Quasi-normal modes and echoes of generalized black hole bounces and their correspondence with shadows

Albert Duran-Cabac\'es; Diego Rubiera-Garcia; Diego S\'aez-Chill\'on G\'omez

REVIEW 2 major objections 4 minor 2 cited by

Quasi-normal modes and echoes of generalized black hole bounces and their correspondence with shadows

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read The paper claims that in a one-parameter family of black bounces, gravitational-wave echoes appear precisely in the horizonless window where multiple photon spheres exist, and that these echoes correspond to extra photon rings in…

desk verdict A concrete echo window in a known black-bounce metric, worth refereeing, but the decoupling of axial fluid perturbations is asserted, not proven. read the letter →

arxiv 2506.10814 v1 pith:MAPZPNJH submitted 2025-06-12 gr-qc

Albert Duran-Cabacés , Diego Rubiera-Garcia , Diego Sáez-Chillón Gómez This is my paper

classification gr-qc MSC 83C5783C35

keywords blackbouncequasi-normalmodesgravitationalwaveechoesRegge-Wheelerpotentialphotonspherewormholeholeshadowtime-domainmethod

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

This paper studies a one-parameter family of spacetimes, the generalized black bounces, that interpolate between regular black holes and traversable wormholes as a single parameter $a$ grows. It derives the axial gravitational perturbation equation for a general spherically symmetric spacetime with an anisotropic fluid source, then solves it numerically in the time domain for this family. The main result is that in the horizonless window $a_I < a \lesssim a_{II}$ the Regge-Wheeler potential develops a multi-peaked structure with wells, so low-frequency modes get trapped and leak out as repeated echoes with decreasing frequency and amplitude. The authors also connect these echoes to the appearance of additional photon rings in simulated thin-disk images, arguing that the usual exponential luminosity falloff of photon rings fails for wormholes.

What carries the argument

The central object is the axial Regge-Wheeler potential $V^{\rm RW}_l(r)$ of Eq. (39), which is the effective barrier in the Schr\"odinger-like equation $(\partial^2_{r_*} + \omega^2 - V)\tilde{Q}=0$ for the perturbation in the tortoise coordinate $r_*$. For the black bounce family this potential has a central maximum plus two symmetric side maxima, creating finite wells near the throat; modes trapped there produce the echoes. On the optical side, the relevant machinery is the eikonal correspondence between quasi-normal mode frequencies and photon-sphere parameters: the real part is tied to the angular velocity and critical impact parameter, while the imaginary part is tied to the Lyapunov exponent, which normally controls the exponential thinning of photon rings.

What would settle it

Solve the coupled axial perturbation system, Eqs. (10)-(12) with the fluid source terms (35)-(37) retained, for $a$ in the echo window and compare the fundamental quasi-normal mode frequency and echo interval with the decoupled results; agreement at numerical precision would confirm the shortcut, while any deviation would invalidate the reported spectrum. A second, observational falsifier: a high signal-to-noise ringdown from a candidate horizonless ultracompact object that shows no echo repetitions, or an echo interval inconsistent with $a \in (a_I, a_{II}]$, would rule out this family as the source.

Watch

Extended reading notes

Core claim

The paper claims that the gravitational-wave ringdown of the generalized black bounce metric with $A(r)=B(r)=1-2Mr^2/(r^2+a^2)^{3/2}$ and $\Sigma(r)=\sqrt{r^2+a^2}$ is governed by a single Regge-Wheeler equation whose potential is multi-peaked exactly when $a$ lies between $a_I = 4M/(3\sqrt{3})$ and $a_{II} = 2\sqrt{5}\,M/5$. In the black-hole range $a \leq a_I$, the fundamental quasi-normal mode's real part increases and its imaginary part approaches zero as the extremal limit is reached, meaning oscillations are faster and longer-lived than for Schwarzschild. For $a_I < a \lesssim a_{II}$ there is no horizon but photon spheres are present; the potential wells between the central and lateral maxima trap modes, producing echoes that repeat the main burst with modulated amplitude and lower frequency. The authors verify this with time-domain evolutions and Prony extraction, and on the imaging side they show that the same objects display extra photon rings whose luminosities do not follow the exponential Lyapunov falloff of black-hole rings.

Load-bearing premise

The results assume the anisotropic fluid's own axial oscillations do not matter for the gravitational-wave signal; if they couple to the metric, the effective potential, the echo interval, and the extracted frequencies would change.

Editorial extensions

If this is right

Within the black-hole range $a \leq a_I$, increasing $a$ raises the real quasi-normal mode frequency and lengthens the damping time, so the ringdown of a regular black bounce is faster and longer-lived than Schwarzschild's.
In the horizonless window $a_I < a \lesssim a_{II}$, every such wormhole should ring with echoes whose repetition interval, amplitude modulation, and downward frequency drift encode the depth and position of the potential wells.
The same wormhole images should show extra photon rings interior to the usual ones, with luminosities that can exceed the exponential falloff predicted by the Lyapunov exponent.
Because the quasi-normal-mode imaging correspondence maps real and imaginary parts to shadow radius and ring falloff, the echo window is exactly the window where imaging deviations from a black hole should also appear.
The correspondence cannot currently match individual echoes to individual photon rings, because echoes are time-dependent while the simulated images are time-averaged.

Reading between the lines

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

Editorial extension: if the decoupling assumption holds, the echo window $(a_I, a_{II}]$ is fixed by the condition that the Regge-Wheeler potential has at least two maxima; a similar criterion could be tested for any horizonless ultracompact metric by counting maxima of its effective potential.
Editorial extension: a practical observational consequence the paper leaves implicit is that a measured echo repetition interval would constrain $a/M$ for this metric family, and the predicted interval could be tabulated as a function of $a$.
Editorial extension: the imaging side suggests that time-averaged photon rings are not a reliable proxy for trapped-mode lifetimes; a time-resolved image sequence of the lensed disk could reveal ring brightness oscillations with the echo period, a testable prediction beyond the paper's static images.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

I'll cut to the chase: the one genuinely new thing here is the echo window for the Lobo et al. generalized black bounce (2) in the horizonless interval a_I < a ≤ a_II. The double-peaked RW potential and the time-domain echo profiles make that physically plausible, and the a=0 endpoint checks against known Schwarzschild QNMs. If I needed a citation for 'these bounces ring and echo in this parameter range,' this paper would be the one.

What it does well: the general axial perturbation derivation in Sec. III is sensible and the identification with the known RW potential is correct; the numerical setup is standard; the imaging part is honest about the fact that the QNM-shadow correspondence does not currently map echo features to specific photon rings. They don't oversell.

The soft spot is the decoupling assumption. In Eq. (38) the source term is proportional to p_parallel(r) x^(3)_lm, and they set x^(3)_lm = 0 because a scalar test field couples to the polar sector. But this background is not a scalar field; it's an anisotropic fluid with p_parallel nonzero at the throat. The stress-test note has this right: you can't invoke the test-field limit for the same fields that build the background. You need to show that the linearized fluid equations force x^(3)_lm = 0, or at least that an axial gravitational wave can exist with the fluid variables unperturbed. For perfect-fluid stars that decoupling is a known result, but for this anisotropic fluid it's not shown, and the example of p_parallel(0) = -1/(8πa²) means the source doesn't vanish numerically if x3 is nonzero. This is a load-bearing gap: if the fluid couples, the effective potential and the echo interval could shift.

Lesser issues: no error bars or convergence checks on the Prony fits—the numbers in Figs. 2, 5, 6 are presented without any stated uncertainty in extraction; the grid step is reported but not swept; and the imaging section is a repackaging of the authors' own ray-tracing code with no new method.

Bottom line: the echo claim is plausible and likely robust, but the paper doesn't establish it as the true gravitational spectrum of this spacetime, only as the spectrum of a truncated proxy. That's fixable. I'd send it to a competent referee, not desk reject. The right referee will ask for the axial fluid perturbation equations and a convergence study. For a reading group it's a decent paper to discuss; not a must-cite for my own QNM work, but I'd cite it if writing about black bounce echoes.

Referee Report

2 major / 4 minor

Summary. The manuscript studies axial perturbations of a generalized black-bounce spacetime (Eq. (2)) that interpolates between regular black holes and traversable wormholes as the parameter a grows. It first derives general axial gravitational perturbation equations for a spherically symmetric metric sourced by an anisotropic fluid, arriving at Eq. (38) with a Regge-Wheeler potential and a fluid source term. The paper then sets the axial fluid variable to zero, reducing to the homogeneous Eq. (40), and computes QNM frequencies by evolving Gaussian initial data with a time-domain scheme. For a in the horizonless window a_I < a <= a_II, it reports echoes produced by the double-barrier structure of the Regge-Wheeler potential. It also computes thin-disk images and argues that additional photon rings in the wormhole regime are the optical counterpart of the echoes.

Significance. If the central modeling assumption is valid, the paper offers a useful single-parameter family in which regular black holes and traversable wormholes are connected, with echo production tied explicitly to the loss of the horizon and the appearance of multiple photon spheres. The general axial perturbation derivation, the successful Schwarzschild benchmark at a=0, and the qualitative explanation of echoes through the multi-peak potential are genuine strengths. The main quantitative claims, however, rest on an unproven decoupling of the axial fluid perturbation and on numerical mode extraction without detailed convergence checks, so the significance is conditional on addressing these points.

major comments (2)

[Sec. III.B, Eqs. (38)-(40)] The homogeneous equation used for all QNM and echo computations is obtained by setting the axial fluid variable x^(3)_lm(r,omega) to zero, but this decoupling is not derived. For the anisotropic-fluid background, p_parallel is generically nonzero (e.g., p_parallel(0)=-1/(8 pi a^2) for the metric (2)), so the source term in Eq. (38) does not vanish automatically. The authors neither show that x^(3)_lm=0 is consistent with the linearized conservation equation (14) nor solve the coupled metric-fluid axial system. Without this, the effective potential (39), the echo interval a_I < a <= a_II, and the frequencies in Figs. 5-6 are modes of a truncated proxy equation rather than established gravitational QNMs of the spacetime. This is load-bearing for the headline echo claim and needs to be fixed, either by demonstrating decoupling or by solving the coupled system.
[Sec. IV.B, Figs. 2, 5, 6] The numerical extraction of QNM frequencies and echo features lacks convergence checks and error estimates. The text states that different values of the step h and domain sizes are used depending on a, but no comparison across resolutions is shown, and the Prony analysis is not described in enough detail (fitting window, number of exponentials N, sensitivity to the Gaussian width and center, observer position, or boundary effects). This matters because the reported behavior near a_I and a_II is non-trivial and partly non-monotonic, with no uncertainty attached. A systematic convergence study plus a frequency-domain check (e.g., direct integration or WKB) for at least several representative a values is needed to support the quantitative claims.

minor comments (4)

[Eq. (40)] The reduced radial equation appears with the wrong sign for the omega^2 term: it should read (partial^2_r* + omega^2 - V_RW_l) phi = 0 to be consistent with Eq. (22) and with the light-cone form used in Eq. (42).
[Fig. 3 caption] The horizontal axis label and units for the tortoise coordinate are not specified in the caption or the text, which makes the multi-peak structure and the inset difficult to interpret quantitatively.
[Sec. I and Fig. 1 caption] There are minor typographical errors, including 'boound' in the introduction and 'Schwarzchild' in the caption of Fig. 1.
[Sec. V] The correspondence between echoes and additional photon rings is stated qualitatively; the paper does not provide a quantitative relation between echo interval or mode frequency and the radii or Lyapunov exponents of the additional rings. This is acknowledged in the conclusions, but a brief quantitative comparison for at least one wormhole example would strengthen the claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the QNM/echo results follow from an explicitly stated truncation of the perturbation equations and are benchmarked against an external Schwarzschild result; self-citations are not load-bearing.

full rationale

The central derivation is self-contained. The metric (2) is substituted into the generally derived axial perturbation equations, leading to Eq. (38) with a fluid source term proportional to the axial fluid variable x̃^(3)_lm. The paper then explicitly states the modeling choice x̃^(3)_lm = 0, invoking a scalar test-field shortcut, and solves the resulting homogeneous Regge-Wheeler equation (40) by time-domain methods. This is an approximation, not a circular reduction: no quantity in Eq. (40) is defined in terms of the echoes or QNM frequencies it later predicts. The parameter a is scanned, not fitted to the output; the frequencies are extracted from the evolved waveform via Prony analysis, and the a = 0 endpoint is compared to known Schwarzschild QNMs from independent references [60,61]. The echo interval a_I < a ≤ a_II arises from the multi-peak structure of the explicitly computed potential (39), not from any pre-imposed echo condition. The QNM-shadow correspondence of Eq. (51) is imported from the external reference [66] and is used for interpretation rather than as the basis of the echo claim. The self-references (e.g., Refs. [69-71]) concern the illustrative imaging setup and are not load-bearing for the central gravitational-wave results; they are used as a tool, not as a source of the predicted mode frequencies. No step reduces to a fit, a renaming, or a self-citation chain, so the paper is not circular in the sense relevant here.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central computation rests on the standard Regge-Wheeler framework and on the Lobo et al. black bounce metric. The only physical free parameter is a, scanned over a range. The numerical setup adds hand-chosen initial data and grid parameters, and the QNM-shadow correspondence is imported from the eikonal limit. No new particles, fields, dimensions, or conserved quantities are introduced.

free parameters (3)

a (family parameter of the black bounce metric) = scanned from 0 to 1.2M; critical values a_I=4/(3 sqrt(3))M and a_II=2 sqrt(5)/5 M
Controls the transition from regular black hole to wormhole and the existence of photon spheres and echoes. Chosen by hand, not fitted.
Gaussian initial data parameters and observer location = sigma=0.25M, v_c=10M, r*_o=10M, with variations in some runs
Numerical setup for the time-domain method; authors state the ringdown should not depend on initial data, but no systematic convergence study is reported.
Numerical grid parameters = h in {0.001, 0.005, 0.01}M; domains up to 1200M
Chosen per parameter range to balance precision and range; discretization is a numerical parameter, not a physical constant.

assumptions (5)

domain assumption Einstein field equations describe the perturbations of the background spacetime
Framework of the paper; the derivation starts with the perturbed Einstein equations in Sec. III A.
domain assumption The Lobo et al. black bounce line element (2) is a valid GR solution with an anisotropic fluid source
Taken from Ref. [34] and used throughout; the source is described by nonlinear electrodynamics plus a phantom scalar according to Ref. [38].
standard math Axial and polar perturbations decouple and the axial sector can be treated in Regge-Wheeler gauge
Standard parity decomposition of spherical perturbations used in Sec. III A.
ad hoc to paper The axial fluid perturbation amplitude x^(3)_lm can be set to zero for the QNM computation
Assumed after Eq. (38) via the scalar test-field shortcut; no proof is given that coupled axial fluid modes do not alter the spectrum.
domain assumption The eikonal QNM-shadow correspondence Eq. (51) remains applicable to horizonless spacetimes with multi-peak potentials
Imported from Ref. [66] and used in Sec. V to connect echoes and photon rings; the authors note caveats and an incomplete identification.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Quasi-normal modes and echoes of generalized black hole bounces and their correspondence with shadows." pith.science (2026). https://pith.science/paper/MAPZPNJH

@misc{pith2026250610814,
  author       = {Pith},
  title        = {Pith review of: Quasi-normal modes and echoes of generalized black hole bounces and their correspondence with shadows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MAPZPNJH}},
  note         = {Machine review of arXiv:2506.10814}
}

abstract

We study the quasi-normal modes (QNMs) of a family of generalized black bounces interpolating between regular black holes and traversable wormhole solutions according to a single extra parameter $a$. Firstly, working with a generic spherically symmetric space-time with arbitrary radial function and an anisotropic fluid matter source, the general equations for the gravitational waves are obtained. Then, we focus on such particular space-time metric and use the time-domain method to find the evolution of the QNMs with respect to the parameter $a$, finding larger frequencies and damped modes as $a$ grows. Furthermore we find that, for a gap in the values of $a$ for which no horizon is present but several photon spheres are, echoes are produced. Such echoes, which come from trapped modes in the potential well that are eventually leaked off for higher frequencies, appear as repetitions of the original wave but with modulated amplitude and decreased frequencies, and study their evolution with $a$. In addition, at the light of the correspondence recently discussed in the literature between QNMs and black hole imaging, we discuss the relation of the features of such echoes with those features (photon rings and shadows) of optical images from thin accretion disks. Despite working with simplified models and settings, our analysis provides useful insights on the usefulness of the correspondence for both gravitational waves and shadows.

Figures

Figures reproduced from arXiv: 2506.10814 by the authors.

**Figure 2.** FIG. 2. Real part (top, red) and imaginary part (bottom, [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 4.** FIG. 4. Dynamical evolution of a Gaussian wavefunction of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5. Real part (top, red) and imaginary part (bottom, [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6. Real part of the trapped modes (labeled by the over [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7. Images of a modified black hole with [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

Discussion (0). Sign in 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

Echoes of a hairy black hole from gravitational decoupling
gr-qc 2026-04 unverdicted novelty 5.0 of 10

Axial perturbations of a gravitationally decoupled hairy black hole develop a double-peak potential that dynamically generates echo-like late-time waveforms in a suitable (α, β) region.
Bondi accretion disk luminosity around neutral and charged Simpson-Visser spacetimes
gr-qc 2025-07 reject novelty 5.0 of 10

Bondi accretion is computed in Simpson-Visser and charged Simpson-Visser spacetimes, but the reported critical (sonic) radii are not supported by the paper's own equations.

Reference graph

Works this paper leans on

73 extracted references · 58 canonical work pages · cited by 2 Pith papers

[1]

Addazi,et al.Prog

A. Addazi,et al.Prog. Part. Nucl. Phys.125(2022), 103948

work page 2022
[2]

Alves Batista,et al.Class

R. Alves Batista,et al.Class. Quant. Grav.42(2025) 032001

work page 2025
[3]

Cardoso and P

V. Cardoso and P. Pani, Living Rev. Rel.22(2019) 4

2019
[4]

R. P. Kerr, Phys. Rev. Lett.11(1963) 237

work page 1963
[5]

E. A. Becerra-Vergara, C. R. Arg¨ uelles, A. Krut, J. A. Rueda and R. Ruffini, Mon. Not. Roy. Astron. Soc. 505(2021) L64

work page 2021
[6]

Bambi, A

C. Bambi, A. Cardenas-Avendano, T. Dauser, J. A. Gar- cia and S. Nampalliwar, Astrophys. J.842(2017) 76

work page 2017
[7]

Carullo,et al.Phys

G. Carullo,et al.Phys. Rev. D98(2018) 104020

work page 2018
[8]

M. Isi, M. Giesler, W. M. Farr, M. A. Scheel and S. A. Teukolsky, Phys. Rev. Lett.123(2019) 111102

work page 2019

Show all 73 references

[9]

Abbottet al.[LIGO Scientific and Virgo], Phys

R. Abbottet al.[LIGO Scientific and Virgo], Phys. Rev. D103(2021) 122002

2021
[10]

Kocherlakotaet al.[Event Horizon Telescope], Phys

P. Kocherlakotaet al.[Event Horizon Telescope], Phys. Rev. D103(2021) 104047

2021
[11]

Akiyamaet al.[Event Horizon Telescope], Astrophys

K. Akiyamaet al.[Event Horizon Telescope], Astrophys. J. Lett.930(2022) L17

2022
[12]

J. M. M. Senovilla and D. Garfinkle, Class. Quant. Grav. 32(2015) 124008

2015
[13]

Ayon-Beato and A

E. Ayon-Beato and A. Garcia, Phys. Rev. Lett.80(1998) 5056

1998
[14]

Carballo-Rubio, F

R. Carballo-Rubio, F. Di Filippo, S. Liberati and M. Visser, Phys. Rev. D101(2020) 084047

2020
[15]

Torres, [arXiv:2208.12713 [gr-qc]]

R. Torres, [arXiv:2208.12713 [gr-qc]]
[16]

C. Lan, H. Yang, Y. Guo and Y. G. Miao, Int. J. Theor. Phys.62(2023) 202

2023
[17]

Bueno, P

P. Bueno, P. A. Cano and R. A. Hennigar, Phys. Lett. B 861(2025) 139260

2025
[18]

Alencar, A

G. Alencar, A. Duran-Cabac´ es, D. Rubiera-Garcia and D. S´ aez-Chill´ on G´ omez, Phys. Rev. D111(2025) 104020

2025
[19]

Eichhorn and A

A. Eichhorn and A. Held, JCAP05(2021) 073

2021
[20]

I. Z. Stefanov, S. S. Yazadjiev and G. G. Gyulchev, Phys. Rev. Lett.104(2010) 251103

2010
[21]

Jusufi, Phys

K. Jusufi, Phys. Rev. D101(2020) 084055

2020
[22]

Pedrotti and S

D. Pedrotti and S. Vagnozzi, Phys. Rev. D110(2024) 084075

2024
[23]

R. A. Konoplya and A. Zhidenko, JCAP09(2024) 068

2024
[24]

R. A. Konoplya and A. Zhidenko, Phys. Lett. B861 (2025) 139288

2025
[25]

G. S. Bisnovatyi-Kogan and O. Y. Tsupko, Phys. Rev. D 105(2022) 064040

2022
[26]

Staelens, D

S. Staelens, D. R. Mayerson, F. Bacchini, B. Ripperda and L. K¨ uchler, Phys. Rev. D107(2023) 124026

2023
[27]

Carballo-Rubio and A

R. Carballo-Rubio and A. Eichhorn, Int. J. Mod. Phys. D33(2024) 2441023

2024
[28]

Murk and I

S. Murk and I. Soranidis, Phys. Rev. D110(2024) 044064

2024
[29]

Cardoso, S

V. Cardoso, S. Hopper, C. F. B. Macedo, C. Palenzuela and P. Pani, Phys. Rev. D94(2016) 084031

2016
[30]

Cardoso and P

V. Cardoso and P. Pani, Nature Astron.1(2017) 586

2017
[31]

Cunha, V.P., C

P. Cunha, V.P., C. Herdeiro, E. Radu and N. Sanchis- Gual, Phys. Rev. Lett.130(2023) 061401

2023
[32]

G. A. Marks, S. J. Staelens, T. Evstafyeva and U. Sper- hake, [arXiv:2504.17775 [gr-qc]]
[33]

Simpson and M

A. Simpson and M. Visser, JCAP02(2019) 042

2019
[34]

F. S. N. Lobo, M. E. Rodrigues, M. V. de Sousa Silva, A. Simpson and M. Visser, Phys. Rev. D103(2021) 084052

2021
[35]

Mazza, E

J. Mazza, E. Franzin and S. Liberati, JCAP04(2021) 082

2021
[36]

Franzin, S

E. Franzin, S. Liberati and V. Vellucci, JCAP01(2024) 020

2024
[37]

H. G. Ellis, J. Math. Phys.14(1973) 104

1973
[38]

K. A. Bronnikov and R. K. Walia, Phys. Rev. D105 (2022) 044039

2022
[39]

Maggiore,Gravitational Waves

M. Maggiore,Gravitational Waves. Vol. 2: Astrophysics and Cosmology,Oxford University Press, 2018

2018
[40]

Regge and J

T. Regge and J. A. Wheeler, Phys. Rev.108(1957) 1063

1957
[41]

F. J. Zerilli, Phys. Rev. D2(1970) 2141

1970
[42]

N. Sago, H. Nakano and M. Sasaki, Phys. Rev. D67 (2003) 104017

2003
[43]

Shoshany, J

B. Shoshany, J. Open Source Softw.6(2021) 3416

2021
[44]

Gravitational Waves. Vol. 1: Theory and Experiments,

M. Maggiore,“Gravitational Waves. Vol. 1: Theory and Experiments,”Oxford University Press, 2007

2007
[45]

X. H. Feng and J. Peng, Phys. Rev. D110(2024) 6

2024
[46]

R. A. Konoplya and O. S. Stashko, Phys. Rev. D111 (2025) 104055

2025
[47]

Berti, V

E. Berti, V. Cardoso and A. O. Starinets, Class. Quant. Grav.26(2009) 163001

2009
[48]

R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys.83 (2011) 793

2011
[49]

Pani, Int

P. Pani, Int. J. Mod. Phys. A28(2013) 1340018

2013
[50]

Franchini and S

N. Franchini and S. H. V¨ olkel, [arXiv:2305.01696 [gr-qc]]

arXiv
[51]

Dutta Roy, J

P. Dutta Roy, J. Das and S. Kar, Eur. Phys. J. Plus134 (2019) 571

2019
[52]

Gundlach, R

C. Gundlach, R. H. Price and J. Pullin, Phys. Rev. D49 (1994) 883

1994
[53]

Molina, P

C. Molina, P. Pani, V. Cardoso and L. Gualtieri, Phys. Rev. D81(2010) 124021

2010
[54]

Chavda, M

A. Chavda, M. Lagos and L. Hui, [arXiv:2412.03435 [gr- qc]]

arXiv
[55]

R. A. Konoplya, D. Ovchinnikov and B. Ahmedov, Phys. Rev. D108(2023) 104054

2023
[56]

Kyutoku, H

K. Kyutoku, H. Motohashi and T. Tanaka, Phys. Rev. D 107(2023) 044012

2023
[57]

H. P. Nollert, Class. Quant. Grav.16(1999) R159

1999
[58]

E. W. Leaver, Phys. Rev. D34(1986) 384. 13

1986
[59]

Berti, V

E. Berti, V. Cardoso, J. A. Gonzalez and U. Sperhake, Phys. Rev. D75(2007), 124017

2007
[60]

L. A. H. Mamani, A. D. D. Masa, L. T. Sanches and V. T. Zanchin, Eur. Phys. J. C82(2022) 897

2022
[61]

Chandrasekhar and S

S. Chandrasekhar and S. L. Detweiler, Proc. Roy. Soc. Lond. A344(1975) 441

1975
[62]

Iyer and C

S. Iyer and C. M. Will, Phys. Rev. D35, 3621 (1987)

1987
[63]

S. E. Gralla, D. E. Holz and R. M. Wald, Phys. Rev. D 100(2019) 024018

2019
[64]

Kocherlakota, L

P. Kocherlakota, L. Rezzolla, R. Roy and M. Wielgus, Phys. Rev. D109(2024) 064064

2024
[65]

Chael, M

A. Chael, M. D. Johnson and A. Lupsasca, Astrophys. J. 918(2021) 6

2021
[66]

Cardoso, A

V. Cardoso, A. S. Miranda, E. Berti, H. Witek and V. T. Zanchin, Phys. Rev. D79(2009) 064016

2009
[67]

Pedrotti and M

D. Pedrotti and M. Calz` a, [arXiv:2504.01909 [gr-qc]]

arXiv
[68]

F. H. Vincent, S. E. Gralla, A. Lupsasca and M. Wielgus, Astron. Astrophys.667(2022) A170

2022
[69]

G. J. Olmo, J. L. Rosa, D. Rubiera-Garcia and D. Saez- Chillon Gomez, Class. Quant. Grav.40(2023) 174002

2023
[70]

L. F. D. da Silva, F. S. N. Lobo, G. J. Olmo and D. Rubiera-Garcia, Phys. Rev. D108(2023) 084055

2023
[71]

De Martino, R

I. De Martino, R. Della Monica and D. Rubiera-Garcia, Phys. Rev. D108(2023) 124054

2023
[72]

G. J. Olmo, D. Rubiera-Garcia and D. S. C. G´ omez, Phys. Lett. B829(2022) 137045

2022
[73]

C. Y. Chen and Y. Yokokura, Phys. Rev. D109(2024) 104058

2024

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Addazi,et al.Prog

A. Addazi,et al.Prog. Part. Nucl. Phys.125(2022), 103948

work page 2022

[2] [2]

Alves Batista,et al.Class

R. Alves Batista,et al.Class. Quant. Grav.42(2025) 032001

work page 2025

[3] [3]

Cardoso and P

V. Cardoso and P. Pani, Living Rev. Rel.22(2019) 4

2019

[4] [4]

R. P. Kerr, Phys. Rev. Lett.11(1963) 237

work page 1963

[5] [5]

E. A. Becerra-Vergara, C. R. Arg¨ uelles, A. Krut, J. A. Rueda and R. Ruffini, Mon. Not. Roy. Astron. Soc. 505(2021) L64

work page 2021

[6] [6]

Bambi, A

C. Bambi, A. Cardenas-Avendano, T. Dauser, J. A. Gar- cia and S. Nampalliwar, Astrophys. J.842(2017) 76

work page 2017

[7] [7]

Carullo,et al.Phys

G. Carullo,et al.Phys. Rev. D98(2018) 104020

work page 2018

[8] [8]

M. Isi, M. Giesler, W. M. Farr, M. A. Scheel and S. A. Teukolsky, Phys. Rev. Lett.123(2019) 111102

work page 2019