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REVIEW 2 major objections 4 minor 96 references

Thermodynamics and Shadows of Kerr black holes endowed with a global monopole charge

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that a Kerr black hole carrying a global monopole charge produces larger, rounder shadows whose angular diameters fall inside the EHT-observed ranges for small values of the monopole parameter.

desk verdict The rotating metric in Eq. (4) does not reduce to the static monopole spacetime in Eq. (1), so the shadow and thermodynamic results are computed for a different, physically mislabeled spacetime. read the letter →

arxiv 2506.02653 v1 pith:2FKR6BQ5 submitted 2025-06-03 gr-qc

Balendra Pratap Singh , Md Sabir Ali This is my paper
classification gr-qc
keywords blackholeshadowglobalmonopoleKerrEventHorizonTelescopeM87SgrA*thermodynamicsNewman-Janisalgorithm
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 extends the Kerr black hole spacetime by adding a global monopole charge $\alpha$, a topological defect left over from a spontaneous symmetry breaking, and derives the thermodynamics and photon-ring shadow of the resulting rotating spacetime. The central result is that the shadow's angular diameter grows with $\alpha$ and shrinks with the spin $a$: for the M87 black hole, diameters between $39\,\mu\mathrm{as}$ and $51\,\mu\mathrm{as}$ occur for $\alpha \in (0,0.08)$ and $a \in (0.7M,0.99M)$, overlapping the EHT value $42 \pm 3\,\mu\mathrm{as}$; for Sgr A*, diameters between $50\,\mu\mathrm{as}$ and $55\,\mu\mathrm{as}$ occur for $\alpha \in (0,0.04)$, inside $48.7 \pm 7\,\mu\mathrm{as}$. If correct, current black-hole images are compatible with a small monopole charge, and future measurements of shadow size and roundness could probe this charge. The paper also finds that the monopole term lowers the Hawking temperature relative to Kerr and changes the local thermodynamic stability.

What carries the argument

The object that carries the argument is the rotating global-monopole metric of Eq. (4), a Kerr-like line element with $\Delta(r) = r^2 - 2Mr + a^2 - 8\pi\alpha^2 r^2$, obtained from the static monopole solution by the Newman-Janis algorithm. This $\Delta$ fixes the horizon radii $r_\pm = (M \pm \sqrt{M^2 + (8\pi\alpha^2 - 1)a^2})/(1 - 8\pi\alpha^2)$ and enters the effective potential $V_{\mathrm{eff}}$ for photon orbits. The shadow then follows from Hamilton-Jacobi/Carter separability, giving impact parameters $\xi$ and $\eta$, celestial coordinates $X$ and $Y$, and the observables shadow radius $R_s$, distortion $\delta_s$, area $A$, and oblateness $D$; the angular diameter is computed as $\theta_s = 2 r_0^{-1}\sqrt{A/\pi}$. The thermodynamic quantities $T$, $C$, and $G$ are built from the same horizon condition $\Delta(r_h)=0$.

What would settle it

Substitute $a=0$ in Eq. (4) and compare directly with Eq. (1): because $g_{tt}$ differs by the $8\pi\alpha^2$ term, the rotating metric is not the rotating counterpart of the stated static spacetime, and repeating the shadow-diameter calculation with a correctly reduced rotating metric would show whether the quoted bounds $\alpha \in (0,0.08)$ and $\alpha \in (0,0.04)$ survive.

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Extended reading notes

Core claim

On its own terms, the paper establishes that a Kerr black hole endowed with a global monopole charge $\alpha$ is a continuous deformation of the Kerr spacetime whose shadow is larger and more circular than the Kerr shadow. The key quantitative claim is that the angular diameter of the shadow grows with $\alpha$ and shrinks with the spin $a$: for $a \in (0.7M, 0.99M)$ and $\alpha \in (0, 0.08)$, the M87 shadow diameter lies between $39\,\mu\mathrm{as}$ and $51\,\mu\mathrm{as}$, overlapping the EHT value $42 \pm 3\,\mu\mathrm{as}$; for Sgr A*, $\alpha \in (0, 0.04)$ gives $50\,\mu\mathrm{as}$ to $55\,\mu\mathrm{as}$, inside $48.7 \pm 7\,\mu\mathrm{as}$. The paper also derives the thermodynamic consequences of the same metric: the monopole lowers the Hawking temperature relative to Kerr, shifts the heat-capacity divergence, and leaves the Gibbs free energy positive, so the rotating monopole black hole is locally stable only in patches and globally unstable. All formulas reduce to Kerr when $\alpha=0$.

Load-bearing premise

The load-bearing premise is that Eq. (4) really is the rotating version of the static global-monopole spacetime, but setting $a=0$ in Eq. (4) gives $g_{tt}=-(1-2M/r)$, whereas Eq. (1) requires $g_{tt}=-(1-2M/r-8\pi\alpha^2)$; the $8\pi\alpha^2$ term drops out of the time-time component of the rotating metric, so the thermodynamic and shadow results inherit that mismatch.

Editorial extensions

If this is right

  • For $a \in (0.7M, 0.99M)$ and $\alpha \in (0, 0.08)$, the predicted M87 angular diameter lies between $39\,\mu\mathrm{as}$ and $51\,\mu\mathrm{as}$, overlapping the EHT measurement $42 \pm 3\,\mu\mathrm{as}$.
  • For the same spin range and $\alpha \in (0, 0.04)$, the predicted Sgr A* diameter lies between $50\,\mu\mathrm{as}$ and $55\,\mu\mathrm{as}$, inside the EHT range $48.7 \pm 7\,\mu\mathrm{as}$.
  • Monopole charge makes the shadow radius and area increase, the distortion decrease, and the oblateness increase, so the shadow looks larger and more circular than the Kerr shadow.
  • The Hawking temperature is lower than Kerr's for the same horizon radius, and the heat capacity diverges at $r_h = a/\sqrt{8\pi\alpha^2 - 1}$, marking a local stability transition.
  • Every result reduces to the standard Kerr spacetime when $\alpha \to 0$, so the model is a one-parameter deformation of Kerr that EHT-type measurements can constrain.

Reading between the lines

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

  • Editorial inference: a corrected rotating metric that keeps the full $1-2M/r-8\pi\alpha^2$ time-time component would likely preserve the qualitative trend of larger, rounder shadows, but the numerical EHT bounds on $\alpha$ could shift and should not be treated as robust until the metric is fixed.
  • Editorial inference: because shadow diameter grows with $\alpha$ and shrinks with spin $a$, a single shadow image cannot cleanly separate monopole charge from spin; full image comparisons or multiple black-hole targets would be needed to break that degeneracy.
  • Editorial inference: the same area-based angular-diameter pipeline could be applied to other topological-defect or exotic-matter spacetimes, and the monotone $\alpha$-$\theta_s$ relation offers a simple null test: if future ngEHT diameters fall below the Kerr prediction, enlargement from a monopole charge would be excluded.
  • Editorial inference: the thermodynamic side predicts a monopole-dependent shift in temperature and in the heat-capacity divergence, so future quasinormal-mode or ringdown observations that constrain $T$ or the stability threshold could independently probe $\alpha$ without using shadows.
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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 / 4 minor

Summary. The manuscript constructs a Kerr-like metric intended to describe a rotating black hole with a global monopole charge, computes its horizon structure and thermodynamic quantities (temperature, heat capacity, Gibbs free energy), derives photon geodesics via the Hamilton-Jacobi/Carter method, computes shadow radius, distortion, area, and oblateness, and compares the angular diameter of the shadow with EHT observations of M87 and Sgr A*. The authors claim that certain ranges of the global monopole charge α and spin a are consistent with EHT data. The central issue is that the rotating metric used throughout, Eq. (4), is not the a=0 limit of the stated static monopole spacetime, so the calculation is performed for a different spacetime.

Significance. Should the metric error be repaired and the calculations redone, the standard observables pipeline could yield useful constraints on a possible global monopole charge. The paper is self-contained and presents explicit analytic expressions and tabulated observables; it also uses the stated masses and distances in Eq. (31) when computing angular diameters, so I do not share the concern that distance and mass are ignored. However, because the starting metric (4) is internally inconsistent with the static model (1)-(2), the thermodynamic and shadow results, including the EHT consistency claims, do not apply to the advertised spacetime. This is a load-bearing defect, not a presentation issue.

major comments (2)
  1. [Sec. II, Eq. (4) vs. Eq. (1)] Setting a=0 in the rotating metric (4) yields g_tt = -(1 - 2M/r) and g_rr = [1 - 2M/r - 8πα^2]^{-1}; the former lacks the -8πα^2 term present in the static monopole metric (1)-(2). Hence Eq. (4) is not the rotating version of Eq. (1), and the spacetime actually studied is not the Kerr black hole with a global monopole charge. This is not a cosmetic issue: the temperature, heat capacity, and Gibbs free energy in Eqs. (7)-(10), the geodesic equations (13)-(18), the impact parameters (21)-(22), the shadow contour (26), and all observables in Tables I-IV are computed from this metric. The claimed EHT constraints on α therefore do not follow from the stated model.
  2. [Sec. VI, Tables III and IV] The abstract states that for α∈(0,0.08) and a∈(0.7M,0.99M) the M87 angular diameter varies from 39 to 51 μas and that the results are consistent with EHT. Table III shows α=0.08 gives θ_s≈50-51 μas, which is outside the quoted EHT value 42±3 μas; even at 3σ the upper bound is 45 μas. Thus only a subrange of the claimed α interval is consistent with M87 observations, and the paper does not determine the claimed upper limit α<0.08. The same issue appears in Table IV for Sgr A*, where α=0.08 gives θ_s≈67-68 μas versus the quoted 48.7±7 μas; the α∈(0,0.04) range is more defensible but is not derived from a fitting or intersection procedure. The consistency claim is therefore not supported by the tables.
minor comments (4)
  1. [Sec. V and Conclusion] The text near Eq. (30) says oblateness D increases with α, and Table II confirms this, but the second-to-last bullet of the Conclusion states 'oblatness D decreases'; the Conclusion bullet should be corrected.
  2. [Fig. 7] The lower-right panel legend entry 'α = 1.2' is presumably a typo for 'α = 0.12'.
  3. [Eq. (9)] The denominator (8πα^2 - 1)r_h^2 - a^2 is negative for all r_h when α < 1/sqrt(8π), so the claimed divergence at r_h = a/sqrt(8πα^2 - 1) has no real solution for the parameter values plotted in Figs. 4-6; the domain of the claim should be stated.
  4. [Table IV] For α=0 and a/M=0.7, a direct evaluation of Eq. (31) using the area in Table II, M=4×10^6 M_sun, and distance 8 kpc gives about 50.5 μas, whereas the table lists 52.459 μas; the discrepancy should be explained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the shadow and thermodynamic results are derived from the written metric and compared with, not fitted to, EHT data.

full rationale

The paper's derivation chain is self-contained in the relevant sense: the static monopole metric (1)-(2) leads to the claimed rotating metric (4)-(5) via the Newman-Janis algorithm as cited to reference [14]; the geodesic equations (13)-(18), effective potential (19), impact parameters (21)-(22), shadow observables (23)-(30), and angular diameters (31) are all algebraic consequences of that metric. The EHT values enter only in Section VI as external comparison data: the parameters alpha and a are scanned over intervals rather than fitted to the observed angular diameters, and the reported ranges of theta_s are statements about the model, not predictions manufactured by construction from the observations. The authors' self-citations (e.g., Refs. [47,48,55,77,78,91]) appear only in the introductory literature survey and are not load-bearing for the metric, geodesic, thermodynamic, or shadow derivations; the central rotating metric is attributed to the external reference [14]. The most substantive concern is that Eq. (4) does not reduce to Eq. (1) when a=0, since g_tt misses the -8 pi alpha^2 term; however, that is an internal correctness or labeling problem about which spacetime is being studied, not a circularity in which a conclusion is assumed in its own derivation. No step satisfies the threshold of reducing to its inputs by definition, fitted-input renaming, or a load-bearing self-citation chain, so the circularity score is 0.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claim rests on the assumed validity of metric (4) as the rotating global monopole spacetime. That assumption fails the basic test of reducing to the static seed (1) when the spin vanishes. The other assumptions are standard for shadow calculations.

free parameters (2)
  • global monopole charge α = scanned over (0, 0.15)
    Model parameter from the metric; not fitted to data, but the ranges in Sec. VI are selected to fall within EHT bounds.
  • spin parameter a = scanned over (0.7M, 0.99M)
    Kerr spin; scanned because a is not independently known for M87 and Sgr A*.
assumptions (3)
  • domain assumption Metric (4) is the rotating global monopole spacetime
    The paper assumes this without proof, citing [14]. It should reduce to (1) for a=0 but does not.
  • standard math Carter separability of the Hamilton-Jacobi equation
    The geodesics are derived using Carter's method, which requires separability; this holds for the metric (4).
  • domain assumption The shadow area angular diameter (31) can be compared directly to EHT ring observations
    The EHT measures the bright photon ring, but the paper treats it as the shadow without a correction factor.

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Cite this review

Pith. "Pith review of Thermodynamics and Shadows of Kerr black holes endowed with a global monopole charge." pith.science (2026). https://pith.science/paper/2FKR6BQ5

@misc{pith2026250602653,
  author       = {Pith},
  title        = {Pith review of: Thermodynamics and Shadows of Kerr black holes endowed with a global monopole charge},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2FKR6BQ5}},
  note         = {Machine review of arXiv:2506.02653}
}
abstract

In this article, we present the thermodynamic and shadow properties of rotating black holes with global monopole charge. The angular diameter of Sgr A$^{*}$ black hole is 48.7 $\pm$ 7 $\mu as$, which is 8 $kps$ far away having a mass of $M = 4.0_{-0.6}^{+1.1} \times 10^6 M\odot$ as observed by Event Horizon Telescope and for the M87 black hole, the observed angular diameter is $\theta_d = 42 \pm 3 \mu$as, which is almost $16$ $Mpc$ far away with a mass of $M = (6.5 \pm 0.7) \times 10^9 M_\odot$. The global monopole charge parameter $\alpha$ strongly affects the shape and size of the black hole shadow. We derived all the necessary equations to obtain the angular diameter of the rotating black hole shadow with the effect of the global monopole charge parameter $\alpha$. For $\alpha$ $\in$ (0, 0.08) with $a$ $\in$ $(0.7 M, 0.99 M)$, the angular diameter of M87 black hole shadow varies from $39$ $\mu as$ to $51$ $\mu as$. The angular diameter of Sgr A$^{*}$ black hole with global monopole charge parameter $\alpha$ $\in$ (0, 0.04) and $a$ $\in$ $(0.7 M, 0.99 M)$, varies from $50$ $\mu as$ to $55$ $\mu as$. For bound values of $\alpha$ and $a$, our results are consistent with the EHT observations.

Figures

Figures reproduced from arXiv: 2506.02653 by the authors.

Figure 1
Figure 1. Variation of black hole horizons for different value [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Variation of black hole horizons for different value [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Variation of black hole horizons for different value [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Variation of temperature with respect to the horiz [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Variation of the heat capacity with respect to the h [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Variation of the Gibbs free energy with respect to t [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: Effective potential of the black hole for different val [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: Plot showing the black hole shadow with the variati [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: Plot showing the black hole shadow with the variati [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: Plot showing the variation of black hole shadow ra [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: Plot showing the variation of black hole shadow ar [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: Variation of angular diameter of M87 black hole wi [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: Variation of angular diameter of Sgr A∗ black hole with black hole spin and global monopole charge parameter B. Sgr A∗ black hole The Sgr A∗ black hole is situated at the heart of our galaxy, the Milky Way [21–26]. The observed mass of this black hole is 4 × 106M⊙ and…

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Reference graph

Works this paper leans on

96 extracted references · 22 canonical work pages

  1. [14]

    Singha and N

    C. Singha and N. Dadhich, [arXiv:2504.16949 [gr-qc]]

    work page arXiv

  2. [1]

    Schwarzschild, Sitzungsber

    K. Schwarzschild, Sitzungsber. Preuss. Akad. Wiss. Ber lin (Math. Phys. ) 1916, 189-196 (1916) [arXiv:physics/9905030 [physics]]

    arXiv 1916

  3. [2]

    Reissner, Annalen der Physik , vol

    H. Reissner, Annalen der Physik , vol. 355, no. 9, pp. 106–120 (1916)

    1916

  4. [3]

    Nordstr¨ om,Koninklijke Nederlandsche Akademie van Wetenschappen Pro ceedings, vol

    G. Nordstr¨ om,Koninklijke Nederlandsche Akademie van Wetenschappen Pro ceedings, vol. 20, pp. 1238–1245 (1918)

    1918

  5. [4]

    Weyl, Annalen der Physik , vol

    H. Weyl, Annalen der Physik , vol. 359, no. 18, pp. 117–145 (1917)

    1917

  6. [5]

    G. B. Jeffery, Proceedings of the Royal Society of London. Series A, Contain ing Papers of a Mathematical and Physical Character , vol. 99, no. 697, pp. 123–134 (1921)

    1921

  7. [6]

    Gravitational field of a spinning mass as an ex ample of algebraically special metrics,

    R. P. Kerr, “Gravitational field of a spinning mass as an ex ample of algebraically special metrics,” Physical Review Letters , vol. 11, no. 5, pp. 237–238 (1963)

    1963

  8. [7]

    Note on the Kerr spinning-pa rticle metric,

    E. T. Newman and A. I. Janis, “Note on the Kerr spinning-pa rticle metric,” Journal of Mathematical Physics , vol. 6, no. 6, pp. 915–917 (1965)

    1965

Show all 96 references
  1. [8]

    Metric of a rotating, charged mass,

    E. T. Newman, R. Couch, K. Chinnapared, A. Exton, A. Praka sh, and R. Torrence, “Metric of a rotating, charged mass,” Journal of Mathematical Physics , vol. 6, no. 6, pp. 918–919 20 (1965)

    1965

  2. [9]

    Gravitational field of a glo bal monopole,

    M. Barriola and A. Vilenkin, “Gravitational field of a glo bal monopole,” Physical Review Letters, vol. 63, no. 4, pp. 341–343 (1989)

    1989

  3. [10]

    Dadhich, K

    N. Dadhich, K. Narayan and U. A. Yajnik, Pramana 50, 307-314 (1998) [arXiv:gr-qc/9703034 [gr-qc]]

    1998 arXiv

  4. [11]

    J. M. Bardeen, B. Carter, and S. W. Hawking, Commun. Math. Phys. 31, 161–170 (1973)

    1973

  5. [12]

    S. W. Hawking, Commun. Math. Phys. 43, 199 (1975), [Erratum: Commun. Math. Phys. 46, 206 (1976)]

    1975

  6. [13]

    J. D. Bekenstein, Phys. Rev. D 7, 2333–2346 (1973)

    1973

  7. [15]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, L1 (2019) [arXiv:1906.11238 [astro-ph.GA]]

    2019 arXiv

  8. [16]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, no.1, L5 (2019) [arXiv:1906.11242 [astro-ph.GA]]

    2019 arXiv

  9. [17]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, no.1, L6 (2019) [arXiv:1906.11243 [astro-ph.GA]]

    2019 arXiv

  10. [18]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, no.1, L3 (2019) [arXiv:1906.11240 [astro-ph.GA]]

    2019 arXiv

  11. [19]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, no.1, L4 (2019) [arXiv:1906.11241 [astro-ph.GA]]

    2019 arXiv

  12. [20]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, no.1, L2 (2019) [arXiv:1906.11239 [astro-ph.IM]]

    2019 arXiv

  13. [21]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, no.2, L15 (2022) [arXiv:2311.08697 [astro-ph.HE]]

    2022 arXiv

  14. [22]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, no.2, L16 (2022) [arXiv:2311.09478 [astro-ph.HE]]

    2022 arXiv

  15. [23]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, no.2, L13 (2022) [arXiv:2311.08679 [astro-ph.HE]]

    2022 arXiv

  16. [24]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, no.2, L14 (2022) [arXiv:2311.09479 [astro-ph.HE]]

    2022 arXiv

  17. [25]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, no.2, L12 (2022) 21 [arXiv:2311.08680 [astro-ph.HE]]

    2022 arXiv

  18. [26]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, no.2, L17 (2022) [arXiv:2311.09484 [astro-ph.HE]]

    2022 arXiv

  19. [27]

    J. L. Synge, Mon. Not. R. Astron. Soc. 131, 463–466 (1966)

    1966

  20. [28]

    J. M. Bardeen, Les Astres Occlus, edited by C. DeWitt and B. S. DeWitt (Gordon and Breach, New York, 1973), pp. 215–240

    1973

  21. [29]

    J. P. Luminet, Astron. Astrophys. 75, 228–235 (1979)

    1979

  22. [30]

    R. A. Cunningham, E. Gabathuler, M. Sproston, and D. G. T aylor, Conf. Proc. C 730508, 298 (1973)

    1973

  23. [31]

    Bozza, Gen

    V. Bozza, Gen. Rel. Grav. 42, 2269-2300 (2010) [arXiv:0911.2187 [gr-qc]]

    2010 arXiv

  24. [32]

    Falcke, F

    H. Falcke, F. Melia and E. Agol, Astrophys. J. Lett. 528, L13 (2000) [arXiv:astro-ph/9912263 [astro-ph]]

    2000 arXiv

  25. [33]

    de Vries, Class

    A. de Vries, Class. Quantum Grav. 17, 123–144 (2000)

    2000

  26. [34]

    Z. Q. Shen, K. Y. Lo, M. C. Liang, P. T. P. Ho and J. H. Zhao, N ature 438, 62 (2005) [arXiv:astro-ph/0512515 [astro-ph]]

    2005 arXiv

  27. [35]

    Yumoto, D

    A. Yumoto, D. Nitta, T. Chiba and N. Sugiyama, Phys. Rev. D 86, 103001 (2012) [arXiv:1208.0635 [gr-qc]]

    2012 arXiv

  28. [36]

    Atamurotov, A

    F. Atamurotov, A. Abdujabbarov and B. Ahmedov, Phys. Re v. D 88, no.6, 064004 (2013)

    2013

  29. [37]

    A. A. Abdujabbarov, L. Rezzolla and B. J. Ahmedov, Mon. N ot. Roy. Astron. Soc. 454, no.3, 2423-2435 (2015) [arXiv:1503.09054 [gr-qc]]

    2015 arXiv

  30. [38]

    P. V. P. Cunha and C. A. R. Herdeiro, Gen. Rel. Grav. 50, no.4, 42 (2018) [arXiv:1801.00860 [gr-qc]]

    2018 arXiv

  31. [39]

    Kumar and S

    R. Kumar and S. G. Ghosh, Astrophys. J. 892, 78 (2020) doi:10.3847/1538-4357/ab77b0 [arXiv:1811.01260 [gr-qc]]

    2020 arXiv

  32. [40]

    Afrin and S

    M. Afrin and S. G. Ghosh, Universe 8, no.1, 52 (2022) [arXiv:2112.15038 [gr-qc]]

    2022 arXiv

  33. [41]

    Hioki and K

    K. Hioki and K. i. Maeda, Phys. Rev. D 80, 024042 (2009) [arXiv:0904.3575 [astro-ph.HE]]

    2009 arXiv

  34. [42]

    Chen and J

    S. Chen and J. Jing, JCAP 05, 023 (2024) [arXiv:2310.06490 [gr-qc]]

    2024 arXiv

  35. [43]

    X. Q. Li, H. P. Yan, X. J. Yue, S. W. Zhou and Q. Xu, JCAP 05, 048 (2024) [arXiv:2401.18066 [gr-qc]]

    2024 arXiv

  36. [44]

    Amarilla, E

    L. Amarilla, E. F. Eiroa and G. Giribet, Phys. Rev. D 81, 124045 (2010) [arXiv:1005.0607 [gr-qc]]. 22

    2010 arXiv

  37. [45]

    Amarilla and E

    L. Amarilla and E. F. Eiroa, Phys. Rev. D 85, 064019 (2012) [arXiv:1112.6349 [gr-qc]]

    2012 arXiv

  38. [46]

    Amarilla and E

    L. Amarilla and E. F. Eiroa, Phys. Rev. D 87, no.4, 044057 (2013) [arXiv:1301.0532 [gr-qc]]

    2013 arXiv

  39. [47]

    M. Amir, B. P. Singh and S. G. Ghosh, Eur. Phys. J. C 78, no.5, 399 (2018) [arXiv:1707.09521 [gr-qc]]

    2018 arXiv

  40. [48]

    B. P. Singh and S. G. Ghosh, Annals Phys. 395, 127-137 (2018) [arXiv:1707.07125 [gr-qc]]

    2018 arXiv

  41. [49]

    Mizuno, Z

    Y. Mizuno, Z. Younsi, C. M. Fromm, O. Porth, M. De Laurent is, H. Olivares, H. Falcke, M. Kramer and L. Rezzolla, Nature Astron. 2, no.7, 585-590 (2018) [arXiv:1804.05812 [astro- ph.GA]]

    2018 arXiv

  42. [50]

    Allahyari, M

    A. Allahyari, M. Khodadi, S. Vagnozzi and D. F. Mota, JCA P 02, 003 (2020) [arXiv:1912.08231 [gr-qc]]

    2020 arXiv

  43. [51]

    Papnoi, F

    U. Papnoi, F. Atamurotov, S. G. Ghosh and B. Ahmedov, Phy s. Rev. D 90, no.2, 024073 (2014) [arXiv:1407.0834 [gr-qc]]

    2014 arXiv

  44. [52]

    Kumar, S

    R. Kumar, S. G. Ghosh and A. Wang, Phys. Rev. D 101, no.10, 104001 (2020) [arXiv:2001.00460 [gr-qc]]

    2020 arXiv

  45. [53]

    Kumar and S

    R. Kumar and S. G. Ghosh, JCAP 07, 053 (2020) [arXiv:2003.08927 [gr-qc]]

    2020 arXiv

  46. [54]

    Kumar, B

    R. Kumar, B. P. Singh and S. G. Ghosh, Annals Phys. 420, 168252 (2020) [arXiv:1904.07652 [gr-qc]]

    2020 arXiv

  47. [55]

    Kumar, B

    R. Kumar, B. P. Singh, M. S. Ali and S. G. Ghosh, Phys. Dark Univ. 34, 100881 (2021) [arXiv:1712.09793 [gr-qc]]

    2021 arXiv

  48. [56]

    B. P. Singh, Annals Phys. 441, 168892 (2022) [arXiv:1711.02898 [gr-qc]]

    2022 arXiv

  49. [57]

    Kumar, A

    R. Kumar, A. Kumar and S. G. Ghosh, Astrophys. J. 896, no.1, 89 (2020) [arXiv:2006.09869 [gr-qc]]

    2020 arXiv

  50. [58]

    Kumar, S

    R. Kumar, S. U. Islam and S. G. Ghosh, Eur. Phys. J. C 80, no.12, 1128 (2020) [arXiv:2004.12970 [gr-qc]]

    2020 arXiv

  51. [59]

    Kumar and S

    R. Kumar and S. G. Ghosh, Class. Quant. Grav. 38, no.8, 8 (2021) [arXiv:2004.07501 [gr-qc]]

    2021 arXiv

  52. [60]

    Kumar, S

    R. Kumar, S. G. Ghosh and A. Wang, Phys. Rev. D 100, no.12, 124024 (2019) [arXiv:1912.05154 [gr-qc]]

    2019 arXiv

  53. [61]

    Kumar, D

    A. Kumar, D. V. Singh and S. Upadhyay, Int. J. Mod. Phys. A 39, no.31, 2450136 (2024) [arXiv:2412.14230 [gr-qc]]

    2024 arXiv

  54. [62]

    B. K. Vishvakarma, D. V. Singh and S. Siwach, Eur. Phys. J . Plus 138, no.6, 536 (2023) [arXiv:2304.14754 [gr-qc]]. 23

    2023 arXiv

  55. [63]

    S. G. Ghosh, R. Kumar and S. U. Islam, JCAP 03, 056 (2021) [arXiv:2011.08023 [gr-qc]]

    2021 arXiv

  56. [64]

    Guo and P

    M. Guo and P. C. Li, Eur. Phys. J. C 80, no.6, 588 (2020) [arXiv:2003.02523 [gr-qc]]

    2020 arXiv

  57. [65]

    Afrin and S

    M. Afrin and S. G. Ghosh, Astrophys. J. 932, no.1, 51 (2022) [arXiv:2110.05258 [gr-qc]]

    2022 arXiv

  58. [66]

    Vagnozzi, R

    S. Vagnozzi, R. Roy, Y. D. Tsai, L. Visinelli, M. Afrin, A . Allahyari, P. Bambhaniya, D. Dey, S. G. Ghosh and P. S. Joshi, et al. Class. Quant. Grav. 40, no.16, 165007 (2023) [arXiv:2205.07787 [gr-qc]]

    2023 arXiv

  59. [67]

    Vagnozzi and L

    S. Vagnozzi and L. Visinelli, Phys. Rev. D 100, no.2, 024020 (2019) [arXiv:1905.12421 [gr-qc]]

    2019 arXiv

  60. [68]

    Afrin, R

    M. Afrin, R. Kumar and S. G. Ghosh, Mon. Not. Roy. Astron. Soc. 504, no.4, 5927-5940 (2021) [arXiv:2103.11417 [gr-qc]]

    2021 arXiv

  61. [69]

    X. J. Gao, T. T. Sui, X. X. Zeng, Y. S. An and Y. P. Hu, Eur. Ph ys. J. C 83, 1052 (2023) [arXiv:2311.11780 [gr-qc]]

    2023 arXiv

  62. [70]

    S. G. Ghosh and R. K. Walia, doi:10.1142/9789811269776 0084 [arXiv:2203.07775 [gr-qc]]

    arXiv

  63. [71]

    S. Li, T. Mirzaev, A. A. Abdujabbarov, D. Malafarina, B. Ahmedov and W. B. Han, Phys. Rev. D 106, no.8, 084041 (2022) [arXiv:2207.10933 [gr-qc]]

    2022 arXiv

  64. [72]

    Sengo, P

    I. Sengo, P. Cunha, V.P., C. A. R. Herdeiro and E. Radu, JC AP 01, 047 (2023) [arXiv:2209.06237 [gr-qc]]

    2023 arXiv

  65. [73]

    W. Liu, D. Wu, X. Fang, J. Jing and J. Wang, JCAP 08, 035 (2024) [arXiv:2406.00579 [gr-qc]]

    2024 arXiv

  66. [74]

    Umarov, O

    D. Umarov, O. Yunusov, F. Atamurotov, A. Abdujabbarov a nd S. G. Ghosh, Chin. Phys. C 49, no.5, 055102 (2025)

    2025

  67. [75]

    Ahmed, S

    F. Ahmed, S. U. Islam and S. G. Ghosh, JHEAp 46, 100350 (2025)

    2025

  68. [76]

    Turakhonov, F

    Z. Turakhonov, F. Atamurotov, S. G. Ghosh and A. Abdujab barov, Phys. Dark Univ. 48, 101880 (2025)

    2025

  69. [77]

    B. P. Singh, Phys. Dark Univ. 42, 101279 (2023) [arXiv:2301.00956 [gr-qc]]

    2023 arXiv

  70. [78]

    B. P. Singh, M. S. Ali and S. G. Ghosh, Commun. Theor. Phys . 77, no.7, 075407 (2025) [arXiv:2207.11907 [gr-qc]]

    2025 arXiv

  71. [79]

    Kumar Walia, S

    R. Kumar Walia, S. G. Ghosh and S. D. Maharaj, Astrophys. J. 939, no.2, 77 (2022) [arXiv:2207.00078 [gr-qc]]

    2022 arXiv

  72. [80]

    C. Liu, T. Zhu, Q. Wu, K. Jusufi, M. Jamil, M. Azreg-A ¨ ınou and A. Wang, Phys. Rev. D 101, no.8, 084001 (2020) [erratum: Phys. Rev. D 103, no.8, 089902 (2021)] [arXiv:2003.00477 [gr-qc]]

    2020 arXiv

  73. [81]

    Brahma, C

    S. Brahma, C. Y. Chen and D. h. Yeom, Phys. Rev. Lett. 126, no.18, 181301 (2021) 24 [arXiv:2012.08785 [gr-qc]]

    2021 arXiv

  74. [82]

    Kumar Walia, JCAP 03, 029 (2023) [arXiv:2207.02106 [gr-qc]]

    R. Kumar Walia, JCAP 03, 029 (2023) [arXiv:2207.02106 [gr-qc]]

    2023 arXiv

  75. [83]

    S. U. Islam, J. Kumar, R. Kumar Walia and S. G. Ghosh, Astr ophys. J. 943, no.1, 22 (2023) [arXiv:2211.06653 [gr-qc]]

    2023 arXiv

  76. [84]

    Afrin, S

    M. Afrin, S. Vagnozzi and S. G. Ghosh, Astrophys. J. 944, no.2, 149 (2023) [arXiv:2209.12584 [gr-qc]]

    2023 arXiv

  77. [85]

    J. Yang, C. Zhang and Y. Ma, Eur. Phys. J. C 83, no.7, 619 (2023) [arXiv:2211.04263 [gr-qc]]

    2023 arXiv

  78. [86]

    B. P. Singh, R. Kumar and S. G. Ghosh, New Astron. 99, 101945 (2023)

    2023

  79. [87]

    Y. Hou, M. Guo and B. Chen, Phys. Rev. D 104, no.2, 024001 (2021) [arXiv:2103.04369 [gr-qc]]

    2021 arXiv

  80. [88]

    Vagnozzi and L

    S. Vagnozzi and L. Visinelli, Res. Notes AAS 6, no.5, 106 (2022) [arXiv:2205.11314 [astro- ph.GA]]

    2022 arXiv

  81. [89]

    Banerjee, S

    I. Banerjee, S. Chakraborty and S. SenGupta, Phys. Rev. D 106, no.8, 084051 (2022) [arXiv:2207.09003 [gr-qc]]

    2022 arXiv

  82. [90]

    A. S. Lemos, J. A. V. Campos and F. A. Brito, Phys. Rev. D 110, no.6, 064079 (2024) [arXiv:2407.04609 [gr-qc]]

    2024 arXiv

  83. [91]

    B. P. Singh, Annals Phys. 470, 169803 (2024) [arXiv:2409.07951 [gr-qc]]

    2024 arXiv

  84. [92]

    Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, New York, 1985)

    S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, New York, 1985)

    1985

  85. [93]

    Carter, Phys

    B. Carter, Phys. Rev. Lett. 26, 331-333 (1971)

    1971

  86. [94]

    A. V. Frolov and V. P. Frolov, Phys. Rev. D 90, no.12, 124010 (2014) [arXiv:1408.6316 [gr-qc]]

    2014 arXiv

  87. [95]

    Black holes,

    J. Bardeen, “Black holes,” in Black Holes , edited by C. DeWitt and B. S. DeWitt (Gordon and Breach, New York, 1973)

    1973

  88. [96]

    Bambi, K

    C. Bambi, K. Freese, S. Vagnozzi and L. Visinelli, Phys. Rev. D 100, no.4, 044057 (2019) [arXiv:1904.12983 [gr-qc]]. 25

    2019 arXiv

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Reviewed August 7, 2026 · model on record in the stance chip above.