Massive particles tunneling from quantum Oppenheimer-Snyder black holes and black hole entropy

Hongwei Tan; Kui Xiao

REVIEW 3 major objections 5 minor 97 references

Massive particles tunneling from quantum Oppenheimer-Snyder black holes and black hole entropy

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

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Pith's one-line read A logarithmic correction to black hole entropy survives when the tunneling particle is massive.

desk verdict A workmanlike extension of the authors' massless tunneling calculation to massive scalars; the log correction is unchanged, but the key α-order integral is not shown and the entropy 'derivation' is really an ansatz. read the letter →

arxiv 2502.05910 v2 pith:VQCWSQSK submitted 2025-02-09 gr-qc hep-th

Hongwei Tan , Kui Xiao This is my paper · ORCID

classification gr-qchep-th MSC 83C5783C45 PACS 04.70.Dy04.60.Pp

keywords blackholetunnelingParikh-WilczekmethodquantumOppenheimer-SnydermassivescalarparticleslogarithmicentropycorrectionloopgravityHawkingradiation

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

Massive scalar particles tunneling out of a quantum Oppenheimer-Snyder black hole acquire a loop-quantum-gravity correction to their emission rate, and that correction translates into a logarithmic term in the black hole entropy: $S_{\rm OS}=S_{\rm Sch}+\frac{\pi\alpha}{2l_p^2}\log\frac{A_{\rm Sch}}{l_p^2}+O(\alpha^2)$. The calculation extends the massless-particle result of the authors' earlier work to massive particles, using the Parikh-Wilczek tunneling picture with a near-horizon massive-particle velocity. The paper's point is that the logarithmic correction survives the switch to massive particles, with the same positive prefactor as in the massless case and in other effective loop-quantum-black-hole results. If true, this strengthens the case that the entropy of quantum Oppenheimer-Snyder black holes follows the universal area-law-plus-logarithm pattern expected from quantum gravity.

What carries the argument

The machinery is a three-step chain. First, the quantum Oppenheimer-Snyder metric is rewritten in Painlevé-Gullstrand coordinates, $ds^2=-\left(1-\frac{2M}{r}+\frac{\alpha M^2}{r^4}\right)d\tilde t^2+2\sqrt{\frac{2M}{r}-\frac{\alpha M^2}{r^4}}\,d\tilde t\,dr+dr^2+r^2d\Omega^2$, which is regular at the horizon and carries the LQG correction. Second, the Parikh-Wilczek emission amplitude $\Gamma\sim e^{-2\,\mathrm{Im}\,A/\hbar}$ is computed with a massive-particle radial velocity $\dot r=\frac{1-\frac{2(M-\omega)}{r}+\frac{\alpha(M-\omega)^2}{r^4}}{2\sqrt{\frac{2(M-\omega)}{r}-\frac{\alpha(M-\omega)^2}{r^4}}}$ (eq. 3.15), obtained from the de Broglie s-wave relation $v_p=v_g/2$. Third, the rate is matched to $\Gamma\sim\exp(\Delta S)$ to read off the entropy; the contour deformation around the horizon pole turns the LQG correction into the logarithmic terms that survive as $\frac{\pi\alpha}{2l_p^2}\log\frac{A_{\rm Sch}}{l_p^2}$.

What would settle it

Repeat the same contour-integral calculation using the relativistic group velocity $v_g=p/E$ (or any Lorentz-invariant dispersion relation) in the Painlevé-Gullstrand q-OS metric; if the logarithmic coefficient in $\Gamma$ is not $\pi\alpha/(2l_p^2)$, the claimed entropy formula does not follow from the stated assumptions. A microstate count of the q-OS horizon producing a different prefactor would also settle the question.

Watch

Extended reading notes

Core claim

The central claim is that when a massive scalar particle of energy $\omega$ tunnels through the horizon of a quantum Oppenheimer-Snyder black hole, the emission rate is $\Gamma\sim\exp\left[-\frac{8\pi M}{l_p^2}\left(\omega-\frac{\omega^2}{2M}\right)-\frac{\pi\alpha}{2l_p^2}\left(\log\frac{A_{\rm Sch}(M)}{l_p^2}-\log\frac{A_{\rm Sch}(M-\omega)}{l_p^2}\right)+O(\alpha^2)\right]$ (eq. 3.25). Using the scheme that identifies a tunneling rate with $\exp(\Delta S)$, the paper derives the entropy $S_{\rm OS}=S_{\rm Sch}+\frac{\pi\alpha}{2l_p^2}\log\frac{A_{\rm Sch}}{l_p^2}+O(\alpha^2)$ (eq. 4.2). Compared with the classical Parikh-Wilczek result, the LQG correction term $\alpha M^2/r^4$ in the metric produces logarithmic terms in the imaginary part of the action. The authors emphasize that the prefactor of the logarithmic correction is positive, consistent with effective loop quantum black hole entropies obtained by other approaches, and they suggest the positive sign could reflect a combination of quantum-gravity and thermal-fluctuation contributions.

Load-bearing premise

The calculation assumes that a massive particle right at the horizon, where it is highly blue-shifted, still obeys the nonrelativistic de Broglie relation $v_p=v_g/2$; if a relativistic relation applies there, the prefactor in the tunneling rate and hence the entropy correction would change.

Editorial extensions

If this is right

The q-OS black hole entropy deviates from the Bekenstein-Hawking area law by a positive logarithmic term; for large horizon area the correction is small but grows as $\log A_{\rm Sch}$.
Massive and massless scalar tunneling give the same form of entropy correction, so the logarithmic term appears to be insensitive to the mass of the emitted particle.
Because the emission rate has the form $\Gamma\sim\exp(\Delta S_{\rm OS})$, the tunneling picture stays consistent with unitary evaporation for this effective LQG black hole.
The positive prefactor distinguishes this result from ordinary LQG state-counting entropies (where the prefactor is negative) and agrees with other effective loop quantum black hole calculations, pointing to a universal sign for effective LQG entropy corrections.

Reading between the lines

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

If the near-horizon massive particle is treated with a relativistic dispersion relation instead of the nonrelativistic $v_p=v_g/2$ rule, the prefactor of the logarithmic term would likely change; the paper's positive $\pi\alpha/(2l_p^2)$ coefficient is not guaranteed by that choice unless a relativistic derivation recovers it.
The same tunneling calculation could be repeated for charged or rotating generalizations of the q-OS metric to test whether the logarithmic coefficient depends on angular momentum or charge.
The entropy formula implies a modified emission temperature through $T=(\partial S/\partial M)^{-1}$; comparing quasinormal-mode or shadow predictions of the q-OS model with those of classical Schwarzschild would provide an observational window on the sign of the correction.

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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

The genuinely new item is the massive-scalar tunneling rate from the q-OS metric, eq. (3.25). Nobody had written that down before; the earlier paper treated massless particles. That is a real technical extension, though the paper's own equations show it lands on the same logarithmic entropy correction, so it is a consistency check within an established program rather than a new phenomenon.

What the paper does well: the Painlevé-Gullstrand setup is clean, the WKB tunneling framework is handled in the standard way, and the back-reaction M→M−ω is carried through consistently. The final emission rate has the expected Parikh–Wilczek form plus a log term, and the sign of the log prefactor is consistent with some effective LQG results. The stress-test note makes a good point that the pole at the shifted horizon may make the π/2 log coefficient robust against changes in the dispersion relation, since any velocity of the form f/(a+b√g) with a+b=2 gives the same residue.

The soft spots are real. The step from (3.22) to (3.23) is the heart of the α-order calculation, and the coefficients 3π/16 and 5π/16 appear without any residue computation shown. That is not a minor omission—the entire new content depends on it. Second, the velocity relation v_p = v_g/2 comes from a nonrelativistic de Broglie argument in ref. [82]; near the horizon the mode is strongly blue-shifted, so the applicability is not obvious. The stress-test note argues the log coefficient may be robust anyway, but that is an external argument, not something the paper demonstrates. Third, the entropy result in section 4 is best read as an ansatz: they define S_OS so that Γ = exp(ΔS_OS). That is a legitimate way to package the emission rate, but calling it a derivation overstates what was computed. The discussion of the positive prefactor and thermal fluctuations is honest and appropriately cautious.

There are also numerous typos and awkward sentences, but nothing that obscures the logic.

Bottom line: this is a modest, incremental paper that would be a useful check for someone working on LQG-corrected black hole thermodynamics. It does not resolve an open problem, and the central integral evaluation needs to be checked by a referee who can do the contour computation explicitly. I would send it to review rather than desk-reject, but I would not cite it in my own work unless I were specifically tracking q-OS tunneling extensions.

Referee Report

3 major / 5 minor

Summary. The manuscript computes the tunneling rate of massive scalar particles from the quantum Oppenheimer-Snyder (q-OS) black hole, whose exterior metric contains the LQG-motivated correction αM^2/r^4. Using the Parikh-Wilczek approach together with the near-horizon de Broglie wave description for massive particles proposed in [82], the authors obtain the emission rate (3.25), which contains the standard Schwarzschild term plus an O(α) logarithmic term. They then invoke the scheme of [1,2] to convert this rate into a black hole entropy formula, S_OS = S_Sch + (πα/(2l_p^2)) log(A_Sch/l_p^2) + O(α^2) (Eq. 4.2), and compare the sign of the logarithmic prefactor with existing LQG results.

Significance. If the calculation is correct, it extends the massless-particle result of [67] to massive scalar particles and provides a concrete, model-dependent prediction for the logarithmic correction to black hole entropy in a LQG-corrected spacetime. The final rate has the expected Parikh-Wilczek/Boltzmann form, and the computation is checkable: although the paper omits the residue evaluation, the quoted α-order coefficients 3π/16 and 5π/16 are consistent with a direct Laurent expansion around the shifted horizon. The entropy formula, however, is inferred from the emission rate rather than independently derived, and the near-horizon velocity relation underlying Eq. (3.15) needs justification. These issues are fixable. The paper is a useful incremental contribution, not a paradigm shift.

major comments (3)

[§3.3, Eqs. (3.22)-(3.23)] The transition from the α-order double integrals in (3.22) to the result (3.23) is not shown. The coefficients 3π/16 and 5π/16 are load-bearing because they combine to produce the π/2 prefactor of the logarithmic term in (3.24), which is the main new result. Please provide the full residue computation for each of the two α-terms, including the iϵ prescription and the Laurent expansion around u = √(2(M-ω')). Without this, an independent reader cannot verify the central claim.
[§4, Eqs. (4.1)-(4.3)] Eq. (4.2) is introduced specifically so that Γ = exp(ΔS_OS) holds; the logarithmic term in S_OS exactly mirrors the logarithmic term in the computed emission rate (3.25). Thus the 'derivation' of the entropy is a relabeling of the emission-rate result rather than an independent computation. Please reframe Section 4 as an inference or consistency check under the scheme of [1,2], and state clearly what physical input, if any, beyond the tunneling rate is needed to establish S_OS.
[§3.2, Eqs. (3.7)-(3.13)] The radial velocity ṙ = f/(2√g) in (3.13) rests on the nonrelativistic relation v_p = v_g/2 for a de Broglie s-wave (3.8). A particle emerging from the near-horizon region is highly blue-shifted, so the validity of this relation is not obvious. Please either provide a relativistic derivation of ṙ or demonstrate that the O(α) logarithmic coefficient is insensitive to the precise velocity-dispersion relation, e.g., by showing that any ṙ = f/(a+b√g) with a+b=2 gives the same pole contribution. This is a load-bearing assumption for the emission rate.

minor comments (5)

[§3.3, Eqs. (3.19)-(3.21)] The integration limits are displayed as 'Z rout rin' and 'Z uout uin' with inconsistent ordering. Please define r_in, r_out, and the orientation of each integral unambiguously.
[Throughout] There are numerous typos and grammatical errors, including 'Oppenheimer-Snyde', 'introdoce', 'emmison', 'ocours', 'countour', 'riginating', and 'Concret examples'. A careful copyedit is needed.
[§3.3, Eq. (3.24)] The assertion that 'M≫ω' is needed for the log term is confusing: the evaluation log(M/(M-ω)) is exact for any M>ω. Please remove or clarify this note.
[§4, last paragraph] The decomposition of the logarithmic prefactor into a_q + a_t = +πα/(2l_p^2) is speculative and not derived. If kept, it should be explicitly labeled as a conjecture.
[§3.3, Eq. (3.22)] The sentence containing 'he positive energy solutions' should read 'the positive energy solutions'.

Circularity Check

1 steps flagged · score 6.0 of 10

The emission-rate computation is self-contained, but the black-hole entropy result is introduced rather than derived: S_OS is defined so that exp(ΔS_OS) reproduces the already-computed rate, making the logarithmic entropy correction a relabeling of the logarithmic term in the emission rate.

self definitional [Section 4, eqs. (4.1)-(4.3)]
"To express the emission rate (3.25) in the formalism of (4.1), we introdoce entropy of the q-OS BH as S_OS = S_Sch + πα/(2l_p^2) log(A_Sch/l_p^2) + O(α^2) ... Finally, the emission rate reads Γ ∼ exp ΔS_OS."

The entropy difference ΔS_OS is not computed independently; S_OS is chosen so that exp(ΔS_OS) exactly reproduces the already-computed emission rate (3.25). Therefore the logarithmic term in (4.2) is identical, by construction, to the logarithmic term already present in the emission rate (3.25). The paper's claim of 'deriving' the black hole entropy is thus a relabeling of the emission-rate result, not a separate first-principles derivation.

full rationale

The central tunneling calculation is self-contained: given the q-OS metric (2.5) and the massive-particle equation of motion (3.15), the emission rate (3.25) is obtained by a direct WKB action computation. The near-horizon velocity relation v_p = v_g/2 (3.8) is a physical approximation rather than a circular input, and it does not affect the pole residue because near the horizon any rdot of the form f/(a+b√g) with a+b=2 gives the same imaginary action. The paper's only substantial circularity is in Section 4, where the entropy is defined to match the computed rate, so the logarithmic entropy correction is not an independent result but a restatement of the emission-rate log term. Overall, the emission-rate claim has independent content, but the headline entropy claim reduces by construction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the effective LQG metric (2.5) imported from ref. [55], the massive-particle kinematics imported from ref. [82], and the entropy ansatz Γ = exp(ΔS) from refs. [1,2]. No free parameters are fitted to data; α is a theory parameter. No new entities are introduced.

free parameters (1)

α (LQG metric correction) = 16√3 π γ^3 l_p^2
Enters the metric (2.5) and controls the size of the logarithmic correction; its numerical value is set by the Barbero-Immirzi parameter γ and is not fitted to data in the paper.

assumptions (5)

domain assumption The q-OS exterior metric (2.5) is the correct effective LQG spacetime obtained by Israel junction conditions with the LQG-deformed Friedmann equation.
Invoked in Sec. 2, eqs. (2.3)-(2.5); taken from ref. [55].
domain assumption The generalized Birkhoff theorem holds for the k=0 q-OS exterior, so the metric is unique and no gravitational radiation is emitted during tunneling.
Invoked in Sec. 3.3 before eq. (3.14), citing ref. [87].
domain assumption Massive particle de Broglie wave obeys the nonrelativistic relation v_p = v_g/2 near the horizon.
Eq. (3.8) in Sec. 3.2, taken from ref. [82]; load-bearing for the equation of motion (3.13).
domain assumption The emission rate can be written as Γ = exp(ΔS) with S the black hole entropy, following refs. [1,2].
Sec. 4, eq. (4.1); this is the scheme that converts the emission rate into an entropy formula.
standard math WKB approximation and geometric optics are valid for the tunneling process.
Used in Secs. 3.2-3.3, standard in the Parikh-Wilczek approach.

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Cite this review

Pith. "Pith review of Massive particles tunneling from quantum Oppenheimer-Snyder black holes and black hole entropy." pith.science (2026). https://pith.science/paper/VQCWSQSK

@misc{pith2026250205910,
  author       = {Pith},
  title        = {Pith review of: Massive particles tunneling from quantum Oppenheimer-Snyder black holes and black hole entropy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VQCWSQSK}},
  note         = {Machine review of arXiv:2502.05910}
}

read the original abstract

In this paper, we investigate the tunneling process of massive scalar particles from a quantum Oppenheimer-Snyder black hole using the tunneling approach proposed by Parikh and Wilczek. We compute the emission rate of these tunneling particles, which includes quantum correction terms compared to the result in classical General Relativity. These correction terms arise from loop quantum gravity effects. Following the scheme outlined in [1, 2], we derive the entropy of the black hole. Consistent with the universal black hole entropy formula in the context of quantum gravity, our findings include a logarithmic correction.

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