Pith. sign in

REVIEW 3 major objections 4 minor 38 references

The paper claims that EMRI gravitational waves around self-dual loop quantum gravity black holes accumulate measurable phase shifts, with a Fisher forecast tightly constraining the quantum parameters P and a0.

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 · deepseek-v4-flash

2026-08-02 01:09 UTC pith:KL43YUNE

load-bearing objection The geodesic and waveform work is competent but conventional; the Fisher forecast mistakes the metric components for what LISA measures, so the headline EMRI constraints don't follow. the 3 major comments →

arxiv 2607.14746 v1 pith:KL43YUNE submitted 2026-07-16 gr-qc

Orbital Dynamics and Gravitational-Wave Signatures of EMRIs in Self-Dual Loop Quantum Gravity Black Holes

Yu Wang , Meilin Liu , Haiguang Xu This is my paper
classification gr-qc PACS 04.30.-w04.70.-s04.60.Pp
keywords loop quantum gravityself-dual black holesextreme mass-ratio inspiralsgravitational wavesorbital dynamicsFisher matrix analysisNewman-Janisparameter estimation
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.

Loop quantum gravity's corrections to black holes could show up in the long inspirals of compact objects around supermassive black holes. The paper shows that in the self-dual LQG spacetime, quantum parameters reshape the effective potential and orbital trajectories; these differences accumulate over thousands of cycles, producing waveforms that depart from Schwarzschild/Kerr. To include astrophysical spin, the static solution is rotated via a standard complex-coordinate transformation. A Fisher information forecast then reports that the spin and the polymeric parameter P are tightly measurable, while the minimal-area parameter a0 remains poorly constrained. If the forecast holds, EMRI observations with space-based detectors would give an observational window into loop quantum gravity.

Core claim

The paper's central claim is that the strong-field imprints of self-dual loop quantum gravity do not stay hidden in the near-horizon region: they feed into the orbital phase and therefore into the gravitational-wave signal over the long EMRI inspiral. Using geodesic evolution and leading-order quadrupole radiation reaction, the authors generate waveforms for both the static and the rotating LQG-corrected geometry and find phase deviations that grow with time. Quantitatively, their Fisher matrix analysis on a geometrically defined observable vector yields fiducial constraints a = 0.902^(+0.0042)_(-0.0041), P = 0.030 ± 0.00015, and a0 = 21.4^(+23)_(-15), leading them to conclude that EMRI obse

What carries the argument

The load-bearing object is the self-dual LQG metric, defined by three functions G(r), F(r), H(r) with two quantum parameters: the polymeric parameter P and minimal-area parameter a0, which reduce to Schwarzschild when both vanish. The rotating version is obtained by applying the revised Newman-Janis algorithm to the static seed metric, giving a Kerr-like geometry with modified K(r), M_eff(r), and Δ(r). The analysis then proceeds through two tools: a phase-cycle decomposition that locks each orbital 2π revolution to a quasi-periodic waveform segment, and a Fisher information matrix built from an observable vector O(θ) of metric components sampled at N radii, whose curvature in parameter space

Load-bearing premise

The forecast assumes that what an EMRI gravitational-wave observation measures is the set of metric components {g_tt, g_tφ, g_φφ, Δ} sampled at N radii, whereas real observations yield a one-dimensional strain time series; if that mapping is wrong, the quoted constraints do not follow.

What would settle it

Repeat the parameter-estimation calculation using the actual gravitational-wave strain waveform h(t) (e.g., a Barack-Cutler-style likelihood) for the same fiducial parameters and a LISA-like noise curve; if the resulting uncertainty on P is orders of magnitude larger than the paper's eq. (68), the central 'measurable imprint' claim is falsified.

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

If this is right

  • LQG corrections accumulate secularly: even tiny metric changes translate into a growing gravitational-wave phase shift over an EMRI's many cycles.
  • The two quantum parameters act differently: P deforms the global potential and orbit, while a0 regularizes the near-horizon region; spin remains the dominant morphological driver.
  • A Fisher forecast on the metric-observable vector gives tight, mildly correlated constraints on a and P, but a0 is essentially unconstrained within the model.
  • The orbital phase acts as an internal clock, so EMRI waveforms can be analyzed as phase-locked per-orbit segments rather than as generic time series.
  • If the forecast holds, EMRI detections by future space-based observatories would provide a direct observational test of loop-quantum-gravity black-hole spacetimes.

Where Pith is reading between the lines

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

  • The Fisher constraints are computed on metric components, not on the actual strain time series that a detector measures; a likelihood built on h(t) with realistic noise could broaden the quoted uncertainties, so the numbers should be read as an optimistic upper bound on measurability.
  • The phase-cycle locking suggests a practical search strategy: template the waveform in azimuthal phase, not time, to isolate secular quantum drift from local strong-field effects; this could be tested with existing EMRI waveform codes.
  • Because the hierarchy (a >> P >> a0) holds, systematic errors in the spin parameter will limit the achievable constraint on P; a0 may remain undetectable even with a detection.
  • The rotating extension is obtained by a complex-coordinate transformation of an LQG-corrected seed; it is not itself derived from an LQG Hamiltonian, so if the true quantum-corrected Kerr geometry differs, the rotating forecasts would need revision.

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

3 major / 4 minor

Summary. The paper studies orbital dynamics and gravitational-wave signals for extreme-mass-ratio inspirals (EMRIs) in a self-dual loop quantum gravity (LQG) black hole spacetime. It first analyzes the static, spherically symmetric geometry with polymeric parameter P and minimal area parameter a0, computing effective potentials, geodesics, and quadrupole waveforms. It then constructs a rotating extension using the Newman–Janis algorithm and studies geodesics, radiation-reaction-driven inspirals, and waveforms. The final section (5.8) performs a Fisher matrix analysis on an observable vector constructed directly from metric components and reports tight constraints on the spin a, the polymeric parameter P, and a0 (Eqs. 67–69), leading to the claim that EMRI observations provide a promising avenue to probe quantum gravity. The quantitative claim rests entirely on this Fisher analysis.

Significance. If the quantitative claim were correct, the paper would be significant: it would demonstrate that LISA-like EMRI observations could sharply constrain loop quantum gravity parameters, a highly nontrivial result. The paper's systematic treatment of geodesic dynamics and waveform morphology in this LQG spacetime is a useful phenomenological contribution, and the rotating metric construction is a reasonable starting point. However, the advertised quantitative result does not follow from the analysis as presented. The Fisher observable is the metric itself, not a measurable detector response, and the fiducial parameter values are internally inconsistent. The paper does not supply reproducible code or machine-checked proofs; its data availability statement reports no associated data.

major comments (3)
  1. [§5.8, Eq. (63)] The observable vector O(θ) is defined as pointwise metric components {g_tt(r_i), g_tφ(r_i), g_φφ(r_i), Δ(r_i)} sampled at N radii. No justification is given that an EMRI gravitational-wave observation measures these quantities. A real LISA observation is a one-dimensional strain time series h(t), related to the metric only through geodesic integration and radiation-reaction evolution. Without a mapping from h(t) to these metric samples, the Fisher matrix in Eq. (64) and the resulting constraints (67)–(69) do not describe measurable parameters. This is the load-bearing premise of the 'promising avenue' conclusion, and it is unsubstantiated.
  2. [§5.8, Eq. (69)] The reported fiducial and 1-σ constraint a0 = 21.4^{+23}_{-15} is inconsistent with the parameter range a0 ∈ [0.01, 0.9] used throughout the orbital and waveform analyses (Figs. 2, 5, 8, 14, 15). The paper never states units or a physical scale for a0, so a value of 21.4 is either a typo or the Fisher analysis is performed at a point far outside the model's intended regime. Additionally, σ in Eq. (64) is never specified, making the quoted uncertainties non-reproducible even within the paper's own framework.
  3. [§5.8, Eq. (64)] The Fisher matrix is computed from derivatives of metric components with respect to the parameters. Since the observable is defined to be the metric components themselves, the 'constraints' largely restate that the metric depends on its parameters. This is close to tautological: it does not quantify how well a realistic EMRI observation can measure (a, P, a0). A valid Fisher analysis must use a waveform model and a detector noise covariance, not pointwise metric values.
minor comments (4)
  1. [§4] The phase-cycle decomposition in Eqs. (24)–(27) is a set of definitions without quantitative content. Equation (27) is never evaluated or used in later sections; this section reads as padding and could be removed or folded into the waveform discussion.
  2. [§5.8, Eqs. (60)–(62)] The functions K(r), M_eff(r), and Δ(r) are defined, but M_eff(r) is never used in the observable vector or the Fisher analysis. Either use M_eff or remove it to avoid confusion.
  3. [§2, Eq. (4)] The parameter a0 is called the 'minimal area parameter,' but its dimensions are unclear: in H(r)=r²+a0²/r², a0 must have dimension length², which is unusual. The paper should explicitly state the physical scale (e.g., a0 ~ ℓ_Pl²) and consistently use values that are small in geometric units.
  4. [Figure captions] Several figure captions do not specify exact parameter values (e.g., Fig. 3 says 'a0 ≠ 0, P ≠ 0' without values). Providing the full parameter set would aid reproducibility.

Circularity Check

1 steps flagged

The Fisher-forecast constraints reduce to the metric's own parameter sensitivity: the observable in Eq. (63) is the metric itself.

specific steps
  1. self definitional [Section 5.8, Eqs. (63)-(69)]
    "The observable vector is defined as O(θ) = {g_tt(r_i), g_tϕ(r_i), g_ϕϕ(r_i), ∆(r_i)}^N_{i=1}, (63) ... The Fisher matrix is then computed as F_ij = 1/σ^2 Σ_k ∂O_k/∂θ_i ∂O_k/∂θ_j, (64) ... The resulting constraints are: a = 0.902^{+0.0042}_{−0.0041}, (67) P = 0.030^{+0.00015}_{−0.00015}, (68) a0 = 21.4^{+23}_{−15}. (69)"

    The observable O(θ) is defined as pointwise metric components g_tt, g_tϕ, g_ϕϕ, ∆ evaluated from the NJA-LQG metric, i.e., as functions of the same parameters θ=(a,P,a0) that the Fisher matrix is supposed to constrain. Eq. (64) then differentiates these components with respect to θ. The Fisher matrix therefore measures only how the metric parameterization is deformed by its own parameters; it contains no link to the EMRI strain observable h(t) computed in Sec. 5.7. The quoted constraints (67)-(69) are a restatement of the model's parameter sensitivity, not a prediction derived from an independent measurement, so the central quantitative claim is circular by construction.

full rationale

The qualitative parts of the paper — geodesic integration, quadrupole waveforms, Newman–Janis rotating extension, and phase decomposition — are self-contained computations from the ansatz metric and are not circular in themselves. However, the paper's headline quantitative claim (that LISA-like EMRI observations can constrain (a,P,a0) to the levels in Eqs. (67)-(69)) rests entirely on the Fisher matrix of Section 5.8. There the 'observable vector' is defined as the metric components themselves, evaluated at radii r_i, and the Fisher matrix is the derivative of those metric components with respect to the parameters. This makes the observable identical to the quantity being constrained: the constraints quantify how strongly θ deforms g_μν, not how well an EMRI waveform measurement can recover θ. No noise covariance, σ value, or mapping from strain h(t) to O(θ) is given, and the fiducial a0=21.4 is inconsistent with the 0.01–0.9 values used elsewhere. Those are correctness problems compounding the circular structure. Because the waveform-modeling sections contain independent content, the score is 6 rather than higher.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central claim rests on adopting the self-dual LQG metric from earlier work, assuming the NJA rotating extension is valid, and on a Fisher analysis whose observable is the metric itself. The quantum parameters P and a0 are hand-chosen inputs; the effective uncertainty σ in the Fisher matrix is unspecified. No new physical entities are introduced; the NJA-LQG metric is a derived spacetime, not an independent entity.

free parameters (3)
  • P (polymeric parameter) = P = 0.030 fiducial in Fisher forecast; 0.01–0.03 in figures
    Hand-chosen quantum correction parameter in the metric (eqs 2–5); not fitted to external data. The Fisher constraints are circularly derived from the metric's own dependence on P.
  • a0 (minimal area parameter) = a0 = 21.4 fiducial in Fisher forecast; 0.01–0.9 in figures
    Hand-chosen quantum scale; dimensionally an area (H(r) = r² + a0²/r²). The fiducial 21.4 in Section 5.8 is inconsistent with the smaller values used elsewhere and is introduced ad hoc.
  • σ (effective uncertainty in Fisher matrix) = not specified
    Introduced in eq (64) without a value or derivation; the entire Fisher forecast depends on this unspecified quantity, making the quoted constraints un-reproducible.
axioms (5)
  • domain assumption The self-dual LQG metric (eqs 1–5) is the correct effective description of quantum-corrected black holes.
    The paper adopts this model from prior literature without deriving it; its physical validity is not established here.
  • domain assumption The Newman–Janis algorithm with K(r) = r²√(F/G) yields a valid rotating spacetime.
    NJA has known ambiguities; the paper assumes the constructed metric is a valid LQG-Kerr spacetime without checking field equations or demonstrating the a→0 limit recovers the static seed (the mass m is never related to M).
  • domain assumption The leading-order quadrupole flux dE/dt = −32/5 µ²r⁴Ω⁶ drives the inspiral.
    Standard Peters–Matthews flux applied to a modified spacetime without checking its validity in the strong-field LQG geometry.
  • ad hoc to paper The Fisher observable O = {g_tt, g_tφ, g_φφ, Δ} sampled at N radii represents what an EMRI gravitational-wave observation measures.
    Section 5.8, eqs (63)–(64). This is the paper's own construct with no argument connecting it to detector data; it guarantees apparent sensitivity to metric parameters.
  • domain assumption The seed mass M and the rotating metric mass m are the same quantity.
    Eqs (28)–(33) introduce m as ADM mass but never state m = M; without this relation the a→0 limit does not obviously reduce to the static self-dual metric, since G(r) is not of the form 1 − 2m/r.

pith-pipeline@v1.3.0-alltime-deepseek · 13648 in / 15863 out tokens · 135813 ms · 2026-08-02T01:09:36.096946+00:00 · methodology

0 comments
read the original abstract

Loop quantum gravity (LQG) predicts quantum modifications to classical black-hole spacetimes, which may leave imprints on the dynamics and gravitational-wave signals of compact objects in the strong-field regime. In this work, we investigate the orbital dynamics and gravitational-wave signatures of extreme mass-ratio inspirals (EMRIs) in a self-dual loop quantum gravity black hole spacetime. We analyze test-particle motion in the static, spherically symmetric self-dual LQG geometry characterized by two quantum parameters: the polymeric parameter $P$ and the minimal area parameter $a_0$. The effective potential and orbital structure are systematically studied, and we quantify the influence of quantum corrections on circular-orbit stability and strong-field orbital behavior. Compared with the classical Schwarzschild spacetime, LQG corrections modify the near-horizon orbital dynamics. Based on the orbital evolution, we construct gravitational-wave waveforms and investigate the impact of quantum corrections on waveform morphology. We find that LQG effects accumulate during the long inspiral phase, leading to noticeable signal deviations from the classical case. To incorporate rotation, we construct a rotating extension of the self-dual spacetime using the Newman--Janis algorithm. The resulting LQG-corrected Kerr geometry is used to analyze orbital motion, revealing the interplay between spin and quantum corrections in strong-field trajectories. Finally, we perform a Fisher matrix analysis to estimate potential constraints on quantum parameters from future space-based gravitational-wave detectors. Our results indicate that EMRI observations provide a promising avenue to probe quantum gravitational effects in black-hole spacetimes.

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

38 extracted references · 2 canonical work pages

  1. [1]

    Ob- servation of Gravitational Waves from a Binary Black Hole Merger,

    B. P. Abbott et al. (LIGO Scientific Collaboration), “Ob- servation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett.116, 061102 (2016)

    2016

  2. [2]

    GWTC-1: A Gravitational-Wave Transient Catalog,

    B. P. Abbott et al., “GWTC-1: A Gravitational-Wave Transient Catalog,” Phys. Rev. X9, 031040 (2019)

    2019

  3. [3]

    Laser Interferometer Space An- tenna,

    P. Amaro-Seoane et al., “Laser Interferometer Space An- tenna,” arXiv:1702.00786

    Pith/arXiv arXiv

  4. [4]

    LISA capture sources: Approx- imate waveforms, signal-to-noise ratios, and parameter estimation accuracy,

    L. Barack and C. Cutler, “LISA capture sources: Approx- imate waveforms, signal-to-noise ratios, and parameter estimation accuracy,” Phys. Rev. D69, 082005 (2004)

    2004

  5. [5]

    Testing general relativity with low-frequency, space- based gravitational-wave detectors,

    J. R. Gair, M. Vallisneri, S. L. Larson, and J. G. Baker, “Testing general relativity with low-frequency, space- based gravitational-wave detectors,” Living Rev. Relativ. 16, 7 (2013)

    2013

  6. [6]

    Black holes, gravitational waves and fundamental physics: a roadmap,

    L. Barack et al., “Black holes, gravitational waves and fundamental physics: a roadmap,” Class. Quantum Grav. 35, 124001 (2018)

    2018

  7. [7]

    Motion of small objects in curved spacetime: An introduction to gravitational self-force,

    A. Pound, “Motion of small objects in curved spacetime: An introduction to gravitational self-force,” Fund. Theor. Phys.179, 399 (2015)

    2015

  8. [8]

    Gravitational waves from merging compact binaries: How accurately can one ex- tract the binary’s parameters from the inspiral wave- form?

    C. Cutler and E. E. Flanagan, “Gravitational waves from merging compact binaries: How accurately can one ex- tract the binary’s parameters from the inspiral wave- form?” Phys. Rev. D49, 2658 (1994)

    1994

  9. [9]

    Probing Planckian corrections with EMRIs,

    A. Maselli et al., “Probing Planckian corrections with EMRIs,” Phys. Rev. Lett.125, 141101 (2020)

    2020

  10. [10]

    Extreme mass ratio inspirals as probes of new gravitational physics,

    L. C. Stein, “Extreme mass ratio inspirals as probes of new gravitational physics,” Phys. Rev. D103, 064033 (2021)

    2021

  11. [11]

    Rovelli,Quantum Gravity, Cambridge University Press (2004)

    C. Rovelli,Quantum Gravity, Cambridge University Press (2004). 17

    2004

  12. [12]

    Background indepen- dent quantum gravity: a status report,

    A. Ashtekar and J. Lewandowski, “Background indepen- dent quantum gravity: a status report,” Class. Quantum Grav.21, R53 (2004)

    2004

  13. [13]

    Thiemann,Modern Canonical Quantum General Rel- ativity, Cambridge University Press (2007)

    T. Thiemann,Modern Canonical Quantum General Rel- ativity, Cambridge University Press (2007)

    2007

  14. [14]

    Modesto, Disappearance of black hole singularity in quantum gravity,Phys

    L. Modesto, Disappearance of black hole singularity in quantum gravity,Phys. Rev. D70, 124009 (2004), doi:10.1103/PhysRevD.70.124009, arXiv:gr-qc/0407097

    Pith/arXiv arXiv 2004

  15. [15]

    Modesto, Loop quantum black hole,Class

    L. Modesto, Loop quantum black hole,Class. Quantum Grav.23, 5587 (2006)

    2006

  16. [16]

    Modesto and I

    L. Modesto and I. Premont-Schwarz, Self-dual black holes in loop quantum gravity: Theory and phenomenol- ogy,Phys. Rev. D80, 064041 (2009)

    2009

  17. [17]

    Modesto, Semiclassical loop quantum black hole,Int

    L. Modesto, Semiclassical loop quantum black hole,Int. J. Theor. Phys.49, 1649 (2010)

    2010

  18. [18]

    Gambini and J

    R. Gambini and J. Pullin, Loop quantization of the Schwarzschild black hole,Phys. Rev. Lett.110(21), 211301 (2013), doi:10.1103/PhysRevLett.110.211301, arXiv:1302.5265 [gr-qc]

    Pith/arXiv arXiv 2013

  19. [19]

    Thiemann and H.A

    T. Thiemann and H.A. Kastrup, Canonical quanti- zation of spherically symmetric gravity in Ashtekar’s selfdual representation,Nucl. Phys. B399, 211 (1993), doi:10.1016/0550-3213(93)90623-W, arXiv:gr- qc/9310012

    arXiv 1993

  20. [20]

    Campiglia, R

    M. Campiglia, R. Gambini, and J. Pullin, Loop quanti- zation of spherically symmetric midi-superspaces,Class. Quantum Gravity24, 3649 (2007), doi:10.1088/0264- 9381/24/14/007, arXiv:gr-qc/0703135

    Pith/arXiv arXiv 2007

  21. [21]

    Bengtsson, Note on Ashtekar’s variables in the spher- ically symmetric case,Class

    I. Bengtsson, Note on Ashtekar’s variables in the spher- ically symmetric case,Class. Quantum Gravity5, L139 (1988), doi:10.1088/0264-9381/5/10/002

    doi:10.1088/0264-9381/5/10/002 1988

  22. [22]

    Bojowald and H.A

    M. Bojowald and H.A. Kastrup, Quantum sym- metry reduction for diffeomorphism invariant theories of connections,Class. Quantum Gravity17, 3009 (2000), doi:10.1088/0264-9381/17/15/311, arXiv:hep- th/9907042

    arXiv 2000

  23. [23]

    Bojowald and R

    M. Bojowald and R. Swiderski, The volume opera- tor in spherically symmetric quantum geometry,Class. Quantum Gravity21, 4881 (2004), doi:10.1088/0264- 9381/21/21/009, arXiv:gr-qc/0407018

    Pith/arXiv arXiv 2004

  24. [24]

    Bojowald and R

    M. Bojowald and R. Swiderski, Spherically symmetric quantum horizons,Phys. Rev. D71, 081501 (2005), doi:10.1103/PhysRevD.71.081501, arXiv:gr-qc/0410147

    Pith/arXiv arXiv 2005

  25. [25]

    Bojowald and R

    M. Bojowald and R. Swiderski, Spherically symmet- ric quantum geometry: Hamiltonian constraint,Class. Quantum Gravity23, 2129 (2006), doi:10.1088/0264- 9381/23/6/015, arXiv:gr-qc/0511108

    Pith/arXiv arXiv 2006

  26. [26]

    Kuchar, Geometrodynamics of Schwarzschild black holes,Phys

    K.V. Kuchar, Geometrodynamics of Schwarzschild black holes,Phys. Rev. D50, 3961 (1994), doi:10.1103/PhysRevD.50.3961, arXiv:gr-qc/9403003

    Pith/arXiv arXiv 1994

  27. [27]

    Perez, Black holes in loop quantum gravity,Rept

    A. Perez, Black holes in loop quantum gravity,Rept. Prog. Phys.80, 126901 (2017)

    2017

  28. [28]

    Bojowald, Black-hole models in loop quantum gravity, Universe6, 125 (2020), doi:10.3390/universe6080125

    M. Bojowald, Black-hole models in loop quantum gravity, Universe6, 125 (2020), doi:10.3390/universe6080125

    work page doi:10.3390/universe6080125 2020

  29. [29]

    Z. Tu, H. Zhang, and J. Wu, Geodesic motion in self- dual loop quantum gravity black holes,Phys. Rev. D 108, 084048 (2023)

    2023

  30. [30]

    Zhang, Y

    L. Zhang, Y. Liu, and Q. Pan, ISCO of self-dual LQG black holes,Class. Quantum Gravity41, 125012 (2024)

    2024

  31. [31]

    R. Wang, K. Lin, and J. Jing, Geodesic structure in self- dual LQG spacetimes,Eur. Phys. J. C84, 567 (2024)

    2024

  32. [32]

    Eccentric extreme mass-ratio inspi- rals: a gateway to probe quantum gravity effects,

    T. Zi and S. Kumar, “Eccentric extreme mass-ratio inspi- rals: a gateway to probe quantum gravity effects,” Eur. Phys. J. C85, 592 (2025), doi:10.1140/epjc/s10052-025- 14330-7

    doi:10.1140/epjc/s10052-025- 2025

  33. [33]

    C. Liu, Z. Zhang, and J. Li, Shadow of rotating LQG black holes,Phys. Rev. D107, 064015 (2023)

    2023

  34. [34]

    P. Chen, Y. Wang, and S. Chen, Quasi-periodic oscil- lations in LQG black hole spacetimes,Class. Quantum Grav.40, 185001 (2023)

    2023

  35. [35]

    Note on the Kerr spinning-particle metric,

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

    1965

  36. [36]

    Erbin, The Newman-Janis algorithm and rotating black holes,Universe3, 13 (2017)

    H. Erbin, The Newman-Janis algorithm and rotating black holes,Universe3, 13 (2017)

    2017

  37. [37]

    Zhang, Q

    X. Zhang, Q. Pan, and J. Jing, Rotating LQG black holes and orbital dynamics,Phys. Rev. D110, 064048 (2024)

    2024

  38. [38]

    M. B. Cruz, C. A. S. Silva and F. A. Brito, Gravita- tional axial perturbations and quasinormal modes of loop quantum black holes, Eur. Phys. J. C79, 157 (2019), doi:10.1140/epjc/s10052-019-6565-2

    work page doi:10.1140/epjc/s10052-019-6565-2 2019