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REVIEW 3 major objections 6 minor 83 references

Influence of external magnetic fields on charged particle motion around a Schwarzschild-like black hole

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read In a Schwarzschild-like black hole immersed in an external magnetic field, charged particles can keep stable circular orbits arbitrarily close to the event horizon, and collisions between these particles and neutral ones falling from…

desk verdict The Section IV rescaling error breaks the paper's main quantitative claims; the original-variable parts are standard, but the ISCO and collision results don't follow from the stated metric. read the letter →

arxiv 2506.23103 v1 pith:LGEDWEWG submitted 2025-06-29 gr-qc

Sojida Mannobova , Gulnisa Abdukayumova , Farruh Atamurotov , Ahmadjon Abdujabbarov , X. Gao , G. Mustafa This is my paper
classification gr-qc MSC 83C5783C1083C22 PACS 04.70.Bw95.30.Sf97.60.Lf
keywords Schwarzschild-likeblackholechargedparticledynamicsISCOexternalmagneticfieldcenter-of-massenergyquasi-periodicoscillationsrelativisticprecessionmodeleffectivepotential
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 studies the motion of charged and neutral test particles around a Schwarzschild-like black hole whose metric carries an extra parameter $a$, with the hole immersed in an external magnetic field. It claims that the magnetic field systematically shrinks the innermost stable circular orbit (ISCO) of charged particles compared with neutral ones, and that for strong fields the stable orbit can approach arbitrarily close to the event horizon. It further claims that a collision between a charged particle on such an ISCO and a neutral particle falling from infinity can yield unbounded center-of-mass energy, so the magnetosphere acts as a natural particle accelerator. The same orbital framework is then used to compute radial and vertical oscillation frequencies, and the relativistic precession model is applied to twin-peak quasi-periodic oscillations with a 3:2 frequency ratio, giving frequencies in the range observable by X-ray missions. If true, the results would link black hole magnetospheres, high-energy particle acceleration, and quasi-periodic X-ray signals to the single parameter pair $(a,b)$.

What carries the argument

The load-bearing object is the dimensionless effective potential $$U=\left(1-$e^{{-a/\rho}}$\right)\left(1+\$beta^{2}$\right), \qquad \$\beta$=\frac{1}{\rho}\left(l+\frac{2ab}{$r_h^{2}$}-b\rho\right),$$ built from the rescaled radial coordinate $\rho=r/r_h$ and the magnetic parameter $b=qB r_h/(2m)$. The ISCO is defined by $U_{\rho}=0$ and $U_{\rho\rho}=0$; solving these two equations gives closed expressions for the magnetic parameter $b_{\pm}$, the angular momentum $l_{\pm}$, and the Lorentz factor $\gamma_{\pm}$ of the orbiting particle. The collision-energy result comes from combining $\gamma_{\pm}$ with the geodesic four-momentum of an infalling neutral particle, and the divergence of $(1-e^{-a/\rho})^{-1/2}$ at the horizon is what drives the center-of-mass energy to infinity. In the QPO section the same circular orbits are fed into the relativistic precession model, with the vertical epicyclic frequency equal to the orbital frequency and the radial frequency obtained from the second derivative of the effective potential.

What would settle it

Recompute the ISCO and collision-energy quantities using the original metric $f(r)=1-2M e^{-a/r}/r$ with $M=1$ and the horizon radius from Eq. (3), without replacing $f$ by $1-e^{-a/\rho}$: if the charged ISCO radius does not tend to zero as $b\to\infty$, or if the center-of-mass energy in Eq. (60) stays finite under the original $f$, the paper's central claim fails. The special case $a=0$ is decisive, since the paper's dimensionless $f$ vanishes while the original is $f=1-1/\rho$.

Watch

Extended reading notes

Core claim

The central claim is that the exponential Schwarzschild-like geometry $f(r)=1-2M e^{-a/r}/r$ combined with an external magnetic field qualitatively changes the location of stable circular orbits and the energy released in particle collisions near the black hole. For neutral particles the ISCO sits at the standard Schwarzschild radius and moves inward as the parameter $a$ grows; for charged particles the effective potential contains a magnetic term proportional to $b=qB/(2m)$, and the ISCO radius decreases monotonically with $b$, approaching the event horizon in the strong-field limit. The paper then derives the center-of-mass energy for a charged ISCO particle colliding with a neutral particle falling from infinity, $$$M^{2}$ = $m^{2}$ + $n^{2}$ + 2m\gamma E\left[(1-$e^{{-a/\rho}}$)^{-1/2} - \frac{v l_z}{\rho}\right],$$ and argues that the divergent first term makes this energy unbounded as the collision point approaches the horizon. Finally, the same circular orbits are used in the relativistic precession model, where the upper twin-peak QPO frequency is the orbital frequency and the lower one is the orbital minus radial frequency, producing 3:2 resonances whose values depend on both $a$ and $b$.

Load-bearing premise

The load-bearing premise is that Section IV's dimensionless metric function $f=1-e^{-a/\rho}$ faithfully represents the original metric $f(r)=1-2M e^{-a/r}/r$ after the rescaling $\rho=r/r_h$ with $M=1$; for $a=0$ the dimensionless function gives $f=0$ instead of $f=1-1/\rho$, and for general $a$ the horizon is not at $\rho=1$, so the ISCO and collision-energy formulas built on this rewriting inherit its inaccuracy.

Editorial extensions

If this is right

  • Charged-particle ISCO radii are systematically smaller than neutral ones and decrease as the magnetic parameter $b$ grows, so magnetized disks can extend closer to the hole than unmagnetized Keplerian disks.
  • In the strong-field limit the stable circular orbit can sit arbitrarily close to the event horizon, giving a regime in which the black hole magnetosphere rather than the vacuum geometry controls the inner edge of the disk.
  • Collisions between charged ISCO particles and neutral particles falling from infinity can reach unbounded center-of-mass energies near the horizon, identifying the magnetosphere as a natural particle accelerator.
  • The relativistic precession model applied to these orbits produces twin-peak QPO frequencies with a 3:2 ratio whose values depend on both $a$ and $b$, offering timing signatures for current and future X-ray missions.
  • The parameter $a$ behaves like a mass reduction: larger $a$ lowers the energy and angular momentum needed for circular orbits and shifts the QPO resonance radius inward.

Reading between the lines

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

  • The qualitative direction of the results, smaller charged ISCOs for stronger magnetic fields and growing collision energy near the horizon, is consistent with the known behavior of charged particles around magnetized Schwarzschild black holes; the exponential parameter $a$ modifies the radii and frequencies rather than creating the effect.
  • Because the dimensionless rewriting in Section IV is inconsistent at $a=0$, the quantitative formulas for ISCO radii, $\gamma$ factors, and collision energies should be re-derived with the exact horizon rescaling before being used to fit X-ray data; the qualitative conclusions are likely to survive, but the numbers may change.
  • The intersection of the upper and lower QPO frequency curves at the 3:2 resonance fixes an orbital radius, so a stellar-mass black hole with measured twin-peak QPOs could in principle constrain the pair $(a,b)$ independently of other observations.
  • The same combination of exponential metric and magnetic parameter could be applied to other regular black hole spacetimes, and a testable extension would be to include plasma effects or radiation reaction to see whether the unbounded collision energy is washed out.
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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 / 6 minor

Summary. This manuscript studies geodesic and charged-particle motion in a Schwarzschild-like spacetime with metric function f(r)=1−2M e^{-a/r}/r, immersed in an external test magnetic field. It derives effective potentials, ISCO conditions, and critical angular momenta for neutral and charged particles; uses a Frolov-type collision setup to argue that charged ISCO particles colliding with neutral infalling particles can produce arbitrarily large center-of-mass energies; and applies the relativistic precession model to compute twin-peak QPO frequencies. The main quantitative results are presented in a dimensionless form in Section IV and summarized in Table I and Figures 12–17.

Significance. The topic is of current interest: magnetized black hole spacetimes as natural particle accelerators and QPO diagnostics are active lines of research. The paper applies standard methods (Hamilton–Jacobi equations, Frolov collision formalism, RP model) and extends them to a regular Schwarzschild-like metric. If the Section IV derivation were valid, the claimed reduction of charged-particle ISCOs toward the horizon and the possibility of unbounded collision energies would be a useful addition to the literature. However, the central dimensionless rewriting in Section IV.A is incorrect, and the paper's headline claims are not supported by its own equations. I find no circularity problem of the fitting type: the paper makes no fit to observational data, and the QPO section is not calibrated. The paper also ships no machine-checked proofs or code; the main deliverable is analytic formulas and figures.

major comments (3)
  1. [Section IV.A, Eqs. (33)-(38)] The dimensionless metric function f = 1 - e^{-a/rho} introduced before Eq. (33) does not follow from Eq. (2). With M=1 and rho = r/r_h, Eq. (2) reads f(rho) = 1 - (2M/r_h) e^{-a/(r_h rho)}/rho, which reduces to 1 - 1/rho for a=0 (using r_h=2M). The paper's f vanishes identically at a=0 and has no real zero for a>0 (f tends to 0 only as rho -> infinity), so the would-be horizon is displaced to infinity. Because Eqs. (37)-(38), (46), (57), and (60) are all derived from this f, the ISCO radii, gamma factors, and collision energies in Table I and Figs. 12-17 are not predictions for the spacetime of Eq. (2). The abstract's claims that charged-particle ISCOs approach the event horizon and that collisions can produce unbounded center-of-mass energies therefore rest on an unphysical effective metric.
  2. [Section IV.A, Eq. (34)] The equation dt/dsigma = E rho/(rho - e^a) is not consistent with f = 1 - e^{-a/rho}. From Eq. (31), dt/dtau = E/f, so with the stated f one would get dt/dsigma = E/(1 - e^{-a/rho}), which does not simplify to E rho/(rho - e^a). The printed form corresponds instead to f = 1 - e^a/rho, which is a different metric function. The dynamical system in Eqs. (33)-(35) is therefore mutually inconsistent as written.
  3. [Section III.A, Eq. (18)] The vector potential A_phi = (1/2) B (r^2 - 2Ma) sin^2 theta is presented as the solution of Maxwell's equation (17), but for a != 0 it is not exact: substituting psi = r^2 - 2Ma into Eq. (17) yields a residual 4Ma(1 - e^{-a/r}). The text says only that a Taylor series expansion was used, without stating the expansion parameter, the order of truncation, or the regime of validity. Since this potential enters the charged-particle effective potential Eq. (27) and hence all charged ISCO and collision results, the accuracy of the approximation at the parameter values used (a/M = 0.1, 0.2 and r approximately r_h) must be quantified before the results can be trusted.
minor comments (6)
  1. [Section IV.A, notation] The symbol "rhea" appears in the definitions of rho, sigma, l, and b; it should be r_h throughout.
  2. [Eqs. (37)-(38)] The notation b^2_+/- is confusing because b is already a dimensionless magnetic parameter; the sign convention for the branches b_+/- and l_+/- should be stated explicitly so the reader can identify which branch corresponds to gamma_+/- in Eq. (46).
  3. [Fig. 1] The label "matric parameter" is a typo; the horizontal axis should be labeled with the dimensionless combination a/M.
  4. [Section III.A, Eq. (18)] The Taylor expansion leading to Eq. (18) should specify the small parameter used and the order of truncation, since the residual of Eq. (17) is not negligible for the parameter values considered later.
  5. [Section V, Eq. (66)] The units conversion should state explicitly that M is measured in solar masses and that Omega is a dimensionless angular frequency; otherwise the numerical frequency values in Hz are not reproducible.
  6. [Section VI, Conclusion] The sentence "the angular component of the magnetic field, i.e., the radial component, increases" is confusing and should be rephrased.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ISCO, collision-energy, and QPO results are derived from external metric and Maxwell inputs through the paper's own equations; self-citations are not load-bearing. The Section IV rescaling error is a correctness defect, not a circular step.

full rationale

The paper's derivations are self-contained and none reduce to their inputs by construction. The spacetime (Eqs. 1-2) and its 'eye of the storm' metric function are taken from external sources (Simpson-Visser [71,74]; Culetu [75,76]); the horizon is obtained from the Lambert-function condition grr=0 (Eq. 3), not assumed. The magnetic 4-potential is obtained by solving Maxwell's equation (Eqs. 14-18) with Wald's external-field ansatz [3]; the charged-particle radial equation and effective potential (Eqs. 26-27) follow from the Hamilton-Jacobi action, and the ISCOs are then solutions of U'=0, U''=0 (Eqs. 30, 33-38), with a and b as free parameters—no parameter is fitted to any observed quantity. The collision energy (Eqs. 47-60) follows from four-momentum conservation and the tetrad normalization (Eqs. 42-46); the unbounded-energy mechanism is the standard divergence of the static-observer redshift factor f^{-1/2} at a horizon, an input property of the metric, not a fitted output. The QPO frequencies (Section V) are computed from the RP model with a stipulated 3:2 resonance and no calibration to observed sources. Self-citations are numerous but not load-bearing: the metric, Maxwell solution, horizon equation, and collision method are external to the authors, while the self-cited items (e.g., [78], [79]) are standard formulas the paper re-derives in-text, so rule 4 applies. The screening note that 'f = 1 − e^{-a/ρ}' (before Eq. 33) is not the correct dimensionless form of f(r)=1−2Me^{-a/r}/r (the 2M/r_h prefactor and 1/ρ factor are dropped; a=0 gives f=0 rather than f=1−1/ρ, and the effective horizon moves to ρ=∞). This makes the quantitative ISCO and unbounded-CM-energy claims in Table I and Figs. 12-17 rest on an effective metric different from Eq. (2), which is a correctness and support defect, not a circular reduction: no equation is identically equal to another by construction, and no fitted parameter is relabeled as a prediction. Per hard rules 1 and 7, the circularity score is therefore 0.

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

No free parameters are fitted to data; the metric parameter a and magnetic parameter b are model inputs from prior literature. The main assumptions are the test-field approximation for the magnetic field, the test-particle approximation, and a dimensionless rewriting of the metric that appears inconsistent with the original spacetime. No new physical entities are introduced.

assumptions (4)
  • domain assumption The spacetime is described by the Schwarzschild-like metric f(r) = 1 - 2M e^{-a/r}/r.
    The metric is taken from Refs. [71,75,76] and is not derived in this paper.
  • ad hoc to paper The external magnetic field is a test field described by A_φ = 1/2 B (r^2 - 2Ma) sin²θ, obtained by a Taylor expansion of the solution of Maxwell's equations.
    This approximate solution is introduced in Section III A without showing the expansion or quantifying its error.
  • domain assumption The particles are treated as test particles; back-reaction of fields and particles on the geometry is neglected.
    Standard in the literature, used throughout the paper.
  • ad hoc to paper The dimensionless rescaling ρ = r/r_h with metric function f = 1 - e^{-a/ρ} used in Section IV is equivalent to the original metric in Eq. (2).
    This equivalence is asserted in Section IV A but is not correct. For a=0 it gives f=0 instead of f=1-1/ρ, so the Schwarzschild limit is not recovered.

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Cite this review

Pith. "Pith review of Influence of external magnetic fields on charged particle motion around a Schwarzschild-like black hole." pith.science (2026). https://pith.science/paper/LGEDWEWG

@misc{pith2026250623103,
  author       = {Pith},
  title        = {Pith review of: Influence of external magnetic fields on charged particle motion around a Schwarzschild-like black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LGEDWEWG}},
  note         = {Machine review of arXiv:2506.23103}
}
read the original abstract

We investigate the dynamics of charged and neutral particles in the vicinity of a Schwarzschild-like black hole immersed in an external magnetic field. We find that the innermost stable circular orbits (ISCOs) for charged particles are systematically smaller than those of neutral particles, demonstrating a fundamental distinction in their orbital dynamics. In the presence of a strong magnetic field, charged particle ISCOs can approach arbitrarily close to the event horizon. We show that collisions between charged particles in ISCOs and neutral particles falling from infinity can produce unbounded center-of-mass energies in the strong-field regime, suggesting the black hole magnetosphere as a natural particle accelerator. Additionally, we apply the relativistic precession model to study quasi-periodic oscillations around the Schwarzschild-like black hole, treating orbital perturbations as coupled harmonic oscillators. Our results provide new insights into high-energy astrophysical processes near magnetized black holes and offer observational signatures through quasi-periodic oscillating frequencies that could be detected by current and future X-ray missions.

Figures

Figures reproduced from arXiv: 2506.23103 by the authors.

Figure 1
Figure 1. FIG. 1: Dependence of horizon radius from matric pa [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The Radial dependence of energy per mass [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4: The Radial dependence of the angular momen [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (11 more)
Figure 6
Figure 6. Figure 6: FIG. 6: The Radial dependence of orbital velocity of [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The magnetic field lines near a Schwarzschild-like black hole for various [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The radial dependence of the effective potential [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Radial dependence of the angular momentum per mass [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Radial dependence of the energy [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 13
Figure 13. Figure 13: FIG. 13: The value of the [PITH_FULL_IMAGE:figures/full_fig_p009_13.png]
Figure 12
Figure 12. Figure 12: FIG. 12: The value of the [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]
Figure 14
Figure 14. Figure 14: FIG. 14: Critical angular momentum as a function of [PITH_FULL_IMAGE:figures/full_fig_p010_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15: Radial dependency of Fundamental frequencies of massive charged particle in external magnetic field: Left [PITH_FULL_IMAGE:figures/full_fig_p012_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16: Upper (blue lines) and Lower (black lines) frequencies with [PITH_FULL_IMAGE:figures/full_fig_p013_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17: Relationship between the upper [PITH_FULL_IMAGE:figures/full_fig_p013_17.png]

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Works this paper leans on

83 extracted references · 56 canonical work pages

  1. [1]

    C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, Misner73 (W. H. Freeman, San Francisco, 1973)

    1973

  2. [2]

    R. D. Blandford and R. L. Znajek, Mon. Not. Roy. Astron. Soc. 179, 433 (1977)

    1977

  3. [3]

    R. M. Wald, Phys. Rev. D 10, 1680 (1974)

    1974

  4. [4]

    J. A. Petterson, Phys. Rev. D 10, 3166 (1974)

    work page 1974

  5. [5]

    J. A. Petterson, Phys. Rev. D 12, 2218 (1975)

    work page 1975

  6. [6]

    Preti, Class

    G. Preti, Class. Quant. Grav. 21, 3433 (2004)

    work page 2004

  7. [7]

    A. A. Abdujabbarov, B. J. Ahmedov, and V . G. Kagra- manova, General Relativity and Gravitation 40, 2515 (2008), arXiv:0802.4349 [gr-qc]

    work page Pith review arXiv 2008

  8. [9]

    Charged particle motion around a quasi-Kerr compact object immersed in an external magnetic field

    B. Narzilloev, A. Abdujabbarov, C. Bambi, and B. Ahme- dov, Phys. Rev. D99, 104009 (2019), arXiv:1902.03414 [gr- qc]

    work page Pith review arXiv 2019

Show all 83 references
  1. [10]

    J. Vrba, J. Rayimbaev, Z. Stuchlik, and B. Ahmedov, Eur. Phys. J. C 83, 854 (2023)

    2023

  2. [11]

    Jawad, F

    A. Jawad, F. Ali, M. Jamil, and U. Debnath, Com- munications in Theoretical Physics 66, 509 (2016), arXiv:1610.07411 [gr-qc]

    2016 arXiv

  3. [12]

    Hussain and M

    S. Hussain and M. Jamil, Phys. Rev. D 92, 043008 (2015), arXiv:1508.02123 [gr-qc]

    2015 arXiv

  4. [13]

    Jamil, S

    M. Jamil, S. Hussain, and B. Majeed, Eur. Phys. J. C 75, 24 (2015), arXiv:1404.7123 [gr-qc]

    2015 arXiv

  5. [14]

    Hussain, S, Hussain, I, and Jamil, M., Eur. Phys. J. C 74, 210 (2014)

    2014

  6. [15]

    De Laurentis, Z

    M. De Laurentis, Z. Younsi, O. Porth, Y. Mizuno, and L. Rezzolla, Phys. Rev. D 97, 104024 (2018), arXiv:1712.00265 [gr-qc]

    2018 arXiv

  7. [16]

    Narzilloev, J

    B. Narzilloev, J. Rayimbaev, S. Shaymatov, A. Abdujab- barov, B. Ahmedov, and C. Bambi, Phys. Rev. D 102, 044013 (2020), arXiv:2007.12462 [gr-qc]

    2020 arXiv

  8. [17]

    Kolo ˇs, A

    M. Kolo ˇs, A. Tursunov, and Z. Stuchl ´ık, Eur. Phys. J. C. 77, 860 (2017), arXiv:1707.02224 [astro-ph.HE]

    2017 arXiv

  9. [18]

    Kov ´aˇr, P

    J. Kov ´aˇr, P . Slan ´y, C. Cremaschini, Z. Stuchl ´ık, V . Karas, and A. Trova, Phys. Rev. D 90, 044029 (2014), arXiv:1409.0418 [gr-qc]

    2014 arXiv

  10. [19]

    A. N. Aliev and D. V . Gal’tsov, Soviet Physics Uspekhi32, 75 (1989)

    1989

  11. [20]

    A. N. Aliev and N. ¨Ozdemir, Mon. Not. R. Astron. Soc. 336, 241 (2002), gr-qc/0208025

    2002 arXiv

  12. [21]

    Shaymatov, F

    S. Shaymatov, F. Atamurotov, and B. Ahmedov, Astro- phys Space Sci 350, 413 (2014)

    2014

  13. [22]

    Abdujabbarov and B

    A. Abdujabbarov and B. Ahmedov, Phys. Rev. D 81, 044022 (2010), arXiv:0905.2730 [gr-qc]

    2010 arXiv

  14. [23]

    Abdujabbarov, B

    A. Abdujabbarov, B. Ahmedov, and A. Hakimov, Phys.Rev. D 83, 044053 (2011), arXiv:1101.4741 [gr-qc]

    2011 arXiv

  15. [24]

    Stuchl´ık and M

    Z. Stuchl´ık and M. Koloˇs, European Physical Journal C76, 32 (2016), arXiv:1511.02936 [gr-qc]

    2016 arXiv

  16. [25]

    Turimov, B

    B. Turimov, B. Ahmedov, A. Abdujabbarov, and C. Bambi, Phys. Rev. D 97, 124005 (2018), arXiv:1805.00005 [gr-qc]

    2018 arXiv

  17. [26]

    Shaymatov, J

    S. Shaymatov, J. Vrba, D. Malafarina, B. Ahmedov, and Z. Stuchl ´ık, Phys. Dark Universe 30, 100648 (2020), arXiv:2005.12410 [gr-qc]

    2020 arXiv

  18. [27]

    Shaymatov, D

    S. Shaymatov, D. Malafarina, and B. Ahmedov, Phys. Dark Univ. 34, 100891 (2021), arXiv:2004.06811 [gr-qc]

    2021 arXiv

  19. [28]

    de Felice and F

    F. de Felice and F. Sorge, Classical and Quantum Gravity 20, 469 (2003)

    2003

  20. [29]

    de Felice, F

    F. de Felice, F. Sorge, and S. Zilio, Classical and Quantum Gravity 21, 961 (2004)

    2004

  21. [30]

    Toshmatov, A

    B. Toshmatov, A. Abdujabbarov, B. Ahmedov, and Z. Stuchl´ık, Astrophys Space Sci 360, 19 (2015)

    2015

  22. [31]

    O. G. Rahimov, A. A. Abdujabbarov, and B. J. Ahme- dov, Astrophysics and Space Science 335, 499 (2011), arXiv:1105.4543 [astro-ph.SR]

    2011 arXiv

  23. [32]

    Davlataliev, B

    A. Davlataliev, B. Narzilloev, I. Hussain, A. Abdujab- barov, and B. Ahmedov, Eur. Phys. J. C84, 694 (2024)

    2024

  24. [33]

    Narzilloev, J

    B. Narzilloev, J. Rayimbaev, A. Abdujabbarov, and C. Bambi, The European Physical Journal C 80, 1074 (2020), arXiv:2005.04752 [gr-qc]

    2020 arXiv

  25. [34]

    Rayimbaev, A

    J. Rayimbaev, A. Abdujabbarov, M. Jamil, B. Ahmedov, and W.-B. Han, Phys. Rev. D102, 084016 (2020)

    2020

  26. [35]

    J. Vrba, A. Abdujabbarov, M. Kolo ˇs, B. Ahmedov, Z. Stuchl ´ık, and J. Rayimbaev, Phys.Rev.D 101, 124039 (2020)

    2020

  27. [36]

    Abdujabbarov, J

    A. Abdujabbarov, J. Rayimbaev, B. Turimov, and F. Ata- murotov, Physics of the Dark Universe30, 100715 (2020)

    2020

  28. [37]

    Davlataliev, F

    A. Davlataliev, F. Atamurotov, A. Abdujabbarov, N. Ju- raeva, and V . Khamidov, Phys. Dark Univ. 46, 101603 (2024)

    2024

  29. [38]

    Rakhimova, F

    G. Rakhimova, F. Atamurotov, F. Javed, A. Abdujabbarov, and G. Mustafa, Nucl. Phys. B 996, 116363 (2023). 16

    2023

  30. [39]

    Mustafa, A

    G. Mustafa, A. Ditta, F. Javed, F. Atamurotov, I. Hus- sain, and B. Ahmedov, Chin. J. Phys. 90, 494 (2024), arXiv:2401.08254 [gr-qc]

    2024 arXiv

  31. [40]

    Ladghami, B

    Y. Ladghami, B. Asfour, A. Bouali, A. Errahmani, and T. Ouali, Phys. Dark Univ. 44, 101470 (2024), arXiv:2403.08991 [hep-th]

    2024 arXiv

  32. [41]

    Wu and S.-W

    S.-P . Wu and S.-W. Wei, Phys. Rev. D 110, 024054 (2024), arXiv:2403.14167 [gr-qc]

    2024 arXiv

  33. [42]

    Ditta, F

    A. Ditta, F. Javed, G. Mustafa, S. K. Maurya, D. Sofuo ˘glu, and F. Atamurotov, Chin. J. Phys. 88, 287 (2024)

    2024

  34. [43]

    Penrose, Nuovo Cimento Rivista Serie 1, 252 (1969)

    R. Penrose, Nuovo Cimento Rivista Serie 1, 252 (1969)

    1969

  35. [44]

    Tursunov, B

    A. Tursunov, B. Juraev, Z. Stuchl ´ık, and M. Kolo ˇs, Phys. Rev. D 104, 084099 (2021), arXiv:2109.10288 [gr-qc]

    2021 arXiv

  36. [45]

    Stuchl ´ık, M

    Z. Stuchl ´ık, M. Kolo ˇs, and A. Tursunov, Universe 7, 416 (2021)

    2021

  37. [46]

    S. M. Wagh and N. Dadhich, Physics Reports 183, 137 (1989)

    1989

  38. [47]

    A. A. Abdujabbarov, B. J. Ahmedov, S. R. Shaymatov, and A. S. Rakhmatov, Astrophys Space Sci 334, 237 (2011), arXiv:1105.1910 [astro-ph.SR]

    2011 arXiv

  39. [48]

    Ba ˜nados, J

    M. Ba ˜nados, J. Silk, and S. M. West, Phys. Rev. Lett. 103, 111102 (2009)

    2009

  40. [49]

    Hackmann, H

    E. Hackmann, H. Nandan, and P . Sheoran, Phys. Lett. B 810, 135850 (2020), arXiv:2006.05045 [gr-qc]

    2020 arXiv

  41. [50]

    O. B. Zaslavskii, Phys. Rev. D 100, 024050 (2019), arXiv:1904.04874 [gr-qc]

    2019 arXiv

  42. [51]

    Stuchl ´ık, M

    Z. Stuchl ´ık, M. Blaschke, and J. Schee, Phys. Rev. D 96, 104050 (2017), arXiv:1711.10890 [gr-qc]

    2017 arXiv

  43. [52]

    Turimov, S

    B. Turimov, S. Shaymatov, and S. Hayitov, Eur. Phys. J. Plus 138, 1022 (2023)

    2023

  44. [53]

    Hejda, J

    F. Hejda, J. P . S. Lemos, and O. B. Zaslavskii, Phys. Rev. D 105, 024014 (2022), arXiv:2109.04477 [gr-qc]

    2022 arXiv

  45. [54]

    Atamurotov, S

    F. Atamurotov, S. Shaymatov, P . Sheoran, and S. Siwach, J. Cosmol. A. P .2021, 045 (2021), arXiv:2105.02214 [gr-qc]

    2021 arXiv

  46. [55]

    G. Z. Babar, F. Atamurotov, S. Ul Islam, and S. G. Ghosh, Phys. Rev. D 103, 084057 (2021), arXiv:2104.00714 [gr-qc]

    2021 arXiv

  47. [56]

    Nosirov, F

    A. Nosirov, F. Atamurotov, G. Rakhimova, A. Abdujab- barov, and S. G. Ghosh, Nucl. Phys. B1005, 116583 (2024)

    2024

  48. [57]

    Davlataliev, J

    A. Davlataliev, J. Rayimbaev, F. Abdulkhamidov, Z. Stuchlik, and A. Abdujabbarov, Phys. Dark Univ. 46, 101590 (2024)

    2024

  49. [58]

    Nosirov, F

    A. Nosirov, F. Atamurotov, G. Rakhimova, and A. Abdu- jabbarov, Eur. Phys. J. Plus138, 846 (2023)

    2023

  50. [59]

    Abdujabbarov, F

    A. Abdujabbarov, F. Atamurotov, N. Dadhich, B. Ahme- dov, and Z. Stuchl ´ık, Eur. Phys. J. C 75, 399 (2015), arXiv:1508.00331 [gr-qc]

    2015 arXiv

  51. [60]

    V . P . Frolov and A. A. Shoom, Phys. Rev. D. 82, 084034 (2010), arXiv:1008.2985 [gr-qc]

    2010 arXiv

  52. [61]

    V . P . Frolov and P . Krtouˇs, Phys. Rev. D 83, 024016 (2011), arXiv:1010.2266 [hep-th]

    2011 arXiv

  53. [63]

    Frolov and I

    V . Frolov and I. Novikov,Black Hole Physics: Basic Concepts and New Developments , Fundamental Theories of Physics (Springer Netherlands, 2012)

    2012

  54. [64]

    Tursunov, M

    A. Tursunov, M. Kolo ˇs, A. Abdujabbarov, B. Ahmedov, and Z. Stuchl´ık, Phys. Rev. D 88, 124001 (2013)

    2013

  55. [65]

    Tursunov, M

    A. Tursunov, M. Kolo ˇs, B. Ahmedov, and Z. Stuchl ´ık, Phys. Rev. D 87, 125003 (2013)

    2013

  56. [66]

    Rayimbaev, K

    J. Rayimbaev, K. F. Dialektopoulos, F. Sarikulov, and A. Abdujabbarov, Eur. Phys. J. C 83, 572 (2023), arXiv:2307.03019 [gr-qc]

    2023 arXiv

  57. [67]

    Abdulkhamidov, P

    F. Abdulkhamidov, P . Nedkova, J. Rayimbaev, J. Kunz, and B. Ahmedov, Phys. Rev. D 109, 104074 (2024), arXiv:2403.08356 [gr-qc]

    2024 arXiv

  58. [68]

    Abdulkhamidov, B

    F. Abdulkhamidov, B. Narzilloev, I. Hussain, A. Abdujab- barov, and B. Ahmedov, Eur. Phys. J. C84, 420 (2024)

    2024

  59. [69]

    Davlataliev, B

    A. Davlataliev, B. Narzilloev, I. Hussain, A. Abdujab- barov, and B. Ahmedov, Phys. Dark Univ. 46, 101569 (2024)

    2024

  60. [70]

    Ditta, F

    A. Ditta, F. Javed, G. Mustafa, F. Atamurotov, and S. Sal- imov, Journal of High Energy Astrophysics43, 51 (2024)

    2024

  61. [71]

    Simpson and M

    A. Simpson and M. Visser, Universe 6, 8 (2019), arXiv:1911.01020 [gr-qc]

    2019 arXiv

  62. [72]

    Simpson and M

    A. Simpson and M. Visser, JCAP 03 (03), 011, arXiv:2111.12329 [gr-qc]

    arXiv

  63. [73]

    Berry, A

    T. Berry, A. Simpson, and M. Visser, Universe 7, 2 (2020), arXiv:2008.13308 [gr-qc]

    2020 arXiv

  64. [74]

    Simpson and M

    A. Simpson and M. Visser, Phys. Rev. D 105, 064065 (2022)

    2022

  65. [75]

    Culetu, Int

    H. Culetu, Int. J. Theor. Phys. 54, 2855 (2015), arXiv:1408.3334 [gr-qc]

    2015 arXiv

  66. [76]

    Culetu, Int

    H. Culetu, Int. J. Mod. Phys. D 24, 1542001 (2015)

    2015

  67. [77]

    R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, Adv. Comput. Math. 5, 329 (1996)

    1996

  68. [78]

    Boboqambarova, B

    M. Boboqambarova, B. Turimov, and A. Abdujab- barov, Mod. Phys. Lett. A 38, 2350071 (2023), [Erratum: Mod.Phys.Lett.A 38, 2350071 (2023)], arXiv:2110.05764 [gr-qc]

    2023 arXiv

  69. [79]

    Narzilloev, J

    B. Narzilloev, J. Rayimbaev, S. Shaymatov, A. Abdujab- barov, B. Ahmedov, and C. Bambi, Phys. Rev. D 102, 104062 (2020), arXiv:2011.06148 [gr-qc]

    2020 arXiv

  70. [80]

    Frolov, Phys

    V . Frolov, Phys. Rev. D85, 024020 (2012), arXiv:1110.6274 [gr-qc]

    2012 arXiv

  71. [81]

    Bardeen, in Proceedings of the 5th International Conference on Gravitation and the Theory of Relativity (1968) p

    J. Bardeen, in Proceedings of the 5th International Conference on Gravitation and the Theory of Relativity (1968) p. 87

    1968

  72. [82]

    Stuchl ´ık and M

    Z. Stuchl ´ık and M. Koloˇs, Astronomy & Astrophysics586, A130 (2016), arXiv:1603.07366 [astro-ph.HE]

    2016 arXiv

  73. [83]

    R. A. Remillard, M. P . Muno, J. E. McClintock, and J. A. Orosz, Astrophys. J. 580, 1030 (2002), arXiv:astro- ph/0202305

    2002

  74. [84]

    Stella, in X-RAY ASTRONOMY: Stellar Endpoints, AGN, and the Diffuse X-ray Background , AIP Conference Pro- ceedings, Vol

    L. Stella, in X-RAY ASTRONOMY: Stellar Endpoints, AGN, and the Diffuse X-ray Background , AIP Conference Pro- ceedings, Vol. 599, edited by N. E. White, G. Malaguti, and G. G. C. Palumbo (2001) pp. 365–376, arXiv:astro- ph/0011395 [astro-ph]

    2001

  75. [85]

    Ingram and S

    A. Ingram and S. Motta, Monthly Notices of the Royal Astronomical Society 444, 2065 (2014), arXiv:1408.0884 [astro-ph.HE]

    2014 arXiv

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