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REVIEW 3 major objections 2 minor 1 cited by

Greybody Factor for massive scalar field in charged black hole

T0 review · 3 major / 2 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Massive scalar fields tunnel out of Reissner-Nordström black holes less readily than massless ones.

desk verdict A modest but competent application of WKB and bound methods to massive scalars in Reissner-Nordström; the monotonicity claim is plausible but the abstract alone doesn't prove it. read the letter →

arxiv 2501.02247 v2 pith:CYJ3FZFR submitted 2025-01-04 gr-qc

Ratchaphat Nakarachinda , Suppawit Polkong , Pitayuth Wongjun This is my paper · ORCID
classification gr-qc MSC 83C5783C4781Q20 PACS 04.70.Dy04.62.+v03.65.Sq
keywords greybodyfactormassivescalarfieldReissner-NordströmblackholeWKBapproximationrigorousboundtransmissionprobabilityeffectivepotentialHawkingradiation
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 sets out to determine how the mass of a scalar field changes its chance of escaping a charged black hole, as measured by the greybody factor. Working in Reissner-Nordström spacetime, the authors derive the transmission probability through the effective potential barrier using the WKB approximation and a rigorous bound method. They find that the two methods agree: the greybody factor decreases as the scalar field mass increases. The physical mechanism is quantum-mechanical: a more massive field couples more strongly to the potential barrier and therefore tunnels through less easily. The rigorous bound is analytic and covers a broader parameter range than the standard WKB expansion.

What carries the argument

The argument runs through the Klein-Gordon equation for a massive scalar field in the Reissner-Nordström background, separated into radial modes that obey a Schrödinger-like equation with an effective potential $V_{\text{eff}}(r)$ depending on the angular momentum, the black hole charge $Q$, and the field mass $m$. The greybody factor is the transmission probability of this potential barrier. The WKB approximation estimates that transmission using phase integrals across the turning points, while the rigorous bound method supplies an analytic estimate that is valid over a broader parameter regime; both methods point to the same inverse mass dependence.

What would settle it

Compute the greybody factor by direct numerical solution of the massive Klein-Gordon equation in Reissner-Nordström spacetime at fixed charge and frequency for two field masses $m_1 < m_2$; if the transmission probability for $m_2$ is not smaller than that for $m_1$ for any such pair, the paper's central claim is wrong. The same test can be run with the paper's own effective potential in a one-dimensional Schrödinger equation.

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Extended reading notes

Core claim

The central claim is that for a massive scalar field in Reissner-Nordström black hole spacetime, the greybody factor is inversely related to the field mass: at fixed charge and frequency, a heavier scalar field has a lower transmission probability through the effective potential. The paper also claims that this behaviour is governed by the height of the potential barrier, so the greybody factor and the potential are directly related in the sense that a higher potential produces a lower greybody factor. Both the WKB approximation and the analytic rigorous bound are reported to reach the same conclusion, with the bound method valid for a wider range of parameters.

Load-bearing premise

Both estimation methods are trustworthy in the parameter range considered, and the effective potential captures the complete barrier a massive scalar field must cross.

Editorial extensions

If this is right

  • Massive fields are suppressed relative to massless fields in the Hawking radiation spectrum of a charged black hole, so the greybody factor directly shapes which particle masses are most likely to be observed.
  • The greybody factor decreases as the height of the effective potential increases, implying that modes with larger angular momentum, which see a higher barrier, should also be more strongly suppressed.
  • The analytic rigorous bound can be applied beyond the regime where WKB is reliable, giving a simple formula for lower-bound transmission in Reissner-Nordström black holes.
  • The mass dependence of the greybody factor, if confirmed, could be used to infer properties such as the charge of the black hole from the relative escape rates of scalar fields with different masses.

Reading between the lines

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

  • If the inverse-mass relationship carries over to other black hole geometries, the ratio of greybody factors for two scalar masses could be used as a probe of the background charge or spin, since the barrier height encodes those parameters.
  • A natural next check would be Kerr spacetime, where the effective potential also depends on the field's azimuthal quantum number; the paper's methods would need to be reworked because superradiant scattering changes the boundary conditions.
  • The paper's quantum-mechanical analogy suggests the same mass suppression should appear in any barrier described by a Schrödinger equation with a mass-dependent height, so the result could be checked in a tabletop wave-packet experiment beyond the WKB regime.
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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 / 2 minor

Summary. The paper, as represented by the abstract, investigates the greybody factor of a massive scalar field in the Reissner-Nordström black-hole spacetime. It uses the Wentzel-Kramers-Brillouin (WKB) approximation and a rigorous bound method to argue that the transmission probability decreases with increasing scalar-field mass, and it states this as an inverse relationship between the greybody factor and the mass. The paper also claims a direct relation between the height of the effective potential and the greybody factor. The full text was not provided to the reviewer; only the abstract was available, so the derivations, boundary conditions, parameter ranges, and numerical checks could not be audited.

Significance. If the full text establishes what the abstract claims, the result would be a useful quantitative statement about massive scalar-field emission from charged black holes, which is relevant to searches for massive dark-matter candidates and to general studies of black-hole greybody factors. The use of two independent methods (WKB and a rigorous bound) is a strength, and the claimed analytic tractability and wider parameter applicability of the bound are valuable if substantiated. However, as received, the manuscript cannot be assessed for correctness: the abstract alone does not contain the derivations, definitions, or parameter ranges needed to verify the central monotonicity claim. The logical gap between a decreasing lower bound and actual monotonicity of the transmission coefficient is a specific technical concern that the full text would need to address.

major comments (3)
  1. [Abstract] The central claim that the greybody factor T decreases monotonically with the scalar-field mass m is not established by the evidence cited in the abstract. A rigorous method that yields a lower bound of the form T >= B(m) with B decreasing in m does not prove that T itself decreases in m; T could lie above B and still increase locally. To support the monotonicity conclusion, the paper must either provide a matching upper bound or benchmark the WKB result against an independent numerical solution over the stated parameter range. No such concrete check is reported in the abstract.
  2. [Abstract] The manuscript as provided to the reviewer contains no equations, no definition of the effective potential, no statement of the WKB order or boundary conditions, and no specification of the parameter ranges for the mass m, charge Q, angular momentum l, or frequency omega. Without these elements, the central claim cannot be checked. In particular, the domain of validity of the WKB approximation (slow variation of the potential between the turning points) and the regime of applicability of the rigorous bound are not stated, so the claim that the bound applies to a wider parameter range is unsupported.
  3. [Abstract] The statement that 'the higher the potential, the lower the greybody factor' is, in a one-dimensional scattering setup, almost a restatement of the fact that a higher barrier suppresses transmission; it is not an independent physical input. The paper should separate this qualitative observation from the quantitative claim of monotonicity in the scalar mass, which is the more substantive result that requires derivation and numerical support.
minor comments (2)
  1. [Abstract] The spelling of 'greybody' is inconsistent: the abstract uses 'greybody factor' in the opening sentence but 'graybody factor' in the final sentence. Please use the same convention throughout.
  2. [Abstract] The phrase 'the higher the potential, the lower the greybody factor' is colloquial; it should be phrased as a precise inequality with the relevant variables and fixed parameters (e.g., l, m, Q, omega) explicitly held constant.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found in the supplied abstract and available text.

full rationale

The visible derivation chain in the abstract consists of two independent computational methods, WKB approximation and rigorous bound, applied to the massive scalar field in the Reissner-Nordström spacetime. The qualitative conclusions (higher effective potential implies lower greybody factor, and larger scalar-field mass implies lower greybody factor) are presented as outputs of those calculations, not as assumptions or fitting inputs. No fitted parameters appear, no equation is defined in terms of the target quantity, and no load-bearing self-citation is quoted in the supplied text. It is possible that the rigorous-bound method is logically fragile if it only establishes a decreasing lower bound, but that would be a correctness or rigor concern, not circularity: the bound is not, on the evidence available, being substituted for the greybody factor itself. The full text was not provided, so the WKB order, boundary conditions, and derivation of the bound could not be audited, but the abstract alone gives no specific reduction of a claimed prediction to its own inputs. Accordingly, no circular step can be exhibited, and the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No fitted parameters or invented entities are identifiable from the abstract. The calculation depends on standard semiclassical and bound methods plus the effective potential barrier picture.

assumptions (3)
  • domain assumption WKB approximation is valid for the considered parameters.
    Abstract lists WKB as a method; no validity conditions or error bounds are given in the abstract.
  • domain assumption The rigorous bound method's inequalities correctly bound the greybody factor.
    Abstract claims a rigorous bound but does not present the inequalities or their derivation.
  • domain assumption Greybody factor is represented as transmission through an effective potential barrier.
    The conclusion that higher potential implies lower greybody factor relies on this standard barrier picture.

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Cite this review

Pith. "Pith review of Greybody Factor for massive scalar field in charged black hole." pith.science (2026). https://pith.science/paper/CYJ3FZFR

@misc{pith2026250102247,
  author       = {Pith},
  title        = {Pith review of: Greybody Factor for massive scalar field in charged black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CYJ3FZFR}},
  note         = {Machine review of arXiv:2501.02247}
}
read the original abstract

The greybody factor of a massive scalar field in the Reissner-Nordstr\"om black hole is investigated using the Wentzel-Kramers-Brillouin (WKB) approximation and rigorous bound methods. We found that the transmission probability and behavior of the potential are directly related in such a way that the higher the potential, the lower the greybody factor. Both methods achieve a similar conclusion, which states that the graybody factor and the mass of the scalar field have an inverse relationship. This can be interpreted in a similar way in quantum mechanics, namely the scalar field with the higher mass will encounter a stronger interaction from the potential and then it is more difficult to penetrate through the potential barrier. The rigorous bound has the advantage of not only being possible to calculate analytically, but also being applicable to a wider range of parameter values compared to the standard WKB approximation.

Figures

Figures reproduced from arXiv: 2501.02247 by the authors.

Figure 1
Figure 1. Left: the potential for various values of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Left: the values of mΦ at which vmax = vmin (solid curve) and vmax = m2 Φ (dashed curve). Right: the potential for various values of q with fixing l = 0 and mΦ = 0.35. 1 5 10 50 100 500 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1 5 10 50 100 500 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1 5 10 50 100 500 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1 5 10 50 100 500 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1 5 10 50 100 500 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1 5 10 50… view at source ↗
Figure 3
Figure 3. The behavior of the potential for varying [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Greybody factor from WKB approximation for both intermediate frequency (IWKB, [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: As illustrated in this figure, the bound associated with [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 5
Figure 5. Figure 5: Lower bounds of greybody factor associated with [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Magnitude of the function X defined in Eq. (29) versus mΦ for various values of l. To check the validity of the rigorous bound method, we compare the bounds with the results approximated by the WKB approximation method. In Figs. 7, it is seen that the bound associated …
Figure 7
Figure 7. Figure 7: Comparison the results obtained from all methods. [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

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Forward citations

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Works this paper leans on

35 extracted references · 9 canonical work pages · cited by 1 Pith paper

  1. [1]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], ``First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,'' Astrophys. J. 875 (2019) no.1, L1 doi:10.3847/2041-8213/ab0ec7 [arXiv:1906.11238 [astro-ph.GA]]

    arXiv 2019

  2. [2]

    B. P. Abbott et al. [LIGO Scientific and Virgo], ``Tests of general relativity with GW150914,'' Phys. Rev. Lett. 116 (2016) no.22, 221101 [erratum: Phys. Rev. Lett. 121 (2018) no.12, 129902] doi:10.1103/PhysRevLett.116.221101 [arXiv:1602.03841 [gr-qc]]

    arXiv 2016

  3. [3]

    Spin of the M87 black hole

    D. Garofalo, ``Spin of the M87 black hole,'' Annalen Phys. 532 (2020) no.4, 1900480 doi:10.1002/andp.201900480 [arXiv:2003.02163 [astro-ph.HE]]

    work page Pith review arXiv 2020

  4. [4]

    F. H. Vincent, M. Wielgus, M. A. Abramowicz, E. Gourgoulhon, J. P. Lasota, T. Paumard and G. Perrin, ``Geometric modeling of M87* as a Kerr black hole or a non-Kerr compact object,'' [arXiv:2002.09226 [gr-qc]]

    arXiv 2002

  5. [5]

    V. I. Dokuchaev and N. O. Nazarova, ``Modeling the motion of a bright spot in jets from black holes M87* and SgrA*,'' [arXiv:2010.01885 [astro-ph.HE]]

    work page Pith review arXiv 2010

  6. [6]

    Black hole shadow to probe modified gravity

    A. Stepanian, S. Khlghatyan and V. G. Gurzadyan, ``Black hole shadow to probe modified gravity,'' Eur. Phys. J. Plus 136 (2021) no.1, 127 doi:10.1140/epjp/s13360-021-01119-2 [arXiv:2101.08261 [gr-qc]]

    work page Pith review arXiv 2021

  7. [7]

    Cornish, D

    N. Cornish, D. Blas and G. Nardini, ``Bounding the speed of gravity with gravitational wave observations,'' Phys. Rev. Lett. 119 (2017) no.16, 161102 doi:10.1103/PhysRevLett.119.161102 [arXiv:1707.06101 [gr-qc]]

    arXiv 2017

  8. [8]

    S. W. Hawking, ``Particle Creation by Black Holes,'' Commun. Math. Phys. 43 (1975), 199-220 [erratum: Commun. Math. Phys. 46 (1976), 206] doi:10.1007/BF02345020

    doi:10.1007/bf02345020 1975

Show all 35 references
  1. [9]

    S. W. Hawking, ``Black Holes and Thermodynamics,'' Phys. Rev. D 13 (1976), 191-197 doi:10.1103/PhysRevD.13.191

    1976 doi

  2. [10]

    Iyer and C

    S. Iyer and C. M. Will, ``Black Hole Normal Modes: A \ WKB\ Approach. 1. Foundations and Application of a Higher Order \ WKB\ Analysis of Potential Barrier Scattering,'' Phys. Rev. D 35 (1987), 3621 doi:10.1103/PhysRevD.35.3621

    1987 doi

  3. [11]

    M. K. Parikh and F. Wilczek, ``Hawking radiation as tunneling,'' Phys. Rev. Lett. 85 (2000), 5042-5045 doi:10.1103/PhysRevLett.85.5042 [arXiv:hep-th/9907001 [hep-th]]

    2000 arXiv

  4. [12]

    H. T. Cho and Y. C. Lin, ``WKB analysis of the scattering of massive Dirac fields in Schwarzschild black hole spacetimes,'' Class. Quant. Grav. 22 (2005), 775-790 doi:10.1088/0264-9381/22/5/001 [arXiv:gr-qc/0411090 [gr-qc]]

    2005 arXiv

  5. [13]

    R. A. Konoplya and A. Zhidenko, ``Passage of radiation through wormholes of arbitrary shape,'' Phys. Rev. D 81 (2010), 124036 doi:10.1103/PhysRevD.81.124036 [arXiv:1004.1284 [hep-th]]

    2010 arXiv

  6. [14]

    Dey and S

    S. Dey and S. Chakrabarti, ``A note on electromagnetic and gravitational perturbations of the Bardeen de Sitter black hole: quasinormal modes and greybody factors,'' Eur. Phys. J. C 79 (2019) no.6, 504 doi:10.1140/epjc/s10052-019-7004-0 [arXiv:1807.09065 [gr-qc]]

    2019 arXiv

  7. [15]

    R. A. Konoplya and A. F. Zinhailo, ``Hawking radiation of non-Schwarzschild black holes in higher derivative gravity: a crucial role of grey-body factors,'' Phys. Rev. D 99 (2019) no.10, 104060 doi:10.1103/PhysRevD.99.104060 [arXiv:1904.05341 [gr-qc]]

    2019 arXiv

  8. [16]

    R. A. Konoplya, A. Zhidenko and A. F. Zinhailo, ``Higher order WKB formula for quasinormal modes and grey-body factors: recipes for quick and accurate calculations,'' Class. Quant. Grav. 36 (2019), 155002 doi:10.1088/1361-6382/ab2e25 [arXiv:1904.10333 [gr-qc]]

    2019 arXiv

  9. [17]

    S. Devi, R. Roy and S. Chakrabarti, ``Quasinormal modes and greybody factors of the novel four dimensional Gauss Bonnet black holes in asymptotically de Sitter space time: scalar, electromagnetic and Dirac perturbations,'' Eur. Phys. J. C 80 (2020) no.8, 760 doi:10.1140/epjc/s...

    2020 arXiv

  10. [18]

    Visser, ``Some general bounds for 1-D scattering,'' Phys

    M. Visser, ``Some general bounds for 1-D scattering,'' Phys. Rev. A 59 (1999), 427-438 doi:10.1103/PhysRevA.59.427 [arXiv:quant-ph/9901030 [quant-ph]]

    1999 arXiv

  11. [19]

    Boonserm and M

    P. Boonserm and M. Visser, ``Bounding the Bogoliubov coefficients,'' Annals Phys. 323 (2008), 2779-2798 doi:10.1016/j.aop.2008.02.002 [arXiv:0801.0610 [quant-ph]]

    2008 arXiv

  12. [20]

    Boonserm, ``Rigorous bounds on Transmission, Reflection, and Bogoliubov coefficients,'' [arXiv:0907.0045 [math-ph]]

    P. Boonserm, ``Rigorous bounds on Transmission, Reflection, and Bogoliubov coefficients,'' [arXiv:0907.0045 [math-ph]]

    arXiv

  13. [21]

    Boonserm, T

    P. Boonserm, T. Ngampitipan and P. Wongjun, ``Greybody factor for black holes in dRGT massive gravity,'' Eur. Phys. J. C 78 (2018) no.6, 492 doi:10.1140/epjc/s10052-018-5975-x [arXiv:1705.03278 [gr-qc]]

    2018 arXiv

  14. [22]

    Boonserm, T

    P. Boonserm, T. Ngampitipan and P. Wongjun, ``Greybody factor for black string in dRGT massive gravity,'' Eur. Phys. J. C 79 (2019) no.4, 330 doi:10.1140/epjc/s10052-019-6827-z [arXiv:1902.05215 [gr-qc]]

    2019 arXiv

  15. [23]

    Barman, ``The Hawking effect and the bounds on greybody factor for higher dimensional Schwarzschild black holes,'' Eur

    S. Barman, ``The Hawking effect and the bounds on greybody factor for higher dimensional Schwarzschild black holes,'' Eur. Phys. J. C 80 (2020) no.1, 50 doi:10.1140/epjc/s10052-020-7613-7 [arXiv:1907.09228 [gr-qc]]

    2020 arXiv

  16. [24]

    Chowdhury and N

    A. Chowdhury and N. Banerjee, ``Greybody factor and sparsity of Hawking radiation from a charged spherical black hole with scalar hair,'' Phys. Lett. B 805 (2020), 135417 doi:10.1016/j.physletb.2020.135417 [arXiv:2002.03630 [gr-qc]]

    2020

  17. [25]

    Kanzi, S

    S. Kanzi, S. H. Mazharimousavi and \.I. Sakall , ``Greybody factors of black holes in dRGT massive gravity coupled with nonlinear electrodynamics,'' Annals Phys. 422 (2020), 168301 doi:10.1016/j.aop.2020.168301 [arXiv:2007.05814 [hep-th]]

    2020

  18. [26]

    Boonserm, S

    P. Boonserm, S. Phalungsongsathit, K. Sansuk and P. Wongjun, Eur. Phys. J. C 83 (2023) no.7, 657 doi:10.1140/epjc/s10052-023-11843-x [arXiv:2305.00459 [gr-qc]]

    2023 arXiv

  19. [27]

    Furuhashi and Y

    H. Furuhashi and Y. Nambu, Prog. Theor. Phys. 112 (2004), 983-995 doi:10.1143/PTP.112.983 [arXiv:gr-qc/0402037 [gr-qc]]

    2004 arXiv

  20. [28]

    H. S. Vieira, V. B. Bezerra and C. R. Muniz, Eur. Phys. J. C 82 (2022) no.10, 932 doi:10.1140/epjc/s10052-022-10908-7 [arXiv:2107.02562 [gr-qc]]

    2022 arXiv

  21. [29]

    J. D. Bekenstein and M. Schiffer, Phys. Rev. D 58 (1998), 064014 doi:10.1103/PhysRevD.58.064014 [arXiv:gr-qc/9803033 [gr-qc]]

    1998 arXiv

  22. [30]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Phys. Rev. Lett. 80 (1998), 5056-5059 doi:10.1103/PhysRevLett.80.5056 [arXiv:gr-qc/9911046 [gr-qc]]

    1998 arXiv

  23. [31]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Phys. Lett. B 464 (1999), 25 doi:10.1016/S0370-2693(99)01038-2 [arXiv:hep-th/9911174 [hep-th]]

    1999 arXiv

  24. [32]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Phys. Lett. B 493 (2000), 149-152 doi:10.1016/S0370-2693(00)01125-4 [arXiv:gr-qc/0009077 [gr-qc]]

    2000 arXiv

  25. [33]

    Pere\ niguez, Phys

    D. Pere\ niguez, Phys. Rev. D 108 (2023) no.8, 084046 doi:10.1103/PhysRevD.108.084046 [arXiv:2302.10942 [gr-qc]]

    2023 arXiv

  26. [34]

    De Felice and S

    A. De Felice and S. Tsujikawa, Phys. Rev. D 109 (2024) no.8, 084022 doi:10.1103/PhysRevD.109.084022 [arXiv:2312.03191 [gr-qc]]

    2024 arXiv

  27. [35]

    merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number orga...

    2010

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