Pith. sign in

REVIEW 4 major objections 7 minor 2 cited by

Long-Lived Quasinormal Modes and Quasi-Resonances around Non-Minimal Einstein-Yang-Mills Black Holes

T0 review · 4 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read This paper establishes that a massive scalar field around non-minimal Einstein-Yang-Mills black holes admits quasi-resonances—arbitrarily long-lived quasinormal modes—only when the spacetime is asymptotically flat, while the…

desk verdict The flat-space massive-scalar QNM results for non-minimal EYM black holes are plausible and new, but the de Sitter tables internally use Λ=0.01 while claiming Λ=0.001, and the abstract states the opposite of the paper's own conclusion. read the letter →

arxiv 2505.08545 v1 pith:6PFE4FTT submitted 2025-05-13 gr-qc

Alexey Dubinsky This is my paper · ORCID
classification gr-qc MSC 83C57
keywords quasinormalmodesmassivescalarfieldnon-minimalEinstein-Yang-Millstheoryquasi-resonancesdeSitterblackholesholeringdowncontinuedfractionmethodWKBapproximation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that the mass of a scalar field changes not just the numerical values but the qualitative fate of black-hole ringing in non-minimal Einstein-Yang-Mills theory. In asymptotically flat spacetimes, pushing the field mass above a critical value drives the damping rate of the fundamental quasinormal mode to zero, producing quasi-resonances—arbitrarily long-lived oscillations whose frequency no longer depends on the non-minimal coupling $\xi$. In asymptotically de Sitter spacetimes, the same field mass suppresses damping but never removes it, so no quasi-resonances appear. In the large-mass regime the frequencies become independent of the multipole number $\ell$, and the paper derives an analytic formula for them. These results matter because long-lived modes are among the most direct observable signatures of near-horizon geometry, and the paper shows that the cosmological background decides whether they can exist at all.

What carries the argument

The argument runs through the radial wave equation in Schrödinger form $d^2\Psi/dr_*^2 + (\omega^2 - V(r))\Psi = 0$, with the effective potential $V(r) = f(r)[\mu^2 + \ell(\ell+1)/r^2 + f'(r)/r]$ built from the non-minimal EYM lapse function $f(r) = 1 + (r^4/(r^4 + 2\xi Q^2))(Q^2/r^2 - 2M/r - \Lambda r^2/3)$. Imposing purely ingoing waves at the horizon and evanescent decay at infinity for massive fields turns the problem into one of solving for complex frequencies $\omega$. Quasi-resonances are identified as the limit where $\mathrm{Im}(\omega) \to 0$. The numerical machinery is a continued-fraction method for the asymptotically flat case, WKB with Padé approximants for the de Sitter case, time-domain integration with Prony extraction for stability checks, and a large-mass expansion of the potential's maximum that yields the analytic formula for the frequencies.

What would settle it

Using an independent method (continued fractions adapted to de Sitter boundary conditions, or time-domain evolution with Prony extraction), compute the fundamental de Sitter frequency for a representative case such as $\Lambda=0.001$, $Q=0.1$, $\xi=0.5$, $\ell=1$, $\mu=1$. If the imaginary part disagrees with the WKB-6/WKB-9 tables by more than their own mutual difference, or if it crosses zero for some $\mu$, then the claimed absence of quasi-resonances in the de Sitter branch collapses; in the flat case, the quasi-resonance claim would be settled by locating the critical $\mu$ at which $\mathrm{Im}(\omega)$ changes sign with high-precision continued-fraction data.

Watch

Extended reading notes

Core claim

The central claim is a sharp split in the spectrum of a massive test scalar field on non-minimal Einstein-Yang-Mills black holes. For $\Lambda = 0$ asymptotically flat solutions, the imaginary part of the fundamental quasinormal frequency decreases with scalar mass $\mu$ and vanishes at a critical mass—specific to the multipole $\ell$ and the black-hole parameters $Q$ and $\xi$—after which the fundamental mode leaves the spectrum and the first overtone takes over; these are the quasi-resonances. For $\Lambda > 0$ asymptotically de Sitter solutions, the damping rate instead saturates at a nonzero constant as $\mu$ grows, so arbitrarily long-lived modes are absent. Near the flat-space quasi-resonant threshold, frequencies computed for different $\xi$ merge, and in the large-$\mu$ limit the dependence on $\ell$ disappears; the paper provides the analytic large-mass expression (23). It also reports that $\ell=0$ perturbations remain stable even when the effective potential has a negative gap, with the late-time signal governed by a slowly decaying pure-imaginary de Sitter mode in near-extremal cases.

Load-bearing premise

The load-bearing premise is that the WKB approximation is accurate for the massive scalar field in the asymptotically de Sitter potentials: the de Sitter frequencies are checked only by comparing sixth- and ninth-order WKB results, so if WKB carries a shared systematic error in this regime, the finite-damping claim for de Sitter black holes is not established.

Editorial extensions

If this is right

  • In asymptotically flat non-minimal EYM spacetimes, a scalar field heavier than a critical value produces an arbitrarily long-lived ringdown, with the first overtone replacing the fundamental mode.
  • In the quasi-resonant limit the quasinormal frequencies stop depending on the non-minimal coupling $\xi$, so the long-lived frequency is fixed by the black-hole parameters and the field mass alone.
  • In asymptotically de Sitter spacetimes the damping rate tends to a nonzero constant as the field mass grows, so no arbitrarily long-lived modes arise regardless of $\mu$.
  • At large mass the spectrum becomes nearly independent of the multipole number $\ell$, and equation (23) gives the frequencies analytically for small $Q$ and $\xi$.
  • Scalar perturbations remain stable even for $\ell=0$ configurations whose effective potential has a negative gap; the observed late-time decay is dominated by a slowly decaying de Sitter mode in near-extremal cases.

Reading between the lines

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

  • If the coupling-independence near quasi-resonance holds beyond the parameter range tested, long-lived frequencies could be used to estimate a black hole's mass and charge without needing to know the non-minimal coupling strength.
  • The flat/dS split suggests that in a universe with a positive cosmological constant, massive fields around these black holes should always decay at a finite rate; an observation of an essentially undamped massive ringdown would then favour an asymptotically flat geometry or a different matter sector.
  • Because the paper notes that de Sitter-branch modes do not obey the null-geodesic/eikonal correspondence, a natural test is to check whether the flat-space quasi-resonant frequencies found here satisfy that correspondence as $\ell \to \infty$.
  • Textual note: the abstract's sentence claiming 'in the de Sitter case, arbitrarily long-lived modes can exist' is reversed relative to the body; the tables and Section V argue the opposite, with quasi-resonances in the flat case and finite damping in de Sitter.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 7 minor

Summary. The manuscript computes quasinormal frequencies of a massive scalar field in the non-minimal Einstein-Yang-Mills black hole backgrounds of Balakin-Lemos-Zayats type, for both asymptotically flat and asymptotically de Sitter cases. It claims that in the flat case the damping rate reaches zero at a critical scalar-field mass, producing quasi-resonances whose frequencies become independent of the non-minimal coupling xi, while in the de Sitter case the damping remains finite for all masses. It further presents WKB-based tables of de Sitter frequencies, an analytic large-mass formula (Eqs. 22-24), and time-domain evolutions arguing stability for l=0 perturbations even when the effective potential has a negative gap.

Significance. If correct, the paper would add a new example of massive-field quasi-resonances in a modified-gravity black hole and provide an explicit analytic large-mass formula containing the cosmological constant and Yang-Mills charge. The use of Leaver's method for the flat case and the time-domain stability check are appropriate, and the analytic formula has the virtue of being directly testable against the tables. However, the numerical de Sitter results are the main quantitative output, and they currently contain an internal parameter inconsistency; the abstract also contradicts the conclusions. The qualitative flat-versus-de Sitter distinction is not new, since the absence of de Sitter quasi-resonances is credited to an earlier analytic proof (Ref. [29]), so the paper's incremental value rests on the numerical tables and analytic formula, which need to be made reliable.

major comments (4)
  1. [Abstract and Section V] The abstract states exactly the opposite of the body's main conclusion: it says that in the de Sitter case arbitrarily long-lived modes can exist, whereas in the asymptotically flat case the damping rate never vanishes completely. Section V concludes the reverse, and Section IV explicitly states that the absence of quasi-resonances in asymptotically de Sitter black holes was proven in Ref. [29]. This contradiction concerns the paper's central claim and must be resolved before publication.
  2. [Section IV, Tables I-IV] The de Sitter frequencies in Tables I-IV are inconsistent with the caption values. For rh=1, Q=0.1, xi=0 and Lambda=0.001, Eq. (20) gives M about 0.5048, and the large-mu peak of V(r)=f[mu^2+l(l+1)/r^2+f'/r] is at rmax about (3M/Lambda)^(1/3) about 11.5 with f(rmax) about 0.869, so WKB predicts Re omega about 0.932 mu. Table I at mu=3.5 quotes Re omega=2.9635, i.e., Re omega/mu about 0.847, which is instead what one obtains with Lambda=0.01 (rmax about 5.3, fmax about 0.717). The same 10% shift in Re omega and roughly factor-3 shift in |Im omega| appears in all four tables, and Eq. (23) with x=sqrt(1-(9M^2 Lambda)^(1/3)) reproduces the tabulated entries only if Lambda=0.01. Thus the tables were either computed at Lambda=0.01 with wrong captions, or with a different potential than stated; the WKB-6/WKB-9 agreement cannot reveal this because both orders use the same input.
  3. [Section IV, Tables I-IV] All de Sitter QNM data are produced solely by the WKB method, and the only stated validation is the agreement between WKB orders 6 and 9, which are both WKB and may share a systematic error. Section IV claims the difference is much smaller than the observed effect, but that comparison cannot validate absolute frequencies. In view of the parameter inconsistency in the preceding comment, the quantitative de Sitter results, including the large-mu constant damping and the apparent l-independence of the frequencies, should be checked with an independent method, such as the Leaver method already used for the flat case or a frequency extraction from time-domain data.
  4. [Section IV, Eqs. (22)-(24)] The analytic large-mass formula is one of the paper's main results, but its derivation is compressed into a single sentence referring to Ref. [78]. The text does not show how the expansion of rmax in powers of 1/mu is organized relative to the smallness of Q and xi, nor how the WKB corrections produce the specific Q^2 and xi terms in Eq. (23). Since the formula is used to assert universal behavior in the quasi-resonance limit, the derivation, or at least a clear statement of the ordering and truncation, should be provided, and the formula's regime of validity stated.
minor comments (7)
  1. [Section III.A, Eq. (7)] The boundary condition for a massive field at infinity is written with sqrt(omega^2-mu^2); for decaying QNMs one needs e^{-sqrt(mu^2-omega^2) r*} for omega<mu, so the sign should be specified explicitly.
  2. [Section III.B, after Eq. (14)] There is a typo 'FIn some cases' that should read 'In some cases'.
  3. [Figures 2, 5, and 6] The axis labels are garbled (e.g., '/Minus', '/LParen1', '/Star'), making the figures unreadable.
  4. [Figures 3 and 4] The captions do not state Lambda=0; since all other figures state Lambda, please specify.
  5. [Tables I-IV] The meaning of 'WKB-6 (m=3)' and 'WKB-9 (m=4)' is not defined; please explain the Pade orders in the text or captions.
  6. [Abstract] The phrase 'quasi normal' should be 'quasinormal'.
  7. [Section V] The conclusion says the threshold for the onset of quasi-resonances is independent of xi, but this is illustrated only for the fundamental l=0 mode in Fig. 4; clarify the scope of this claim.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central claims are independent numerical and asymptotic results, not fits or self-citation chains.

full rationale

I walked the derivation chain from the metric and wave equation to the computed quasinormal frequencies and the large-mass formula. The effective potential (Eq. 6), the Leaver continued-fraction method (Eqs. 14-16), the WKB/Pade framework (Eq. 18), and the analytic large-mu expansion (Eqs. 22-23) form a self-contained chain. Eq. (23) is an asymptotic expansion of the WKB expression around the potential maximum, not a fit or a renamed input, so the claimed universal independence of xi in the quasi-resonance limit follows from the derived leading terms rather than being imposed. The statement that asymptotically de Sitter black holes lack quasi-resonances is supported by the cited analytic proof [29], which is by Konoplya and Zhidenko and is not the present author's own prior work, so it is independent external evidence. Several self-citations appear ([43], [65], [80], [93], [104], [107]), but they support numerical techniques, background results, or extensions, and none carries the central claim by itself. The WKB-6 versus WKB-9 agreement is an internal consistency check rather than a circular reduction. Any concern about the de Sitter tabulated values, such as possible inconsistency between the stated Lambda and the implied Lambda, is a correctness or reproducibility issue, not circularity, because the prediction is not equal to its input by construction. No step in the paper reduces a claimed prediction to a fitted parameter or to the paper's own unverified assertion.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities. The model parameters (M, Q, xi, Lambda, mu) are theory inputs from the Lagrangian and metric, not free parameters fitted to data. The main unproved premises are the validity of the test-field approximation, the WKB reliability for the de Sitter spectra, and the validity of the 1/mu expansion behind Eq. (23).

assumptions (4)
  • domain assumption The metric (Eq. 3) is an exact black hole solution of the non-minimal EYM action (Eq. 1) with a Wu-Yang magnetic monopole.
    Adopted from prior work [25,26]; not re-derived here.
  • domain assumption The scalar field is a test field, so its backreaction on the metric is neglected.
    Standard in QNM computations; stated implicitly by using the Klein-Gordon equation on a fixed background.
  • domain assumption The WKB approximation with two turning points is valid for the massive scalar field effective potential in the de Sitter cases computed in Tables I-IV.
    The paper argues the potential has a single peak; the only numerical support is agreement between WKB-6 and WKB-9, not an independent exact method.
  • ad hoc to paper The large-mass expansion for the position of the potential maximum in powers of 1/mu, with Q and xi small, converges and the resulting WKB formula (Eq. 23) is accurate.
    The derivation is sketched in one paragraph; no explicit error estimate or numerical verification is provided.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Long-Lived Quasinormal Modes and Quasi-Resonances around Non-Minimal Einstein-Yang-Mills Black Holes." pith.science (2026). https://pith.science/paper/6PFE4FTT

@misc{pith2026250508545,
  author       = {Pith},
  title        = {Pith review of: Long-Lived Quasinormal Modes and Quasi-Resonances around Non-Minimal Einstein-Yang-Mills Black Holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6PFE4FTT}},
  note         = {Machine review of arXiv:2505.08545}
}
read the original abstract

Using accurate computational methods, we compute the quasinormal frequencies of a massive scalar field propagating near a black hole in the framework of non-minimal Einstein-Yang-Mills theory with a non-zero cosmological constant. We show that increasing the mass of the scalar field significantly decreases the damping rate of the quasinormal modes for both asymptotically flat and de Sitter black holes. However, in the de Sitter case, arbitrarily long-lived modes can exist, whereas in the asymptotically flat case, the damping rate never vanishes completely. In the limit of quasi-resonances, we observe a kind of universal behavior where the frequencies do not depend on the coupling constant. Applying the time-domain integration of perturbation equations we show that even when the effective potential has a negative gap, the scalar field is stable and the perturbations decay in time. In the regime of large mass of the field we obtain the analytic formula for quasinormal modes.

Figures

Figures reproduced from arXiv: 2505.08545 by the authors.

Figure 1
Figure 1. FIG. 1. The parametric range allowing for existence of a blac [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Effective potentials for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Real and imaginary parts of QNMs for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Real and imaginary parts of QNMs for [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Effective potentials for [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Effective potentials for [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

  1. Gravitational perturbations of a regular T-duality inspired black hole: Quasinormal modes, excitation factors, and time-domain evolution

    gr-qc 2026-07 conditional novelty 4.0 of 10

    Gravitational quasinormal-mode frequencies of a T-duality-inspired regular black hole are computed, showing the zero-point length makes the ringdown oscillate faster and alters damping non-monotonically.

  2. Bonanno-Reuter regular black hole: quasi-resonances, grey-body factors and absorption cross-sections of a massive scalar field

    gr-qc 2025-10 conditional novelty 4.0 of 10

    Damping of massive scalar quasinormal modes decreases with field mass in the Bonanno-Reuter black hole, and grey-body factors shift to higher frequencies.

Reference graph

Works this paper leans on

123 extracted references · 3 canonical work pages · cited by 2 Pith papers

  1. [29]

    Bambi, Rev

    C. Bambi, Rev. Mod. Phys. 89, 025001 (2017), arXiv:1509.03884 [gr-qc]

    arXiv 2017

  2. [78]

    J. G. Rosa and S. R. Dolan, Phys. Rev. D 85, 044043 (2012), arXiv:1110.4494 [hep-th]

    arXiv 2012

  3. [1]

    0.1 0 .219004 − 0.197391i 0.217054 − 0.196784i 0.693%

  4. [2]

    0.5 0 .421281 − 0.036498i 0.422884 − 0.038726i 0.649%

  5. [3]

    0.845301 − 0.041298i 0.845280 − 0.041305i 0.00257%

    1. 0.845301 − 0.041298i 0.845280 − 0.041305i 0.00257%

  6. [4]

    1.5 1 .267256 − 0.041800i 1.267255 − 0.041799i 0.00008%

  7. [5]

    1.689323 − 0.041974i 1.689322 − 0.041974i 0.00001%

    2. 1.689323 − 0.041974i 1.689322 − 0.041974i 0.00001%

  8. [6]

    2.5 2 .111441 − 0.042053i 2.111441 − 0.042053i 2.3 × 10− 6%

Show all 123 references
  1. [7]

    2.533588 − 0.042096i 2.533588 − 0.042096i 0%

    3. 2.533588 − 0.042096i 2.533588 − 0.042096i 0%

  2. [8]

    0.844181 − 0.041255i 0.844162 − 0.041259i 0.00232% 1.5 1 .5 1 .265589 − 0.041749i 1.265588 − 0.041748i 0.00006% 1.5 2

    3.5 2 .955754 − 0.042122i 2.955754 − 0.042122i 0% 1.5 0 .1 0 .220577 − 0.192122i 0.217081 − 0.191494i 1.21% 1.5 0 .5 0 .422034 − 0.037838i 0.422213 − 0.038734i 0.216% 1.5 1 . 0.844181 − 0.041255i 0.844162 − 0.041259i 0.00232% 1.5 1 .5 1 .265589 − 0.041749i 1.265588 − 0.041748i...

  3. [9]

    0.1 0 .571010 − 0.184714i 0.570971 − 0.184662i 0.0109%

  4. [10]

    0.5 0 .626010 − 0.152171i 0.625863 − 0.152179i 0.0228%

  5. [11]

    0.903983 − 0.018856i 0.893933 − 0.038788i 2.47%

    1. 0.903983 − 0.018856i 0.893933 − 0.038788i 2.47%

  6. [12]

    1.5 1 .289036 − 0.039274i 1.288996 − 0.039251i 0.00361%

  7. [13]

    1.704894 − 0.040816i 1.704895 − 0.040818i 0.00015%

    2. 1.704894 − 0.040816i 1.704895 − 0.040818i 0.00015%

  8. [14]

    2.5 2 .123644 − 0.041376i 2.123644 − 0.041376i 3.6 × 10− 6%

  9. [15]

    2.543647 − 0.041646i 2.543647 − 0.041646i 0%

    3. 2.543647 − 0.041646i 2.543647 − 0.041646i 0%

  10. [16]

    0.894470 − 0.031586i 0.892114 − 0.038408i 0.806% 1.5 1 .5 1 .287172 − 0.039286i 1.287132 − 0.039259i 0.00372% 1.5 2

    3.5 2 .964320 − 0.041800i 2.964320 − 0.041800i 0% 1.5 0 .1 0 .566338 − 0.181388i 0.566289 − 0.181329i 0.0129% 1.5 0 .5 0 .621775 − 0.149053i 0.621579 − 0.149087i 0.0310% 1.5 1 . 0.894470 − 0.031586i 0.892114 − 0.038408i 0.806% 1.5 1 .5 1 .287172 − 0.039286i 1.287132 − 0.039259...

  11. [17]

    B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 116, 241103 (2016), arXiv:1606.04855 [gr-qc]

    2016 arXiv

  12. [18]

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

    2016 arXiv

  13. [19]

    B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 119, 141101 (2017), arXiv:1709.09660 [gr-qc]

    2017 arXiv

  14. [20]

    B. P. Abbott et al. (LIGO Scientific, VIRGO), Phys. Rev. Lett. 118, 221101 (2017), [Erra- tum: Phys. Rev. Lett.121,no.12,129901(2018)], arXiv:1706.01812 [gr-qc]

    2017 arXiv

  15. [21]

    B. P. Abbott et al. (LIGO Scientific, Virgo), Astrophys. J. 851, L35 (2017), arXiv:1711.05578 [astro-ph.HE]

    2017 arXiv

  16. [22]

    Berti, V

    E. Berti, V. Cardoso, and A. O. Starinets, Class. Quant. Grav. 26, 163001 (2009), arXiv:0905.2975 [gr-qc]

    2009 arXiv

  17. [23]

    S. V. Bolokhov and M. Skvortsova, (2025), arXiv:2504.05014 [gr-qc]

    2025 arXiv

  18. [24]

    K. D. Kokkotas and B. G. Schmidt, Living Rev. Rel. 2, 2 (1999), arXiv:gr-qc/9909058 [gr-qc]

    1999 arXiv

  19. [25]

    R. A. Konoplya and A. Zhi- denko, Rev. Mod. Phys. 83, 793 (2011), arXiv:1102.4014 [gr-qc]

    2011 arXiv

  20. [26]

    Goddi et al., Int

    C. Goddi et al., Int. J. Mod. Phys. D 26, 1730001 (2016), arXiv:1606.08879 [astro-ph.HE]

    2016 arXiv

  21. [27]

    Akiyama et al

    K. Akiyama et al. (Event Horizon Tele- scope), Astrophys. J. Lett. 875, L6 (2019), arXiv:1906.11243 [astro-ph.GA]

    2019 arXiv

  22. [28]

    Akiyama et al

    K. Akiyama et al. (Event Horizon Tele- scope), Astrophys. J. Lett. 875, L1 (2019), arXiv:1906.11238 [astro-ph.GA]

    2019 arXiv

  23. [30]

    Giesler et al

    M. Giesler et al. , Phys. Rev. D 111, 084041 (2025), arXiv:2411.11269 [gr-qc]

    2025 arXiv

  24. [31]

    Giesler, M

    M. Giesler, M. Isi, M. A. Scheel, and S. Teukolsky, Phys. Rev. X 9, 041060 (2019), arXiv:1903.08284 [gr-qc]

    2019 arXiv

  25. [32]

    R. A. Konoplya and A. Zhidenko, JHEAp 44, 419 (2024), arXiv:2209.00679 [gr-qc]

    2024 arXiv

  26. [33]

    A. F. Zinhailo, Phys. Lett. B 853, 138682 (2024), arXiv:2403.06867 [gr-qc]

    2024 arXiv

  27. [34]

    R. A. Konoplya and O. S. Stashko, (2024), arXiv:2408.02578 [gr-qc]

    2024 arXiv

  28. [35]

    Stuchlík and A

    Z. Stuchlík and A. Zhidenko, (2025), arXiv:2503.06775 [gr-qc]

    2025 arXiv

  29. [36]

    S. V. Bolokhov, Phys. Rev. D 110, 024010 (2024), arXiv:2311.05503 [gr-qc]

    2024 arXiv

  30. [37]

    R. A. Konoplya, D. Ovchinnikov, and B. Ahmedov, Phys. Rev. D 108, 104054 (2023), arXiv:2307.10801 [gr-qc]

    2023 arXiv

  31. [38]

    A. F. Zinhailo, (2024), 10.13140/RG.2.2.26785.01124

    2024

  32. [39]

    Zhang, H

    D. Zhang, H. Gong, G. Fu, J.-P. Wu, and Q. Pan, Eur. Phys. J. C 84, 564 (2024), arXiv:2402.15085 [gr-qc]

    2024 arXiv

  33. [40]

    Stashko, Phys

    O. Stashko, Phys. Rev. D 110, 084016 (2024), arXiv:2407.07892 [gr-qc]

    2024 arXiv

  34. [41]

    A. B. Balakin, J. P. S. Lemos, and A. E. Zayats, Phys. Rev. D 93, 024008 (2016), arXiv:1512.02653 [gr-qc]

    2016 arXiv

  35. [42]

    A. B. Balakin and A. E. Zayats, Phys. Lett. B 644, 294 (2007), arXiv:gr-qc/0612019

    2007 arXiv

  36. [43]

    B. C. Lütfüoğlu, (2025), arXiv:2504.18482 [gr-qc]

    2025 arXiv

  37. [44]

    D. J. Gogoi and S. Ponglert- sakul, Eur. Phys. J. C 84, 652 (2024), arXiv:2402.06186 [gr-qc]

    2024 arXiv

  38. [45]

    R. A. Konoplya and A. V. Zhidenko, Phys. Lett. B 609, 377 (2005), arXiv:gr-qc/0411059

    2005 arXiv

  39. [46]

    Ohashi and M.-a

    A. Ohashi and M.-a. Sakagami, Class. Quant. Grav. 21, 3973 (2004), arXiv:gr-qc/0407009

    2004 arXiv

  40. [47]

    R. A. Konoplya and A. Zhi- denko, Phys. Rev. D 97, 084034 (2018), arXiv:1712.06667 [gr-qc]

    2018 arXiv

  41. [48]

    S. S. Seahra, C. Clarkson, and R. Maartens, Phys. Rev. Lett. 94, 121302 (2005), arXiv:gr-qc/0408032

    2005 arXiv

  42. [49]

    R. A. Konoplya and A. Zhi- denko, Phys. Lett. B 853, 138685 (2024), arXiv:2307.01110 [gr-qc]

    2024 arXiv

  43. [50]

    Afzal et al

    A. Afzal et al. (NANOGrav), Astrophys. J. Lett. 951, L11 (2023), [Erratum: Astro- phys.J.Lett. 971, L27 (2024), Erratum: Astrophys.J. 971, L27 (2024)], arXiv:2306.16219 [astro-ph.HE]

    2023 arXiv

  44. [51]

    T. V. Fernandes, D. Hilditch, J. P. S. Lemos, and V. Cardoso, Phys. Rev. D 105, 044017 (2022), arXiv:2112.03282 [gr-qc]

    2022 arXiv

  45. [52]

    A. F. Zinhailo, Eur. Phys. J. C 78, 992 (2018), arXiv:1809.03913 [gr-qc]

    2018 arXiv

  46. [53]

    M. S. Churilova, R. A. Konoplya, and 12 A. Zhidenko, Phys. Lett. B 802, 135207 (2020), arXiv:1911.05246 [gr-qc]

    2020 arXiv

  47. [54]

    Koyama and A

    H. Koyama and A. Tomimatsu, Phys. Rev. D 65, 084031 (2002), arXiv:gr-qc/0112075

    2002 arXiv

  48. [55]

    Jing, Phys

    J. Jing, Phys. Rev. D 72, 027501 (2005), arXiv:gr-qc/0408090

    2005 arXiv

  49. [56]

    Moderski and M

    R. Moderski and M. Rogatko, Phys. Rev. D 64, 044024 (2001), arXiv:gr-qc/0105056

    2001 arXiv

  50. [57]

    R. A. Konoplya, A. Zhidenko, and C. Molina, Phys. Rev. D 75, 084004 (2007), arXiv:gr-qc/0602047

    2007 arXiv

  51. [58]

    G. W. Gibbons and M. Ro- gatko, Phys. Rev. D 77, 044034 (2008), arXiv:0801.3130 [hep-th]

    2008 arXiv

  52. [59]

    Dubinsky, EPL 147, 19003 (2024), arXiv:2403.01883 [gr-qc]

    A. Dubinsky, EPL 147, 19003 (2024), arXiv:2403.01883 [gr-qc]

    2024 arXiv

  53. [60]

    K. D. Kokkotas, R. A. Konoplya, and A. Zhidenko, Phys. Rev. D 83, 024031 (2011), arXiv:1011.1843 [gr-qc]

    2011 arXiv

  54. [61]

    R. A. Konoplya and R. D. B. Fontana, Phys. Lett. B 659, 375 (2008), arXiv:0707.1156 [hep-th]

    2008 arXiv

  55. [62]

    Wu and R

    C. Wu and R. Xu, Eur. Phys. J. C 75, 391 (2015), arXiv:1507.04911 [gr-qc]

    2015 arXiv

  56. [63]

    Davlataliev, B

    A. Davlataliev, B. Narzilloev, I. Hussain, A. Abdujabbarov, and B. Ahmedov, (2024), arXiv:2412.09464 [gr-qc]

    2024 arXiv

  57. [64]

    E. W. Leaver, Proc. Roy. Soc. Lond. A 402, 285 (1985)

    1985

  58. [65]

    E. W. Leaver, Phys. Rev. D 34, 384 (1986)

    1986

  59. [66]

    Gundlach, R

    C. Gundlach, R. H. Price, and J. Pullin, Phys. Rev. D 49, 883 (1994), arXiv:gr-qc/9307009

    1994 arXiv

  60. [67]

    Momennia, Phys

    M. Momennia, Phys. Rev. D 106, 024052 (2022), arXiv:2204.03259 [gr-qc]

    2022 arXiv

  61. [68]

    Skvortsova, Grav

    M. Skvortsova, Grav. Cosmol. 30, 68 (2024), arXiv:2311.02729 [gr-qc]

    2024 arXiv

  62. [69]

    Skvortsova, Grav

    M. Skvortsova, Grav. Cosmol. 30, 279 (2024), arXiv:2405.15807 [gr-qc]

    2024 arXiv

  63. [70]

    Aneesh, S

    S. Aneesh, S. Bose, and S. Kar, Phys. Rev. D 97, 124004 (2018), arXiv:1803.10204 [gr-qc]

    2018 arXiv

  64. [71]

    B. C. Lütfüoğlu, (2025), arXiv:2503.16087 [gr-qc]

    2025 arXiv

  65. [72]

    M. S. Churilova, R. A. Konoplya, Z. Stuch- lik, and A. Zhidenko, JCAP 10, 010 (2021), arXiv:2107.05977 [gr-qc]

    2021 arXiv

  66. [73]

    Malik, EPL 147, 69001 (2024), arXiv:2410.04306 [gr-qc]

    Z. Malik, EPL 147, 69001 (2024), arXiv:2410.04306 [gr-qc]

    2024 arXiv

  67. [74]

    Malik, Int

    Z. Malik, Int. J. Theor. Phys. 64, 30 (2025)

    2025

  68. [75]

    Rostworowski, Acta Phys

    A. Rostworowski, Acta Phys. Polon. B 38, 81 (2007), arXiv:gr-qc/0606110

    2007 arXiv

  69. [76]

    Nollert, Phys

    H.-P. Nollert, Phys. Rev. D 47, 5253 (1993)

    1993

  70. [77]

    Zhidenko, Phys

    A. Zhidenko, Phys. Rev. D 74, 064017 (2006), arXiv:gr-qc/0607133

    2006 arXiv

  71. [79]

    R. A. Konoplya and A. Zhidenko, Phys. Rev. D 76, 084018 (2007), [Erra- tum: Phys.Rev.D 90, 029901 (2014)], arXiv:0707.1890 [hep-th]

    2007 arXiv

  72. [80]

    Kanti, R

    P. Kanti, R. A. Konoplya, and A. Zhidenko, Phys. Rev. D 74, 064008 (2006), arXiv:gr-qc/0607048

    2006 arXiv

  73. [81]

    Dubinsky, (2024), 10.13140/RG.2.2.35132.24961, arXiv:2405.13552 [gr-qc]

    A. Dubinsky, (2024), 10.13140/RG.2.2.35132.24961, arXiv:2405.13552 [gr-qc]

    2024

  74. [82]

    B. F. Schutz and C. M. Will, Astrophys. J. Lett. 291, L33 (1985)

    1985

  75. [83]

    Iyer and C

    S. Iyer and C. M. Will, Phys. Rev. D 35, 3621 (1987)

    1987

  76. [84]

    R. A. Konoplya, Phys. Rev. D 68, 024018 (2003), arXiv:gr-qc/0303052

    2003 arXiv

  77. [85]

    Matyjasek and M

    J. Matyjasek and M. Opala, Phys. Rev. D 96, 024011 (2017), arXiv:1704.00361 [gr-qc]

    2017 arXiv

  78. [86]

    R. A. Konoplya, Gen. Rel. Grav. 34, 329 (2002), arXiv:gr-qc/0109096

    2002 arXiv

  79. [87]

    A. F. Zinhailo, Eur. Phys. J. C 79, 912 (2019), arXiv:1909.12664 [gr-qc]

    2019 arXiv

  80. [88]

    Xiong, P

    W. Xiong, P. Liu, C.-Y. Zhang, and C. Niu, Phys. Rev. D 106, 064057 (2022), arXiv:2112.12523 [gr-qc]

    2022 arXiv

  81. [89]

    Skvortsova, Fortsch

    M. Skvortsova, Fortsch. Phys. 72, 2400132 (2024), arXiv:2405.06390 [gr-qc]

    2024 arXiv

  82. [90]

    Hamil and B

    B. Hamil and B. C. Lütfüoğlu, Chin. Phys. C 49, 025107 (2025), arXiv:2410.09551 [gr-qc]

    2025 arXiv

  83. [91]

    Liu and H

    H. Liu and H. Guo, Phys. Rev. D 110, 104058 (2024), arXiv:2409.00320 [gr-qc]

    2024 arXiv

  84. [92]

    R. A. Konoplya and E. Ab- dalla, Phys. Rev. D 71, 084015 (2005), arXiv:hep-th/0503029

    2005 arXiv

  85. [93]

    Kodama, R

    H. Kodama, R. A. Konoplya, and A. Zhi- denko, Phys. Rev. D 81, 044007 (2010), arXiv:0904.2154 [gr-qc]

    2010 arXiv

  86. [94]

    Malik, (2025), arXiv:2504.12570 [gr-qc]

    Z. Malik, (2025), arXiv:2504.12570 [gr-qc]

    2025

  87. [95]

    R. A. Konoplya, A. F. Zinhailo, and Z. Stuchlik, Phys. Rev. D 102, 044023 (2020), arXiv:2006.10462 [gr-qc]

    2020 arXiv

  88. [96]

    Dubinsky and A

    A. Dubinsky and A. Zin- hailo, Eur. Phys. J. C 84, 847 (2024), arXiv:2404.01834 [gr-qc]

    2024 arXiv

  89. [97]

    B. C. Lütfüoğlu, (2025), arXiv:2504.09323 [gr-qc]

    2025 arXiv

  90. [98]

    R. D. B. Fontana, P. A. González, E. Papantonopou- los, and Y. Vásquez, Phys. Rev. D 103, 064005 (2021), arXiv:2011.10620 [gr-qc]

    2021 arXiv

  91. [99]

    S. V. Bolokhov, Eur. Phys. J. C 84, 634 (2024), arXiv:2404.09364 [gr-qc]

    2024 arXiv

  92. [100]

    D. V. Gal’tsov and A. A. Matiukhin, Class. Quant. Grav. 9, 2039 (1992)

    1992

  93. [101]

    Zhidenko, Class

    A. Zhidenko, Class. Quant. Grav. 21, 273 (2004), arXiv:gr-qc/0307012

    2004 arXiv

  94. [102]

    R. A. Konoplya, Phys. Lett. B 823, 136734 (2021), arXiv:2109.01640 [gr-qc]

    2021 arXiv

  95. [103]

    R. A. Konoplya and A. Zhidenko, JHEP 06, 037 (2004), arXiv:hep-th/0402080

    2004 arXiv

  96. [104]

    R. A. Konoplya and A. Zhi- denko, Phys. Rev. D 95, 104005 (2017), arXiv:1701.01652 [hep-th]

    2017 arXiv

  97. [105]

    R. A. Konoplya and A. Zhi- denko, Phys. Rev. D 89, 024011 (2014), arXiv:1309.7667 [hep-th]

    2014 arXiv

  98. [106]

    M. A. Cuyubamba, R. A. Konoplya, and A. Zhidenko, Phys. Rev. D 93, 104053 (2016), arXiv:1604.03604 [gr-qc]

    2016 arXiv

  99. [107]

    Dyatlov, Commun

    S. Dyatlov, Commun. Math. Phys. 306, 119 (2011), arXiv:1003.6128 [math.AP]

    2011 arXiv

  100. [108]

    Dyatlov, Annales Henri Poincare 13, 1101 (2012), arXiv:1101.1260 [math.AP]

    S. Dyatlov, Annales Henri Poincare 13, 1101 (2012), arXiv:1101.1260 [math.AP]

    2012 arXiv

  101. [109]

    Dubinsky, Mod

    A. Dubinsky, Mod. Phys. Lett. A 39, 2450108 (2024), arXiv:2404.18004 [gr-qc]

    2024 arXiv

  102. [110]

    R. A. Konoplya, A. Khrabustovskyi, J. Kříž, and A. Zhidenko, JCAP 04, 062 (2025), 13 arXiv:2501.16134 [gr-qc]

    2025 arXiv

  103. [111]

    R. A. Konoplya and A. Zhidenko, JCAP 11, 028 (2022), arXiv:2210.04314 [gr-qc]

    2022 arXiv

  104. [112]

    Lopez-Ortega, Gen

    A. Lopez-Ortega, Gen. Rel. Grav. 38, 1565 (2006), arXiv:gr-qc/0605027

    2006 arXiv

  105. [113]

    Lopez-Ortega, Gen

    A. Lopez-Ortega, Gen. Rel. Grav. 39, 1011 (2007), arXiv:0704.2468 [gr-qc]

    2007 arXiv

  106. [114]

    Lopez-Ortega, Gen

    A. Lopez-Ortega, Gen. Rel. Grav. 44, 2387 (2012), arXiv:1207.6791 [gr-qc]

    2012 arXiv

  107. [115]

    Cardoso, A

    V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. Zanchin, Phys. Rev. D 79, 064016 (2009), arXiv:0812.1806 [hep-th]

    2009 arXiv

  108. [116]

    R. A. Konoplya, Phys. Lett. B 838, 137674 (2023), arXiv:2210.08373 [gr-qc]

    2023 arXiv

  109. [117]

    R. A. Konoplya and A. F. Zin- hailo, Eur. Phys. J. C 80, 1049 (2020), arXiv:2003.01188 [gr-qc]

    2020 arXiv

  110. [118]

    S. V. Bolokhov, Phys. Lett. B 856, 138879 (2024), arXiv:2310.12326 [gr-qc]

    2024 arXiv

  111. [119]

    Malik, Int

    Z. Malik, Int. J. Mod. Phys. A 40, 2450132 (2025)

    2025

  112. [120]

    Dubinsky, Phys

    A. Dubinsky, Phys. Dark Univ. 46, 101657 (2024), arXiv:2405.08262 [gr-qc]

    2024 arXiv

  113. [121]

    R. A. Konoplya and A. Zhidenko, JCAP 09, 068 (2024), arXiv:2406.11694 [gr-qc]

    2024 arXiv

  114. [122]

    Malik, JCAP 04, 042 (2025), arXiv:2412.19443 [gr-qc]

    Z. Malik, JCAP 04, 042 (2025), arXiv:2412.19443 [gr-qc]

    2025 arXiv

  115. [123]

    Dubinsky, (2024), arXiv:2412.00625 [gr-qc]

    A. Dubinsky, (2024), arXiv:2412.00625 [gr-qc]

    2024

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.