Pith. sign in

REVIEW 5 major objections 5 minor 5 cited by

Observational features of the rotating Bardeen black hole surrounded by perfect fluid dark matter

T0 review · 5 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A rotating Bardeen black hole wrapped in perfect fluid dark matter would cast shadow and redshift patterns distinct from Kerr's.

desk verdict Solid incremental parameter survey of Bardeen-PFDM disk images, but the missing horizon check and unphysical figures make the current version unreliable. read the letter →

arxiv 2411.11680 v1 pith:6OCHZ4HW submitted 2024-11-18 astro-ph.HE

Ke-Jian He , Guo-Ping Li , Chen-Yu Yang , Xiao-Xiong Zeng This is my paper · ORCID
classification astro-ph.HE
keywords blackholeshadowrotatingBardeenperfectfluiddarkmatterraytracingthinaccretiondiskredshiftfactorgravitationallensinghorizon-scaleimaging
topics Dark Matter
open problems Dark Matter
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 uses ray tracing to predict what a rotating Bardeen black hole surrounded by perfect fluid dark matter would look like, under two background-light models: a celestial sphere and a thin accretion disk. It claims that the shadow contour, the brightness pattern of direct and lensed disk images, and the redshift distribution all respond in characteristic ways to the black hole's spin, magnetic charge, dark matter parameter, and the observer's inclination. In particular, higher inclination pushes the observed flux of the disk into the lower half of the image, larger dark matter strength enlarges both shadow and lensed-image regions, and redshift dominates at low inclination while blueshift appears only at high inclination. Because these features differ from those of a Kerr black hole, the paper argues they could serve as observational fingerprints for distinguishing this exotic spacetime from general-relativistic black holes.

What carries the argument

The machinery is backward ray tracing of null geodesics, performed in the frame of a zero angular momentum observer (ZAMO). The photon trajectories are integrated in the rotating Bardeen–PFDM metric (Eqs. 10–13), with the shadow boundary fixed by the photon-sphere conditions $R(r)=0$, $R'(r)=0$, and the observed intensity computed via a radiative-transfer formula $I_{\nu_o} = \sum_n F_n g_n^3 J_n$ that sums over the $n$ intersections of each ray with the equatorial thin disk. The redshift factor $g_n = \nu_o/\nu_n$ is split into circular-orbit and plunging-orbit pieces inside the ISCO, which is what produces the distinct red ring around the inner shadow.

What would settle it

A direct test would be to check whether the rotating metric (10)–(13) actually solves the Einstein–Maxwell equations with the PFDM energy-momentum tensor (6)–(7); if the field equations fail or the energy conditions are violated over the parameter range, the predicted images do not describe any real black hole. Observationally, high-resolution 230 GHz images of M87* or Sagittarius A* that do not show the predicted inclination-dependent flux concentration toward the lower half of the image, or that show a circular shadow where the model demands a D-shape for the same spin and inclination, would falsify the model's uniqueness claim.

Watch

Extended reading notes

Core claim

The central discovery asserted is that the observable appearance of the rotating Bardeen black hole surrounded by perfect fluid dark matter is a sensitive function of the spacetime parameters. For a celestial background, the shadow is disk-like at low spin and becomes D-shaped at high spin or high inclination, with its size growing as the absolute value of the dark matter parameter increases while magnetic charge mostly only distorts the shape. For a thin accretion disk, the image consists of direct, lensed, and photon-ring components; as the observer's inclination increases, their flux concentrates toward the lower half of the image, and the inner shadow deforms from circular to arched. The redshift-factor maps show that at small angles gravitational redshift dominates and no blueshift is visible, while at large angles Doppler shifts appear with blueshift on the left and redshift on the right; increasing spin, magnetic charge, or the absolute dark matter parameter suppresses both redshift and blueshift. The paper concludes these parameter-dependent patterns could provide a way to distinguish this model from other black hole spacetimes.

Load-bearing premise

The rotating Bardeen–PFDM metric, taken from the literature via the Newman–Janis algorithm, correctly describes a physical black hole surrounded by perfect-fluid dark matter, including the parameter ranges used (spin up to 0.99, dark matter parameter down to −0.9).

Editorial extensions

If this is right

  • Higher observer inclination compresses direct and lensed disk images toward the lower half of the image plane, so equatorial views of such a black hole would look markedly asymmetric.
  • Larger absolute dark matter parameter enlarges the shadow, the photon-ring radius, and the lensed-image region, while magnetic charge has a comparatively small effect on size.
  • Redshift dominates the spectral pattern at low inclination; blueshift appears only at large inclination, with the split left/right at 83 degrees.
  • Increasing spin, magnetic charge, or the absolute dark matter parameter attenuates both redshift and blueshift, meaning Doppler signatures weaken in stronger dark-matter halos.
  • The transition from a circular to a D-shaped shadow as spin or inclination rises is a direct observable discriminator against Schwarzschild and Kerr black holes.

Reading between the lines

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

  • The simulated images imply that a single high-inclination horizon-scale observation could test the flux-concentration-to-lower-half prediction, though real observations require the assumed disk emission profile to match the model's emissivity.
  • If the rotating metric obtained by the Newman–Janis algorithm is not a genuine solution of the coupled Einstein–nonlinear-electrodynamics–PFDM system, the predicted images lose their physical basis; checking the metric against the field equations or energy conditions would be the decisive test.
  • The same ray-tracing pipeline could be applied to other regular black hole metrics (Hayward, Ayón-Beato–García) surrounded by dark matter to see whether the claimed distinguishing patterns are unique to Bardeen–PFDM or generic.
  • Because the paper normalizes the fudge factor to unity and fixes emissivity coefficients, the quantitative brightness predictions are model-dependent; the shape and redshift trends are more robust than absolute flux levels.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The manuscript uses backward ray-tracing to compute shadow images, thin-disk images, and redshift maps for a rotating Bardeen black hole surrounded by perfect fluid dark matter. Two background light-source models are considered: a celestial sphere and an optically/geometrically thin accretion disk. The central claims are that the shadow morphology, the direct/lensed emission structure, and the redshift distribution depend on the spin parameter a, magnetic charge G, dark matter parameter α, and observer inclination in characteristic ways that could distinguish this spacetime from other black hole models.

Significance. The topic is timely and the numerical setup is standard: a ZAMO screen, backward null geodesic integration, and a decomposition into direct, lensed, and photon-ring contributions are appropriate. The paper extends the existing shadow-only analysis of Ref. [75] to accretion-disk images and redshift maps, which is a useful step for EHT-style model comparisons. However, the predictive value of the results is currently limited by several concrete issues: some displayed parameter sets do not describe a black hole, the printed geodesic equations are internally inconsistent, the intensity maps contain unphysical negative values, and the emissivity law is tuned for visual appeal rather than derived from a physical model. With corrections, the qualitative parameter trends could still be of interest, but as presented the quantitative claims are not reliable.

major comments (5)
  1. [II, III; Eq. (16), Fig. 3(c)] The paper never verifies that the parameter combinations used in the shadow and image plots admit an event horizon. For M=1 and (a,G,α)=(0.99,0.3,-0.01), the function Δr in Eq. (16) is positive for all r>0 (for example, Δr(1)≈0.18 and Δr(2)≈1.01), so this spacetime is a naked singularity rather than a black hole. Fig. 3(c) nevertheless presents a "black hole shadow" for exactly these parameters. The authors must restrict all plots to the horizon-existence region of the parameter space, or explicitly state that some images describe naked singularities; otherwise the abstract's black-hole interpretation is not supported.
  2. [II; Eqs. (19) and (21)] The printed separated Hamilton-Jacobi equations are internally inconsistent and cannot be used to reproduce the ray tracing. Eq. (21) contains cos θ where the standard Carter separation requires cos^2 θ, and the combination of terms in Eq. (19) does not match the subsequent definitions of R(r) and Θ(θ) in Eqs. (20) and (21). Since the images are obtained by integrating Eqs. (22)-(25), the manuscript should present the correct geodesic equations or state that the numerical integration uses corrected versions; as printed, the core numerical method is not reproducible.
  3. [IV.B; Fig. 5] Several intensity maps contain unphysical negative observed intensities. For example, Fig. 5(i) has colorbar values down to -4×10^6 and Fig. 5(j) down to -1.5×10^5. Because each term in Eq. (41) is a product of non-negative factors (fudge factor, g_n^3, and emissivity J_n), the observed intensity cannot be negative. This indicates an error in the ray-tracing or visualization pipeline and invalidates the quantitative flux comparisons in Figs. 5-10 unless corrected.
  4. [IV.A; Eq. (47)] The emissivity profile for the accretion-disk images is chosen with η1=-1/2 and η2=-2 "in order to achieve a more visually appealing effect that aligns with the 230 GHz image," rather than derived from a physical radiative model. The claimed parameter dependences of the direct and lensed fluxes are therefore not robust observables; they may be artifacts of this tuning. The authors should justify the choice from a disk model or show that the qualitative conclusions are stable under different emissivity profiles.
  5. [V; Eq. (45)] The redshift-factor formulas contain an apparent normalization error: the text states that for asymptotically flat spacetimes e→0 as r_o→∞, but the ZAMO quantities in Eq. (31) imply e=ϖ+bσ→1 in that limit (g_tφ→0, g_tt→-1, g_φφ→r^2 sin^2θ). With e=0, Eq. (43) would predict a vanishing redshift factor for all emitters, contradicting the nonzero maps in Figs. 11-12. The sign of the bσ term also appears inconsistent with Eq. (33), where the local energy is ϖE−σL. The redshift computations need to be corrected and re-run.
minor comments (5)
  1. [Abstract] The abstract uses "dark matter parameter a" where the dark matter parameter is denoted α throughout the paper; the spin parameter is a, so the notation should be made consistent.
  2. [III and figure captions] The text refers to "the dark matter parameter a = −0.5" when the notation is α, and Fig. 5's caption says the rotation parameters are "α=0, 0.1, 0.5 and 0.99" instead of a, with "α=−05" missing a decimal point. These notation errors should be corrected.
  3. [I] There is a typo "rotating Bardeen black hole black hole" in the introduction that should be fixed.
  4. [References] Reference [75] is given as "G. S. M and S. Das" with an incomplete author list; the full citation should be completed.
  5. [IV.B] The manuscript does not report the numerical resolution, integration tolerances, or convergence tests for the ray-tracing images, and it does not explicitly state that M=1 is used. Adding this information is necessary for the reader to assess the accuracy of the intensity and redshift maps.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the shadow, disk images, and redshift maps are derived from an externally sourced metric via standard geodesic ray tracing.

full rationale

The paper's central outputs (shadow contours, thin-disk images, and redshift-factor maps) follow from (i) the rotating Bardeen–PFDM metric in Eqs. (10)–(13), imported from Refs. [77,78] (not the authors' own work), and (ii) standard backward ray-tracing equations (22)–(25) and the radiative-transfer formulas (39)–(41). The emissivity profile in Eq. (47), with η1 = -1/2 and η2 = -2, is adopted from Ref. [33] as an illustrative input rather than fitted to the paper's own predictions, so the intensity maps are model outputs rather than reproductions of their inputs. The authors' prior papers appear only in a survey list of thin-disk imaging studies (Refs. [34-40,46,47]) and are not load-bearing for the derivation. The redshift-factor conclusions follow directly from Eqs. (43)-(46) and the computed orbital frequencies, not from an imported uniqueness claim. A possible concern that some displayed parameter combinations may lack an event horizon, since Δr = 0 in Eq. (16) is never explicitly checked for the plotted cases, is a spacetime-validity/correctness issue rather than a circularity in the derivation chain.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The paper's central observable predictions rest on the assumed Bardeen-PFDM metric, the Newman-Janis rotating extension, the separability of null geodesics, and a thin-disk emission model whose emissivity is tuned to produce appealing EHT-like images. All of these are inputs from prior work; the paper adds parameter scans rather than new physical content.

free parameters (6)
  • emissivity coefficients η1, η2 = η1 = -1/2, η2 = -2
    Chosen in Eq. (47) to produce 'a more visually appealing effect that aligns with the 230 GHz image'; these are not derived from a physical accretion model and they set the absolute intensity images.
  • fudge factor Fn = 1
    All fudge factors are normalized to 1 in Section IV.A, which suppresses photon-ring luminosity differences.
  • spin parameter a = 0, 0.1, 0.5, 0.99
    Scanned input values for spacetime spin, chosen by hand for the plots and not constrained by data.
  • magnetic charge G = 0.01, 0.1, 0.2, 0.3
    Scanned input values for the non-linear electrodynamics charge.
  • dark matter parameter α = -0.9 to -0.01
    Scanned input values for PFDM intensity; only negative values are considered.
  • accretion disk radii and observer distance = r_o = 100M, r_d1 = 20M, r_d2 = r_h
    Choice of simulation domain, which affects image scale and outer disk truncation.
assumptions (6)
  • domain assumption The static Bardeen solution with f(r) = 1 - 2Mr^2/(r^2+G^2)^(3/2) + (α/r) ln(r/|α|) is a valid solution of the Einstein-nonlinear electrodynamics equations coupled to PFDM.
    Taken from Refs. [56,73,76]; no independent derivation is given in this paper.
  • domain assumption The rotating metric in Eqs. (10)-(13), obtained via the Newman-Janis algorithm, is the correct rotating extension of the static solution.
    Taken from Refs. [77,78]; the paper does not verify energy conditions or the validity of the Newman-Janis generated metric.
  • standard math Hamilton-Jacobi separation of null geodesics holds with a Carter constant κ for this metric.
    Assumed in Eqs. (19)-(21); the printed form of Θ(θ) is dimensionally suspicious and is not the standard Kerr-like Carter equation.
  • domain assumption The accretion disk is optically and geometrically thin, electrically neutral, lies on the equatorial plane, and follows circular or plunging geodesics partitioned by ISCO.
    Model adopted from Ref. [33]; no self-consistent disk physics is developed.
  • domain assumption The radiative transfer can be simplified to I = Σ F_n g_n^3 J_n with F_n = 1 and no absorption.
    Requires an optically thin disk and fudge factors set to 1, as used in Eq. (41).
  • ad hoc to paper The thin-disk emissivity is log J = η1 ε^2 + η2 ε with η1 = -1/2 and η2 = -2, chosen to visually match the 230 GHz EHT image.
    This emissivity is not derived from disk physics; it is tuned 'to achieve a more visually appealing effect' in Eq. (47).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Observational features of the rotating Bardeen black hole surrounded by perfect fluid dark matter." pith.science (2026). https://pith.science/paper/6OCHZ4HW

@misc{pith2026241111680,
  author       = {Pith},
  title        = {Pith review of: Observational features of the rotating Bardeen black hole surrounded by perfect fluid dark matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6OCHZ4HW}},
  note         = {Machine review of arXiv:2411.11680}
}
abstract

By employing ray-tracing techniques, we investigate the shadow images of rotating Bardeen black holes surrounded by perfect fluid dark matter. In this work, two models are considered for the background light source, namely the celestial light source model and the thin accretion disk model. Regarding the celestial light source, the investigation focuses on the impact of variations in relevant parameters and observed inclination on the contour and size of the shadow. For the thin accretion disk model, the optical appearance of a black hole is evidently contingent upon the radiative properties exhibited by the accretion disk, as well as factors such as observed inclination and relevant parameters governing spacetime. With an increasing observation inclination, the observed flux of direct and lensed images of the accretion disk gradually converge towards the lower region of the image, while an increase in the dark matter parameter $a$ significantly expands the region encompassing both direct and lensed images. Furthermore, the predominant effect is redshift at lower observation angles, whereas the blueshift effect only becomes apparent at higher observation angles. Simultaneously, the increase in the observation inclination will amplify the redshift effect, whereas an increase in the magnetic charge $\mathcal{G}$, rotation parameter $a$ and the absolute value of dark matter parameter $\alpha$ will attenuate the redshift effect observed in the image. These observations of a rotating Bardeen black hole surrounded by perfect fluid dark matter could provide a convenient way to distinguish it from other black hole models.

Figures

Figures reproduced from arXiv: 2411.11680 by the authors.

Figure 1
Figure 1. FIG. 1: The shadow of the rotating Bardeen black hole surrounded by PFDM is observed for different values [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The shadow of the rotating Bardeen black hole surrounded by PFDM is observed for different values [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The shadow of the rotating Bardeen black hole surrounded by PFDM is observed for different values [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The shadow of the rotating Bardeen black hole surrounded by PFDM is observed for different [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: In the thin accretion disk model, the images of a rotating Bardeen black holes surrounded by [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The observed flux distribution of direct and lensed images of the accretion disk. The colors yellow, [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: In the thin accretion disk model, the images of a rotating Bardeen black holes surrounded by [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The observed flux distribution of direct and lensed images of the accretion disk. The colors yellow, [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: In the thin accretion disk model, the images of a rotating Bardeen black holes surrounded by [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: The observed flux distribution of direct and lensed images of the accretion disk. The colors yellow, [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: The redshift factor distribution in direct images of accretion disks. The redshift factor is visually [PITH_FULL_IMAGE:figures/full_fig_p027_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: The redshift factor distribution in lensed images of accretion disks. The redshift factor is visually [PITH_FULL_IMAGE:figures/full_fig_p028_12.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Novel extended inner shadow in images of Johannsen-Psaltis black holes with thin accretion disks

    gr-qc 2026-07 conditional novelty 6.0 of 10

    JP black holes with non-closed horizons produce an 'extended inner shadow' distinct from the ordinary inner shadow, and all image scales (inner shadow, photon ring, peak positions) grow monotonically with |ϵ3|.

  2. The shadow and accretion disk images of the rotation loop quantum black bounce

    gr-qc 2025-02 reject novelty 4.0 of 10

    Simulated shadows and disk images of the rotating LQG black bounce show parameter-dependent shadow contraction, Doppler asymmetry, and hat-like lensed image morphology.

  3. Holographic Einstein Ring of Quantum Corrected AdS-Reissner-Nordstrom Black Holes in Kiselev Spacetime

    gr-qc 2025-01 reject novelty 4.0 of 10

    The holographic Einstein ring radius of the quantum-corrected AdS-Reissner-Nordstrom black hole in Kiselev spacetime decreases with the quantum correction, equation of state, temperature, and chemical potential, incre...

  4. Optical appearance of the Konoplya-Zhidenko rotating non-Kerr black hole surrounded by a thin accretion disk

    astro-ph.HE 2025-01 conditional novelty 4.0 of 10

    For a Konoplya-Zhidenko rotating non-Kerr black hole, increasing the deformation parameter enlarges and roundens the inner shadow and changes how direct versus lensed images separate at high viewing angles.

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

    gr-qc 2025-01 conditional novelty 4.0 of 10

    For a rotating hairy black hole, thin-disk flux, temperature, luminosity, and ray-traced images deviate increasingly from Kerr as spin grows, especially in the inner disk.

Reference graph

Works this paper leans on

84 extracted references · 42 canonical work pages · cited by 5 Pith papers

  1. [75]

    G. S. M and S. Das, [arXiv:2201.01484 [gr-qc]]

    work page Pith review arXiv

  2. [1]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett.875, L1 (2019)

    2019

  3. [2]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett.875, no.1, L2 (2019)

    2019

  4. [3]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett.875, no.1, L3 (2019)

    2019

  5. [4]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett.875, no.1, L4 (2019)

    2019

  6. [5]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett.875, no.1, L5 (2019)

    work page 2019

  7. [6]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett.875, no.1, L6 (2019)

    work page 2019

  8. [7]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett.930, no.2, L12 (2022)

    work page 2022

Show all 84 references
  1. [8]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett.930, no.2, L13 (2022)

    2022

  2. [9]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett.930, no.2, L14 (2022)

    2022

  3. [10]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett.930, no.2, L15 (2022)

    2022

  4. [11]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett.930, no.2, L16 (2022)

    2022

  5. [12]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett.930, no.2, L17 (2022)

    2022

  6. [13]

    J. L. Synge, Mon. Not. Roy. Astron. Soc.131, no.3, 463-466 (1966)

    1966

  7. [14]

    Bardeen, Proceedings, Ecole d’Eté de Physique Théorique: Les Astres Occlus: Les Houches, France, August, 1972, pp

    J.M. Bardeen, Proceedings, Ecole d’Eté de Physique Théorique: Les Astres Occlus: Les Houches, France, August, 1972, pp. 215-240 (1973)

    1973

  8. [15]

    P. V. P. Cunha and C. A. R. Herdeiro, Gen. Rel. Grav.50, no.4, 42 (2018)

    2018

  9. [16]

    Perlick and O

    V. Perlick and O. Y. Tsupko, Phys. Rept.947, 1-39 (2022)

    2022

  10. [17]

    X. Liu, S. Chen and J. Jing, Sci. China Phys. Mech. Astron.65, no.12, 120411 (2022)

    2022

  11. [18]

    M. Wang, S. Chen and J. Jing, Commun. Theor. Phys.74, no.9, 097401 (2022)

    2022

  12. [19]

    S. Chen, J. Jing, W. L. Qian and B. Wang, Sci. China Phys. Mech. Astron.66, no.6, 260401 (2023)

    2023

  13. [20]

    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)

    2023

  14. [21]

    Zhong, Z

    Z. Zhong, Z. Hu, H. Yan, M. Guo and B. Chen, Phys. Rev. D104, no.10, 104028 (2021)

    2021

  15. [22]

    Zhang, H

    Z. Zhang, H. Yan, M. Guo and B. Chen, Phys. Rev. D107, no.2, 024027 (2023)

    2023

  16. [23]

    Huang, Z

    J. Huang, Z. Zhang, M. Guo and B. Chen, Phys. Rev. D109, no.12, 124062 (2024) 31

    2024

  17. [24]

    P. C. Li, M. Guo and B. Chen, Phys. Rev. D101, no.8, 084041 (2020)

    2020

  18. [25]

    R. D. Blandford and R. L. Znajek, Mon. Not. Roy. Astron. Soc.179, 433-456 (1977)

    1977

  19. [26]

    Narayan, M

    R. Narayan, M. D. Johnson and C. F. Gammie, Astrophys. J. Lett.885, no.2, L33 (2019)

    2019

  20. [27]

    X. X. Zeng, H. Q. Zhang and H. Zhang, Eur. Phys. J. C80, no.9, 872 (2020)

    2020

  21. [28]

    Heydari-Fard, M

    M. Heydari-Fard, M. Heydari-Fard and N. Riazi, Int. J. Mod. Phys. D32, no.13, 2350088 (2023)

    2023

  22. [29]

    S. E. Gralla, D. E. Holz and R. M. Wald, Phys. Rev. D100, no. 2, 024018 (2019)

    2019

  23. [30]

    X. X. Zeng and H. Q. Zhang, Eur. Phys. J. C80, no.11, 1058 (2020)

    2020

  24. [31]

    J. Peng, M. Guo and X. H. Feng, Chin. Phys. C45, no.8, 085103 (2021)

    2021

  25. [32]

    Zhang, Y

    Z. Zhang, Y. Hou and M. Guo, Chin. Phys. C48, no.8, 085106 (2024)

    2024

  26. [33]

    Y. Hou, Z. Zhang, H. Yan, M. Guo and B. Chen, Phys. Rev. D106, no.6, 064058 (2022)

    2022

  27. [34]

    K. J. He, S. C. Tan and G. P. Li, Eur. Phys. J. C82, no.1, 81 (2022)

    2022

  28. [35]

    G. P. Li and K. J. He, Eur. Phys. J. C81, no.11, 1018 (2021)

    2021

  29. [36]

    X. X. Zeng, K. J. He and G. P. Li, Sci. China Phys. Mech. Astron.65, no.9, 290411 (2022)

    2022

  30. [37]

    S. Guo, K. J. He, G. R. Li and G. P. Li, Class. Quant. Grav.38, no.16, 165013 (2021)

    2021

  31. [38]

    X. X. Zeng, G. P. Li and K. J. He, Nucl. Phys. B974, 115639 (2022)

    2022

  32. [39]

    G. P. Li and K. J. He, JCAP06, 037 (2021)

    2021

  33. [40]

    K. J. He, S. Guo, S. C. Tan and G. P. Li, Chin. Phys. C46, no.8, 085106 (2022)

    2022

  34. [41]

    X. J. Gao, T. T. Sui, X. X. Zeng, Y. S. An and Y. P. Hu, Eur. Phys. J. C83, 1052 (2023)

    2023

  35. [42]

    Y. H. Cui, S. Guo, Y. X. Huang, Y. Liang and K. Lin, Eur. Phys. J. C84, no.8, 772 (2024)

    2024

  36. [43]

    K. Wang, C. J. Feng and T. Wang, Eur. Phys. J. C84, no.5, 457 (2024)

    2024

  37. [44]

    S. Hu, D. Li, C. Deng, X. Wu and E. Liang, JCAP04, 089 (2024)

    2024

  38. [45]

    Y. Meng, X. M. Kuang, X. J. Wang, B. Wang and J. P. Wu, Eur. Phys. J. C84, no.3, 305 (2024)

    2024

  39. [46]

    X. X. Zeng, K. J. He, G. P. Li, E. W. Liang and S. Guo, Eur. Phys. J. C82, no.8, 764 (2022)

    2022

  40. [47]

    X. X. Zeng, M. I. Aslam and R. Saleem, Eur. Phys. J. C83, no.2, 129 (2023)

    2023

  41. [48]

    Zhang, S

    Z. Zhang, S. Chen and J. Jing, JCAP09, 027 (2024)

    2024

  42. [49]

    Zhang, Y

    Z. Zhang, Y. Hou, M. Guo and B. Chen, JCAP05, 032 (2024)

    2024

  43. [50]

    Gjorgjieski, J

    K. Gjorgjieski, J. Kunz and P. Nedkova, Eur. Phys. J. C84, no.3, 286 (2024)

    2024

  44. [51]

    Cognola, R

    G. Cognola, R. Myrzakulov, L. Sebastiani and S. Zerbini, Phys. Rev. D88, no.2, 024006 (2013)

    2013

  45. [52]

    Zhang, J

    C. Zhang, J. Lewandowski, H. Li and Y. Ma, Phys. Rev. D99, no.12, 124012 (2019)

    2019

  46. [53]

    M Bardeen

    J. M Bardeen. InProc. Int. Conf. GR5, Tbilisi , volume 174 (1968)

    1968

  47. [54]

    Borde, Phys

    A. Borde, Phys. Rev. D50, 3692-3702 (1994)

    1994

  48. [55]

    Borde, Phys

    A. Borde, Phys. Rev. D55, 7615-7617 (1997)

    1997

  49. [56]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Phys. Lett. B493, 149-152 (2000)

    2000

  50. [57]

    S. A. Hayward, Phys. Rev. Lett.96, 031103 (2006)

    2006

  51. [58]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Phys. Rev. Lett.80, 5056-5059 (1998)

    1998

  52. [59]

    Berej, J

    W. Berej, J. Matyjasek, D. Tryniecki and M. Woronowicz, Gen. Rel. Grav.38, 885-906 (2006)

    2006

  53. [60]

    Y. H. Khan, S. Upadhyay and P. A. Ganai, Mod. Phys. Lett. A36, no.24, 2130023 (2021). 32

    2021

  54. [61]

    Fernando and J

    S. Fernando and J. Correa, Phys. Rev. D86, 064039 (2012)

    2012

  55. [62]

    Ghaffarnejad and H

    H. Ghaffarnejad and H. Niad, Int. J. Theor. Phys.55, no.3, 1492-1505 (2016)

    2016

  56. [63]

    S. Zhou, J. Chen and Y. Wang, Int. J. Mod. Phys. D21, 1250077 (2012)

    2012

  57. [64]

    Guo and Y

    Y. Guo and Y. G. Miao, Nucl. Phys. B980, 115839 (2022)

    2022

  58. [65]

    Zwicky, Helv

    F. Zwicky, Helv. Phys. Acta6, 110-127 (1933)

    1933

  59. [66]

    Holmberg Annals of the Observatory of Lund, 6, 1-173 (1973 )

    E. Holmberg Annals of the Observatory of Lund, 6, 1-173 (1973 )

    1973

  60. [67]

    Smith, Astrophys

    S. Smith, Astrophys. J.83, 23-30 (1936)

    1936

  61. [68]

    P. A. R. Adeet al. [Planck], Astron. Astrophys.571, A16 (2014)

    2014

  62. [69]

    Jusufi, M

    K. Jusufi, M. Jamil, P. Salucci, T. Zhu and S. Haroon, Phys. Rev. D100, no.4, 044012 (2019)

    2019

  63. [70]

    R. A. Konoplya, Phys. Lett. B795, 1-6 (2019)

    2019

  64. [71]

    V. V. Kiselev, Class. Quant. Grav.20, 1187-1198 (2003)

    2003

  65. [72]

    Rahaman, K

    F. Rahaman, K. K. Nandi, A. Bhadra, M. Kalam and K. Chakraborty, Phys. Lett. B694, 10-15 (2011)

    2011

  66. [73]

    M. H. Li and K. C. Yang, Phys. Rev. D86, 123015 (2012)

    2012

  67. [74]

    Z. Xu, J. Wang and X. Hou, Class. Quant. Grav.35, no.11, 115003 (2018)

    2018

  68. [76]

    H. X. Zhang, Y. Chen, T. C. Ma, P. Z. He and J. B. Deng, Chin. Phys. C45, no.5, 055103 (2021)

    2021

  69. [77]

    Azreg-Aïnou, Phys

    M. Azreg-Aïnou, Phys. Rev. D90, no.6, 064041 (2014)

    2014

  70. [78]

    Azreg-Aïnou, Eur

    M. Azreg-Aïnou, Eur. Phys. J. C74, no.5, 2865 (2014)

    2014

  71. [79]

    P. V. P. Cunha, C. A. R. Herdeiro, E. Radu and H. F. Runarsson, Phys. Rev. Lett.115, no.21, 211102 (2015)

    2015

  72. [80]

    F. Long, S. Chen, M. Wang and J. Jing, Eur. Phys. J. C80, no.12, 1180 (2020)

    2020

  73. [81]

    Z. Hu, Z. Zhong, P. C. Li, M. Guo and B. Chen, Phys. Rev. D103, no.4, 044057 (2021)

    2021

  74. [82]

    Chael, M

    A. Chael, M. D. Johnson and A. Lupsasca, Astrophys. J.918, no.1, 6 (2021)

    2021

  75. [83]

    R. W. Lindquist, Annals Phys.37, no.3, 487-518 (1966)

    1966

  76. [84]

    Hadar, M

    S. Hadar, M. D. Johnson, A. Lupsasca and G. N. Wong, Phys. Rev. D103, no.10, 104038 (2021)

    2021

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.