Thin Accretion Disk Around Rotating Hairy Black Hole: Radiative Property and Optical Appearance

Xiao-Kan Guo; Zhen Li

REVIEW 3 major objections 6 minor 61 references

Thin Accretion Disk Around Rotating Hairy Black Hole: Radiative Property and Optical Appearance

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A rotating hairy black hole's thin accretion disk radiates brighter and hotter than a Kerr disk of the same mass and spin, with the gap growing at high spin and toward the inner disk.

desk verdict Useful thin-disk radiative calculation for a hairy Kerr-like metric, undermined by an algebraic error in the photon-trajectory expansion that feeds the ray-traced images. read the letter →

arxiv 2501.01018 v1 pith:L6FVVOG6 submitted 2025-01-02 gr-qc

Zhen Li , Xiao-Kan Guo This is my paper · ORCID

classification gr-qc

keywords rotatinghairyblackholegravitationaldecouplingthinaccretiondiskradiativefluxtemperatureprofiledifferentialluminosityraytracingbolometricimage

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 asks whether thin accretion disks can reveal that black holes carry hair beyond mass and spin. It studies the rotating hairy black hole obtained by gravitational decoupling and computes the disk's radiative flux, temperature, differential luminosity, and ray-traced bolometric image, comparing everything with the Kerr metric. It claims the deviations from Kerr become significant for rapid rotation and in the inner disk, where a larger hair parameter δ and a smaller hair-entropy parameter h0 raise the flux, temperature, luminosity, and image brightness. If this is right, high-spin sources observed with horizon-scale resolution could distinguish hairy black holes from Kerr black holes.

What carries the argument

The load-bearing object is the rotating hairy black hole metric with $\Delta = r^2 + a^2 - 2Mr + \delta r^2 e^{-r/(M - h_0/2)}$, which reduces to the Kerr metric at $\delta=0$. The argument runs through circular timelike geodesics, whose angular velocity, energy, and angular momentum feed the thin-disk flux integral; through the radial potential $R(r)$ for null geodesics, whose four largest roots classify photon trajectories and define the photon-shell critical curve; and through backward ray tracing, which inverts the geodesic integrals to place source flux on the image plane with a redshift factor $\chi = 1/[u^t(1-\lambda\Omega)]$. The hairy parameters $\delta$ and $h_0$ enter every stage through $\Delta$, which is why the disk's inner edge, brightness, and image structure all shift together.

What would settle it

Numerically solve $R(r)=0$ over a grid of hairy parameters $\delta>0$, $h_0<2M$, and spin up to the critical value, and ask whether any real root outside the event horizon is missing from the four largest roots used here; if one is, the photon-shell classification and the resulting bolometric images are incomplete.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes a parameter map: for the rotating hairy metric with $\Delta = r^2 + a^2 - 2Mr + \delta r^2 e^{-r/(M - h_0/2)}$, the event horizon, ISCO radius, and photon-shell boundary all shrink with spin faster than they do in Kerr, and the ISCO moves inward as $\delta$ grows and $h_0$ shrinks. Using the thin-disk flux formula, it finds that the radial profiles of $F(r)$, $T(r) \propto F(r)^{1/4}$, and $dL_\infty/d\ln r$ all rise above the Kerr values in the inner region, with the largest enhancement for $(\delta=1, h_0=1)$ at $a=0.8$; at $a=0.2$ the profiles are nearly indistinguishable from Kerr. Ray-traced bolometric images then show a smaller apparent horizon, stronger azimuthal dragging, higher brightness, and a more compact appearance for the hairy black hole at high spin. The paper's central claim is that these combined radiative and image deviations are the observational signature of the hairy geometry.

Load-bearing premise

The image calculation assumes that the four largest roots of the radial potential $R(r)=0$ capture every photon path that matters, with all other roots inside the horizon or complex, and the paper states this without proof for the hairy metric; if another real root lay outside the horizon, some photon trajectories would turn elsewhere and the ray-traced images would change.

Editorial extensions

If this is right

If the central claim is right, a rapidly spinning hairy black hole with $(\delta=1, h_0=1)$ will present a thin-disk image that is brighter and more compact than a Kerr image at the same mass and spin.
The inner edge of the disk moves inward as $\delta$ grows and $h_0$ shrinks, so inner-disk flux and temperature are the most sensitive probes of hair.
At low spin the two geometries are nearly degenerate in flux, temperature, luminosity, and image, concentrating the detectable signature in high-spin systems.
The redshift distribution on the direct image changes visibly with hair parameters at high spin and high inclination, which would alter observed spectral line shapes.

Reading between the lines

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

Editorial inference: because $\delta$ and $h_0$ push the orbital radii in opposite directions, a single flux measurement can only constrain a degenerate combination of the two; separating them needs independent observables such as spectral line profiles or quasi-periodic oscillations.
Editorial inference: the same modified $\Delta$ should also change inner-disk oscillation frequencies and fluorescent iron-line profiles, offering non-imaging tests of the same hairy geometry.
Editorial inference: the unproved root classification can be checked directly by a numerical search for real roots of $R(r)=0$ outside the horizon across the hairy parameter space; if any additional real root appears, the lensing-band structure of these images would need revision.
Editorial inference: the high-spin dependence suggests an observational strategy of targeting known high-spin accreting black holes, since low-spin sources cannot discriminate the models.

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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

Short version: the radiative-property part of this paper is a solid, routine extension of the Novikov-Thorne machinery to the rotating hairy black hole metric of Contreras, Ovalle and Casadio. The optical-appearance half rests on a radial potential that is incorrectly expanded in Eq. (14)-(15), and the ray-traced images inherit that error.

What is new: this is the first computation of thin-disk flux, temperature, luminosity, and bolometric images specifically for this hairy rotating metric, with the ISCO computed numerically. The standard formulas are applied faithfully, and the figures are clear. The qualitative conclusion that deviations from Kerr grow with spin and in the inner disk is plausible and, for the radiative quantities, the calculation appears self-consistent.

The soft spot is serious. Expanding R(r) from Eq. (7) with the stated Δ gives, for photons,

R = r^4 + [a^2 - η - λ^2 - δ e^{-r/(M-h0/2)}(η+(λ-a)^2)] r^2 + 2M(η+(λ-a)^2) r - a^2 η.

The hair term belongs in the coefficient of r^2. Eq. (15) instead sets B = 2(M - δ r^2 e^{-...})(η+(λ-a)^2), which puts a spurious -2δ(η+(λ-a)^2) r^3 e^{-...} into the r term. This is not a typo-level issue: Sec. II.B uses the four largest roots of this R to classify photon trajectories, and Sec. IV uses that classification to assign rays to direct or lensed bands. The integral kernels in Eq. (30)-(31) appear to use the correct modified Δ, so the printed R and the ray-tracing kernels are internally inconsistent. If the code used the printed B, the images show a different spacetime; if it used the correct R, the manuscript's core ray-tracing equations are wrong and the results are not reproducible from the paper.

The assertion that most additional roots lie inside the horizon or appear as complex conjugates is also unproven; with the corrected R you have a transcendental equation, and the four-root classification is not guaranteed. The deviations from Kerr are not quantified anywhere, so 'significant' is a visual impression, not a number.

The radiative sections stand alone and are likely correct. The optical part needs reworking. I recommend sending this to a serious referee, with the clear instruction to check the null-geodesic algebra; if the authors correct Eq. (15) and redo the ray tracing, the paper becomes a useful template. As it stands, I would only cite the radiative results, and even then with a caveat.

Referee Report

3 major / 6 minor

Summary. This paper studies the radiative properties and optical appearance of a geometrically thin, optically thick accretion disk around a rotating hairy black hole obtained by gravitational decoupling. Using the Page-Thorne model, the authors numerically compute the radiative flux, temperature, and differential luminosity as functions of radius for several values of the spin a and hairy parameters (δ, h0), and compare with Kerr. They then adapt the aart ray-tracing code to produce bolometric images of the disk, including direct and lensed bands, for a subset of parameters. The paper's central claim is that deviations from the Kerr predictions become significant for rapidly rotating black holes or in the inner region of the disk.

Significance. If the results are correct, the paper would provide concrete predictions for distinguishing this hairy black hole model from Kerr using continuum disk observations and future black hole imaging, extending existing shadow-only studies to full disk images. The use of standard Page-Thorne formulas and the public aart code is a practical strength, and the paper addresses a timely observational question. However, the ray-tracing section rests on an incorrect polynomial expansion of the radial potential, and the root-classification assertion is unproven, so the significance of the image results is currently compromised.

major comments (3)

[II.B, Eqs. (14)-(15)] Expanding Eq. (7) for photons with Δ = r² + a² − 2Mr + δ r² e^{−r/(M−h0/2)} gives R(r) = r⁴ + (a²−η−λ²)r² + 2M S r − a²η − δ S r² e^{−r/(M−h0/2)}, where S = η + (λ−a)². The hair term multiplies r², not r³. The printed B = 2(M − δ r² e^{−r/(M−h0/2)})S in Eq. (15) therefore yields a spurious −2δ S r³ e^{−r/(M−h0/2)} contribution to the coefficient of r. Since the root classification in Sec. II.B and the ray-tracing implementation in Sec. IV rely on this polynomial form, the equations as printed do not represent the stated spacetime. Please correct the expansion and clarify whether the numerical code uses the printed B or the exact R(r); if it uses the exact R(r), the manuscript's ray-tracing equations must be made consistent so that the results are reproducible.
[II.B, after Eq. (14)] The statement that 'most of the additional roots lie deep inside the horizon and appear as complex conjugates' is asserted without proof or numerical demonstration. For the corrected R(r), the exponential factor makes R(r)=0 a transcendental equation, and there is no guarantee that only the four largest real roots matter for photon trajectories outside the horizon. Because the classification of direct and lensed bands in Sec. IV depends on this root structure, please provide a concrete verification, such as a numerical survey over the parameter space used in Figs. 5–7, showing that no additional real roots with r > r+ contribute.
[Abstract and Secs. III-IV] The central claim that deviations from Kerr are 'significant' in the rapid-rotation case or in the inner disk region is supported only by visual inspection of Figs. 2–7. No quantitative measure, such as fractional differences in flux, temperature, luminosity, or image brightness, is given. Please add a quantitative statistic or a percentage-deviation plot to substantiate the key claim.

minor comments (6)

[III, text after Fig. 3] The word 'observationas' should be 'observations'.
[Captions of Figs. 5-7] The phrase 'inclination angels' should be 'inclination angles'.
[IV.A, text after Eq. (25)] The word 'reversely' should be 'conversely'.
[II, below Eq. (2)] The relation h0 = λh is confusing because λ is used later for the energy-scaled angular momentum; please clarify the notation.
[Eq. (22)] The fourth-root notation is ambiguous; use \sqrt[4]{F(r)/\sigma} instead of a bare superscript 4.
[V, Conclusion] The sentence about a related study 'coming soon' is unconventional for a published paper; consider removing it or citing the work once it is available.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the disk observables are derived from the assumed hairy metric by explicit geodesic and ray-tracing calculations.

full rationale

The derivation is self-contained conditional on the assumed rotating hairy metric (Eq. 2) taken from gravitational decoupling [12]. Photon geodesics are obtained from the standard Hamilton-Jacobi separation with R(r) and Theta(theta) (Eqs. 7-8), and the disk flux, temperature, luminosity, and ray-traced images follow by explicit numerical integration of Eqs. (20), (22), (23), and (35). No parameter is fitted to the quantities being predicted; the ISCO and photon-shell radii are computed from the same R(r), and the Kerr comparison uses the standard delta=0 limit. The self-citations (Refs. 26, 27, 56) are contextual references to related hairy-black-hole studies and do not supply any load-bearing input to the derivation. The unproven statement that extra roots of R(r)=0 lie deep inside the horizon or appear as complex conjugates is a potential correctness gap in the photon-trajectory classification, but it is not circular: the classification does not presuppose the paper's conclusions about disk images. Therefore no circular step meeting the quoted-evidence standard is present.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central results depend on the prior hairy metric and the standard thin-disk model. The only new parameters chosen for this study are δ and h0, which are scanned rather than fitted. No new entities are introduced.

free parameters (2)

δ (decoupling/deformation parameter) = 0.5, 1.0
Chosen by hand in Figs. 1-4; controls the strength of the hairy deviation from Kerr.
h0 (primary hair parameter) = 1.0, 1.5
Chosen by hand; constrained only by asymptotic flatness (h0 ≤ 2M).

assumptions (4)

domain assumption The rotating hairy black hole metric (Eqs. 1-2) is a valid spacetime used as the background for geodesic and disk calculations.
The metric is taken from Ref. [12] without derivation or discussion of the energy conditions; the paper's results inherit its validity.
standard math The geodesic equations separate with the same conserved quantities as Kerr, with R(r) and Θ(θ) in Eqs. (7)-(8).
The metric has the same algebraic structure as Kerr with a radial mass function, for which Hamilton-Jacobi separation is standard; the paper uses it without proof.
ad hoc to paper Photon trajectories can be classified using only the four largest roots of R(r) = 0.
The paper asserts that additional roots lie deep inside the horizon or appear as complex conjugates (Sec. II.B), but no proof is given; this underpins the ray-tracing image construction.
domain assumption The thin disk model assumptions: geometrically thin, optically thick, circular equatorial orbits, thermal equilibrium, sub-Eddington accretion rate.
Stated in Sec. III following Novikov-Thorne; the radiative flux formula (Eq. 20) relies on these assumptions.

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Cite this review

Pith. "Pith review of Thin Accretion Disk Around Rotating Hairy Black Hole: Radiative Property and Optical Appearance." pith.science (2026). https://pith.science/paper/L6FVVOG6

@misc{pith2026250101018,
  author       = {Pith},
  title        = {Pith review of: Thin Accretion Disk Around Rotating Hairy Black Hole: Radiative Property and Optical Appearance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L6FVVOG6}},
  note         = {Machine review of arXiv:2501.01018}
}

read the original abstract

The gravitational decoupling method systematically generates hairy modifications to the solutions in general relativity due to new gravitational sources. In view of the recent advances in astronomical observations, these hairy solutions are expected to be testable in the near-term observations. In this paper, we study the radiative property and optical appearance of the thin accretion disk around the rotating hairy black holes obtained by gravitational decoupling. We numerically compute the radiative flux, temperature, and differential luminosity of the thin accretion disk, and we also show its bolometric image by the ray-tracing method. By comparing with the results for the Kerr metric, we found that the deviations of the observational properties of the thin accretion disk from those of Kerr metric becomes significant in the rapid rotating case, or in the inner region of the disk. These results guide the observational investigations on the rotating hairy black hole.

Figures

Figures reproduced from arXiv: 2501.01018 by the authors.

**Figure 2.** FIG. 2. The radiative flux per accretion rate of thin accretion [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. The temperature of thin accretion disk [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5. Comparison of direct image (or [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6. Comparison of redshift factor on the direct image for Kerr black holes (left column) and hairy black holes with hairy [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7. Comparison of black hole images for Kerr black holes (left column) and rotating hairy black holes with ( [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

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Works this paper leans on

61 extracted references · 53 canonical work pages

[1]

B. Carter. Phys. Rev. Lett. 26, 331 (1971)

work page 1971
[2]

S. W. Hawking. Communications in Mathematical Physics. 25, 152 (1972)

work page 1972
[3]

D. C. Robinson. Phys. Rev. Lett. 34, 905 (1975)

work page 1975
[4]

P. O. Mazur. Journal of Physics A Mathematical General. 15, 3173 (1982)

work page 1982
[5]

J. D. Bekenstein. Phys. Rev. D. 51, 6608 (1995)

work page 1995
[6]

X. Y. Chew, D. h. Yeom and J. L. Bl´ azquez-Salcedo, Phys. Rev. D. 108, 044020 (2023)

work page 2023
[7]

X. Y. Chew and K. G. Lim, Phys. Rev. D. 109, 064039 (2024)

work page 2024
[8]

X. Y. Chew and D. h. Yeom, Phys. Rev. D. 110, 044036 (2024)

work page 2024

Show all 61 references

[9]

X. Y. Chew and Y. S. Myung, Phys. Rev. D. 110, 044011 (2024)

2024
[10]

C. A. R. Herdeiro, E. Radu. Int. J. Mod. Phys. D, 24, 1542014 (2015)

2015
[11]

Ovalle, R

J. Ovalle, R. Casadio, E. Contreras and A. Sotomayor. Phys. Dark Univ. 31, 100744 (2021)

2021
[12]

Contreras, J

E. Contreras, J. Ovalle, R. Casadio. Phys. Rev. D, 103, 044020 (2021)

2021
[13]

J. Ovalle. Phys. Rev. D 95, 104019 (2017)

2017
[14]

J. Ovalle. Phys. Lett. B 788, 213 (2019)

2019
[15]

S. U. Islam, S. G. Ghosh. Phys. Rev. D 103 124052 (2021)

2021
[16]

Afrin, R

M. Afrin, R. Kumar and S. G. Ghosh. Mon. Not. Roy. Astron. Soc. 504, 5927 (2021)

2021
[17]

Shahzadi, M

M. Shahzadi, M. Koloˇ s, Z. Stuchl ´ ık and Y. Habib, Eur. Phys. J. C. 82, 407 (2022)

2022
[18]

S. G. Ghosh and M. Afrin, Contribution to: MG16, 1167- 1178 (2023)

2023
[19]

R. T. Cavalcanti, R. C. de Paiva and R. da Rocha. Eur. Phys. J. Plus 137, 1185 (2022)

2022
[20]

Tang and Z

M. Tang and Z. Xu, JHEP. 12, 125 (2022)

2022
[21]

Vagnozzi, et al

S. Vagnozzi, et al. Class. Quant. Grav.40, 165007 (2023)

2023
[22]

S. K. Jha and A. Rahaman. [arXiv:2205.06052]

arXiv
[23]

Y. Meng, X. M. Kuang, X. J. Wang, B. Wang and J. P. Wu, Eur. Phys. J. C 84, 305 (2024)

2024
[24]

Vertogradov, M

V. Vertogradov, M. Misyura and P. Bambhaniya, [arXiv:2403.16743 [gr-qc]]

arXiv
[25]

Mahapatra and I

S. Mahapatra and I. Banerjee. [arXiv:2208.05796]

arXiv
[26]

Li, Phys

Z. Li, Phys. Lett. B 841, 137902 (2023)

2023
[27]

Li and F

Z. Li and F. Yuan, Phys. Rev. D 108, 024039 (2023)

2023
[28]

Vertogradov and D

V. Vertogradov and D. Kudryavcev, arXiv:2303.01413

arXiv
[29]

Edelson et al

R. Edelson et al. Astrophys. J. 870, 123 (2019)

2019
[30]

W. J. Guo et al. Astrophys. J. 929, 19 (2022)

2022
[31]

Sathyaprakash R., et al. MNRAS. 511, 5346 (2022)

2022
[32]

Shen et al

Y. Shen et al. Astrophys. J, Supplement Series. 276, 26 (2024)

2024
[33]

Akiyama et al

K. Akiyama et al. (Event Horizon Telescope), Astro- phys.J. Lett. 875, L1 (2019)

2019
[34]

Akiyama et al

K. Akiyama et al. (Event Horizon Telescope), Astro- phys.J. Lett. 930, L15 (2022)

2022
[35]

S. E. Gralla and A. Lupsasca. Phys. Rev. D. 101, 044031 (2020)

2020
[36]

S. E. Gralla, A. Lupsasca and D. P. Marrone, Phys. Rev. D. 102, 124004 (2020)

2020
[37]

I. D. Novikov, K. S. Thorne, Black Holes (Les Astres Occlus), New York, 343 (1973)

1973
[38]

D. N. Page, K. S. Thorne, Astrophys. J. 191, 499 (1974)

1974
[39]

N. I. Shakura and R. A. Sunyaev. A&A. 24, 337 (1973)

1973
[40]

Kurmanov, et al

E. Kurmanov, et al. Astrophys. J. 925, 210 (2022)

2022
[41]

Boshkayev, et al

K. Boshkayev, et al. Eur. Phys. J. Plus. 139, 273 (2024)

2024
[42]

Boshkayev, et al

K. Boshkayev, et al. Eur. Phys. J. C. 84, 230 (2024)

2024
[43]

Ravanal, G

Y. Ravanal, G. G´ omez and N. Cruz, Phys. Rev. D.108, 8 (2023)

2023
[44]

Mustafa, et al

G. Mustafa, et al. Eur. Phys. J. C. 84, 690 (2024)

2024
[45]

Patra, B

S. Patra, B. R. Majhi and S. Das, JCAP. 060, 01 (2024)

2024
[46]

L. A. S´ anchez, Eur. Phys. J. C.84, 635 (2024)

2024
[47]

Y. H. Jiang and T. Wang, Phys. Rev. D. 110, 103009 (2024)

2024
[48]

Layeghi, J

O. Layeghi, J. Ghanbari and M. Moeen Moghaddas, [arXiv:2409.02955 [gr-qc]]

arXiv
[49]

H. B. Zheng, M. Q. Wu, G. P. Li and Q. Q. Jiang,[arXiv:2411.10315 [gr-qc]]

arXiv
[50]

Y. Hou, Z. Zhang, H. Yan, M. Guo and B. Chen, Phys. Rev. D. 106, 064058 (2022)

2022
[51]

J. Peng, M. Guo and X. H. Feng, Chin. Phys. C. 45, 085103 (2021)

2021
[52]

Zhang, Y

Z. Zhang, Y. Hou, M. Guo and B. Chen, JCAP. 05, 032 (2024)

2024
[53]

C. Y. Yang, M. I. Aslam, X. X. Zeng and R. Saleem, [arXiv:2411.11807 [astro-ph.HE]]. 12

arXiv
[54]

K. J. He, G. P. Li, C. Y. Yang and X. X. Zeng,[arXiv:2411.11680 [astro-ph.HE]]

arXiv
[55]

X. X. Zeng, M. I. Aslam and R. Saleem, Eur. Phys. J. C. 83, 129 (2023)

2023
[56]

Li,[arXiv:2412.08447 [gr-qc]]

Z. Li,[arXiv:2412.08447 [gr-qc]]

arXiv
[57]

Zhang, G

D. Zhang, G. Fu, X. J. Wang, Q. Pan, X. M. Kuang and J. P. Wu,[arXiv:2412.20450 [gr-qc]]

arXiv
[58]

J. M. Bardeen, W. H. Press and S. A. Teukolsky, Astro- phys. J. 178, 347 (1972)

1972
[59]

H. Chen, X. Y. Chew and W. Fan, [arXiv:2411.00565 [gr-qc]]

arXiv
[60]

C. T. Cunningham and J. M. Bardeen, Astrophys. J. 183, 237 (1973)

1973
[61]

Cardenas-Avendano, A

A. Cardenas-Avendano, A. Lupsasca, H. Zhu, Phys. Rev. D. 107, 043030, (2023). arXiv:2211.07469

2023 arXiv

Pith tools

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

[1] [1]

B. Carter. Phys. Rev. Lett. 26, 331 (1971)

work page 1971

[2] [2]

S. W. Hawking. Communications in Mathematical Physics. 25, 152 (1972)

work page 1972

[3] [3]

D. C. Robinson. Phys. Rev. Lett. 34, 905 (1975)

work page 1975

[4] [4]

P. O. Mazur. Journal of Physics A Mathematical General. 15, 3173 (1982)

work page 1982

[5] [5]

J. D. Bekenstein. Phys. Rev. D. 51, 6608 (1995)

work page 1995

[6] [6]

X. Y. Chew, D. h. Yeom and J. L. Bl´ azquez-Salcedo, Phys. Rev. D. 108, 044020 (2023)

work page 2023

[7] [7]

X. Y. Chew and K. G. Lim, Phys. Rev. D. 109, 064039 (2024)

work page 2024

[8] [8]

X. Y. Chew and D. h. Yeom, Phys. Rev. D. 110, 044036 (2024)

work page 2024