Pith. sign in

REVIEW 2 major objections 4 minor 48 references

A Hernquist dark matter halo softens superradiance around a rotating black hole while deepening scalar binding, with all corrections set by ρ₀r₀³.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · grok-4.5

2026-07-13 06:43 UTC pith:XXXCRKS6

load-bearing objection Solid exact static seed plus clean AAM spectra on a new NJA metric; the rotating stress-energy is asserted rather than fully checked, so treat the geometry as a useful background rather than a proven Einstein solution. the 2 major comments →

arxiv 2607.08796 v1 pith:XXXCRKS6 submitted 2026-07-08 gr-qc hep-th

Novel Kerr-Hernquist Black Hole: Quasibound State, Scalar Cloud, Bomb, Superradiant Scattering

David Senjaya This is my paper
classification gr-qc hep-th PACS 04.70.-s04.62.+v95.35.+d97.60.Lf
keywords Kerr–Hernquist black holeHernquist dark matter haloquasibound statesscalar cloudblack hole bombsuperradiant scatteringanalytical asymptotic matchingNewman–Janis algorithm
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds an exact static black hole inside a Hernquist dark-matter halo and then rotates it with the Newman–Janis algorithm, producing a Kerr–Hernquist geometry that is consistent with the Einstein equations for an anisotropic fluid. In that background a massive scalar field is solved by analytical asymptotic matching. The resulting quasibound spectrum keeps its hydrogen-like shape, but every correction is controlled by the single combination ρ₀r₀³. Raising halo density or scale radius deepens the potential well (stronger binding), lowers the critical boson mass needed for a scalar cloud, suppresses the growth of co-rotating superradiant instabilities, speeds the decay of counter-rotating modes, and shrinks both the height and the frequency window of the superradiant amplification factor. The same geometry therefore unifies quasibound states, scalar clouds, black-hole bombs and energy extraction, and shows that a realistic galactic halo leaves concrete, calculable imprints on all of them.

Core claim

The Hernquist halo preserves the hydrogenic structure of the quasibound spectrum while shifting every frequency and rate through the single combination ρ₀r₀³: denser or more extended halos strengthen binding, lower the critical mass for scalar-cloud formation, suppress co-rotating growth rates, accelerate counter-rotating decay, and reduce both the magnitude and the frequency range of superradiant amplification.

What carries the argument

Analytical asymptotic matching of near-horizon hypergeometric solutions to far-zone confluent-hypergeometric solutions, which yields closed-form expressions for the complex quasibound frequencies and the amplification factor Z, both controlled by the halo-modified parameters A and ξ(r_H).

Load-bearing premise

That applying the Newman–Janis algorithm to the static Hernquist seed still produces a metric that solves Einstein’s equations with a physically acceptable rotating anisotropic-fluid stress-energy tensor.

What would settle it

Compute the Einstein tensor of the constructed Kerr–Hernquist metric and check whether its projections onto the orthonormal tetrad reproduce a consistent anisotropic-fluid T_μν whose density matches the Hernquist profile; any mismatch falsifies the geometry and all derived spectra.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Quasibound frequencies and instability rates of galactic-centre black holes acquire a measurable shift proportional to ρ₀r₀³.
  • The critical boson mass for scalar-cloud formation is lowered, so lighter fields become unstable in denser or larger halos.
  • Co-rotating black-hole bombs grow more slowly and extract less rotational energy once a Hernquist halo is present.
  • The superradiant frequency window shrinks, reducing the range of waves that can be amplified by the black hole.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the ρ₀r₀³ corrections survive full numerical evolution, continuous-wave gravitational-wave searches for ultralight bosons around Sgr A* or M87* must include an environmental systematic.
  • The same matching procedure can be repeated for other Dehnen profiles (NFW, Burkert, …) to test whether the suppression of amplification is universal or profile-dependent.
  • A halo-induced reduction of the amplification factor may leave an imprint on the stochastic gravitational-wave background sourced by a population of spinning black holes in galaxies.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The manuscript constructs an exact static Schwarzschild–Hernquist black hole by solving the Einstein equations with an anisotropic fluid whose density is the Hernquist profile (Eqs. 6–15), then generates a rotating counterpart via the Newman–Janis algorithm (metric (28)). On this background it separates the massive Klein–Gordon equation and, in the low-frequency/slow-rotation regime, matches near-horizon hypergeometric solutions to far-zone confluent-hypergeometric solutions. The matching yields an analytic quasibound-state spectrum (Eq. 90), a critical scalar mass for cloud formation (Eq. 100), the black-hole-bomb condition, and the superradiant amplification factor Z. The central claim is that the halo corrections are controlled by the combination ρ₀r₀³: they deepen the binding energy, lower m_crit, suppress co-rotating growth rates, accelerate counter-rotating decay, and shrink both the magnitude and frequency window of superradiant amplification.

Significance. If the rotating geometry is a genuine Einstein solution, the paper supplies a clean, fully analytic framework that unifies quasibound states, scalar clouds, black-hole bombs and superradiant scattering for a rotating black hole immersed in a realistic dark-matter halo. The hydrogen-like spectrum with explicit ρ₀r₀³ corrections and the closed-form expressions for m_crit and Z are concrete, falsifiable predictions that can be compared with numerical or observational studies of environmental effects on ultralight bosons. The static seed is derived self-consistently from the Einstein equations, and the subsequent AAM analysis is standard and transparent; these are genuine strengths. The principal open question is whether the Newman–Janis metric continues to solve the field equations with a physically acceptable rotating fluid.

major comments (2)
  1. Appendix D only lists the coordinate components of G_μν for metric (28), introduces an orthonormal tetrad, and asserts that the projections “immediately yield” the stress-energy components. It never exhibits the explicit functional forms of ρ, p_r, p_θ, p_φ, never verifies that they reduce to the known static Hernquist density and pressures when a→0, and never checks ∇_μ T^μ_ν=0 or the energy conditions. Because every subsequent spectral formula (Eqs. 90, 100, 117) is derived on this geometry, the physical status of the Kerr–Hernquist solution remains unproven. The authors should either supply the missing verification or clearly reframe the metric as a phenomenological NJA construction.
  2. The weak-halo expansions used for the horizon radius and ξ(r_H) (Eqs. 93–97) assume ρ₀ r₀³ ≪ 1 and are then inserted into the expressions for m_crit and ω_c. The abstract and the discussion of §§4.1–4.2 present the ρ₀ r₀³ corrections as general. The manuscript should either derive the corresponding quantities for finite halo strength or explicitly restrict the claimed phenomenology to the weak-halo regime.
minor comments (4)
  1. Section 5 repeatedly refers to a “Dehnen dark matter halo” while the body of the paper treats the Hernquist profile (α,β,γ)=(1,4,1). The terminology should be made consistent.
  2. The product that appears after Eq. (84) is written with an empty product symbol; the intended range j=1 oℓ should be restored for readability.
  3. Several self-citations to the author’s earlier exact-QBS papers are appropriate for the method, but a brief comparison with existing numerical or semi-analytic results for Kerr quasibound states (even in vacuum) would help the reader gauge the accuracy of the AAM approximation.
  4. The notation m_ℓ for the azimuthal number is non-standard and easily confused with the scalar mass m; a conventional m would improve clarity.

Circularity Check

0 steps flagged

No load-bearing circularity: spectra follow from AAM on the NJA metric with free halo parameters; only minor non-essential self-citations of the author's prior AAM applications.

full rationale

The derivation chain is self-contained and non-circular. The static seed is obtained by direct integration of the Einstein equations with the given Hernquist density (Eqs. 10–15), yielding an exact f(r). The rotating metric is generated by the Newman–Janis algorithm (Sec. 3) and asserted to solve the Einstein equations via tetrad projection of Gμν (Appendix D); whether that verification is complete is a correctness question, not a circularity. All subsequent results—the QBS frequency (Eq. 90), the critical mass for scalar clouds (Eq. 100), the instability condition (Eq. 92), and the amplification factor Z (Eq. 115)—are obtained by asymptotic matching of the Klein–Gordon radial equation on that fixed background. The halo parameters ρ₀ and r₀ enter only as external inputs that shift A and ξ(r_H); they are never fitted to data, nor is any spectral quantity defined in terms of itself. Self-citations (Refs. [4–8]) merely document the author’s earlier uses of the same AAM technique on different metrics; they supply no uniqueness theorem, no ansatz, and no numerical input that forces the present formulae. Consequently the central claims reduce neither by definition nor by self-citation chain to their inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 1 invented entities

The central claims rest on classical GR plus the standard low-frequency matching technique; the only non-standard ingredients are the anisotropic-fluid model for the Hernquist halo and the assumption that Newman–Janis preserves a consistent Einstein solution. No free parameters are fitted to data; ρ₀ and r₀ remain free environmental inputs.

axioms (4)
  • domain assumption Einstein equations with anisotropic fluid T^μ_ν = diag(−ρ_DM, p_r, p_t, p_t) and the Schwarzschild-like condition g_tt g_rr = −1 imply p_r = −ρ_DM and the integral expression for f(r).
    Used from §2 onward to obtain the exact static seed (Eqs. 6–11).
  • ad hoc to paper The Newman–Janis complexification of the static seed yields a stationary axisymmetric metric that continues to solve Einstein’s equations with a suitably rotated anisotropic stress-energy tensor.
    Invoked in §3; verification is claimed only by projection onto a tetrad in Appendix D.
  • domain assumption In the regime mM ≪ 1, |ω|M ≪ 1, ma ≪ 1 the angular equation reduces to spherical harmonics (λ = ℓ(ℓ+1)) and the radial equation admits matched hypergeometric / confluent-hypergeometric solutions.
    Standard AAM premise stated in §4.1; controls the entire spectral analysis.
  • domain assumption The Hernquist density profile ρ_DM = ρ₀ (r/r₀)^−1 (1 + r/r₀)^−3 is an adequate model for the galactic dark-matter halo surrounding the black hole.
    Chosen in §1–2; all environmental corrections are proportional to the combination ρ₀ r₀³ that this profile produces.
invented entities (1)
  • Kerr–Hernquist geometry (metric (28)) no independent evidence
    purpose: Provides the axisymmetric background on which all scalar dynamics are computed.
    Constructed by NJA from the exact static seed; no independent observational confirmation is offered beyond the formal Einstein-equation check in Appendix D.

pith-pipeline@v1.1.0-grok45 · 22712 in / 2989 out tokens · 30017 ms · 2026-07-13T06:43:33.932088+00:00 · methodology

0 comments
read the original abstract

We present a novel rotating black hole solution surrounded by a Hernquist dark matter halo, obtained by applying the Newman--Janis algorithm to the exact Schwarzschild--Hernquist spacetime. The resulting Kerr--Hernquist geometry provides an axisymmetric background for investigating scalar-field dynamics in realistic dark matter environments. Using the analytical asymptotic matching method, we derive the quasibound-state spectrum, identify the conditions for scalar cloud formation and the black hole bomb instability, and obtain an analytic expression for the superradiant scattering amplification factor. We show that the halo preserves the hydrogen-like structure of the quasibound-state spectrum while introducing corrections governed by the combination $\rho_0 r_0^3$. Increasing the halo density and scale radius enhances the scalar-field binding energy, lowers the critical field mass for scalar cloud formation, suppresses the growth rate of the superradiant instability for co-rotating modes ($m_\ell>0$), and accelerates the decay of counter-rotating modes ($m_\ell<0$). Furthermore, the dark matter halo reduces both the magnitude and frequency range of superradiant amplification, thereby weakening energy extraction from the black hole. These results demonstrate that the Kerr--Hernquist geometry provides a unified framework for studying quasibound states, scalar clouds, black hole bombs, and superradiant scattering, while revealing how a Hernquist dark matter halo leaves observable imprints on the spectrum and stability of rotating black holes.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

48 extracted references · 2 canonical work pages · 1 internal anchor

  1. [1]

    Rees, M.J.: Black hole models for active galactic nuclei. Annu. Rev. Astron. Astrophys.22, 471 (1984) https://doi.org/10.1146/annurev.aa.22.090184.002351

    doi:10.1146/annurev.aa.22.090184.002351 1984

  2. [2]

    Kormendy, J., Richstone, D.: Inward bound—the search for supermassive black holes in galactic nuclei. Annu. Rev. Astron. Astrophys.33, 581 (1995) https: //doi.org/10.1146/annurev.aa.33.090195.003053

    doi:10.1146/annurev.aa.33.090195.003053 1995

  3. [3]

    Nature562, 51 (2018) https://doi.org/10.1038/s41586-018-0542-z

    Bertone, G., Tait, T.M.P.: A new era in the search for dark matter. Nature562, 51 (2018) https://doi.org/10.1038/s41586-018-0542-z

    doi:10.1038/s41586-018-0542-z 2018

  4. [4]

    The European Physical Journal C84(2024) https://doi.org/10.1140/epjc/ s10052-024-12422-4

    Senjaya, D.: Exact massless scalar quasibound states of the ernst black hole. The European Physical Journal C84(2024) https://doi.org/10.1140/epjc/ s10052-024-12422-4

    doi:10.1140/epjc/ 2024

  5. [5]

    Senjaya, D.: The Kerr–Bumblebee exact massive and massless scalar quasibound states and Hawking radiation. Eur. Phys. J. C84(4), 424 (2024) https://doi.org/ 10.1140/epjc/s10052-024-12794-7

    doi:10.1140/epjc/s10052-024-12794-7 2024

  6. [6]

    Senjaya, D.: Scalar quasibound states in Einstein-Maxwell-Bumblebee black holes with non-minimal Maxwell-Bumblebee coupling. Nucl. Phys. B1022, 117226 (2026) https://doi.org/10.1016/j.nuclphysb.2025.117226

    doi:10.1016/j.nuclphysb.2025.117226 2026

  7. [7]

    Senjaya, D.: Spectroscopy of Einstein–Skyrme gravitational atom: exact solution, black hole bomb, and Hawking radiation. Eur. Phys. J. C86(4), 343 (2026) https://doi.org/10.1140/epjc/s10052-026-15614-2

    doi:10.1140/epjc/s10052-026-15614-2 2026

  8. [8]

    Senjaya, D., Ponglertsakul, S.: The spectroscopy of Kerr–Einstein–Maxwell- dilaton-axion: exact quasibound states, scalar cloud, horizon’s Boson statistics and superradiant. Eur. Phys. J. C85(3), 352 (2025) https://doi.org/10.1140/ epjc/s10052-025-14014-2 arXiv:2501.08788 [gr-qc]

    Pith/arXiv arXiv 2025

  9. [9]

    Xavier, S.V.M.C.B., Junior, H.C.D.L., Crispino, L.C.B.: Shadows of black holes with dark matter halo. Phys. Rev. D107, 064040 (2023) https://doi.org/10. 1103/PhysRevD.107.064040

    2023

  10. [10]

    Cardoso, V., Destounis, K., Duque, F.,et al.: Black holes in galaxies: Environ- mental impact on gravitational-wave generation and propagation. Phys. Rev. D 105, 061501 (2022) https://doi.org/10.1103/PhysRevD.105.L061501

    doi:10.1103/physrevd.105.l061501 2022

  11. [11]

    Konoplya, R.A.: Shadow of a black hole surrounded by dark matter. Phys. Lett. B795, 1 (2019) https://doi.org/10.1016/j.physletb.2019.05.043

    doi:10.1016/j.physletb.2019.05.043 2019

  12. [12]

    Astrophys

    Konoplya, R.A., Zhidenko, A.: Solutions of the einstein equations for a black hole surrounded by a galactic halo. Astrophys. J.933, 166 (2022) https://doi.org/10. 3847/1538-4357/ac76bc 25

    2022

  13. [13]

    Hou, X., Xu, Z., Zhou, M.,et al.: Black hole shadow of sgr a* in dark matter halo. J. Cosmol. Astropart. Phys.2018, 015 (2018) https://doi.org/10.1088/ 1475-7516/2018/07/015

    2018

  14. [14]

    Yang, Y., Liu, D., ¨Ovg¨ un, A.,et al.: Black hole surrounded by the pseudo- isothermal dark matter halo. Eur. Phys. J. C84, 63 (2024) https://doi.org/10. 1140/epjc/s10052-024-12412-6

    2024

  15. [15]

    Liang, X., Hu, Y.-P., Wu, C.-H.,et al.: Thermodynamics and evaporation of perfect fluid dark matter black hole in phantom background. Eur. Phys. J. C83, 1009 (2023) https://doi.org/10.1140/epjc/s10052-023-12200-8

    doi:10.1140/epjc/s10052-023-12200-8 2023

  16. [16]

    Carvalho, I.D.D., Alencar, G., Muniz, C.R.: Thermodynamics of static and sta- tionary black holes in einstein–gauss–bonnet gravity with dark matter. Phys. Dark Univ.42, 101290 (2023) https://doi.org/10.1016/j.dark.2023.101290

    doi:10.1016/j.dark.2023.101290 2023

  17. [17]

    Anjum, A., Afrin, M., Ghosh, S.G.: Investigating effects of dark matter on photon orbits and black hole shadows. Phys. Dark Univ.40, 101195 (2023) https://doi. org/10.1016/j.dark.2023.101195

    doi:10.1016/j.dark.2023.101195 2023

  18. [18]

    Pantig, R.C., ¨Ovg¨ un, A.: Dark matter effect on the weak deflection angle by black holes at the center of milky way and m87 galaxies. Eur. Phys. J. C82, 1 (2022) https://doi.org/10.1140/epjc/s10052-022-10319-8

    doi:10.1140/epjc/s10052-022-10319-8 2022

  19. [19]

    Stuchl´ ık, Z., Vrba, J.: Supermassive black holes surrounded by dark matter mod- eled as anisotropic fluid: epicyclic oscillations and their fitting to observed qpos. J. Cosmol. Astropart. Phys.2021, 059 (2021) https://doi.org/10.1088/1475-7516/ 2021/11/059

    doi:10.1088/1475-7516/ 2021

  20. [20]

    Fortschr

    Pantig, R.C., ¨Ovg¨ un, A.: Black hole in quantum wave dark matter. Fortschr. Phys.71, 2200164 (2023) https://doi.org/10.1002/prop.202200164

    doi:10.1002/prop.202200164 2023

  21. [21]

    ¨Ovg¨ un, A., Sese, L.J.F., Pantig, R.C.: Constraints via the event horizon tele- scope for black hole solutions with dark matter under the generalized uncertainty principle minimal length scale effect. Ann. Phys.536, 2300390 (2024) https: //doi.org/10.1002/andp.202300390

    doi:10.1002/andp.202300390 2024

  22. [22]

    Monthly Notices of the Royal Astronomical Society278, 488–502 (1996) https: //doi.org/10.1093/mnras/278.2.488

    Zhao, H.: Analytical dynamical models for double-power-law galactic nuclei. Monthly Notices of the Royal Astronomical Society278, 488–502 (1996) https: //doi.org/10.1093/mnras/278.2.488

    doi:10.1093/mnras/278.2.488 1996

  23. [23]

    Monthly Notices of the Royal Astronomical Society265, 250–256 (1993) https: //doi.org/10.1093/mnras/265.2.250

    Dehnen, W.: A family of potential-density pairs for spherical galaxies and bulges. Monthly Notices of the Royal Astronomical Society265, 250–256 (1993) https: //doi.org/10.1093/mnras/265.2.250

    doi:10.1093/mnras/265.2.250 1993

  24. [24]

    Senjaya, D., Sereewat, P.: Black hole in cored plummer dark matter environment: 26 Novel solution, light ring, shadow, lensing, lyapunov exponent, eikonal quasinor- mal modes and thermodynamics-phase transition. Phys. Dark Univ.52, 102264 (2026) https://doi.org/10.1016/j.dark.2026.102264

    doi:10.1016/j.dark.2026.102264 2026

  25. [25]

    Benkrane, A.: Entropy analysis of dark matter halo structures. Phys. Dark Univ. 50, 102130 (2025) https://doi.org/10.1016/j.dark.2025.102130

    doi:10.1016/j.dark.2025.102130 2025

  26. [26]

    Jumaniyozov, S.: Thermodynamic fluctuations and radiation properties around Schwarzschild black holes immersed in Hernquist dark matter halo. Eur. Phys. J. C85(11), 1267 (2025) https://doi.org/10.1140/epjc/s10052-025-15025-9

    work page doi:10.1140/epjc/s10052-025-15025-9 2025

  27. [27]

    Furuhashi, H., Nambu, Y.: Instability of massive scalar fields in Kerr-Newman space-time. Prog. Theor. Phys.112, 983–995 (2004) https://doi.org/10.1143/ PTP.112.983 arXiv:gr-qc/0402037

    Pith/arXiv arXiv 2004

  28. [28]

    Hod, S.: Stationary resonances of rapidly-rotating Kerr black holes. Eur. Phys. J. C73(4), 2378 (2013) https://doi.org/10.1140/epjc/s10052-013-2378-x arXiv:1311.5298 [gr-qc]

    Pith/arXiv arXiv doi:10.1140/epjc/s10052-013-2378-x 2013

  29. [30]

    Huang, Y., Liu, D.-J.: Scalar clouds and the superradiant instability regime of Kerr-Newman black hole. Phys. Rev. D94(6), 064030 (2016) https://doi.org/10. 1103/PhysRevD.94.064030 arXiv:1606.08913 [gr-qc]

    Pith/arXiv arXiv 2016

  30. [31]

    Physical Review D90(6) (2014) https://doi.org/10.1103/physrevd

    Azreg-A¨ ınou, M.: Generating rotating regular black hole solutions without com- plexification. Physical Review D90(6) (2014) https://doi.org/10.1103/physrevd. 90.064041

    doi:10.1103/physrevd 2014

  31. [32]

    Drake, S.P., Turolla, R.: The application of the newman–janis algorithm in obtaining interior solutions of the kerr metric. Class. Quantum Grav.14, 1883 (1997) https://doi.org/10.1088/0264-9381/14/7/021

    doi:10.1088/0264-9381/14/7/021 1997

  32. [33]

    Brauer, O., Camargo, H.A., Socolovsky, M.: Newman–janis algorithm revisited. Int. J. Theor. Phys.54, 302 (2015) https://doi.org/10.1007/s10773-014-2225-3

    doi:10.1007/s10773-014-2225-3 2015

  33. [34]

    Lombardo, D.J.C.: The newman–janis algorithm, rotating solutions and einstein– born–infeld black holes. Class. Quantum Grav.21, 1407 (2004) https://doi.org/ 10.1088/0264-9381/21/6/009

    doi:10.1088/0264-9381/21/6/009 2004

  34. [35]

    Kim, J.-H.: Single kerr–schild metric for taub–nut instanton. Phys. Rev. D111, 021703 (2025) https://doi.org/10.1103/PhysRevD.111.L021703

    doi:10.1103/physrevd.111.l021703 2025

  35. [36]

    Abbas, G., Ali, R.H., Mustafa, G.: Thermodynamical analysis with extended phase transition of ads hairy black hole in gravitational decoupling theory. Phys. 27 Scr.99, 045025 (2024) https://doi.org/10.1088/1402-4896/ad340a

    doi:10.1088/1402-4896/ad340a 2024

  36. [37]

    Black Hole Shadows Modelling in Extended gravity: Rotation Accounting and coupled effects

    Alexeyev, S., Zenin, O., Baiderin, A.: Black hole shadows modelling in extended gravity: Rotation accounting and coupled effects. arXiv preprint (2025) https: //doi.org/10.31857/S0044451025040030 2503.17280

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.31857/s0044451025040030 2025

  37. [38]

    Astropart

    Jafarzade, K., Shaymatov, S., Jamil, M.: Shadows and optical appearances of black holes in r2 gravity. Astropart. Phys.168, 103100 (2025) https://doi.org/ 10.1016/j.astropartphys.2025.103100

    doi:10.1016/j.astropartphys.2025.103100 2025

  38. [39]

    Fazzini, F.: Effective kerr geometry from loop quantum gravity. Phys. Rev. D 111, 046025 (2025) https://doi.org/10.1103/PhysRevD.111.046025

    doi:10.1103/physrevd.111.046025 2025

  39. [40]

    arXiv preprint (2025) https://doi

    Li, Q.-Q., Zhang, Y., Iminniyaz, H.: Rotating and non-linear magnetic-charged black hole with an anisotropic matter field. arXiv preprint (2025) https://doi. org/10.48550/arXiv.2501.15983 2501.15983

    Pith/arXiv arXiv doi:10.48550/arxiv.2501.15983 2025

  40. [41]

    arXiv preprint (2025) https://doi.org/10

    Fathi, M., Sekhmani, Y.: Shadows aspect of rotating black holes in the ein- stein–ads su(n)-nonlinear sigma model. arXiv preprint (2025) https://doi.org/10. 48550/arXiv.2503.02179 2503.02179

    Pith/arXiv arXiv 2025

  41. [42]

    Zahid, M., Yunusov, O., Shen, C.,et al.: Shadows and quasinormal modes of rotating black holes in horndeski theory: Parameter constraints using eht observations of m87* and sgr a*. Phys. Dark Univ.47, 101734 (2025) https: //doi.org/10.1016/j.dark.2024.101734

    doi:10.1016/j.dark.2024.101734 2025

  42. [43]

    arXiv preprint (2025) https://doi.org/10.48550/arXiv.2501.01308 2501.01308

    Raza, M.A., Zubair, M., Atamurotov, F.,et al.: Influence of quantum correction on kerr black hole in effective loop quantum gravity via shadows and eht results. arXiv preprint (2025) https://doi.org/10.48550/arXiv.2501.01308 2501.01308

    doi:10.48550/arxiv.2501.01308 2025

  43. [44]

    Azreg-A¨ ınou, M.: Generating rotating regular black hole solutions without complexification. Phys. Rev. D90, 064041 (2014) https://doi.org/10.1103/ PhysRevD.90.064041

    2014

  44. [45]

    Yang, H., Miao, Y.-G.: Superradiance of massive scalar particles around rotating regular black holes*. Chin. Phys. C47(7), 075101 (2023) https://doi.org/10. 1088/1674-1137/accdc7 arXiv:2211.15130 [gr-qc]

    Pith/arXiv arXiv 2023

  45. [46]

    Van Nostrand, London (1968)

    Bell, W.W.: Special Functions for Scientists and Engineers. Van Nostrand, London (1968)

    1968

  46. [47]

    Frank W. J. Olver, R.F.B. Daniel W. Lozier, Clark, C.W.: NIST Handbook of Mathematical Functions, 1st edn. Cambridge University Press, ??? (2010)

    2010

  47. [48]

    Tata Mcgraw hill„ New delhi (1991) 28

    Simmons, G.F.: Differential Equations with Application and Historical Notes, 2nd edn. Tata Mcgraw hill„ New delhi (1991) 28

    1991

  48. [49]

    Dover books on math- ematics

    Bell, W.W.: Special Functions for Scientists and Engineers. Dover books on math- ematics. Dover Publications, ??? (2004). https://books.google.co.th/books?id= Jj5pXGTZIKkC 29

    2004