mathrm{BH}}$. The cosmic string enters the relation through the factor $(1-4\mu)$ in the horizon area and, after the Stefan–Boltzmann law is used, leaves a residual dependence on $\mu$ through that same factor in the denominator, so the tension genuinely modifies the entropy evolution rather than merely rescaling the total area.

Load-bearing premise

The derivation assumes the classical Bekenstein–Hawking area law $S_{\mathrm{BH}}=A/4$ still holds for the loop quantum-corrected black hole with a cosmic string, a property borrowed by analogy from a Hayward-black-hole calculation and not derived from the metric itself.

Editorial extensions

If this is right

  • For a large-mass loop quantum-corrected black hole, the cosmic-string tension $\mu$ and the quantum-gravity parameter $\alpha$ both enter the coefficient relating interior entropy to horizon entropy, so the string changes the quantitative evolution law.
  • The total entropy $S_1=S_{\Sigma'}+S_{\mathrm{BH}}$ increases throughout Hawking evaporation, so the second law of thermodynamics is satisfied in this model.
  • In the limits $\alpha\to0$ and $\mu\to0$, the relation reduces to the Schwarzschild result $dS_{\Sigma'}=-\sqrt{3}\pi^2/(240\sigma)\,dS_{\mathrm{BH}}$, recovering earlier work as a special case.
  • The derivation is restricted to large masses, where interior and horizon are in thermal equilibrium; for small masses the temperature identification breaks down and the relation is not established.

Reading between the lines

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

  • A direct consistency test is to derive the horizon entropy from the first law $dM=T_H\,dS$ using the paper's own temperature (16) and mass–radius relation (22); if the result differs from the area-law expression $2\pi(1-4\mu)r_+dr_+$ unless $\alpha=0$, the central relation (26) would need modification.
  • Within the model, the leftover $1/(1-4\mu)$ in the coefficient means the string's effect is not just a common rescaling of both entropies: two holes with the same mass and $\alpha$ but different tensions should show different ratios of interior-to-horizon entropy change.
  • The same volume-entropy machinery could be applied to small masses by introducing a non-equilibrium interior temperature, which would allow the evolution to be followed through the Planck-scale remnant stage.
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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

2 major / 5 minor

Summary. This paper studies a loop quantum-corrected Schwarzschild black hole pierced by an infinitely straight cosmic string. Using the Christodoulou-Rovelli interior volume formula and assuming a massless scalar field interior entropy proportional to T_H^3 V, it derives the evolution relation between interior entropy and Bekenstein-Hawking entropy during Hawking radiation, Eq. (26), and the total variation whose derivative with respect to advanced time is claimed positive, Eq. (29). The calculations from Eq. (14) through Eq. (23) are internally consistent and the Schwarzschild limits are recovered. The load-bearing input is the classical area law S_BH = A/4 for the quantum-corrected string spacetime, introduced between Eqs. (24) and (25).

Significance. If Eq. (26) were correct, the paper would provide a concrete example of a cosmic string changing the relative rate of interior-entropy growth versus horizon-entropy loss while preserving the second law. The paper is transparent, uses no fitted parameters, and correctly reproduces known Schwarzschild and mu = 0 limits. However, the claimed novelty is the (1 - 4 mu) dependence in Eq. (26), and this dependence disappears when the Stefan-Boltzmann law is written consistently with the first law for the energy E = (1 - 4 mu) M. The paper's intended result is therefore not established.

major comments (2)
  1. [Section III, Eq. (24)-(25)] The area-law input is inconsistent with the first-law framework stated in the same section. The text says that for an asymptotic observer the energy E is not equal to the metric mass M and uses dS_BH = dE/T_H. The standard conical-spacetime identification that makes this reduce to S_BH = A/4 in the alpha = 0 limit is E = (1 - 4 mu) M. With that identification, Eqs. (16) and (22) give dS_BH = dE/T_H = 2 pi (1 - 4 mu) r_+^4 / (r_+^3 - alpha M) dr_+, which differs from Eq. (25) by the factor r_+^3 / (r_+^3 - alpha M) = 1 + O(alpha M / r_+^3). Since Eq. (26) is obtained by dividing Eq. (23) by Eq. (25), the alpha-dependent coefficient in Eq. (26) is not established unless a consistent entropy is derived from the first law of metric (11) or the result is explicitly truncated to leading order in alpha. The Hayward analogy, reference [58], is for a different metric and does not close this gap.
  2. [Section III, Eq. (20) and Eq. (24)] The Stefan-Boltzmann law is written for the metric mass M, but the first law is written for the energy E. If E = (1 - 4 mu) M, energy conservation with luminosity sigma T_H^4 A and horizon area A = (1 - 4 mu) 4 pi r_+^2 gives (1 - 4 mu) dM/dv = -sigma T_H^4 A, i.e., dM/dv = -sigma T_H^4 4 pi r_+^2, not Eq. (20). Combining this consistent form with Eq. (19) yields dS_Sigma' = -[pi^2 M r_+ sqrt(27 M^2 - 16 alpha) / (180 sigma (r_+^3 - alpha M))] dS_BH, which contains no (1 - 4 mu)^{-1} factor. Thus the cosmic-string dependence advertised in the abstract vanishes under the standard identification. If instead M in Eq. (20) is intended to be the energy, then Eq. (24) should be dS_BH = dM/T_H and the area law would not hold. In either case, Eqs. (20) and (24)-(25) are mutually inconsistent.
minor comments (5)
  1. [Section III, Eq. (14)] The statement that the maximal hypersurface is at r_v = 3M(v)/2 is asserted without derivation; one line showing that d[-r^4 f(r)]/dr = 0 gives r = 3M/2 would help the reader.
  2. [Section III, Eq. (17)] The rescaled temperature tilde T_H is not defined; please define it explicitly and state that it is dimensionless.
  3. [Section III, Eq. (24)] The energy E is never explicitly defined; the standard expression E = (1 - 4 mu) M should be stated before Eq. (24), since the first law depends on it.
  4. [Abstract] The large-mass restriction should appear in the abstract, because the derivation of Eq. (19) relies on quasi-static evaporation and thermal equilibrium between the interior and the horizon.
  5. [General] There are grammatical errors and ambiguous notations (e.g., 'The obtained results shown that', 'It is still unknown that how do cosmic strings influence', and the dual use of alpha as a constant and as a function through Eq. (4)); an editorial pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation combines independently sourced formulas and recovers known limits; the area-law concern is a consistency issue, not a circularity.

full rationale

The paper's derivation chain is self-contained in the sense that no target result is embedded in its inputs. The interior volume formula (14) is obtained by applying the Christodoulou-Rovelli maximal-volume prescription to the loop-quantum-corrected cosmic-string metric; the interior entropy formula is imported from ref. [2]; the Hawking temperature (16) is computed from the metric by surface gravity; the Stefan-Boltzmann law (20) is an external physical input; and the Bekenstein-Hawking entropy-area law (25) is imported from classical general relativity and from ref. [58]. Equation (26) is then obtained by algebraically combining these ingredients, dividing dS_Sigma' by dS_BH. No parameter is fitted to the quantity being predicted, and no step defines the output in terms of itself. The limiting cases (27) and (28) are external anchors, not inputs that force the general result. The paper also explicitly declares that it ignores LQG-modified entropy formulas, so there is no hidden reuse of a modified entropy as if it were a prediction. The area-law concern raised by the skeptic is a physical-consistency objection: using dE/T_H with E = (1-4mu)M and the paper's own temperature and mass-radius relations gives a different dS_BH than the assumed area law. That is a correctness risk about whether the imported area law is compatible with the effective metric, but it is not circularity, because the paper does not derive Eq. (25) from Eq. (24) and does not use the first law to obtain the relation it then calls a prediction. No self-citation is load-bearing, no uniqueness theorem is imported from the author's prior work, and no fitted input is renamed as a prediction. The appropriate circularity score is therefore 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No numbers are fitted in this paper. The result depends on physical inputs (α from loop quantum gravity, μ the cosmic string tension) and on imported thermodynamic formulas (interior entropy, area law, Stefan-Boltzmann). The most consequential assumption is the classical area law S=A/4 for a spacetime where the first law is not satisfied; it enters at Eq (25) and propagates to the central relation (26).

assumptions (6)
  • domain assumption Interior entropy is proportional to the interior volume: SΣ = π² T³ V/45 (Eq 2).
    Imported from ref [2]; assumed to hold for the massless scalar field in the conical black hole interior. This is the foundation of the interior entropy concept but is not derived in the paper.
  • domain assumption Interior temperature of the black hole equals the Hawking temperature during evaporation for large mass.
    Stated in Section III after Eq (17); required to use Eq (2) with TH and to identify dT ~ 0. Breaks down for small mass, a limitation the paper acknowledges.
  • domain assumption Quasi-static approximation: over an infinitesimal time interval the black hole mass can be treated as constant and dTH ~ 0.
    Adopted from ref [26], used in Eqs (18)-(19) to write V approximately equal to a constant times v. This is the leading-order approximation that makes the derivation tractable.
  • ad hoc to paper The classical Bekenstein-Hawking entropy-area law SBH = A/4 remains valid for the loop quantum-corrected black hole with cosmic string.
    The paper asserts this based on an analogy with Hayward black holes (refs [56,58]) and explicitly ignores LQG-modified entropy formulas (refs [49-52]). It is not verified against the first law for this metric.
  • domain assumption Mass loss of the black hole follows the Stefan-Boltzmann law dM/dv = -σ TH⁴ A (Eq 20) with a constant σ ~ 10^-5.
    Standard QFT result used as input; graybody factors are absorbed into σ. The area A includes the cosmic string deficit angle factor (1 - 4μ).
  • domain assumption The background spacetimes: the loop quantum-corrected Schwarzschild metric of ref [31] and the cosmic string piercing along the symmetry axis with deficit angle δ = 8πμ (Eqs 3, 11, 12).
    These metrics are the starting point; the paper does not derive them.

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Cite this review

Pith. "Pith review of Evolution of interior entropy of a loop quantum-corrected black hole pierced by a cosmic string." pith.science (2026). https://pith.science/paper/4GVW22GD

@misc{pith2026250100422,
  author       = {Pith},
  title        = {Pith review of: Evolution of interior entropy of a loop quantum-corrected black hole pierced by a cosmic string},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4GVW22GD}},
  note         = {Machine review of arXiv:2501.00422}
}
read the original abstract

The concept of interior entropy of black hole provides a possible way to deal with information problem. We derive evolution relation between interior entropy and Bekenstein-Hawking entropy for a loop quantum-corrected black hole with large mass pierced by an infinitely straight cosmic string during Hawking radiation. We find that the present cosmic string can influence the evolution relation between these two types of entropy under black hole evaporation and the second law of thermodynamics is satisfied for the total variation of these two types of entropy.

Figures

Figures reproduced from arXiv: 2501.00422 by the authors.

Figure 1
Figure 1. FIG. 1: Outer horizon [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

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Reviewed August 10, 2026 · model on record in the stance chip above.