Comment on "Quantum tunneling from Schwarzschild black hole in non-commutative gauge theory of gravity"

A. A. Ara\'ujo Filho; Iarley P. Lobo

REVIEW 4 major objections 5 minor 19 references

Comment on "Quantum tunneling from Schwarzschild black hole in non-commutative gauge theory of gravity"

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A recent quantum-tunneling calculation uses a wrong black-hole horizon: its horizon equation has only complex roots, so the emission-rate results are invalidated.

desk verdict The comment's horizon critique is undone by a sign error in its own Eq. (4); with the correct sign the truncated 1/g_rr has a real root near r≈2.0075, so the all-complex-roots argument collapses, though the singular-perturbation worry about the truncated horizon remains worth a referee's attention. read the letter →

arxiv 2507.22965 v1 pith:S4Q2K33N submitted 2025-07-30 gr-qc hep-th

A. A. Araújo Filho , Iarley P. Lobo This is my paper · ORCID

classification gr-qchep-th PACS 04.70.Dy04.60.-m

keywords non-commutativegaugetheoryofgravityquantumtunnelingHawkingradiationeventhorizonSchwarzschildblackholeSeiberg-WittenmapPainlevé-Gullstrandcoordinatesparticlecreation

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 comment takes issue with a recent calculation of particle creation via quantum tunneling from a Schwarzschild black hole in non-commutative gauge theory of gravity. The authors' central point is that the event horizon used in that work is not a solution of the metric the work itself writes down: solving $1/g_{rr}=0$ for the order-$\Theta^2$ metric gives, for $M=1$ and $\Theta=0.01$, ten roots and no real positive one. Since the tunneling action is evaluated by integrating around the horizon pole, an incorrect or missing real horizon changes the whole emission spectrum. The comment then supplies a corrected metric under a different twist and derives a corrected tunneling rate, while noting that the usual surface-gravity definition of temperature is not well defined in this setting. The issue matters because the horizon location is the load-bearing input for black-hole radiation predictions, and a wrong horizon invalidates the derived spectrum.

What carries the argument

The load-bearing object is the radial metric component $g_{rr}$ of the non-commutative Schwarzschild solution and the horizon condition $1/g_{rr}=0$. The argument's mechanism is simple: if the only roots of that condition are complex, the supposed real horizon at $r_h^{\mathrm{NC}}$ does not exist in the metric being used, and the contour integral for the tunneling amplitude, which is taken around the pole at the modified horizon, has no physical pole to encircle. The replacement calculation uses the $\partial_r\wedge\partial_\theta$ twist and the Seiberg-Witten map to build the corrected metric, then repeats the Painlev\'e-Gullstrand tunneling computation for the corrected radial function.

What would settle it

Compute the zeros of the untruncated or resummed inverse radial metric for $M=1$, $\Theta=0.01$. If a real root near $r=2M$ appears when the expansion is not truncated before solving, the claim that the original horizon is not a solution would fail; if no real root appears in the full expression, the comment's central objection is confirmed.

Watch

Extended reading notes

Core claim

The paper's central claim is that the event horizon in the commented work was incorrectly determined: for the order-$\Theta^2$ metric used there, the equation $1/g_{rr}=0$ has no real positive root when $M=1$ and $\Theta=0.01$; all ten solutions are complex. The real horizon radius $r_h^{\mathrm{NC}} = r_h\left(1+\frac{3}{8}(\Theta/r_h)^2\right)$ that the original paper quotes therefore does not follow from its own truncated metric. An expansion of the outermost root has leading imaginary terms $2M + \frac{3i\Theta}{4} + O(\Theta^{3/2})$, so the original truncation omitted the leading imaginary contribution and used the wrong factor for the real correction. Because the tunneling calculation integrates around the pole at the horizon, all subsequent results must be revised. The comment goes on to recompute the metric and the tunneling probability using the $\partial_r\wedge\partial_\theta$ twist, obtaining $\mathrm{Im}S = 2\pi\omega(2M-\omega) + \frac{25}{32}\pi\Theta^2[\ln M - \ln(M-\omega)]$, and observes that the same horizon error appears in the companion papers [1--4].

Load-bearing premise

The argument assumes that truncating $1/g_{rr}$ at order $\Theta^2$ and then solving for zeros is legitimate at the would-be horizon $r\approx 2M$; if the expansion is not uniform there, the absence of real roots in the truncated polynomial may be an artifact rather than a disproof of the original real horizon.

Editorial extensions

If this is right

The original tunneling rate, particle number density, and any derived temperature in the commented paper should be recomputed, because every one of those quantities is evaluated at the horizon that the comment finds to be missing or incorrect.
If the order-$\Theta^2$ configuration genuinely has no real horizon, the object may be a regular, horizonless spacetime, and the tunneling picture of Hawking radiation needs to be replaced or reinterpreted rather than merely corrected.
The corrected first-order result is $\Gamma(\omega,\Theta)\sim \exp\left[-4\pi\omega(2M-\omega)-\frac{25}{16}\pi\Theta^2(\ln M - \ln(M-\omega))\right]$, which shows the non-commutative correction suppressing particle creation as $\Theta$ grows.
Under the $\partial_r\wedge\partial_\theta$ twist, the surface gravity is not well defined, so standard geometric black-hole thermodynamics does not apply to this non-commutative background.
The same horizon error is reported in the cited companion papers [1--4], which therefore also need scrutiny.

Reading between the lines

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

I infer that the all-complex answer should be checked across a range of $M$ and $\Theta$; the single example ($M=1$, $\Theta=0.01$) is suggestive but does not by itself show the original horizon formula fails for all parameter values.
I infer that the discrepancy between the original factor $3/8$ and the corrected factor $5/32$ is a sign of twist-dependence: non-commutative corrections are not unique until the deformation prescription is fixed, so comparing predictions across papers requires comparing the chosen twist rather than just the value of $\Theta$.
I infer that if the deformed spacetime is horizonless, the quantum-tunneling method loses its defining boundary condition, and particle creation should instead be modeled as radiation from a regular compact object; testing this requires the full non-perturbative metric.

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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

Candid take: the comment's central claim does not survive contact with its own Eq. (4). You start from Eq. (2b) and want 1/g_rr up to O(Θ²). Direct inversion gives a − Θ²Ca², with a = 1 − 2M/r, plus a sign: Eq. (4) prints a plus where the algebra gives a minus. That sign is decisive. For M=1, Θ=0.01, the printed equation has no real roots; the corrected one crosses zero near r≈2.0075. So the headline assertion — that the truncated horizon equation has only complex roots and therefore Eq. (3) from Touati-Zaim is wrong — is itself supported by an internal inconsistency. The comparison in Eq. (5) inherits the same problem.

The comment does a few things well. Flagging that Touati-Zaim never justified their horizon radius is fair game; the Θ² corrections in the metric blow up at r=2M, so a truncated expansion at the horizon is a singular perturbation, and a real horizon at r = 2M[1 + (3/8)(Θ/r_h)²] does not follow automatically from the series. That point stands even after the sign fix. And the ∂r∧∂θ corrected tunneling calculation in Secs. II-III is a real piece of work — it imports the Juric-Kumara-Pozar metric, runs the standard Parikh-Wilczek machinery, and gets a concrete rate with a log correction that suppresses emission. It is new, though not deep, and the statement that surface gravity is ill-defined under that twist is worth taking seriously.

Soft spots besides the sign error: the corrected calculation quietly switches to a different non-commutative twist instead of fixing the original one, so it never resolves the complex-horizon problem it raises; the tunneling result itself treats r=2(M−ω′) as a real pole, ignoring the complexification complaint from Sec. I; and the claim that non-commutativity implies a regular black hole without a physical horizon needs an argument, not just the observation that the roots are complex.

Verdict: the letter is honest in intent and shows the authors know the tunneling machinery, but a signing error sits on the main load-bearing claim. That is exactly what a referee should catch. Send it to review, but the all-complex-roots argument and Eq. (5) need to be redone; the corrected-tunneling section can probably survive on its own.

Referee Report

4 major / 5 minor

Summary. This manuscript is a Comment on Touati and Zaim, Phys. Lett. B 848 (2024) 138335. It claims that the event horizon of the non-commutative Schwarzschild solution used in that paper was incorrectly determined: expanding 1/g_rr to second order in the non-commutativity parameter Θ is said to give only complex roots, so the real horizon radius in Eq. (3) is declared wrong and the subsequent quantum-tunneling calculation is said to be invalid. The Comment then proposes a corrected metric using a ∂r∧∂θ twist following Ref. [10], recomputes the Painlevé–Gullstrand tunneling rate, and argues that the same horizon error was repeated in Refs. [1–4].

Significance. A successful Comment of this sort would be valuable: it would correct a published result, flag a systematic error in a series of papers, and bring the more recent construction of Ref. [10] into the discussion. The manuscript is transparent in presenting the explicit root computation for M=1, Θ=0.01, which is a checkable claim, and it makes a falsifiable prediction for the corrected emission rate. However, the root computation contains a sign error, and the corrected tunneling calculation is not actually performed with the corrected horizon. As it stands, the paper does not establish its central claims; the material is potentially salvageable in revision.

major comments (4)

[§I, Eqs. (2b)–(4)] The small-Θ expansion of 1/g_rr in Eq. (4) has the wrong sign. Writing Eq. (2b) as g_rr = a^{-1} + Θ² C with a=1−2M/r and C=M(12M²+Mr(S−14)−r²(5+S))/(8r²(2M−r)³), S=√(1−2M/r), the reciprocal is a − Θ² C a² + O(Θ⁴). Using a²=(r−2M)²/r² and (2M−r)³=−(r−2M)³ gives a + Θ² M P/(8r⁴(r−2M)), where P=12M²−14Mr−5r²+r(M−r)S. Equation (4), with denominator (2M−r), equals a − Θ² M P/(8r⁴(r−2M)). For M=1, Θ=0.01 the correct sign gives a real positive root near r≈2.0075; the ten complex roots listed in §I are an artifact of the sign error. The claim that the 'mere presence of these complex solutions' disproves Eq. (3) is therefore not valid.
[§I, Eq. (4) and following] Even with the sign corrected, the truncated equation cannot support the conclusion drawn. The Θ² coefficient of 1/g_rr diverges as (r−2M)^{-1} at r=2M (and the coefficient in Eq. (2b) as (r−2M)^{-3}), so the expansion in Θ² is not uniform in a neighbourhood of the horizon. The corrected root at r−2M≈0.75Θ lies in the regime where the Θ² term and the leading term are of the same order, and the O(Θ⁴) terms are not controlled. Thus the root of the truncated polynomial is not a reliable prediction of the exact horizon, and the statements in §I about the absence of a real horizon, or about a 'regular black hole lacking a physical event horizon', are unsupported.
[§III, Eqs. (22)–(25)] The corrected tunneling calculation is not consistent with the stated pole choice. In Eq. (24) both Θ² terms contain a factor 1/(r−2(M−ω′)), and they are multiplied by 1/(1−√(2(M−ω′)/r))², which behaves like 16(M−ω′)²/(r−2(M−ω′))² near r=2(M−ω′). The Θ² integrand therefore has a triple pole at the Schwarzschild pole and zero residue there. The logarithmic correction in Eq. (25) cannot arise from contour integration around r=2(M−ω′); a nonzero Θ² contribution would require locating the Θ-dependent pole of the full integrand, which is not done. Consequently the emission rate (26) and density (27) do not follow from the calculation as presented.
[§II, Eq. (19)] The relationship between the horizon critique and the 'corrected solution' is not established. If Eq. (19) is the metric that should replace Eq. (2b), then the event horizon should be determined from the corrected g_rr, and the demonstration that Eq. (3) is wrong should be carried out for that metric; the paper instead analyses a truncated polynomial obtained from the old metric (2b). The comment also states that the same problem was repeated in Refs. [1–4], but no explicit check of those papers' horizon equations is provided. These gaps leave the scope of the claimed correction unclear.

minor comments (5)

[§I, Eq. (5)] The expansion for r10 contains terms of order Θ, Θ^{3/2}, and Θ²; the fractional power is not explained and appears to be an artifact of expanding a complex root of Eq. (4).
[Abstract] The phrase 'the same issue have been repeated' needs grammatical correction, and the claim that the issue was repeated in Refs. [1–4] is not demonstrated in the body beyond the invalid §I argument.
[References] References [14] and [16] are the same paper (Chaichian, Tureanu, and Zet, Phys. Lett. B 660 (2008) 573) and should not be listed twice.
[Figure 1] The caption does not specify the values of M and Θ used for each curve, and the axes are not visibly labelled in the printed figure.
[§III, Eq. (22)] The notation in the Painlevé–Gullstrand form conflates g(r) with g_rr(r,Θ); the definitions of f, h, and g should be stated explicitly before Eq. (22).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the paper's central horizon critique is derived from the target paper's own metric equation, and the self-citations are methodological or support only the separate corrected calculation.

full rationale

The paper's primary claim — that Touati and Zaim's Eq. (3) horizon radius is incorrect — is generated from the target's own metric component Eq. (2b) by computing 1/g_rr and examining its roots (Sec. I). This is a direct use of the target's input, not a quantity defined in terms of the comment's conclusion. The corrected calculation in Secs. II–III does rely on the framework of Refs. [10,11–15] and on the metric (19) attributed to the authors' companion paper [17], but those citations support the alternative tunneling computation, not the main negative claim. The authors' own Refs. [7–9] are invoked only as methodological precedents for the tunneling formalism and are not load-bearing. The apparent sign error in Eq. (4) identified by external scrutiny would be a computational correctness problem, not a circular reduction: the logic still runs from the target's equations to the comment's conclusion. Applying the hard rules, there is no exhibited step where a prediction is equivalent to its input by definition or where a fitted parameter is renamed as a prediction. The central critique is therefore independent of the comment's own prior work, and the self-citations do not raise the circularity score above the negligible range.

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

No free parameters are fitted and no new entities are introduced. The analysis relies on the original quoted metric, the validity of a small-Theta expansion near the horizon, the corrected metric from Ref [10] with a partial_r wedge partial_theta twist, and the standard Parikh-Wilczek tunneling method.

assumptions (4)

domain assumption The metric components (2a)-(2d), quoted from Ref [5], are the correct starting point for the horizon analysis.
The comment's root computation is only meaningful if the quoted metric is what the original paper actually used.
domain assumption The order-Theta-squared truncated expression for 1/g_rr can be used to determine the event horizon.
The series expansion is taken at fixed small Theta and evaluated at r approximately 2M despite singular terms; this is the weakest premise.
ad hoc to paper The metric (19) from Ref [10], built with a partial_r wedge partial_theta twist, is the appropriate corrected non-commutative Schwarzschild metric.
The paper imports this metric to replace the original calculation, but the original used a different twist.
domain assumption The Parikh-Wilczek tunneling method applies: the imaginary part of the action for a massless particle is computed via contour integration around the shifted pole r=2(M-omega').
The emission rate formula Eq. (26) depends on this standard method and on energy conservation M to M-omega'.

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Cite this review

Pith. "Pith review of Comment on "Quantum tunneling from Schwarzschild black hole in non-commutative gauge theory of gravity"." pith.science (2026). https://pith.science/paper/S4Q2K33N

@misc{pith2026250722965,
  author       = {Pith},
  title        = {Pith review of: Comment on "Quantum tunneling from Schwarzschild black hole in non-commutative gauge theory of gravity"},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S4Q2K33N}},
  note         = {Machine review of arXiv:2507.22965}
}

read the original abstract

The particle creation via quantum tunneling was recently calculated for the Schwarzschild non-commutative black hole solution in Ref. [Phys. Lett. B 848 (2024) 138335, e-Print: 2310.02445 [gr-qc]]. Nevertheless, it contains inconsistencies in the calculations that need to be properly corrected. In particular, the event horizon was incorrectly determined in that work, which affected all the subsequent calculations. Moreover, the same issue have been repeated elsewhere by the same authors in Refs. [1-4].

Figures

Figures reproduced from arXiv: 2507.22965 by the authors.

**Figure 1.** The particle creation density n(ω, Θ) is plotted as a function of the frequency ω, with curves corresponding to distinct values of the non–commutative parameter Θ. In addition, the particle number density can be expressed through the tunneling rate as follows: n(ω, Θ) = Γ(ω, Θ) 1 − Γ(ω, Θ) = 1 exp 25 16πΘ2 [ln M − ln(M − ω)] + 4πω(2M − ω) [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [10]

Constructing noncommutative black holes,

T. Juri´ c, A. N. Kumara, and F. Poˇ zar, “Constructing noncommutative black holes,” Nucl. Phys. B, vol. 1017, p. 116950, 2025. 10

work page 2025

[2] [1]

Schwarzschild black hole surrounded by a cavity and phase transition in the non-commutative gauge theory of gravity,

A. Touati and S. Zaim, “Schwarzschild black hole surrounded by a cavity and phase transition in the non-commutative gauge theory of gravity,” Astropart. Phys., vol. 161, p. 102988, 2024

work page 2024

[3] [2]

Geodesic equation in non-commutative gauge theory of gravity*,

A. Touati and S. Zaim, “Geodesic equation in non-commutative gauge theory of gravity*,” Chin. Phys. C, vol. 46, no. 10, p. 105101, 2022

work page 2022

[4] [3]

On Modified First Law of Black hole Thermodynamics in The Non- Commutative Gauge Theory,

A. Touati and S. Zaim, “On Modified First Law of Black hole Thermodynamics in The Non- Commutative Gauge Theory,” 5 2022

work page 2022

[5] [4]

Thermodynamic properties of Schwarzschild black hole in non- commutative gauge theory of gravity,

A. Touati and S. Zaim, “Thermodynamic properties of Schwarzschild black hole in non- commutative gauge theory of gravity,” Annals Phys., vol. 455, p. 169394, 2023

work page 2023

[6] [5]

Quantum tunneling from Schwarzschild black hole in non- commutative gauge theory of gravity,

A. Touati and S. Zaim, “Quantum tunneling from Schwarzschild black hole in non- commutative gauge theory of gravity,” Phys. Lett. B, vol. 848, p. 138335, 2024

work page 2024

[7] [6]

Quantum gravitational corrections to particle creation by black holes,

X. Calmet, S. D. H. Hsu, and M. Sebastianutti, “Quantum gravitational corrections to particle creation by black holes,” Phys. Lett. B, vol. 841, p. 137820, 2023

work page 2023

[8] [7]

Particle production induced by a Lorentzian non-commutative spacetime

A. A. Ara´ ujo Filho, “Particle production induced by a Lorentzian non-commutative space- time,” arXiv preprint arXiv:2502.19366, 2025

work page Pith review arXiv 2025