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REVIEW 2 major objections 4 minor 1 cited by

Rotation plus a magnetic field can quench superradiance and create horizon-decaying scalar clouds that isolated Kerr black holes cannot support.

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2026-07-12 11:34 UTC pith:KGX6PF2O

load-bearing objection Clean weak-field calculation of a magnetic horizon gap that produces Type-II evanescent scalar clouds with Im(ω)=0, but the advertised Kerr-BR geometry is reduced to Kerr plus test field throughout. the 2 major comments →

arxiv 2606.27958 v3 pith:KGX6PF2O submitted 2026-06-26 gr-qc

Horizon-Evanescent Scalar Clouds from Coupled Rotation and Magnetic Fields around Black Holes

Hengyu Xu , Haowei Chen , Shao-Jun Zhang This is my paper
classification gr-qc
keywords scalar cloudssuperradianceKerr-Bertotti-Robinsonmagnetized black holeshorizon gapType-II cloudscharged massive scalar
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.

Isolated spinning black holes can host stationary scalar clouds only when a bosonic field is perfectly synchronized with the horizon, so that the net energy flux vanishes. This paper shows that when the same black hole sits in an external magnetic field, the magnetic coupling opens a finite frequency band below that kinematic threshold in which the near-horizon wavenumber becomes purely imaginary. Inside that band the physical boundary condition changes from a propagating ingoing wave to a regular exponentially decaying state, the superradiant flux is forced to zero, and a new class of horizon-decaying (Type-II) scalar clouds appears. The Kerr-Bertotti-Robinson geometry is used as a clean, separable laboratory in which both the classical synchronized clouds and the new horizon-decaying clouds can be constructed analytically and numerically. The result implies that magnetized astrophysical environments can support a richer spectrum of stationary bosonic configurations than the vacuum Kerr spacetime alone.

Core claim

Magnetic coupling in the Kerr-Bertotti-Robinson geometry produces a strictly positive horizon gap, so that inside a finite frequency band below the usual kinematic threshold the horizon wavenumber is purely imaginary; the physical boundary condition becomes a regular exponentially decaying state, the superradiant flux is quenched, and a new class of horizon-decaying (Type-II) scalar clouds exists, distinct from classical synchronized Type-I clouds.

What carries the argument

The positive horizon gap (ω_c^{2} − D = a^{2} B^{2} q^{2} r₊^{2} / 16 M^{2} > 0), which forces the near-horizon wavenumber to become imaginary inside a finite band and thereby converts the horizon boundary condition from a propagating wave into an exponentially decaying state.

Load-bearing premise

The whole construction is done under a weak-magnetic-field approximation that neglects the back-reaction of the magnetic field on the spacetime curvature.

What would settle it

A numerical or analytic calculation of the same charged massive scalar in a magnetized Kerr geometry that retains O(B^{2}) curvature terms, checking whether the positive horizon gap and the zero-flux Type-II clouds survive when B is no longer small.

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

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 paper studies a charged massive scalar field on the Kerr–Bertotti–Robinson (Kerr-BR) geometry and claims that black-hole rotation together with an external magnetic field produces a positive near-horizon gap. After mapping the radial Klein–Gordon equation to a Schrödinger-like form, the authors show that the horizon wavenumber becomes purely imaginary inside a finite frequency band below the usual kinematic threshold ω_c. In that band the physical horizon boundary condition is a regular exponentially decaying state rather than a propagating ingoing wave, the superradiant flux is quenched (Im(ω)=0), and a new class of horizon-decaying (Type-II) scalar clouds appears alongside the classical synchronized (Type-I) clouds. The argument is developed analytically via matched asymptotic expansions (Appendix C) that exhibit an exact imaginary-phase cancellation, and is illustrated by numerical shooting solutions for both branches (Sec. V, Appendix D). All concrete calculations are performed under the weak-field truncation B≪1.

Significance. If the positive-gap mechanism and the resulting Type-II clouds survive under a consistent treatment of the background, the work identifies a qualitatively new stationary bosonic configuration that is absent in isolated Kerr systems and that is realized by the joint action of rotation and magnetic coupling. The algebraic reduction of the discriminant ω_c²−D to the perfect square a²B²q²r₊²/(16M²)>0 (Eqs. 7–8) and the subsequent phase-cancellation proof that forces Im(δω)=0 (Appendix C) are clean and parameter-free within the adopted approximation. The numerical profiles of both Type-I and Type-II clouds, including overtones, supply concrete spatial structure. These results would enlarge the known taxonomy of scalar clouds and motivate further study of magnetized astrophysical environments.

major comments (2)
  1. The abstract and introduction present the Kerr-BR geometry as the laboratory, yet every concrete result (tortoise coordinate, V_eff, α², discriminant, matched asymptotics, shooting) is obtained after the explicit weak-B truncation that replaces the full Kerr-BR metric functions by their Kerr limits and retains only a test magnetic potential (Sec. II, paragraph preceding App. A; App. A). Because the gap itself is an O(B²) effect, it is not guaranteed a priori that the same algebraic simplification survives once the retained O(B²) curvature corrections of the exact Type-D Kerr-BR metric are restored. The authors themselves flag the ultra-strong regime as open (Sec. VI); the same consistency issue already applies inside the weak-B window used for all reported results. A short calculation that restores the leading O(B²) metric corrections to the near-horizon expansion of V_eff (or an explici
  2. The numerical Type-II clouds (Figs. 5–7) are shown only at the single frequency ω=ω_c. The analytic quenching band is the finite interval ω₁≤ω≤min{ω₂,ω_f}. At least one representative solution strictly inside the open interval (or a short scan of Im(ω) across the band) should be supplied so that the existence of a continuum of horizon-decaying stationary states, rather than only the mid-band point, is demonstrated.
minor comments (4)
  1. Notation for the horizon function switches between Δ and Δ₀ without a clear statement of when the B-dependent terms are dropped; a single consistent definition would help.
  2. In several figure captions the phrase “gradual linear growth” for the near-horizon Type-II profiles is potentially confusing; a brief remark that this is the visual appearance of a very small κ_h would clarify the plots.
  3. The bound-state constraints of Appendix B are exhaustive but dense; a short summary sentence in the main text pointing to the most relevant (m>0, Bq>0) case would improve readability.
  4. A few typographical inconsistencies appear (e.g., “Schr¨ odinger”, missing spaces around some equation numbers); a light copy-edit pass would remove them.

Circularity Check

0 steps flagged

No circularity: the positive horizon gap, Im(ω)=0 quenching, and Type-II existence follow from direct algebraic evaluation of the near-horizon dispersion plus matched asymptotics/shooting on the KG equation; nothing is fitted or self-defined into existence.

full rationale

The load-bearing chain is: (i) weak-B reduction of Kerr-BR + charged massive KG (Sec. II, App. A) yields an effective potential whose near-horizon limit is −k_h^{2} = −[(ω−ω_c)^{2}−(ω_c^{2}−D)]; (ii) explicit substitution produces the algebraic identity ω_c^{2}−D = a^{2}B^{2}q^{2} r_{+}^{2}/(16M^{2}) > 0 (Eqs. 7–8), opening a real band [ω_{1},ω_{2}]; (iii) inside that band α^{2} < 0 so α becomes pure imaginary, the factor α·[k]^{2l+1} cancels the imaginary phase of the far-region wavenumber, forcing δω real (App. C, Eqs. C12–C13); (iv) numerical shooting simply solves the same radial ODE subject to the resulting decaying horizon BC and finds discrete (µ_s,B) pairs that also decay at infinity (App. D). No parameter is fitted to data and then re-used as a prediction; the resonance condition is the standard Detweiler matching, not a self-citation; the sole self-citation ([46]) concerns thermodynamics of the background and is never invoked for the gap or the clouds. The weak-B approximation is an explicit modeling choice whose domain of validity the authors themselves flag (Sec. VI); that is a correctness caveat, not a circular reduction of the claimed derivation to its inputs. Hence score 0.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 2 invented entities

The central claim rests on the exact Kerr-BR solution, the charged massive Klein-Gordon equation, the weak-B truncation that restores separability, standard LNRF horizon boundary conditions, and the validity of matched asymptotic expansions for the complex frequency. No free parameters are fitted to data; numerical values (M = 1, a = 0.5, q = 0.1–0.3) are illustrative. The only invented classification is the Type-I / Type-II dichotomy used to label the two branches.

free parameters (2)
  • illustrative spin a/M = 0.5
    Fixed by hand for all numerical profiles and phase diagrams; the analytic gap formula itself is independent of this choice but the concrete wave-function shapes are not.
  • illustrative charge q = 0.1–0.3
    Chosen for plots; the existence of the positive gap requires only B q ≠ 0, yet quantitative decay constants κ_h depend on the product.
axioms (4)
  • domain assumption Kerr-Bertotti-Robinson metric is an exact Type-D Einstein-Maxwell solution that reduces to Kerr when B → 0
    Invoked throughout Sec. II and used to guarantee separability of the Klein-Gordon equation; taken from Podolský & Ovcharenko (2025).
  • ad hoc to paper Weak-magnetic-field approximation B ≪ 1 permits truncation of O(B²) curvature back-reaction and reduction of the angular equation to a perturbed spheroidal equation
    Explicitly stated before Eq. (A3) and in Sec. VI; without it the radial and angular equations do not decouple in the form used for matching and shooting.
  • domain assumption Physical horizon boundary condition is the one that is ingoing in the locally non-rotating frame (Bardeen-Press-Teukolsky)
    Appendix B; determines the sign choice that converts an imaginary wavenumber into an exponentially decaying rather than growing solution.
  • standard math Matched asymptotic expansion of Detweiler / Furuhashi-Nambu type yields the leading imaginary frequency shift
    Appendix C; standard technique for low-frequency massive scalar modes on Kerr-like backgrounds.
invented entities (2)
  • Type-II (horizon-decaying / horizon-evanescent) scalar clouds no independent evidence
    purpose: Label the new stationary bound states that exist inside the magnetic quenching band and decay exponentially at the horizon
    Classification introduced by the authors; no independent observational handle is provided beyond the linear frequency-domain solutions themselves.
  • Positive horizon gap criterion no independent evidence
    purpose: Abstract the condition ω_c² − D > 0 under which the horizon wavenumber becomes imaginary below the kinematic threshold
    Presented as a broader sufficient criterion realized explicitly by Kerr-BR; currently verified only inside this geometry and the weak-B truncation.

pith-pipeline@v1.1.0-grok45 · 21454 in / 3294 out tokens · 39725 ms · 2026-07-12T11:34:27.758241+00:00 · methodology

0 comments
read the original abstract

We show that black-hole rotation and an external magnetic field can jointly generate a qualitatively new class of scalar cloud. Using the Kerr-Bertotti-Robinson geometry as a separable laboratory for magnetized rotating black holes, we study a charged massive scalar field and map the radial Klein-Gordon equation into a one-dimensional Schr\"{o}dinger-like form. The magnetic coupling shifts the near-horizon dispersion relation and realizes a positive horizon gap: a sufficient near-horizon criterion under which the horizon wavenumber becomes purely imaginary in a finite frequency band below the usual kinematic synchronization frequency. In this band the physical horizon boundary condition is no longer a propagating ingoing wave, but a regular exponentially decaying state. This rotation--magnetic-field mechanism quenches the superradiant flux and supports horizon-decaying scalar clouds (Type-II), distinct from the usual synchronized propagating clouds (Type-I). Matched asymptotic expansions and numerical shooting solutions are used to exhibit both branches and their spatial profiles. Thus the Kerr-Bertotti-Robinson solution is not an isolated curiosity, but an explicit realization of a broader positive-gap criterion for stationary bosonic configurations absent in isolated Kerr systems.

Figures

Figures reproduced from arXiv: 2606.27958 by Haowei Chen, Hengyu Xu, Shao-Jun Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1. Left: Parametric phase diagram in the ( [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Radial wavefunctions [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Comparison of different overtone numbers ( [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The permissible parameter space for classical synchronized scalar clouds at [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Radial wavefunctions [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison of different overtone modes ( [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The permissible parameter space for horizon-decaying scalar clouds strictly evaluated at [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗

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

  1. Massless scalar scattering by Kerr-Bertotti-Robinson black holes:transparent-end channels and superradiance

    gr-qc 2026-07 accept novelty 7.0

    In the transparent-end model, scalar superradiance on Kerr–BR black holes operates only when both the horizon frequency gate and the outer propagation gate are open, and the propagation gate closes the window at BM≈0....

Reference graph

Works this paper leans on

54 extracted references · 7 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Gravitational collapse: The role of general relativity,

    R. Penrose, “Gravitational collapse: The role of general relativity,”Riv. Nuovo Cim.1(1969) 252–276. 23

    1969

  2. [2]

    Reversible and irreversible transformations in black-hole physics,

    D. Christodoulou, “Reversible and irreversible transformations in black-hole physics,”Physical Review Letters25no. 25, (1970) 1596

    1970

  3. [3]

    Introducing the black hole,

    R. Ruffini and J. A. Wheeler, “Introducing the black hole,”Physics Today24no. 1, (1971) 30–41

    1971

  4. [4]

    Generation of Waves by a Rotating Body,

    Y. B. Zeldovich, “Generation of Waves by a Rotating Body,”Soviet Journal of Experimental and Theoretical Physics Letters14(1971) 180

    1971

  5. [5]

    Amplification of Cylindrical Electromagnetic Waves Reflected from a Rotating Body,

    Y. B. Zeldovich, “Amplification of Cylindrical Electromagnetic Waves Reflected from a Rotating Body,”Soviet Journal of Experimental and Theoretical Physics35(1972) 1085

    1972

  6. [6]

    Interpretation of gravitational-wave observations,

    C. W. Misner, “Interpretation of gravitational-wave observations,”Physical Review Letters28no. 15, (1972) 994

    1972

  7. [7]

    Amplification of waves during reflection from a rotating

    A. A. Starobinskii, “Amplification of waves during reflection from a rotating ”black hole”,”Sov. Phys. JETP37no. 1, (1973) 28–32

    1973

  8. [8]

    Extraction of energy and charge from a black hole,

    J. D. Bekenstein, “Extraction of energy and charge from a black hole,”Physical Review D7no. 4, (1973) 949

    1973

  9. [9]

    Perturbations of a rotating black hole. i. fundamental equations for gravitational, electromagnetic, and neutrino-field perturbations,

    S. A. Teukolsky, “Perturbations of a rotating black hole. i. fundamental equations for gravitational, electromagnetic, and neutrino-field perturbations,”The Astrophysical Journal185(1973) 635–648

    1973

  10. [10]

    Floating orbits, superradiant scattering and the black-hole bomb,

    W. H. Press and S. A. Teukolsky, “Floating orbits, superradiant scattering and the black-hole bomb,” Nature238no. 5361, (1972) 211–212

    1972

  11. [11]

    On quantum resonances in stationary geometries,

    T. Damour, N. Deruelle, and R. Ruffini, “On quantum resonances in stationary geometries,”Lett. Nuovo Cim.15no. 8, (1976) 257–262

    1976

  12. [12]

    Instabilities of massive scalar perturbations of a rotating black hole,

    T. J. M. Zouros and D. M. Eardley, “Instabilities of massive scalar perturbations of a rotating black hole,”Annals of Physics118no. 1, (1979) 139–155

    1979

  13. [13]

    Klein-gordon equation and rotating black holes,

    S. Detweiler, “Klein-gordon equation and rotating black holes,”Physical Review D22no. 10, (1980) 2323

    1980

  14. [14]

    Instability of massive scalar fields in kerr-newman spacetime,

    H. Furuhashi and Y. Nambu, “Instability of massive scalar fields in kerr-newman spacetime,”Progress of Theoretical Physics112no. 6, (2004) 983–995

    2004

  15. [15]

    Black-hole bomb and superradiant instabilities,

    V. Cardoso, O. J. C. Dias, J. P. S. Lemos, and S. Yoshida, “Black-hole bomb and superradiant instabilities,”Physical Review D70no. 4, (2004) 044039

    2004

  16. [16]

    Instability of the massive klein-gordon field on the kerr spacetime,

    S. R. Dolan, “Instability of the massive klein-gordon field on the kerr spacetime,”Physical Review D 76no. 8, (2007) 084001

    2007

  17. [17]

    Superradiant instabilities of rotating black holes in the time domain,

    S. R. Dolan, “Superradiant instabilities of rotating black holes in the time domain,”Physical Review D87no. 12, (2013) 124026

    2013

  18. [18]

    Superradiant instabilities in astrophysical systems,

    H. Witek, V. Cardoso, A. Ishibashi, and U. Sperhake, “Superradiant instabilities in astrophysical systems,”Physical Review D87no. 4, (2013) 043513

    2013

  19. [19]

    Stationary scalar clouds around rotating black holes,

    S. Hod, “Stationary scalar clouds around rotating black holes,”Physical Review D86no. 10, (2012) 104026

    2012

  20. [20]

    Kerr-newman black holes with stationary charged scalar clouds,

    S. Hod, “Kerr-newman black holes with stationary charged scalar clouds,”Physical Review D90 no. 2, (2014) 024051. 24

    2014

  21. [21]

    Kerr black holes with scalar hair,

    C. A. R. Herdeiro and E. Radu, “Kerr black holes with scalar hair,”Physical Review Letters112 no. 22, (2014) 221101

    2014

  22. [22]

    Construction and physical properties of kerr black holes with scalar hair,

    C. A. R. Herdeiro and E. Radu, “Construction and physical properties of kerr black holes with scalar hair,”Classical and Quantum Gravity32no. 14, (2015) 144001

    2015

  23. [23]

    Kerr-newman scalar clouds,

    C. L. Benone, L. C. B. Crispino, C. A. R. Herdeiro, and E. Radu, “Kerr-newman scalar clouds,” Physical Review D90no. 10, (2014) 104024

    2014

  24. [24]

    Scalar clouds around Kerr–Sen black holes,

    Y. Huang, D.-J. Liu, X.-H. Zhai, and X.-Z. Li, “Scalar clouds around Kerr–Sen black holes,”Class. Quant. Grav.34no. 15, (2017) 155002,arXiv:1706.04441 [gr-qc]

    Pith/arXiv arXiv 2017

  25. [25]

    Black-hole bomb and photon-mass bounds,

    P. Pani, V. Cardoso, L. Gualtieri, E. Berti, and A. Ishibashi, “Black-hole bomb and photon-mass bounds,”Physical Review Letters109no. 13, (2012) 131102

    2012

  26. [26]

    Superradiant instability and backreaction of massive vector fields around kerr black holes,

    W. E. East and F. Pretorius, “Superradiant instability and backreaction of massive vector fields around kerr black holes,”Phys. Rev. Lett.119(Jul, 2017) 041101. https://link.aps.org/doi/10.1103/PhysRevLett.119.041101

    doi:10.1103/physrevlett.119.041101 2017

  27. [27]

    Kerr black holes with proca hair,

    C. Herdeiro, E. Radu, and H. R´ unarsson, “Kerr black holes with proca hair,”Classical and Quantum Gravity33no. 15, (2016) 154001

    2016

  28. [28]

    Stationary scalar clouds supported by rapidly-rotating acoustic black holes in a photon-fluid model,

    S. Hod, “Stationary scalar clouds supported by rapidly-rotating acoustic black holes in a photon-fluid model,”Physical Review D103no. 8, (2021) 084003

    2021

  29. [29]

    Acoustic black-hole bombs and scalar clouds in a photon-fluid model,

    M. Ciszak and F. Marino, “Acoustic black-hole bombs and scalar clouds in a photon-fluid model,” Physical Review D103no. 4, (2021) 045004

    2021

  30. [30]

    Superradiance: New Frontiers in Black Hole Physics,

    R. Brito, V. Cardoso, and P. Pani, “Superradiance: New Frontiers in Black Hole Physics,”Lect. Notes Phys.906(2015) pp.1–237,arXiv:1501.06570 [gr-qc]

    Pith/arXiv arXiv 2015

  31. [31]

    Electromagnetic extraction of energy from kerr black holes,

    R. D. Blandford and R. L. Znajek, “Electromagnetic extraction of energy from kerr black holes,” Monthly Notices of the Royal Astronomical Society179no. 3, (1977) 433–456

    1977

  32. [32]

    Black hole in a uniform magnetic field,

    R. M. Wald, “Black hole in a uniform magnetic field,”Physical Review D10no. 6, (1974) 1680

    1974

  33. [33]

    Kerr black holes in a magnetic universe,

    F. J. Ernst, “Kerr black holes in a magnetic universe,”Journal of Mathematical Physics17no. 1, (1976) 54–56

    1976

  34. [34]

    Pure magnetic and magic electric geons,

    M. A. Melvin, “Pure magnetic and magic electric geons,”Physics Letters8no. 1, (1964) 65–68

    1964

  35. [35]

    Ergoregions in magnetised black hole spacetimes,

    G. W. Gibbons, A. H. Mujtaba, and C. N. Pope, “Ergoregions in magnetised black hole spacetimes,” Classical and Quantum Gravity30no. 12, (2013) 125008

    2013

  36. [36]

    Thermodynamics of Regular Accelerating Black Holes,

    M. Astorino, “Thermodynamics of Regular Accelerating Black Holes,”Phys. Rev. D95no. 6, (2017) 064007,arXiv:1612.04387 [gr-qc]

    Pith/arXiv arXiv 2017

  37. [37]

    Insights from melvin–kerr–newman spacetimes,

    I. Booth, M. Hunt, A. Palomo-Lozano, and H. K. Kunduri, “Insights from melvin–kerr–newman spacetimes,”Classical and Quantum Gravity32no. 23, (2015) 235025

    2015

  38. [38]

    Revival of penrose process for astrophysical applications,

    S. M. Wagh, S. V. Dhurandhar, and N. Dadhich, “Revival of penrose process for astrophysical applications,”Atrophys. J.301(1986) 1018

    1986

  39. [39]

    The energetics of black holes in electromagnetic fields by the penrose process,

    S. M. Wagh and N. Dadhich, “The energetics of black holes in electromagnetic fields by the penrose process,”Phys. Rept.183no. 4, (1989) 137–192. 25

    1989

  40. [40]

    Magnetized black hole as a gravitational lens,

    R. A. Konoplya, “Magnetized black hole as a gravitational lens,”Phys. Lett. B644(2007) 219–223, arXiv:gr-qc/0608066

    Pith/arXiv arXiv 2007

  41. [41]

    Magnetic field creates strong superradiant instability,

    R. A. Konoplya, “Magnetic field creates strong superradiant instability,”Phys. Lett. B666(2008) 283–287,arXiv:0801.0846 [hep-th]

    Pith/arXiv arXiv 2008

  42. [42]

    Superradiant instability of black holes immersed in a magnetic field,

    R. Brito, V. Cardoso, and P. Pani, “Superradiant instability of black holes immersed in a magnetic field,”Physical Review D89no. 10, (2014) 104045

    2014

  43. [43]

    Black holes, stationary clouds and magnetic fields,

    N. M. Santos and C. A. R. Herdeiro, “Black holes, stationary clouds and magnetic fields,”Physics Letters B815(2021) 136142

    2021

  44. [44]

    Kerr black hole in a uniform Bertotti-Robinson magnetic field: An exact solution,

    J. Podolsk´ y and H. Ovcharenko, “Kerr black hole in a uniform Bertotti-Robinson magnetic field: An exact solution,”arXiv preprint arXiv:2507.05199(2025)

    arXiv 2025

  45. [45]

    Uniform electromagnetic field in the theory of general relativity,

    B. Bertotti, “Uniform electromagnetic field in the theory of general relativity,”Phys. Rev.116(Dec,

  46. [46]

    1331–1333.https://link.aps.org/doi/10.1103/PhysRev.116.1331

    doi:10.1103/physrev.116.1331

  47. [47]

    Thermodynamics of Kerr-Bertotti-Robinson black hole,

    L. Hu, R.-G. Cai, and S.-J. Wang, “Thermodynamics of Kerr-Bertotti-Robinson black hole,” arXiv:2603.18821 [gr-qc]

    arXiv

  48. [48]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun,Handbook of mathematical functions with formulas, graphs, and mathematical tables, vol. 55. US Government printing office, 1964

    1964

  49. [49]

    C. M. Bender and S. A. Orszag,Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory. Springer Science & Business Media, 1999

    1999

  50. [50]

    W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,Numerical recipes 3rd edition: The art of scientific computing. Cambridge university press, 2007

    2007

  51. [51]

    Black Hole in an External Magnetic Field,

    D. V. Galtsov and V. I. Petukhov, “Black Hole in an External Magnetic Field,”Zh. Eksp. Teor. Fiz. 74(1978) 801–818

    1978

  52. [52]

    Energy extraction via magnetic reconnection in magnetized black holes,

    S.-J. Zhang, “Energy extraction via magnetic reconnection in magnetized black holes,”arXiv preprint arXiv:2405.16941(2024)

    Pith/arXiv arXiv 2024

  53. [53]

    Physics in ultra-strong magnetic fields,

    R. C. Duncan, “Physics in ultra-strong magnetic fields,”AIP Conf. Proc.526no. 1, (2000) 830, arXiv:astro-ph/0002442

    Pith/arXiv arXiv 2000

  54. [54]

    Rotating black holes: locally nonrotating frames, energy extraction, and scalar synchrotron radiation,

    J. M. Bardeen, W. H. Press, and S. A. Teukolsky, “Rotating black holes: locally nonrotating frames, energy extraction, and scalar synchrotron radiation,”The Astrophysical Journal178(1972) 347–370

    1972