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arxiv: 2605.18249 · v1 · pith:GFOF6C5Anew · submitted 2026-05-18 · 🌀 gr-qc

Quasinormal modes of a rotating loop quantum black hole

Zhongzhinan Dong , Shulan Li , Dan Zhang , Jian-Pin Wu This is my paper

Pith reviewed 2026-05-20 09:19 UTC · model grok-4.3

classification 🌀 gr-qc
keywords quasinormal modesloop quantum gravityrotating black holesscalar perturbationsquantum correctionscontinued fraction methodovertone spectrum
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The pith

Quantum corrections in rotating loop quantum black holes reduce both oscillation frequencies and damping rates of scalar perturbations for all fundamental modes.

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

This paper studies the ringing behavior of an effective rotating black hole that incorporates loop quantum gravity corrections. It builds the spacetime metric from a spherical quantum model and applies the continued fraction technique to find the complex frequencies of a massless scalar field. The central result is that stronger quantum corrections make the real part of the frequency smaller and the damping weaker across all lowest modes, so the perturbations oscillate more slowly and last longer. Rotation adds further structure: it produces a frequency crossover for non-spinning perturbations that disappears when angular momentum is present, and it shifts the location of overtone outbursts to smaller quantum parameters. These patterns supply concrete predictions for how quantum gravity might appear in the ringdown stage of gravitational-wave signals.

Core claim

The paper establishes that, on the effective rotating loop quantum black hole background obtained from a covariant spherical solution via the improved Newman-Janis algorithm, every fundamental quasinormal mode of a massless scalar field exhibits a monotonic drop in both real frequency and damping rate as the quantum correction parameter is increased; rotation further modulates the spectrum by inducing crossovers or suppressions that depend on the angular quantum numbers of the perturbation, while the overtone sector preserves the characteristic quantum signatures of the non-rotating case but displaces them toward weaker corrections.

What carries the argument

The effective rotating loop quantum black hole metric, constructed from the covariant spherical model by the improved Newman-Janis algorithm, which serves as the background on which the continued-fraction computation of scalar quasinormal modes is performed.

If this is right

  • Larger quantum corrections produce slower oscillations and weaker damping for every fundamental scalar mode.
  • Rotation induces a real-frequency crossover for spherically symmetric perturbations that is suppressed once orbital angular momentum is added.
  • In the overtone sector rotation moves the quantum-induced outbursts to smaller values of the correction parameter.
  • Nonzero azimuthal angular momentum further increases frequencies and decreases damping, producing spectral inversions among higher overtones.

Where Pith is reading between the lines

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

  • If the metric is realistic, quantum gravity effects would lengthen the ringdown phase of mergers involving spinning black holes, altering the expected gravitational-wave template.
  • The same construction could be used to compute electromagnetic or gravitational perturbation spectra and check whether the reported trends persist.
  • Comparison with full numerical relativity simulations that include loop quantum corrections would test whether the effective metric reproduces the dominant late-time behavior.
  • The shift of overtone features to weaker quantum parameters suggests that even modest loop quantum effects could become visible in high-overtone data if rotation is present.

Load-bearing premise

The effective metric obtained by applying the improved Newman-Janis algorithm to the spherical loop quantum model accurately captures the essential quantum-gravity features once rotation is included.

What would settle it

A gravitational-wave observation showing that the damping time of the ringdown signal from a spinning black hole shortens rather than lengthens as the inferred quantum correction grows would falsify the monotonic reduction.

Figures

Figures reproduced from arXiv: 2605.18249 by Dan Zhang, Jian-Pin Wu, Shulan Li, Zhongzhinan Dong.

Figure 1
Figure 1. Figure 1: FIG. 1: QNFs of the fundamental modes ( [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: QNFs of the fundamental modes ( [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: QNFs of the fundamental modes ( [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: QNFs of scalar field perturbations ( [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: QNFs of scalar field perturbations ( [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: QNFs of scalar field perturbations ( [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
read the original abstract

We investigate the quasinormal modes of a massless scalar field on an effective rotating loop quantum black hole background, constructed from a covariant spherical model via an improved Newman-Janis algorithm. Using the continued fraction method, we compute the spectrum for both fundamental and overtone modes, and systematically analyze how the frequencies depend on the quantum correction, spin, and angular structure of the perturbation. For all fundamental modes, increasing the quantum gravity correction monotonically reduces both the oscillation frequency and the damping rate, signaling slower oscillations and prolonged decay. Rotation imprints a nontrivial modulation: for a spherically symmetric perturbation, the real frequency displays a crossover as the spin grows, whereas this feature is suppressed once angular momentum is turned on; further activating the azimuthal component enhances the frequency and reduces the damping even more strongly. In the overtone sector, the rotating solution retains the hallmark quantum gravitational signatures of the spherical case - overtone outbursts and non-monotonic evolution - with rotation shifting these phenomena to weaker quantum corrections. Nonzero orbital angular momentum suppresses the outbursts, while the azimuthal degree of freedom boosts the frequency, giving rise to novel spectral inversions among higher overtones. Our results confirm that the effective rotating metric captures essential loop quantum gravity features, providing clear theoretical benchmarks for black hole spectroscopy and future gravitational-wave observations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper computes quasinormal modes of a massless scalar field on an effective rotating loop quantum black hole background obtained by applying an improved Newman-Janis algorithm to a covariant spherical LQG solution. Using the continued fraction method, it reports spectra for fundamental and overtone modes and finds that increasing the quantum correction parameter monotonically reduces both the real oscillation frequency and the damping rate for all fundamental modes. Rotation introduces nontrivial modulations, including a crossover in real frequency for spherically symmetric perturbations that is suppressed with angular momentum; overtones retain spherical LQG signatures such as outbursts but shifted to weaker quantum corrections, with additional inversions among higher overtones.

Significance. If the effective metric construction is reliable, the results supply concrete, falsifiable benchmarks for how loop quantum gravity corrections modify black-hole ringdown signals, which could inform future gravitational-wave spectroscopy. The systematic parameter study covering spin, quantum correction, and angular structure of the perturbation, together with the use of the standard continued-fraction method, constitutes a clear strength. The work is therefore potentially useful for the community provided the central modeling assumption is placed on firmer ground.

major comments (2)
  1. [Background construction paragraph] Background construction paragraph (and abstract): the strongest claim—that increasing the quantum correction monotonically reduces both Re(ω) and |Im(ω)| for every fundamental mode—rests on the improved Newman-Janis algorithm preserving the essential LQG features of the underlying spherical solution. No explicit check is supplied that the resulting rotating metric satisfies the effective Einstein equations with the LQG term or that the quantum correction remains localized without spurious frame-dragging contributions. This verification is load-bearing for the reported monotonic trends and for the assertion that the metric “captures essential loop quantum gravity features.”
  2. [Results on fundamental modes] Results section on fundamental modes: the reported crossover in real frequency for spherically symmetric perturbations as spin increases is presented as a nontrivial imprint of rotation, yet the text does not quantify the numerical convergence or error bars on the continued-fraction roots at the crossover point. Without such diagnostics it is unclear whether the crossover is a genuine physical feature or an artifact of truncation in the continued-fraction expansion.
minor comments (2)
  1. Notation for the quantum correction parameter is introduced without an explicit range or normalization convention in the abstract; a brief statement of the interval explored would improve readability.
  2. Figure captions for the frequency plots should explicitly state the range of the quantum parameter and the fixed values of spin and multipole indices used in each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which have helped us improve the presentation and clarify key aspects of our work. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

  1. Referee: Background construction paragraph (and abstract): the strongest claim—that increasing the quantum correction monotonically reduces both Re(ω) and |Im(ω)| for every fundamental mode—rests on the improved Newman-Janis algorithm preserving the essential LQG features of the underlying spherical solution. No explicit check is supplied that the resulting rotating metric satisfies the effective Einstein equations with the LQG term or that the quantum correction remains localized without spurious frame-dragging contributions. This verification is load-bearing for the reported monotonic trends and for the assertion that the metric “captures essential loop quantum gravity features.”

    Authors: We agree that an explicit verification of the effective Einstein equations for the rotating metric would provide stronger support for the construction. The improved Newman-Janis algorithm is applied following the standard procedure that preserves the essential quantum-corrected functions from the spherical covariant LQG solution, reducing exactly to that solution in the non-rotating limit. We have added a dedicated paragraph in Section II of the revised manuscript justifying this approach, emphasizing the localization of quantum corrections near the horizon and the absence of spurious frame-dragging terms by construction. We have also moderated the language in the abstract and background section, replacing the direct assertion with a statement that the metric provides an effective description capturing essential LQG features. A complete derivation of the rotating effective equations lies beyond the present scope but is noted as a direction for future work. revision: partial

  2. Referee: Results section on fundamental modes: the reported crossover in real frequency for spherically symmetric perturbations as spin increases is presented as a nontrivial imprint of rotation, yet the text does not quantify the numerical convergence or error bars on the continued-fraction roots at the crossover point. Without such diagnostics it is unclear whether the crossover is a genuine physical feature or an artifact of truncation in the continued-fraction expansion.

    Authors: We thank the referee for highlighting the need for numerical diagnostics. In the revised manuscript we have added a new subsection discussing the convergence properties of the continued-fraction method. We increased the truncation order systematically and verified that the location and existence of the crossover remain stable, with relative variations in the real frequency below 0.1 percent across truncation levels. Error estimates derived from the convergence tests are now reported for the frequencies near the crossover point, and these are indicated on the relevant figures. These checks confirm that the crossover is a robust physical feature arising from the interplay between rotation and the quantum correction rather than a truncation artifact. revision: yes

Circularity Check

0 steps flagged

No significant circularity; QNM spectra computed independently on constructed metric

full rationale

The derivation proceeds by first adopting a known covariant spherical LQG metric, applying the improved Newman-Janis algorithm to obtain an effective rotating background, and then numerically solving the massless scalar wave equation via the continued-fraction method. The reported monotonic dependence of Re(ω) and |Im(ω)| on the quantum parameter is an output of that numerical procedure, not a re-expression of the input metric parameters or a fitted quantity. Self-citations for the spherical seed metric and the NJ algorithm are present but do not carry the central claim; the spectra remain falsifiable against external benchmarks and are not forced by definition or by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the validity of the effective metric construction and the applicability of the continued fraction method to the wave equation on that background; the quantum correction parameter is treated as an input whose value is varied rather than derived.

free parameters (1)
  • quantum correction parameter
    Controls the strength of loop quantum gravity effects in the effective metric and is varied to study monotonic trends.
axioms (1)
  • domain assumption The improved Newman-Janis algorithm applied to the covariant spherical model yields a physically acceptable rotating metric that retains essential loop quantum gravity features.
    Invoked in the construction of the background spacetime used for all subsequent calculations.
invented entities (1)
  • effective rotating loop quantum black hole metric no independent evidence
    purpose: Background geometry incorporating loop quantum gravity corrections for a rotating case
    Constructed via the improved Newman-Janis algorithm; no independent observational or theoretical verification is provided in the abstract.

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Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages · 22 internal anchors

  1. [1]

    1: QNFs of the fundamental modes (n= 0) for the scalar field perturbation with l= 0 andm= 0, as a function of the loop quantum parameterr 0

    20 r0 |Im(ω)| a = 0 a = 0.1 a = 0.2 a = 0.3 a = 0.4 FIG. 1: QNFs of the fundamental modes (n= 0) for the scalar field perturbation with l= 0 andm= 0, as a function of the loop quantum parameterr 0. Left panel: real part Re(ω) of the QNFs; right panel: absolute value of the imaginary part|Im(ω)|. The curves are for the rotation parametera= 0,0.1,0.2,0.3,0....

    work page

  2. [2]

    C. Lan, H. Yang, Y. Guo, and Y.-G. Miao,Regular Black Holes: A Short Topic Review, Int. J. Theor. Phys.62(2023), no. 9 202, [arXiv:2303.11696]

    work page arXiv 2023

  3. [3]

    Black Holes in Loop Quantum Gravity

    A. Perez,Black Holes in Loop Quantum Gravity, Rept. Prog. Phys.80(2017), no. 12 126901, [arXiv:1703.09149]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  4. [4]

    Loop quantum gravity and black hole singularity

    L. Modesto,Loop quantum gravity and black hole singularity, in 17th SIGRAV Conference, 1, 2007.hep-th/0701239

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  5. [5]

    Quantum Nature of the Big Bang

    A. Ashtekar, T. Pawlowski, and P. Singh,Quantum nature of the big bang, Phys. Rev. Lett. 96(2006) 141301, [gr-qc/0602086]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  6. [6]

    Quantum Nature of the Big Bang: Improved dynamics

    A. Ashtekar, T. Pawlowski, and P. Singh,Quantum Nature of the Big Bang: Improved dynamics, Phys. Rev. D74(2006) 084003, [gr-qc/0607039]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  7. [7]

    Quantum Nature of the Big Bang: An Analytical and Numerical Investigation

    A. Ashtekar, T. Pawlowski, and P. Singh,Quantum Nature of the Big Bang: An Analytical and Numerical Investigation. I., Phys. Rev. D73(2006) 124038, [gr-qc/0604013]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  8. [8]

    Loop Quantum Cosmology, Boundary Proposals, and Inflation

    M. Bojowald and K. Vandersloot,Loop quantum cosmology, boundary proposals, and inflation, Phys. Rev. D67(2003) 124023, [gr-qc/0303072]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  9. [9]

    Ashtekar and E

    A. Ashtekar and E. Bianchi,A short review of loop quantum gravity, Rept. Prog. Phys.84 (2021), no. 4 042001, [arXiv:2104.04394]. 23

    work page arXiv 2021

  10. [10]

    Dynamical coherent states and physical solutions of quantum cosmological bounces

    M. Bojowald,Dynamical coherent states and physical solutions of quantum cosmological bounces, Phys. Rev. D75(2007) 123512, [gr-qc/0703144]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  11. [11]

    Inflation from Quantum Geometry

    M. Bojowald,Inflation from quantum geometry, Phys. Rev. Lett.89(2002) 261301, [gr-qc/0206054]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  12. [12]

    Space-Time Structure of Loop Quantum Black Hole

    L. Modesto,Semiclassical loop quantum black hole, Int. J. Theor. Phys.49(2010) 1649–1683, [arXiv:0811.2196]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  13. [13]

    Loop quantum black hole

    L. Modesto,Loop quantum black hole, Class. Quant. Grav.23(2006) 5587–5602, [gr-qc/0509078]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  14. [14]

    Loop quantization of spherically symmetric midi-superspaces

    M. Campiglia, R. Gambini, and J. Pullin,Loop quantization of spherically symmetric midi-superspaces, Class. Quant. Grav.24(2007) 3649–3672, [gr-qc/0703135]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  15. [15]

    Gambini, J

    R. Gambini, J. Olmedo, and J. Pullin,Spherically symmetric loop quantum gravity: analysis of improved dynamics, Class. Quant. Grav.37(2020), no. 20 205012, [arXiv:2006.01513]

    work page arXiv 2020

  16. [16]

    C. G. Boehmer and K. Vandersloot,Loop Quantum Dynamics of the Schwarzschild Interior, Phys. Rev. D76(2007) 104030, [arXiv:0709.2129]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  17. [17]

    Phenomenological loop quantum geometry of the Schwarzschild black hole

    D.-W. Chiou,Phenomenological loop quantum geometry of the Schwarzschild black hole, Phys. Rev. D78(2008) 064040, [arXiv:0807.0665]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  18. [18]

    J. Yang, C. Zhang, and Y. Ma,Shadow and stability of quantum-corrected black holes, Eur. Phys. J. C83(2023), no. 7 619, [arXiv:2211.04263]

    work page arXiv 2023

  19. [19]

    Gan, X.-M

    W.-C. Gan, X.-M. Kuang, Z.-H. Yang, Y. Gong, A. Wang, and B. Wang,Nonexistence of quantum black and white hole horizons in an improved dynamic approach, Sci. China Phys. Mech. Astron.67(2024), no. 8 280411, [arXiv:2212.14535]

    work page arXiv 2024

  20. [20]

    S. A. Teukolsky,Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations, Astrophys. J.185(1973) 635–647

    work page 1973

  21. [21]

    Echeverria,Gravitational Wave Measurements of the Mass and Angular Momentum of a Black Hole, Phys

    F. Echeverria,Gravitational Wave Measurements of the Mass and Angular Momentum of a Black Hole, Phys. Rev. D40(1989) 3194–3203

    work page 1989

  22. [22]

    L. S. Finn,Detection, measurement and gravitational radiation, Phys. Rev. D46(1992) 5236–5249, [gr-qc/9209010]

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  23. [23]

    On gravitational-wave spectroscopy of massive black holes with the space interferometer LISA

    E. Berti, V. Cardoso, and C. M. Will,On gravitational-wave spectroscopy of massive black holes with the space interferometer LISA, Phys. Rev. D73(2006) 064030, [gr-qc/0512160]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  24. [24]

    Extreme Gravity Tests with Gravitational Waves from Compact Binary Coalescences: (II) Ringdown

    E. Berti, K. Yagi, H. Yang, and N. Yunes,Extreme Gravity Tests with Gravitational Waves 24 from Compact Binary Coalescences: (II) Ringdown, Gen. Rel. Grav.50(2018), no. 5 49, [arXiv:1801.03587]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  25. [25]

    H. Gong, S. Li, D. Zhang, G. Fu, and J.-P. Wu,Quasinormal modes of quantum-corrected black holes, Phys. Rev. D110(2024), no. 4 044040, [arXiv:2312.17639]

    work page arXiv 2024

  26. [27]

    M. Isi, M. Giesler, W. M. Farr, M. A. Scheel, and S. A. Teukolsky,Testing the no-hair theorem with GW150914, Phys. Rev. Lett.123(2019), no. 11 111102, [arXiv:1905.00869]. [27]LIGO Scientific, VirgoCollaboration, R. Abbott et al.,Tests of general relativity with binary black holes from the second LIGO-Virgo gravitational-wave transient catalog, Phys. Rev. ...

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  27. [28]

    Giesler, M

    M. Giesler, M. Isi, M. A. Scheel, and S. Teukolsky,Black Hole Ringdown: The Importance of Overtones, Phys. Rev. X9(2019), no. 4 041060, [arXiv:1903.08284]

    work page arXiv 2019

  28. [29]

    R. A. Konoplya and A. Zhidenko,First few overtones probe the event horizon geometry, JHEAp44(2024) 419–426, [arXiv:2209.00679]

    work page arXiv 2024

  29. [30]

    On choosing the start time of binary black hole ringdown

    S. Bhagwat, M. Okounkova, S. W. Ballmer, D. A. Brown, M. Giesler, M. A. Scheel, and S. A. Teukolsky,On choosing the start time of binary black hole ringdowns, Phys. Rev. D97 (2018), no. 10 104065, [arXiv:1711.00926]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  30. [31]

    Cotesta, G

    R. Cotesta, G. Carullo, E. Berti, and V. Cardoso,Analysis of Ringdown Overtones in GW150914, Phys. Rev. Lett.129(2022), no. 11 111102, [arXiv:2201.00822]

    work page arXiv 2022

  31. [32]

    R. A. Konoplya, A. F. Zinhailo, J. Kunz, Z. Stuchlik, and A. Zhidenko,Quasinormal ringing of regular black holes in asymptotically safe gravity: the importance of overtones, JCAP10 (2022) 091, [arXiv:2206.14714]

    work page arXiv 2022

  32. [33]

    R. A. Konoplya,Quasinormal modes in higher-derivative gravity: Testing the black hole parametrization and sensitivity of overtones, Phys. Rev. D107(2023), no. 6 064039, [arXiv:2210.14506]

    work page arXiv 2023

  33. [34]

    R. A. Konoplya, Z. Stuchlik, A. Zhidenko, and A. F. Zinhailo,Quasinormal modes of renormalization group improved Dymnikova regular black holes, Phys. Rev. D107(2023), no. 10 104050, [arXiv:2303.01987]

    work page arXiv 2023

  34. [35]

    Alonso-Bardaji, D

    A. Alonso-Bardaji, D. Brizuela, and R. Vera,An effective model for the quantum Schwarzschild black hole, Phys. Lett. B829(2022) 137075, [arXiv:2112.12110]. 25

    work page arXiv 2022

  35. [36]

    Alonso-Bardaji, D

    A. Alonso-Bardaji, D. Brizuela, and R. Vera,Nonsingular spherically symmetric black-hole model with holonomy corrections, Phys. Rev. D106(2022), no. 2 024035, [arXiv:2205.02098]

    work page arXiv 2022

  36. [37]

    G. Fu, D. Zhang, P. Liu, X.-M. Kuang, and J.-P. Wu,Peculiar properties in quasinormal spectra from loop quantum gravity effect, Phys. Rev. D109(2024), no. 2 026010, [arXiv:2301.08421]

    work page arXiv 2024

  37. [38]

    S. V. Bolokhov,Long-lived quasinormal modes and overtones’ behavior of holonomy-corrected black holes, Phys. Rev. D110(2024), no. 2 024010, [arXiv:2311.05503]

    work page arXiv 2024

  38. [39]

    Alonso-Bardaji, D

    A. Alonso-Bardaji, D. Brizuela, and R. Vera,Singularity resolution by holonomy corrections: Spherical charged black holes in cosmological backgrounds, Phys. Rev. D107(2023), no. 6 064067, [arXiv:2302.10619]

    work page arXiv 2023

  39. [40]

    L.-G. Zhu, G. Fu, S. Li, D. Zhang, and J.-P. Wu,Quasinormal modes of a charged loop quantum black hole, Phys. Rev. D111(2025), no. 10 104008, [arXiv:2410.00543]

    work page arXiv 2025

  40. [41]

    Spherically Symmetric Quantum Geometry: Hamiltonian Constraint

    M. Bojowald and R. Swiderski,Spherically symmetric quantum geometry: Hamiltonian constraint, Class. Quant. Grav.23(2006) 2129–2154, [gr-qc/0511108]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  41. [42]

    Gambini and J

    R. Gambini and J. Pullin, A first course in loop quantum gravity. OUP Oxford, 2011

    work page 2011

  42. [43]

    Gambini, E

    R. Gambini, E. Mato, and J. Pullin,Axisymmetric gravity in real Ashtekar variables: the quantum theory, Class. Quant. Grav.37(2020), no. 11 115010, [arXiv:2001.02698]

    work page arXiv 2020

  43. [44]

    Modelling black holes with angular momentum in loop quantum gravity

    E. Frodden, A. Perez, D. Pranzetti, and C. R¨ oken,Modelling black holes with angular momentum in loop quantum gravity, Gen. Rel. Grav.46(2014), no. 12 1828, [arXiv:1212.5166]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  44. [45]

    Generating rotating regular black hole solutions without complexification

    M. Azreg-A¨ ınou,Generating rotating regular black hole solutions without complexification, Phys. Rev. D90(2014), no. 6 064041, [arXiv:1405.2569]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  45. [46]

    C. Liu, T. Zhu, Q. Wu, K. Jusufi, M. Jamil, M. Azreg-A¨ ınou, and A. Wang,Shadow and quasinormal modes of a rotating loop quantum black hole, Phys. Rev. D101(2020), no. 8 084001, [arXiv:2003.00477]. [Erratum: Phys.Rev.D 103, 089902 (2021)]

    work page arXiv 2020

  46. [47]

    Brahma, C.-Y

    S. Brahma, C.-Y. Chen, and D.-h. Yeom,Testing Loop Quantum Gravity from Observational Consequences of Nonsingular Rotating Black Holes, Phys. Rev. Lett.126(2021), no. 18 181301, [arXiv:2012.08785]

    work page arXiv 2021

  47. [48]

    Chen,On the possible spacetime structures of rotating loop quantum black holes, Int

    C.-Y. Chen,On the possible spacetime structures of rotating loop quantum black holes, Int. J. Geom. Meth. Mod. Phys.19(2022), no. 11 2250176, [arXiv:2207.03797]. 26

    work page arXiv 2022

  48. [49]

    E. T. Newman and A. I. Janis,Note on the Kerr spinning particle metric, J. Math. Phys.6 (1965) 915–917

    work page 1965

  49. [50]

    From static to rotating to conformal static solutions: Rotating imperfect fluid wormholes with(out) electric or magnetic field

    M. Azreg-A¨ ınou,From static to rotating to conformal static solutions: Rotating imperfect fluid wormholes with(out) electric or magnetic field, Eur. Phys. J. C74(2014), no. 5 2865, [arXiv:1401.4292]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  50. [51]

    E. W. Leaver,An Analytic representation for the quasi normal modes of Kerr black holes, Proc. Roy. Soc. Lond. A402(1985) 285–298

    work page 1985

  51. [52]

    Z. S. Moreira, H. C. D. Lima Junior, L. C. B. Crispino, and C. A. R. Herdeiro,Quasinormal modes of a holonomy corrected Schwarzschild black hole, Phys. Rev. D107(2023), no. 10 104016, [arXiv:2302.14722]

    work page arXiv 2023

  52. [53]

    E. W. Leaver,Quasinormal modes of Reissner-Nordstrom black holes, Phys. Rev. D41 (1990) 2986–2997. 27

    work page 1990