pith. sign in

arxiv: 2606.11728 · v1 · pith:LGZMRDYJnew · submitted 2026-06-10 · 🌀 gr-qc

Periodic orbits as probes of charged loop quantum gravity black holes through gravitational waves

Abolhassan Mohammadi , Arun Kumar , Hongwei Tan , Sushant G. Ghosh This is my paper

Pith reviewed 2026-06-27 09:19 UTC · model grok-4.3

classification 🌀 gr-qc
keywords loop quantum gravityblack holesgravitational wavesextreme-mass-ratio inspiralsperiodic orbitspolymerization parametercharged spacetimes
0
0 comments X

The pith

Gravitational waves from periodic orbits around charged loop quantum gravity black holes exceed the sensitivity of planned space-based detectors.

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

This paper examines how test particles move in the spacetime of charged black holes modified by loop quantum gravity effects, where the central singularity is replaced by a transition surface whose size is fixed by the area gap. It classifies the resulting periodic orbits and computes the gravitational waves they would emit during extreme-mass-ratio inspirals using the quadrupole formula. The waveforms depend on the LQG polymerization parameter, and their characteristic strains peak in the millihertz frequency band at levels above the noise curves of LISA, Taiji, and TianQin. A sympathetic reader would care because this suggests that future gravitational-wave observations could directly constrain the polymerization parameter in the strong-field regime near black holes.

Core claim

In the spacetime of a charged LQG black hole with polymerization parameter fixed by the area-gap condition, the effective potential admits innermost stable circular orbits and marginally bound orbits. Periodic orbits are labeled by integers (z, w, v) according to the rational ratio of azimuthal to radial frequencies, and the gravitational waveforms they produce in the quadrupole approximation display a zoom-whirl structure whose amplitude and phase vary with the charge parameter Q and the LQG correction. The resulting characteristic strains lie in the millihertz band and surpass the projected sensitivities of LISA, Taiji, and TianQin for all examined values of Q.

What carries the argument

The zoom-whirl taxonomy of periodic orbits, defined by the triple (z, w, v) and the frequency ratio q = ω_φ / ω_r - 1, which determines the topology and the shape of the emitted gravitational waveform.

If this is right

  • The polarization waveforms in time and frequency domain depend on the LQG parameter.
  • Characteristic strain peaks in millihertz for all Q.
  • Future observations can constrain the LQG polymerization parameter in strong-field regime.
  • Orbit topology shapes the waveform morphology.

Where Pith is reading between the lines

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

  • This approach could be extended to spinning LQG black holes to see if the constraints improve.
  • Comparison with waveforms from other modified gravity theories might allow distinguishing LQG effects.
  • Detection would provide a direct test of the area-gap condition in curved spacetime.

Load-bearing premise

The quadrupole approximation is sufficient to capture the leading gravitational-wave emission from these extreme-mass-ratio inspirals, and the polymerization parameter is fixed solely by the area-gap condition without additional model dependence.

What would settle it

A precise measurement of the characteristic strain from an EMRI around a black hole with known charge that falls below the projected LISA sensitivity curve or shows no dependence on the expected LQG corrections would falsify the claim that these signals can constrain the polymerization parameter.

Figures

Figures reproduced from arXiv: 2606.11728 by Abolhassan Mohammadi, Arun Kumar, Hongwei Tan, Sushant G. Ghosh.

Figure 1
Figure 1. Figure 1: FIG. 1. The figure displays the behavior of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The shaded colored regions show the allowed param [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The figure displays the behavior of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The plot presents the behavior of the rational fre [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The plot displays different periodic orbits around the LQG black hole classified by the topological non-negative integer [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: shows h+ and h× for three representative pe￾riodic orbits: (z, w, v) = (1, 2, 0), (2, 2, 1), and (3, 2, 2). We fix Q = 0.4 and E = 0.96 for all three. Each wave￾form has the characteristic zoom–whirl structure, a well– known signature of strong–field EMRIs [23, 26, 55, 83]. This structure has recently been studied in various quantum–corrected black hole spacetimes [87, 104, 105]. The pattern follows the tw… view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The correlation between the orbital segments and the waveform is shown for (1 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The figure illustrates the effect of the charge parameter, or the effect of the polymerization parameter [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: shows the frequency–domain spectra for the (3, 2, 2) orbit at different Q values. In all cases, the main spectral peak falls in the millihertz band. This is consis￾tent with the expected frequency range of EMRIs around supermassive black holes, whose characteristic frequen￾cies scale as f ∼ (M/106 M⊙) −1 mHz. This puts the sig￾nal right in the sensitivity windows of LISA, Taiji, and TianQin [21, 59–61]. Th… view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The characteristic strain is displayed for three pe [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The characteristic strain for the (3 [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
read the original abstract

Gravitational waves from extreme-mass-ratio inspirals (EMRI) provide a direct probe of the strong-field geometry of black holes. Motivated by this, we study the motion of test particles and the resulting gravitational wave emission in the spacetime of a charged black hole inspired by loop quantum gravity (LQG), where the classical singularity is replaced by a smooth transition surface arising from the LQG polymerization, in which its radius is set by the LQG area gap condition. As a result, the polymerization parameter $\delta_b$ is uniquely determined by the mass $M$ and charge parameter $Q$, so that all cases examined in this work contain LQG correction. By constructing the effective potential, the innermost stable circular orbit (ISCO) and the marginally bound orbit (MBO) are determined. Periodic orbits are classified using the Levin-Perez-Giz zoom-whirl taxonomy, showing how the orbit topology shapes the waveform, so that each closed trajectory is labeled by the triple integer $(z, w, v)$ and located through the rational frequency ratio $q = \omega_\phi/\omega_r - 1$. Within the quadrupole approximation, the gravitational waveforms for an EMRIs are estimated, and the resulting polarizations are obtained in the time-domain and frequency-domain. The resulting polarizations in the time-domain exhibit a zoom-whirl morphology, with the waveform amplitude and phase dependent on the LQG parameter. The characteristic strain peaks in the millihertz band for all values of the charge parameter $Q$, and they exceed the projected sensitivities of LISA, Taiji, and TianQin, suggesting that future observations could place meaningful constraints on the LQG polymerization parameter in the strong-field regime.

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 / 0 minor

Summary. The paper studies test-particle motion in the spacetime of charged LQG black holes with the polymerization parameter δ_b fixed uniquely by the area-gap condition in terms of M and Q. It constructs the effective potential to locate the ISCO and MBO, classifies periodic orbits via the Levin-Perez-Giz (z,w,v) zoom-whirl taxonomy using the rational frequency ratio q = ω_φ/ω_r − 1, generates time- and frequency-domain waveforms in the quadrupole approximation, and reports that the characteristic strain peaks in the millihertz band and exceeds the projected sensitivities of LISA, Taiji and TianQin for all values of Q, thereby suggesting that future observations could constrain the LQG parameter.

Significance. If the radiation modeling step is validated, the work supplies a parameter-free (given M and Q) set of waveform predictions for periodic orbits in an LQG-inspired geometry that could be confronted with space-based detector data. The unique fixing of δ_b by the area-gap condition without additional fitting parameters is a clear strength that avoids circularity in the model.

major comments (2)
  1. [Abstract] Abstract: the claim that characteristic strains exceed LISA/Taiji/TianQin sensitivities (and thereby constrain δ_b) is obtained from waveforms computed via the quadrupole formula applied to geodesic motion in the effective LQG metric. No error estimate, convergence test, or comparison against a Teukolsky-like calculation in the modified background is supplied to show that LQG curvature corrections leave the leading radiation amplitude and far-zone propagation unchanged at the quoted level. This assumption is load-bearing for the detectability conclusion.
  2. [Abstract] Abstract: the frequency ratio q = ω_φ/ω_r − 1 is asserted to label the (z,w,v) orbits and to produce the claimed zoom-whirl morphology, yet no explicit verification is given that this relation continues to hold once the LQG effective potential (rather than the classical one) is used to define the frequencies.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that characteristic strains exceed LISA/Taiji/TianQin sensitivities (and thereby constrain δ_b) is obtained from waveforms computed via the quadrupole formula applied to geodesic motion in the effective LQG metric. No error estimate, convergence test, or comparison against a Teukolsky-like calculation in the modified background is supplied to show that LQG curvature corrections leave the leading radiation amplitude and far-zone propagation unchanged at the quoted level. This assumption is load-bearing for the detectability conclusion.

    Authors: We acknowledge that the waveforms rely on the quadrupole approximation applied to geodesics in the effective LQG metric, without a full Teukolsky-style perturbation analysis or explicit error estimates for curvature corrections. This is a standard leading-order approach in EMRI studies of modified spacetimes when higher-order calculations are not yet available. The LQG corrections enter through the background metric, and the detectability estimates are intended as indicative for the millihertz band. In the revised manuscript we will add an explicit discussion of this limitation, including a statement that the conclusions on detector sensitivity are preliminary and would benefit from future waveform modeling beyond the quadrupole formula. revision: partial

  2. Referee: [Abstract] Abstract: the frequency ratio q = ω_φ/ω_r − 1 is asserted to label the (z,w,v) orbits and to produce the claimed zoom-whirl morphology, yet no explicit verification is given that this relation continues to hold once the LQG effective potential (rather than the classical one) is used to define the frequencies.

    Authors: The frequencies ω_φ and ω_r are obtained by integrating the geodesic equations using the effective potential derived from the LQG metric. The rational ratio q is then applied exactly as in the classical Levin-Perez-Giz taxonomy to identify closed orbits. Because the frequencies are computed directly from the LQG potential, the relation holds by construction for the periodic orbits we classify. We will add a clarifying sentence in the revised text (near the definition of q) stating that the taxonomy is applied to frequencies extracted from the modified potential, together with a brief note confirming consistency with the zoom-whirl morphology observed in the waveforms. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper fixes the LQG polymerization parameter δ_b uniquely via the area-gap condition from M and Q (abstract), then computes geodesics, periodic orbits via Levin-Perez-Giz taxonomy, and GW polarizations via the quadrupole formula applied to those orbits. No parameters are fitted to the target strain data and renamed as predictions; the detectability claim follows directly from the model metric and standard radiation formula without self-definition or self-citation load-bearing steps. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the LQG polymerization model taken from prior literature, the standard geodesic equation in GR, and the quadrupole formula for gravitational waves; no new free parameters are introduced because δ_b is fixed by M and Q.

axioms (2)
  • domain assumption The LQG polymerization replaces the classical singularity with a transition surface whose radius is set by the area-gap condition, uniquely determining δ_b from M and Q.
    This is the foundational modification invoked to ensure every examined case contains LQG corrections.
  • domain assumption The quadrupole approximation suffices for the leading gravitational-wave emission from extreme-mass-ratio inspirals.
    This is the standard radiation-reaction formula used to generate the time- and frequency-domain polarizations.

pith-pipeline@v0.9.1-grok · 5851 in / 1526 out tokens · 24006 ms · 2026-06-27T09:19:23.215673+00:00 · methodology

discussion (0)

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

Forward citations

Cited by 1 Pith paper

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

  1. Gravitational Wave Signatures from Periodic Orbits around a Non--commutative Schwarzschild Black Hole

    gr-qc 2026-06 unverdicted novelty 5.0

    Non-commutative corrections move periodic orbits inward, lower the energy needed for a given orbit shape, and generate gravitational waves with phase shifts plus higher amplitude; a bound Θ/M² < 0.014 is extracted fro...

Reference graph

Works this paper leans on

108 extracted references · 42 linked inside Pith · cited by 1 Pith paper

  1. [1]

    zoom–whirl

    That’s what we expect for a flat spacetime, and it meansEis just the energy per unit mass measured by an observer at infinity [73, 74]. The valueE= 1 marks the boundary between bound and unbound motion. IfE >1, the particle is unbound and can fly off to infinity. IfE <1, it follows a bound trajectory. Stable bound orbits sit between the ISCO and the MBO, ...

  2. [2]

    B. P. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. Lett.116, 061102 (2016), arXiv:1602.03837 [gr-qc]

    Pith/arXiv arXiv 2016

  3. [3]

    B. P. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. D93, 122003 (2016), arXiv:1602.03839 [gr-qc]

    Pith/arXiv arXiv 2016

  4. [4]

    B. P. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. Lett.116, 241102 (2016), arXiv:1602.03840 [gr-qc]

    Pith/arXiv arXiv 2016

  5. [5]

    B. P. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. Lett.116, 131103 (2016), arXiv:1602.03838 [gr-qc]

    Pith/arXiv arXiv 2016

  6. [6]

    Ashtekar, New J

    A. Ashtekar, New J. Phys.7, 198 (2005), arXiv:gr- qc/0410054

    arXiv 2005

  7. [7]

    Thiemann, Int

    T. Thiemann, Int. J. Mod. Phys. A23, 1113 (2008). 14

    2008

  8. [8]

    Rovelli, Class

    C. Rovelli, Class. Quant. Grav.32, 124005 (2015), arXiv:1506.00927 [gr-qc]

    Pith/arXiv arXiv 2015

  9. [9]

    Modesto, Int

    L. Modesto, Int. J. Theor. Phys.49, 1649 (2010), arXiv:0811.2196 [gr-qc]

    Pith/arXiv arXiv 2010

  10. [10]

    Ashtekar, J

    A. Ashtekar, J. Olmedo, and P. Singh, Phys. Rev. Lett. 121, 241301 (2018), arXiv:1806.00648 [gr-qc]

    Pith/arXiv arXiv 2018

  11. [11]

    Ashtekar, J

    A. Ashtekar, J. Olmedo, and P. Singh, Class. Quant. Grav.37, 205017 (2020), arXiv:2001.08833 [gr-qc]

    arXiv 2020

  12. [12]

    Modesto, J

    L. Modesto, J. W. Moffat, and P. Nicolini, Phys. Lett. B695, 397 (2011), arXiv:1010.0680 [gr-qc]

    Pith/arXiv arXiv 2011

  13. [13]

    Corichi and P

    A. Corichi and P. Singh, Class. Quant. Grav.33, 055006 (2016), arXiv:1506.08015 [gr-qc]

    Pith/arXiv arXiv 2016

  14. [14]

    Ashtekar, J

    A. Ashtekar, J. Olmedo, and P. Singh, Phys. Rev. D 98, 126003 (2018), arXiv:1806.02406 [gr-qc]

    Pith/arXiv arXiv 2018

  15. [15]

    Bodendorfer, F

    N. Bodendorfer, F. M. Mele, and J. M¨ unch, Class. Quant. Grav.38, 095002 (2021), arXiv:1912.00774 [gr- qc]

    arXiv 2021

  16. [16]

    Bodendorfer, F

    N. Bodendorfer, F. M. Mele, and J. M¨ unch, Phys. Lett. B819, 136390 (2021), arXiv:1911.12646 [gr-qc]

    arXiv 2021

  17. [17]

    Elizaga Navascu´ es, A

    B. Elizaga Navascu´ es, A. Garc´ ıa-Quismondo, and G. A. Mena Marug´ an, Phys. Rev. D106, 043531 (2022), arXiv:2208.00425 [gr-qc]

    arXiv 2022

  18. [18]

    Elizaga Navascu´ es, A

    B. Elizaga Navascu´ es, A. Garc´ ıa-Quismondo, and G. A. Mena Marug´ an, Phys. Rev. D106, 063516 (2022), arXiv:2207.04677 [gr-qc]

    arXiv 2022

  19. [19]

    Alonso-Bardaji, D

    A. Alonso-Bardaji, D. Brizuela, and R. Vera, Phys. Rev. D106, 024035 (2022), arXiv:2205.02098 [gr-qc]

    arXiv 2022

  20. [20]

    Alonso-Bardaji, D

    A. Alonso-Bardaji, D. Brizuela, and R. Vera, Phys. Lett. B829, 137075 (2022), arXiv:2112.12110 [gr-qc]

    arXiv 2022

  21. [21]

    Z. S. Moreira, H. C. D. Lima Junior, L. C. B. Crispino, and C. A. R. Herdeiro, Phys. Rev. D107, 104016 (2023), arXiv:2302.14722 [gr-qc]

    arXiv 2023

  22. [22]

    P. A. Seoaneet al.(LISA), Living Rev. Rel.26, 2 (2023), arXiv:2203.06016 [gr-qc]

    Pith/arXiv arXiv 2023

  23. [23]

    Babak, J

    S. Babak, J. Gair, A. Sesana, E. Barausse, C. F. Sop- uerta, C. P. L. Berry, E. Berti, P. Amaro-Seoane, A. Pe- titeau, and A. Klein, Phys. Rev. D95, 103012 (2017), arXiv:1703.09722 [gr-qc]

    Pith/arXiv arXiv 2017

  24. [24]

    Barack and C

    L. Barack and C. Cutler, Phys. Rev. D69, 082005 (2004), arXiv:gr-qc/0310125

    Pith/arXiv arXiv 2004

  25. [25]

    J. R. Gair, M. Vallisneri, S. L. Larson, and J. G. Baker, Living Rev. Rel.16, 7 (2013), arXiv:1212.5575 [gr-qc]

    Pith/arXiv arXiv 2013

  26. [26]

    H. A. Borges, I. P. R. Baranov, F. C. Sobrinho, and S. Carneiro, Class. Quant. Grav.41, 05LT01 (2024), arXiv:2310.01560 [gr-qc]

    arXiv 2024

  27. [27]

    Levin and G

    J. Levin and G. Perez-Giz, Phys. Rev. D77, 103005 (2008), arXiv:0802.0459 [gr-qc]

    Pith/arXiv arXiv 2008

  28. [28]

    Levin, Class

    J. Levin, Class. Quant. Grav.26, 235010 (2009), arXiv:0907.5195 [gr-qc]

    Pith/arXiv arXiv 2009

  29. [29]

    Levin and B

    J. Levin and B. Grossman, Phys. Rev. D79, 043016 (2009), arXiv:0809.3838 [gr-qc]

    Pith/arXiv arXiv 2009

  30. [30]

    Bambhaniya, D

    P. Bambhaniya, D. N. Solanki, D. Dey, A. B. Joshi, P. S. Joshi, and V. Patel, Eur. Phys. J. C81, 205 (2021), arXiv:2007.12086 [gr-qc]

    arXiv 2021

  31. [31]

    Rana and A

    P. Rana and A. Mangalam, Class. Quant. Grav.36, 045009 (2019), arXiv:1901.02730 [gr-qc]

    Pith/arXiv arXiv 2019

  32. [32]

    Misra and J

    V. Misra and J. Levin, Phys. Rev. D82, 083001 (2010), arXiv:1007.2699 [gr-qc]

    Pith/arXiv arXiv 2010

  33. [33]

    Lin and X.-M

    H.-Y. Lin and X.-M. Deng, Eur. Phys. J. C83, 311 (2023)

    2023

  34. [34]

    C. Liu, C. Ding, and J. Jing, Commun. Theor. Phys. 71, 1461 (2019), arXiv:1804.05883 [gr-qc]

    arXiv 2019

  35. [35]

    G. Z. Babar, A. Z. Babar, and Y.-K. Lim, Phys. Rev. D96, 084052 (2017), arXiv:1710.09581 [gr-qc]

    Pith/arXiv arXiv 2017

  36. [36]

    Yao and X

    J.-T. Yao and X. Li, Phys. Rev. D108, 084067 (2023)

    2023

  37. [37]

    Lin and X.-M

    H.-Y. Lin and X.-M. Deng, Universe8, 278 (2022)

    2022

  38. [38]

    C. Gong, T. Zhu, R. Niu, Q. Wu, J.-L. Cui, X. Zhang, W. Zhao, and A. Wang, Phys. Rev. D105, 044034 (2022), arXiv:2112.06446 [gr-qc]

    arXiv 2022

  39. [39]

    Z. C. S. Chan and Y.-K. Lim, Gen. Rel. Grav.57, 35 (2025), arXiv:2502.03082 [gr-qc]

    arXiv 2025

  40. [40]

    R. Wang, F. Gao, and H. Chen, Annals Phys.447, 169167 (2022)

    2022

  41. [41]

    Haroon and T

    S. Haroon and T. Zhu, Phys. Rev. D112, 044046 (2025), arXiv:2502.09171 [gr-qc]

    arXiv 2025

  42. [42]

    Z.-Y. Tu, T. Zhu, and A. Wang, Phys. Rev. D108, 024035 (2023), arXiv:2304.14160 [gr-qc]

    arXiv 2023

  43. [43]

    Zhou and Y

    T.-Y. Zhou and Y. Xie, Eur. Phys. J. C80, 1070 (2020)

    2020

  44. [44]

    Alloqulov, S

    M. Alloqulov, S. Shaymatov, B. Ahmedov, and T. Zhu, (2025), arXiv:2508.05245 [gr-qc]

    arXiv 2025

  45. [45]

    Z.-L. Wei, J. Zhang, Y. Xie, and P.-L. Yin, Eur. Phys. J. C85, 698 (2025)

    2025

  46. [46]

    Wang, Y.-P

    C.-H. Wang, Y.-P. Zhang, T. Zhu, and S.-W. Wei, (2025), arXiv:2508.20558 [gr-qc]

    arXiv 2025

  47. [47]

    Yang, Y.-P

    S. Yang, Y.-P. Zhang, T. Zhu, L. Zhao, and Y.-X. Liu, JCAP01, 091 (2025), arXiv:2407.00283 [gr-qc]

    arXiv 2025

  48. [48]

    L. Zhao, M. Tang, and Z. Xu, Eur. Phys. J. C85, 36 (2025), arXiv:2411.01979 [gr-qc]

    arXiv 2025

  49. [49]

    Jiang, M

    H. Jiang, M. Alloqulov, Q. Wu, S. Shaymatov, and T. Zhu, Phys. Dark Univ.46, 101627 (2024)

    2024

  50. [50]

    Alloqulov, T

    M. Alloqulov, T. Xamidov, S. Shaymatov, and B. Ahmedov, Eur. Phys. J. C85, 798 (2025), arXiv:2504.05236 [gr-qc]

    arXiv 2025

  51. [51]

    S. Zare, T. Zhu, L. M. Nieto, S. Lu, and H. Hassan- abadi, (2025), arXiv:2510.05166 [gr-qc]

    arXiv 2025

  52. [52]

    H. Gong, S. Long, X.-J. Wang, Z. Xia, J.-P. Wu, and Q. Pan, (2025), arXiv:2509.23318 [gr-qc]

    Pith/arXiv arXiv 2025

  53. [53]

    Li and X.-M

    Y.-Z. Li and X.-M. Kuang, (2025), arXiv:2509.07333 [gr-qc]

    arXiv 2025

  54. [54]

    W. Deng, S. Long, Q. Tan, and J. Jing, (2025), arXiv:2510.24468 [gr-qc]

    arXiv 2025

  55. [55]

    Kumar, A

    A. Kumar, A. Mohammadi, and S. G. Ghosh, (2026), arXiv:2605.07187 [gr-qc]

    Pith/arXiv arXiv 2026

  56. [56]

    Babak, H

    S. Babak, H. Fang, J. R. Gair, K. Glampedakis, and S. A. Hughes, Phys. Rev. D75, 024005 (2007), [Erratum: Phys.Rev.D 77, 04990 (2008)], arXiv:gr- qc/0607007

    arXiv 2007

  57. [57]

    J. R. Gair and K. Glampedakis, Phys. Rev. D73, 064037 (2006), arXiv:gr-qc/0510129

    Pith/arXiv arXiv 2006

  58. [58]

    Qiao, Z.-W

    X. Qiao, Z.-W. Xia, Q. Pan, H. Guo, W.-L. Qian, and J. Jing, JCAP03, 006 (2025), arXiv:2408.10022 [gr-qc]

    arXiv 2025

  59. [59]

    Yang, Y.-P

    S. Yang, Y.-P. Zhang, L. Zhao, and Y.-X. Liu, Eur. Phys. J. C86, 35 (2026), arXiv:2509.24835 [gr-qc]

    arXiv 2026

  60. [60]

    Amaro-Seoaneet al.(LISA), (2017), arXiv:1702.00786 [astro-ph.IM]

    P. Amaro-Seoaneet al.(LISA), (2017), arXiv:1702.00786 [astro-ph.IM]

    Pith/arXiv arXiv 2017

  61. [61]

    Hu and Y.-L

    W.-R. Hu and Y.-L. Wu, Natl. Sci. Rev.4, 685 (2017)

    2017

  62. [62]

    Luoet al.(TianQin), Class

    J. Luoet al.(TianQin), Class. Quant. Grav.33, 035010 (2016), arXiv:1512.02076 [astro-ph.IM]

    Pith/arXiv arXiv 2016

  63. [63]

    Immirzi, Nucl

    G. Immirzi, Nucl. Phys. B Proc. Suppl.57, 65 (1997), arXiv:gr-qc/9701052

    Pith/arXiv arXiv 1997

  64. [64]

    K. A. Meissner, Class. Quant. Grav.21, 5245 (2004), arXiv:gr-qc/0407052

    Pith/arXiv arXiv 2004

  65. [65]

    Ghosh and P

    A. Ghosh and P. Mitra, Phys. Rev. D71, 027502 (2005), arXiv:gr-qc/0401070

    Pith/arXiv arXiv 2005

  66. [66]

    Bojowald, Living Rev

    M. Bojowald, Living Rev. Rel.8, 11 (2005), arXiv:gr- qc/0601085

    arXiv 2005

  67. [67]

    Tibrewala, Class

    R. Tibrewala, Class. Quant. Grav.29, 235012 (2012), arXiv:1207.2585 [gr-qc]. 15

    Pith/arXiv arXiv 2012

  68. [68]

    Gambini, E

    R. Gambini, E. M. Capurro, and J. Pullin, Phys. Rev. D91, 084006 (2015), arXiv:1412.6055 [gr-qc]

    Pith/arXiv arXiv 2015

  69. [69]

    Gambini, F

    R. Gambini, F. Ben´ ıtez, and J. Pullin, Universe8, 526 (2022), arXiv:2102.09501 [gr-qc]

    arXiv 2022

  70. [70]

    I. P. R. Baranov, H. A. Borges, F. C. Sobrinho, and S. Carneiro, Class. Quant. Grav.42, 085012 (2025), arXiv:2407.21076 [gr-qc]

    arXiv 2025

  71. [71]

    Pigozzo, F

    C. Pigozzo, F. S. Bacelar, and S. Carneiro, Class. Quant. Grav.38, 045001 (2021), arXiv:2001.03440 [gr- qc]

    arXiv 2021

  72. [72]

    Bodendorfer, F

    N. Bodendorfer, F. M. Mele, and J. M¨ unch, Class. Quant. Grav.36, 195015 (2019), arXiv:1902.04542 [gr- qc]

    arXiv 2019

  73. [73]

    Kumar, Q

    A. Kumar, Q. Wu, T. Zhu, and S. G. Ghosh, Phys. Dark Univ.52, 102305 (2026), arXiv:2511.17975 [gr-qc]

    arXiv 2026

  74. [74]

    Chandrasekhar,The mathematical theory of black holes(1985)

    S. Chandrasekhar,The mathematical theory of black holes(1985)

    1985

  75. [75]

    R. M. Wald,General Relativity(Chicago Univ. Pr., Chicago, USA, 1984)

    1984

  76. [76]

    C. W. Misner, K. S. Thorne, and J. A. Wheeler,Grav- itation(W. H. Freeman, San Francisco, 1973)

    1973

  77. [77]

    Nasereldin and K

    S. Nasereldin and K. Lake, (2019), arXiv:1902.05129 [gr-qc]

    Pith/arXiv arXiv 2019

  78. [78]

    Glampedakis, S

    K. Glampedakis, S. A. Hughes, and D. Kennefick, Phys. Rev. D66, 064005 (2002), arXiv:gr-qc/0205033

    Pith/arXiv arXiv 2002

  79. [79]

    J. M. Bardeen, W. H. Press, and S. A. Teukolsky, As- trophys. J.178, 347 (1972)

    1972

  80. [80]

    Grossman and J

    R. Grossman and J. Levin, Phys. Rev. D79, 043017 (2009), arXiv:0811.3798 [gr-qc]

    Pith/arXiv arXiv 2009

Showing first 80 references.