Pith. sign in

REVIEW 3 major objections 4 minor 4 cited by

The Spectroscopy of Kerr-Einstein-Maxwell-Dilaton-Axion: Exact Quasibound States, Scalar Cloud, Horizon's Boson Statistics and Superradiance

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read The paper derives exact quasibound-state frequencies for scalar fields around Kerr-EMDA black holes.

desk verdict Solid exact QBS derivation for Kerr-EMDA, but the scalar cloud section drops a nonvanishing imaginary term and the Hawking flux integral is mishandled, so the paper needs major revision before its headline claims can stand. read the letter →

arxiv 2501.08788 v1 pith:UQ45PBFN submitted 2025-01-15 gr-qc

David Senjaya , Supakchai Ponglertsakul This is my paper · ORCID
classification gr-qc MSC 83C5733E10 PACS 04.70.-s04.62.+v
keywords Kerr-EMDAblackholequasiboundstatesKlein-GordonequationconfluentHeunfunctionscalarcloudsuperradianceHawkingradiationgravitationalatom
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 derives exact solutions of the massive scalar Klein-Gordon equation in the Kerr-EMDA black hole background, a rotating charged spacetime from Einstein-Maxwell-dilaton-axion theory. Its central claim is that bound states of the scalar field have quantized complex frequencies given by a single exact condition (Eq. 55), obtained by imposing regularity at infinity and purely ingoing waves at the horizon. From that spectrum the paper derives a light-scalar cloud mass-spin relation, a horizon boson distribution and Hawking flux, and an amplification factor for superradiant scattering. A sympathetic reading takes this as the first exact scalar spectroscopy of the rotating EMDA black hole, reducing to the ordinary Kerr case when the dilaton parameter $D=0$.

What carries the argument

The load-bearing object is the confluent Heun differential equation and its polynomial condition $\delta/\alpha+(\beta+\gamma)/2+1=-n_r$, produced after a change of radial variable $y=(r-r_+)/\delta_r$ to the normal form. The polynomial condition is what converts the exact radial solution into the discrete quantization condition (Eq. 55), and the same radial solution is the seed for the scalar cloud, the horizon distribution function, and the amplification factor. In the superradiance section the engine is the asymptotic matching of near-horizon and far-region solutions expressed as hypergeometric functions, with the coefficients of the two independent far-region branches defining the reflection coefficient and hence the amplification factor.

What would settle it

A direct test is to solve Eq. (62) without dropping any terms for the parameter values in Fig. 5; its imaginary part is $\omega_a r_s$, which is nonzero whenever the black hole rotates, so no real scalar mass $\mu$ can satisfy the equation. This calculation alone decides whether stationary scalar clouds exist in this model, independent of any numerical scheme.

Watch

Extended reading notes

Core claim

The central discovery is that the Klein-Gordon equation for a massive scalar field in the Kerr-EMDA spacetime separates completely, and the radial factor is a confluent Heun function, a special-function family generalizing hypergeometric functions with one additional regular singular point. Requiring the radial wave to be normalizable at infinity and purely ingoing at the event horizon truncates the confluent Heun polynomial and fixes the complex eigenfrequencies $\omega$ through Eq. (55), where $\omega_a=m_\ell a/[r_+(r_+-2D)+a^2]$ is the horizon angular velocity. The same exact waveform is then used in three further claims: at $\mathrm{Re}(\omega)=\omega_a$ and $\mathrm{Im}(\omega)=0$, the light-mass limit gives a scalar-cloud mass-spin relation; linearizing the radial solution near the horizon gives a bosonic emission spectrum and a Hawking energy flux expressed as a polylogarithm; and matching near-horizon and far-zone hypergeometric solutions gives the superradiance amplification factor $Z_{\ell,m_\ell,a}$.

Load-bearing premise

The scalar-cloud results assume that the term $i\omega_a r_s$ can simply be set to zero when the scalar frequency is taken to equal the horizon's angular velocity, but that term does not contain the scalar mass and is nonzero for any rotating black hole, so the plotted cloud curves do not follow from the exact solution.

Editorial extensions

If this is right

  • Equation (55) yields the complete set of complex quasibound frequencies for massive scalar perturbations of Kerr-EMDA black holes, reducing to the Kerr spectrum when $D=0$ and to the massless expressions (56)--(57) when $\omega_0=0$.
  • In the ultralight limit the frequencies take the hydrogenic 'gravitational atom' form $\omega\approx\omega_0(1-\omega_0^2/(8n^2))$ with an imaginary part controlled by $m_\ell a$, so the regime relevant to superradiant instabilities is reproduced.
  • The near-horizon boson distribution is Bose-Einstein-like and turns into exponential growth when $\omega<\omega_a$; the associated Hawking flux vanishes in the extremal limit for all azimuthal modes.
  • The amplification factor $Z_{\ell,m_\ell,a}$ is positive only in the window $\omega_0<\omega<\omega_a$, vanishes at the superradiance threshold $\omega=\omega_a$, and grows with both the spin parameter $a$ and the charge-related dilaton parameter $D$.

Reading between the lines

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

  • If the exact spectrum (55) survives direct numerical integration of the radial equation, it gives a benchmark for computing quasinormal modes and greybody factors in EMDA, including regimes where perturbative methods fail.
  • The scalar-cloud inconsistency suggests that stationary clouds in this theory may need a charged scalar or an altered resonance condition; a corrected derivation would either modify the mass-spin curve or rule out neutral clouds for rotating EMDA black holes.
  • The superradiance amplification factor is derived to first order in $a\omega$; comparing it with a full numerical solution across the $(a,Q,\omega_0)$ parameter space would quantify the error of the asymptotic-matching approximation and could sharpen the threshold frequency.
  • The horizon boson distribution, being built from the exact radial solution, may be used to estimate spontaneous scalar pair-production rates and to connect the quasibound spectrum with black-hole evaporation endpoints.
Share X Bluesky LinkedIn Reddit HN

Formalized claims in Lean

  1. Claim #1: The central discovery is that the Klein-Gordon equation for a massive scalar field in the Kerr-EMDA spacetime separates completely, and the radial factor is a confluent Heun function, a special-function family generalizing hypergeometric functions with one additional regular singular point. Requiring the radial wave to be normalizable at infinity and purely ingoing at the event horizon truncates t

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript studies a massive scalar field in the Kerr-EMDA black hole spacetime. It separates the Klein-Gordon equation, solves the radial equation in terms of confluent Heun functions, and derives a quantization condition for quasibound-state frequencies, Eq. (55). The authors then use this condition to derive scalar cloud configurations (Sec. 4), a boson distribution near the horizon and a Hawking flux expressed as a polylogarithm (Sec. 5), and a superradiance amplification factor via asymptotic matching (Sec. 6). The abstract and conclusions advertise four headline results: exact eigensolutions, a scalar cloud mass-spin relation, horizon boson statistics/Hawking flux, and superradiance amplification.

Significance. The exact reduction of the Klein-Gordon equation in this rotating, charged, dilaton-axion background is a potentially useful technical contribution, and the gravitational-atom limit, Eqs. (58)-(59), provides a concrete, testable prediction. The derivation of the confluent Heun solution is self-contained and does not rely on fitted parameters. However, the scalar-cloud and Hawking-flux sections contain inconsistencies that invalidate two of the four advertised results. Because these sections are central to the paper's title and abstract, the significance of the manuscript in its current form is substantially reduced.

major comments (3)
  1. [Section 4, Eqs. (62)-(66)] The scalar cloud derivation is internally inconsistent. Substituting Re(ω)=ω_a and Im(ω)=0 into the exact quasibound-state quantization condition (55) produces Eq. (62), in which the left-hand side has an explicit term +iω_a r_s while the right-hand side is the real number -n. For a>0 and mℓ>0, ω_a = mℓ a/[r+(r+−2D)+a²] is positive, so this imaginary term is nonzero, independent of μ, and does not vanish in the light-mass limit μr_s→0. The series (63) is not a solution of Eq. (62): its imaginary part is O(μ³r_s³), whereas the uncancelled imaginary term is order one for fixed black hole parameters. Omitting this term 'for light scalar fields' is an additional assumption that contradicts the cloud condition Im(ω)=0, not a consequence of the expansion. Moreover, the quasibound-state quantization condition was derived using the purely ingoing horizon boundary condition, which is not the correct boundary condition for stationary scalar clouds; the cloud should instead be obtained from solutions regular at the horizon. Consequently, Eq. (66) and the scalar cloud curves in Fig. 5 do not follow from the exact solution, and the claimed analytic mass-spin relation for scalar clouds is unsupported.
  2. [Section 5, Eqs. (75)-(76) and Fig. 6] For mℓ>0 the integral in Eq. (75) is divergent: the integrand has a simple pole at ω=ω_a, so the flux is logarithmically divergent. The closed form (76) follows from expanding the denominator in a geometric series, which converges only when the exponent 4πmℓ a/δr is negative, i.e., for mℓ<0 (and mℓ=0 as a limiting case). For mℓ>0 the polylogarithm Li_2(e^{4πmℓ a/δr}) is not the value of the integral but an analytic continuation of a divergent series; the statement that positive-mℓ fluxes are 'complex-valued' and hence discarded is not a valid treatment. The claim that the Hawking flux vanishes at extremality for all mℓ is therefore not established, and Fig. 6 omits the superradiant modes that are central to the paper's topic. A complete treatment would include the greybody factor, which typically resolves the threshold divergence.
  3. [Section 5, Eq. (74)] In Eq. (74) the boson distribution is written as 1/|1−e^ζ|. For ζ<0 (i.e., ω<ω_a) this equals 1/(1−e^ζ), which is positive and grows as ω→ω_a. The standard Damour-Ruffini result for a rotating black hole is 1/(e^ζ−1), which is negative in the superradiant band; the negative value is the usual signature of superradiant amplification. The absolute value changes the sign of the occupation number and is not derived from the normalization condition (73); its use should be justified, and the relation to the standard spectrum should be clarified. As written, the 'exponential growth' of the distribution in the superradiant regime is an artifact of the absolute value.
minor comments (4)
  1. [Throughout] There are numerous typographical errors and malformed expressions, e.g., 'obatining' in the Conclusions, 'Klien-Gordon' before Eq. (77), and the broken radical notation in Eq. (58).
  2. [Section 6, Eq. (81)] The symbol ∆ω is introduced in Eq. (81) but never explicitly defined; please define it or remove it.
  3. [Figure 5] The captions of Fig. 5 do not specify which parameters are varied for each curve; the left/right descriptions in the text are ambiguous and should be clarified.
  4. [References] Reference [59] lists 'Phys. Rev. D 1719, 012019 (2021)', which is not a valid volume; please verify the citation.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central derivation is self-contained, and the paper's many self-citations are contextual rather than load-bearing.

full rationale

The paper's main chain is a direct analytical calculation: the Klein-Gordon equation is separated on the Kerr-EMDA background, the radial equation is put in normal form, identified with the confluent Heun equation, and the quantization condition (47)/(55) follows from the polynomial condition of the Heun function. No spectral parameter is fitted to the target frequencies, and the quasibound-state spectrum is not assumed as an input. The 'gravitational atom' expressions (58)-(59) are expansions of the derived quantization condition, so they are not circular even though they agree with known results in [8,43]. The scalar-cloud section does impose the defining cloud conditions Re(ω)=ω_a and Im(ω)=0 from the cited literature, and the mass-spin relation (66) is then obtained by expanding the resulting equation; this is a calculation under stated resonance conditions rather than a hidden re-insertion of the answer. The paper invokes many earlier works by the same authors, but the Heun formalism, the polynomial condition, and the asymptotic-matching formulas are either derived in the appendices or traced to standard references [50,61,62], so the self-citations are not the load-bearing justification. The scalar-cloud derivation does contain a non-circularity defect: the imaginary term iω_a r_s in Eq. (62) is of the same order as the retained real terms and is dropped by fiat to obtain (66), so Fig. 5 may not follow from the exact quantization condition. That is a mathematical consistency issue, not a circularity, and therefore does not raise the circularity score. Overall, the central derivation is self-contained and independent of fitted values, so the circularity burden is low.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper relies on the known Kerr-EMDA metric, the test-field Klein-Gordon equation, standard special-function properties, and the Damour-Ruffini method. It introduces no new entities or fitted parameters. The nonstandard assumptions are the omission of the imaginary term in the scalar cloud condition and the exclusion of positive-mℓ Hawking flux modes.

assumptions (7)
  • domain assumption The Kerr-EMDA metric (3) is an exact solution of the action (1).
    The paper takes the metric from García et al. [33] and does not re-derive it.
  • domain assumption The massive scalar field is a test field obeying the minimally coupled Klein-Gordon equation (8).
    Backreaction of the scalar on the geometry is neglected.
  • standard math The confluent Heun polynomial condition (B10) with integer n characterizes the quasibound state spectrum.
    Standard property of Heun functions used to obtain the quantization condition (48).
  • standard math The angular separation constant λ is given by the perturbative spheroidal harmonic expansion (15) to O(σ).
    The paper uses this expansion for small σ, so the exact solution carries an approximation when a≠0.
  • standard math The Damour-Ruffini analytic continuation gives the boson distribution (74).
    The method is cited from [23] and applied directly.
  • ad hoc to paper The imaginary term iωa rs in Eq. (62) may be omitted for light scalar fields.
    This omission is necessary to obtain the scalar cloud relation (64), but the term is independent of μ and does not vanish as μ→0.
  • ad hoc to paper Positive mℓ Hawking flux modes can be discarded when the polylogarithm becomes complex.
    The paper excludes these modes because the integral (75) diverges at the superradiant threshold, without a physical regulator.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Spectroscopy of Kerr-Einstein-Maxwell-Dilaton-Axion: Exact Quasibound States, Scalar Cloud, Horizon's Boson Statistics and Superradiance." pith.science (2026). https://pith.science/paper/UQ45PBFN

@misc{pith2026250108788,
  author       = {Pith},
  title        = {Pith review of: The Spectroscopy of Kerr-Einstein-Maxwell-Dilaton-Axion: Exact Quasibound States, Scalar Cloud, Horizon's Boson Statistics and Superradiance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UQ45PBFN}},
  note         = {Machine review of arXiv:2501.08788}
}
read the original abstract

In the present study, we investigate the quasibound states, scalar cloud and superradiance of relativistic scalar fields bound to a rotating black hole in Einstein-Maxwell-Dilaton-Axion theory (Kerr-EMDA). We present the exact eigensolutions of the governing Klein-Gordon equation in the black hole background. By imposing boundary conditions on the quasibound states, we are able to find the exact complex quasibound state frequencies of the corresponding radial wave functions in terms of the confluent Heun polynomial. Considering light scalar field limit of the obtained solution, we investigate the scalar-black hole resonance configuration known as the scalar cloud. In addition, we obtain analytic relation between light scalar mass and black hole spin for scalar cloud. We explore a boson distribution function by linearly expanding the radial wave function near the black hole's event horizon. Moreover, by applying the Damour-Ruffini method, this allows us to calculate the Hawking radiation flux. In the final section, we consider propagating wave in a slowly rotating Kerr-EMDA black hole for bosons having much larger Compton wavelength comparing to the size of rotating black hole. This condition allows us to use the asymptotic matching technique to calculate the amplification factor for scalar fields in the Kerr-EMDA black hole. We present the dependence of amplification factor on black hole parameters by graphical analysis.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 4 Pith papers

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

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

    gr-qc 2026-07 conditional novelty 6.0 of 10

    A Kerr–Hernquist black hole preserves a hydrogen-like scalar quasibound spectrum corrected by ρ₀r₀³, which lowers the scalar-cloud mass threshold and suppresses co-rotating superradiant growth while narrowing the ampl...

  2. A Novel Kerr-like Black Hole in a General Double Power Law Dark Matter Environment: Geometry, Spectroscopy, and Energy Extraction

    gr-qc 2026-08 reject novelty 5.0 of 10

    The paper presents a Newman-Janis rotating black hole in a double power law dark matter halo, with analytic scalar spectra, but the claimed singularity removal for gamma <= 2 is not established because the ring singul...

  3. The Spectroscopy of the 2+1 Dimensional Analog Black Hole in Photon-Fluid Model

    gr-qc 2025-09 conditional novelty 5.0 of 10

    For the 2+1D photon-fluid analog black hole, the paper derives exact quasibound states, Hawking radiation, superradiance amplification, and greybody factors, finding superradiance for ̟ < ω < m_l Ω_H.

  4. Tuning A Rotating Black Hole Spectrum with Dark Matter Halo: Quasibound States, Scalar Cloud, Black Hole Bomb and Superradiant Scattering

    gr-qc 2026-06 unverdicted novelty 4.0 of 10

    For a rotating black hole in a Dehnen dark matter halo, the scalar quasibound spectrum is the Kerr hydrogenic spectrum with a halo-shifted effective mass, and the superradiant window narrows through a halo-modified ho...

Reference graph

Works this paper leans on

60 extracted references · 14 canonical work pages · cited by 4 Pith papers

  1. [1]

    Lei, Y.-H., Yang, Z.-H., Kuang, X.-M.: Scalar field pertu rbation around a rotating hairy black hole: quasinormal modes, quasibound states and superradiant insta- bility. Eur. Phys. J. C 84(4), 438 (2024) https://doi.org/10.1140/epjc/s10052- 024-12814-6 arXiv:2310.05190 [gr-qc]

    arXiv 2024

  2. [2]

    Siqueira, P.H.C., Richartz, M.: Quasinormal modes, qua sibound states, scalar clouds, and superradiant instabilities of a Kerr-li ke black hole. Phys. Rev. D 106(2), 024046 (2022) https://doi.org/10.1103/PhysRevD.106.024046 arXiv:2205.00556 [gr-qc]

    arXiv 2022

  3. [3]

    Noda, S., Motohashi, H.: Spectroscopy of Kerr-AdS5 spac etime with the Heun function: Quasinormal modes, greybody factor, and eva poration. Phys. Rev. D 106(6), 064025 (2022) https://doi.org/10.1103/PhysRevD.106.064025 arXiv:2206.07721 [gr-qc]

    arXiv 2022

  4. [5]

    Physics Letters B 848, 138373 (2024) https://doi

    Senjaya, D.: Exact analytical quasibound states of a sca lar particle around a reissner-nordstr¨ om black hole. Physics Letters B 848, 138373 (2024) https://doi. org/10.1016/j.physletb.2023.138373

    arXiv 2024

  5. [6]

    Unive rse 9(5), 205 (2023) https://doi.org/10.3390/universe9050205 arXiv:2107.02065 [gr-qc]

    Vieira, H.S.: Quasibound States, Stability and Wave Fun ctions of the Test Fields in the Consistent 4D Einstein–Gauss–Bonnet Gravity. Unive rse 9(5), 205 (2023) https://doi.org/10.3390/universe9050205 arXiv:2107.02065 [gr-qc]

    arXiv 2023

  6. [7]

    Senjaya, D.: The Kerr–Bumblebee exact massive and massl ess scalar quasibound states and Hawking radiation. Eur. Phys. J. C 84(4), 424 (2024) https://doi.org/ 10.1140/epjc/s10052-024-12794-7

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

  7. [8]

    Huang, Y., Zhang, H.: Quasibound states of charged dilat onic black holes. Phys. Rev. D 103(4), 044062 (2021) https://doi.org/10.1103/PhysRevD.103.044062 29 arXiv:2012.12778 [gr-qc]

    arXiv 2021

  8. [9]

    Unive rse 9(5), 205 (2023) https://doi.org/10.3390/universe9050205

    Vieira, H.S.: Quasibound states, stability and wave fun ctions of the test fields in the consistent 4d einstein–gauss–bonnet gravity. Unive rse 9(5), 205 (2023) https://doi.org/10.3390/universe9050205

    doi:10.3390/universe9050205 2023

Show all 60 references
  1. [10]

    Vieira, H.S., Bezerra, V.B., Muniz, C.R.: Instability of the charged massive scalar field on the Kerr–Newman black hole spacetime. Eur. Ph ys. J. C 82(10), 932 (2022) https://doi.org/10.1140/epjc/s10052-022-10908-7 arXi v:2107.02562 [gr-qc]

    2022 arXiv

  2. [11]

    Physics Letters B 849, 138414 (2024) https:// doi.org/10.1016/j.physletb.2023.138414

    Senjaya, D.: Exact massive and massless scalar quasibo und states around a charged lense-thirring black hole. Physics Letters B 849, 138414 (2024) https:// doi.org/10.1016/j.physletb.2023.138414

    2024

  3. [12]

    Journal of High Energy Astrophy sics 41, 61–66 (2024) https://doi.org/10.1016/j.jheap.2024.01.004

    Senjaya, D.: Exact scalar quasibound states solutions of f(r) theory’s static spheri- cally symmetric black hole. Journal of High Energy Astrophy sics 41, 61–66 (2024) https://doi.org/10.1016/j.jheap.2024.01.004

    2024 doi

  4. [13]

    The European Physical Journal C 84 (2024) https://doi.org/10.1140/epjc/s10052- 024-12422-4

    Senjaya, D.: Exact massless scalar quasibound states o f the ernst black hole. The European Physical Journal C 84 (2024) https://doi.org/10.1140/epjc/s10052- 024-12422-4

    2024 doi

  5. [14]

    Senjaya, D.: Exact massive and massless scalar quasibo und states solutions of the Einstein–Maxwell-dilaton (EMD) black hole. Eur. Phys. J. C 84(3), 229 (2024) https://doi.org/10.1140/epjc/s10052-024-12600-4

    2024 doi

  6. [15]

    Senjaya, D.: Exact phonon quasibound states around an o ptical black hole. Eur. Phys. J. C 84(4), 388 (2024) https://doi.org/10.1140/epjc/s10052-024-12755-0

    2024 doi

  7. [16]

    Senjaya, D.: Relativistic scalar fields canonical quan tization in Ein- stein–Yang–Mills–Higgs’s rotating black hole space-time . Eur. Phys. J. C 84(7), 747 (2024) https://doi.org/10.1140/epjc/s10052-024-13106-9

    2024 doi

  8. [17]

    Senjaya, D., Burikham, P., Harko, T.: The exact relativ istic scalar quasibound states of the dyonic Kerr–Sen black hole: quantized energy, and Hawking radi- ation. Eur. Phys. J. C 84(8), 857 (2024) https://doi.org/10.1140/epjc/s10052- 024-13225-3 arXiv:2405.15219 [gr-qc]

    2024 arXiv

  9. [18]

    Vieira, H.S., Destounis, K., Kokkotas, K.D.: Analog Sc hwarzschild black holes of Bose-Einstein condensates in a cavity: Quasinormal mode s and quasibound states. Phys. Rev. D 107(10), 104038 (2023) https://doi.org/10.1103/PhysRevD. 107.104038 arXiv:2301.11480 [gr-qc]

    2023 arXiv

  10. [19]

    Hod, S.: Stationary Scalar Clouds Around Rotating Blac k Holes. Phys. Rev. D 86, 104026 (2012) https://doi.org/10.1103/PhysRevD.86.129902 arXiv:121 1.3202 [gr-qc]. [Erratum: Phys.Rev.D 86, 129902 (2012)] 30

    2012 doi

  11. [20]

    : Kerr-Newman scalar clouds

    Benone, C.L., Crispino, L.C.B., Herdeiro, C., Radu, E. : Kerr-Newman scalar clouds. Phys. Rev. D 90(10), 104024 (2014) https://doi.org/10.1103/PhysRevD. 90.104024 arXiv:1409.1593 [gr-qc]

    2014 arXiv

  12. [21]

    Annals Phys

    Senjaya, D., Ponglertsakul, S.: The extreme Reissner– Nordstr¨ om Black Hole: New exact solutions to the Klein–Gordon equation with minimal c oupling. Annals Phys. 473, 169898 (2025) https://doi.org/10.1016/j.aop.2024.169898

    2025

  13. [22]

    Hod, S.: Spatially regular charged black holes support ing charged massive scalar clouds. Phys. Rev. D 109(6), 064074 (2024) https://doi.org/10.1103/PhysRevD. 109.064074 arXiv:2401.07907 [gr-qc]

    2024 arXiv

  14. [23]

    Damour, T., Ruffini, R.: Black-hole evaporation in the kl ein-sauter-heisenberg- euler formalism. Phys. Rev. D 14, 332 (1976)

    1976

  15. [24]

    Franzin, E., Liberati, S., Oi, M.: Superradiance in Ker r-like black holes. Phys. Rev. D 103(10), 104034 (2021) https://doi.org/10.1103/PhysRevD.103.104034 arXiv:2102.03152 [gr-qc]

    2021 arXiv

  16. [25]

    Khodadi, M., Talebian, A., Firouzjahi, H.: Black Hole S uperradiance in f(R) Gravities (2020) arXiv:2002.10496 [gr-qc]

    2020 arXiv

  17. [26]

    Chen, C., Ge, F., Pan, Q., Jing, J.: Scalar perturbation around rotating Einstein- bumblebee BTZ black holes under Robin boundary conditions: Quasinormal modes and superradiance. Phys. Lett. B 861, 139228 (2025) https://doi.org/10. 1016/j.physletb.2024.139228

    2025

  18. [27]

    Huang, Y., Liu, D.-J.: Scalar clouds and the superradia nt instability regime of Kerr-Newman black hole. Phys. Rev. D 94(6), 064030 (2016) https://doi.org/10. 1103/PhysRevD.94.064030 arXiv:1606.08913 [gr-qc]

    2016 arXiv

  19. [28]

    JHEP 07, 003 (2024) https://doi.org/10.1007/ JHEP07(2024)003 arXiv:2404.02977 [hep-th]

    Wu, Z.-H., Lu, H.: Superradiant instability of charged extremal black holes in Einstein-Born-Infeld gravity. JHEP 07, 003 (2024) https://doi.org/10.1007/ JHEP07(2024)003 arXiv:2404.02977 [hep-th]

    2024 arXiv

  20. [29]

    Jha, S.K., Khodadi, M., Rahaman, A., Sheykhi, A.: Super radiant energy extrac- tion from rotating hairy Horndeski black holes. Phys. Rev. D 107(8), 084052 (2023) https://doi.org/10.1103/PhysRevD.107.084052 arXiv:2212.13051 [gr-qc]

    2023 arXiv

  21. [31]

    Alexander, S., Gabadadze, G., Jenks, L., Yunes, N.: Bla ck hole superradiance in dynamical Chern-Simons gravity. Phys. Rev. D 107(8), 084016 (2023) https:// doi.org/10.1103/PhysRevD.107.084016 arXiv:2201.02220 [gr-qc] 31

    2023 arXiv

  22. [32]

    Richarte, M.G., Martins, E.L., Fabris, J.C.: Scatteri ng and absorption of a scalar field impinging on a charged black hole in the Einstein-Maxwe ll-dilaton the- ory. Phys. Rev. D 105(6), 064043 (2022) https://doi.org/10.1103/PhysRevD.105. 064043 arXiv:2111.01595 [gr-qc]

    2022 arXiv

  23. [33]

    Garc ´ ıa, A., Galtsov, D., Kechkin, O.: Class of station ary axisymmetric solutions of the einstein-maxwell-dilaton-axion field equations. Ph ys. Rev. Lett. 74, 1276– 1279 (1995) https://doi.org/10.1103/PhysRevLett.74.1276

    1995 doi

  24. [34]

    Ganguly, C., SenGupta, S.: Penrose process in a charged axion–dilaton cou- pled black hole. Eur. Phys. J. C 76(4), 213 (2016) https://doi.org/10.1140/epjc/ s10052-016-4058-0 arXiv:1401.6826 [hep-th]

    2016 arXiv

  25. [35]

    Sahoo, S.K., Yadav, N., Banerjee, I.: Imprints of Einst ein-Maxwell-dilaton-axion gravity in the observed shadows of Sgr A* and M87*. Phys. Rev. D 109(4), 044008 (2024) https://doi.org/10.1103/PhysRevD.109.044008 arXiv:23 05.14870 [gr-qc]

    2024 doi

  26. [36]

    Press, W., Teukolsky, S.: Perturbations of a rotating b lack hole. ii. dynamical sta- bility of the kerr metric. The Astrophysical Journal 185, 649–674 (1973) https:// doi.org/10.1086/152445

    1973 doi

  27. [37]

    Berti, E., Cardoso, V., Casals, M.: Eigenvalues and eig enfunctions of spin- weighted spheroidal harmonics in four and higher dimension s. Phys. Rev. D 73, 024013 (2006) https://doi.org/10.1103/PhysRevD.73.024013

    2006 doi

  28. [38]

    Berti, E., Cardoso, V., Casals, M.: Erratum: Eigenvalu es and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dime nsions [phys. rev. d 73, 024013 (2006)]. Phys. Rev. D 73, 109902 (2006) https://doi.org/10.1103/ PhysRevD.73.109902

    2006

  29. [39]

    Cho, H.T., Cornell, A.S., Doukas, J., Naylor, W.: Asymp totic iteration method for spheroidal harmonics of higher-dimensional Ker r-(A)dS black holes. Phys. Rev. D 80, 064022 (2009) https://doi.org/10.1103/PhysRevD.80.064022 arXiv:0904.1867 [gr-qc]

    2009 arXiv

  30. [40]

    Suzuki, H., Takasugi, E., Umetsu, H.: Perturbations of Kerr-de Sitter black hole and Heun’s equations. Prog. Theor. Phys. 100, 491–505 (1998) https://doi.org/ 10.1143/PTP.100.491 arXiv:gr-qc/9805064

    1998 arXiv

  31. [41]

    Ponglertsakul, S., Gwak, B.: Massive scalar perturbat ions on Myers-Perry–de Sitter black holes with a single rotation. Eur. Phys. J. C 80(11), 1023 (2020) https://doi.org/10.1140/epjc/s10052-020-08616-1 arXi v:2007.16108 [gr-qc]

    2020 arXiv

  32. [42]

    Proceedings of the Royal Society of L ondon

    Breuer, R.A., Ryan, M.P., Waller, S.M.: Some propertie s of spin-weighted spheroidal harmonics. Proceedings of the Royal Society of L ondon. A. Mathe- matical and Physical Sciences 358, 71–86 (1977) 32

    1977

  33. [43]

    Yang, H., Miao, Y.-G.: Superradiance of massive scalar particles around rotating regular black holes*. Chin. Phys. C 47(7), 075101 (2023) https://doi.org/10. 1088/1674-1137/accdc7 arXiv:2211.15130 [gr-qc]

    2023 arXiv

  34. [44]

    Rowan, D.J., Stephenson, G.: The Klein-Gordon Equatio n in a Kerr-Newman Background Space. J. Phys. A 10, 15–23 (1977) https://doi.org/10.1088/0305- 4470/10/1/009

    1977 doi

  35. [45]

    Garc ´ ıa, G., Salgado, M.: Regular scalar charged cloud s around a reissner- nordstrom black hole and no-hair theorems. Phys. Rev. D 104, 064054 (2021) https://doi.org/10.1103/PhysRevD.104.064054

    2021 doi

  36. [46]

    Richartz, M., Rosa, J.a.L., Berti, E.: Bounds on the mas s of superradiantly unstable scalar fields around Kerr black holes (2024) arXiv:2405.01003 [gr-qc]

    2024 arXiv

  37. [47]

    Herdeiro, C.A.R., Radu, E.: Kerr black holes with scala r hair. Phys. Rev. Lett. 112, 221101 (2014) https://doi.org/10.1103/PhysRevLett.112.221101 arXiv:1403.2757 [gr-qc]

    2014 arXiv

  38. [48]

    Furuhashi, H., Nambu, Y.: Instability of massive scala r 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

    2004 arXiv

  39. [49]

    Hod, S.: Stationary resonances of rapidly-rotating Ke rr black holes. Eur. Phys. J. C 73(4), 2378 (2013) https://doi.org/10.1140/epjc/s10052-013-2378-x arXiv:1311.5298 [gr-qc]

    2013 arXiv

  40. [50]

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

    2010

  41. [51]

    Sakalli, I.: Analytical solutions in rotating linear d ilaton black holes: res- onant frequencies, quantization, greybody factor, and Haw king radiation. Phys. Rev. D 94(8), 084040 (2016) https://doi.org/10.1103/PhysRevD.94.084040 arXiv:1606.00896 [gr-qc]

    2016 arXiv

  42. [52]

    Other the sis (2003)

    Musiri, S.: Aspects of black hole scattering. Other the sis (2003)

    2003

  43. [53]

    Brito, R., Cardoso, V., Pani, P.: Superradiance: New Fr ontiers in Black Hole Physics. Lect. Notes Phys. 906, 1–237 (2015) https://doi.org/10.1007/978-3-319- 19000-6 arXiv:1501.06570 [gr-qc]

    2015 arXiv

  44. [54]

    Daniel Baumann, H.S.C., Porto, R.A.: Probing ultralig ht bosons with binary black holes. Phys. Rev. D 99, 044001 (2019)

    2019

  45. [55]

    Baumann, D., Chia, H.S., Porto, R.A.: Probing ultralig ht bosons with binary black holes. Phys. Rev. D 99, 044001 (2019) https://doi.org/10.1103/PhysRevD. 99.044001 33

    2019 doi

  46. [56]

    Journal of Cosmology and Astroparticle Physics 2019(12), 006 (2019) https:// doi.org/10.1088/1475-7516/2019/12/006

    Baumann, D., Chia, H.S., Stout, J., Haar, L.: The spectr a of gravitational atoms. Journal of Cosmology and Astroparticle Physics 2019(12), 006 (2019) https:// doi.org/10.1088/1475-7516/2019/12/006

    2019 doi

  47. [57]

    Baumann, D., Bertone, G., Stout, J., Tomaselli, G.M.: I onization of gravitational atoms. Phys. Rev. D 105, 115036 (2022) https://doi.org/10.1103/PhysRevD.105. 115036

    2022 doi

  48. [58]

    Journal of High Energy Astrophy sics 40, 49–54 (2023) https://doi.org/10.1016/j.jheap.2023.10.002

    Senjaya, D.: Exact analytical quasibound states of a sc alar particle around a slowly rotating black hole. Journal of High Energy Astrophy sics 40, 49–54 (2023) https://doi.org/10.1016/j.jheap.2023.10.002

    2023 doi

  49. [59]

    Senjaya, D., Rivera, A.S.: Canonical quantization of n eutral and charged static black hole as a gravitational atom. Phys. Rev. D 1719, 012019 (2021)

    2021

  50. [60]

    Tata Mcgraw hill„ New delhi (1991)

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

    1991

  51. [61]

    C larendon Press, ??? (1995)

    Ronveaux, A.: Heun’s Differential Equations, 1st edn. C larendon Press, ??? (1995)

    1995

  52. [62]

    Dover books on mathematics

    Bell, W.W.: Special Functions for Scientists and Engin eers. Dover books on mathematics. Dover Publications, ??? (2004). https://books.google.co.th/books? id=Jj5pXGTZIKkC 34

    2004

Pith tools

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