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REVIEW 2 major objections 5 minor 69 references

On a loop quantum black bounce, scalar waves echo only when the effective potential forms a well between two barriers; regular-black-hole configurations show a single decaying ringdown.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 01:35 UTC pith:4N2DJY4T

load-bearing objection Competent but numerically under-documented QNM/echo catalogue for a new black-bounce spacetime; physics is plausible, echo evidence not yet verified. the 2 major comments →

arxiv 2607.25738 v1 pith:4N2DJY4T submitted 2026-07-28 gr-qc

Echoes and quasinormal modes for static loop quantum black bounces

Huajie Gong , Guoyang Fu , Shulan Li , Jian-Pin Wu , Qin Tan , Qiyuan Pan This is my paper
classification gr-qc PACS 04.70.-s04.30.-w04.60.Pp
keywords loop quantum black bouncequasinormal modesechoesregular black holestraversable wormholesscalar perturbationseffective potentialfinite difference method
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies massless scalar-field ringing on the static loop quantum black bounce (LQBB) spacetime, a two-parameter family that can describe either a regular black hole or a traversable wormhole. Its central claim is that the shape of the effective potential controls the whole late-time waveform: in the regular-black-hole regime the potential is a single barrier, so the perturbation decays monotonically and produces no echoes, while in traversable-wormhole configurations a double-barrier potential with a well produces repeated echoes whose spacing and strength track the width and depth of that well. The paper also computes fundamental quasinormal frequencies with two independent methods, Prony and direct integration, and finds good agreement, with decay slowing as the bounce parameter r_b or the quantum parameter alpha grows in the black-hole case and non-monotonic frequency behaviour in the wormhole case. This gives a concrete scalar-wave diagnostic that distinguishes quantum-corrected regular black holes from traversable wormholes and ties echo observables directly to the horizon-versus-throat structure.

Core claim

For massless scalar perturbations with multipole index l=1 on the static LQBB geometry, the paper finds that the effective potential in the tortoise coordinate r_* has different topologies in the two regimes. In the regular-black-hole (RBH) regime it is a positive single barrier and the time-domain signal is an ordinary damped ringdown with no echo; increasing r_b or alpha makes the ringdown decay more slowly. In the traversable-wormhole regime, for small enough r_b and alpha, the potential develops a well bounded by two barriers and the waveform shows repeated echo pulses; as either parameter grows the well shallows, the echoes weaken and eventually cease. The fundamental quasinormal freque

What carries the argument

The central object is the effective potential V(r) for scalar perturbations, plotted against the tortoise coordinate r_*, defined by dr_*/dr = 1/f(r) with f(r) = 1 - 2M/sqrt(r^2+r_b^2) + alpha^2 M^2/(r^2+r_b^2)^2. Its topology—single positive barrier for regular black holes, double barrier surrounding a potential well for traversable wormholes—determines whether echoes appear. The paper uses a finite-difference time-domain solver to produce waveforms and the Prony and direct-integration methods to extract fundamental complex frequencies; for wormholes it imposes the even-parity throat condition dPhi/dr_* = 0 at r_* = 0.

Load-bearing premise

The central claims assume the finite-difference time-domain integrations are numerically converged on the displayed time windows; the paper fixes only the CFL ratio Delta t / Delta r_* = 1/2 and does not state the individual step sizes or the Gaussian initial-data parameters a and b, so the late-time echoes and small imaginary parts of the wormhole quasinormal frequencies could in principle be discretization artifacts.

What would settle it

Rerun the time-domain integrations with Delta t and Delta r_* halved while keeping Delta t / Delta r_* = 1/2, and also with different Gaussian widths and centres: if the echo amplitudes near 10^-6 to 10^-7 and the tiny imaginary parts such as Im omega approximately -0.002 for alpha = 1.5, r_b = 1.5 do not stabilize, the echo claim is a numerical artifact; comparing with a spectral or frequency-domain computation of the same wormhole modes would settle it independently.

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

If this is right

  • If the central claim is right, one scalar-wave ringdown can label a compact object: repeated late-time pulses imply a traversable wormhole with a potential well, while a clean damped sinusoid points to the regular-black-hole branch (or a wormhole whose well is too shallow to echo).
  • In the wormhole regime the echo spacing decreases as r_b grows at fixed alpha, so a measured echo interval would give a direct estimate of the bounce/throat scale.
  • In the RBH branch the paper predicts longer-lived ringdown as alpha or r_b increases, while in the wormhole branch the fundamental frequency depends non-monotonically on the parameters, so extracting both the real and imaginary parts could help pin the two model parameters.
  • The agreement between Prony and direct-integration results for the fundamental modes supports the claim that the reported complex frequencies are physical rather than artifacts of one numerical scheme.
  • The authors suggest the same analysis can be extended to electromagnetic and gravitational perturbations and to the rotating LQBB spacetime, which is the natural route toward observational tests.

Where Pith is reading between the lines

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

  • The late-time echo amplitudes shown in the paper sit near 10^-6 to 10^-7, a range where a second-order finite-difference scheme can produce spurious oscillations; because no convergence study is given, the quantitative echo amplitudes and spacings should be treated as provisional until reproduced on finer grids and with an independent solver.
  • If the echoes are real, the repetition rate encodes the round-trip travel time across the potential well, so an observed echo train could be inverted to estimate the wormhole throat size and the quantum parameter without a full model fit.
  • The non-monotonic wormhole QNF behaviour hints at mode interactions between the even- and odd-parity sectors; since only even-parity modes are computed, an odd-parity calculation could change the predicted echo pattern.
  • A parameter-free check would be to compare the echo spacing read off the waveform with the round-trip tortoise time 2 * integral of dr/f(r) across the well; agreement would validate the numerical echoes, disagreement would expose them as artifacts.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies massless scalar perturbations (l=1) of the static loop quantum black bounce (LQBB) spacetime. It computes the effective potential, evolves perturbations in the time domain with the finite difference method (FDM), and extracts fundamental quasinormal frequencies (QNFs) using the Prony method and direct integration method (DIM). The central claim is a clean distinction in ringdown behavior: RBH configurations with a single-peak effective potential show no echoes, while selected traversable wormhole configurations with a potential well between two barriers produce clear echoes. The paper also reports parameter trends: in the RBH regime increasing r_b or α slows the decay, while wormhole QNFs are non-monotonic in the model parameters. The model and numerical methods are standard, and the effective potential and boundary conditions are written out explicitly.

Significance. If the numerical results are converged, the paper provides a concrete example in which scalar wave echoes distinguish regular black holes from traversable wormholes in a quantum-gravity-inspired spacetime, with the mechanism traced to the single-barrier vs. double-barrier shape of the effective potential. The manuscript has clear strengths: no quantity is fitted to the target data, the potential and metric are explicit, and two independent QNF extraction methods agree in the non-echo sectors. However, the headline echo claim rests entirely on time-domain waveforms that lack convergence and boundary-independence evidence, and the wormhole modes that would explain the echoes are not corroborated by the Prony method. Thus the primary distinguishing claim is not yet fully supported and requires additional numerical validation.

major comments (2)
  1. [Appendix A, Eqs. (A3)-(A5); Sec. IV, Figs. 6-7] The echo claim hinges on FDM waveforms at amplitudes of order 10^-6 to 10^-9. The manuscript gives only the CFL ratio Δt/Δr_*=1/2 and does not report Δt, Δr_*, the radial grid boundaries, the Gaussian initial-data parameters a and b, or any convergence test. At these late-time amplitudes, numerical reflections from finite boundaries, grid dispersion, or an under-resolved potential can mimic physical echoes. Please provide the actual grid parameters, the boundary treatment, and a convergence study (e.g., halving Δt and Δr_* and comparing waveforms), and show that the echo arrival times and amplitudes are stable and independent of boundary placement.
  2. [Sec. V, Tables II and III] In the echo cases (α=1, r_b=1.9,2.0; α=1.5, r_b=1.5), the Prony column is literally 'echo' — no QNF is extracted — while DIM reports a long-lived complex frequency. Therefore the two-method agreement, which is emphasized as a validation, does not cover the modes relevant to the echo claim. The long-lived wormhole QNFs (e.g., 0.314188 - 0.0069937i and 0.292600 - 0.0022223i) are predicted only by DIM. Please confirm these modes with an independent spectral analysis of the echo train (e.g., FFT or Prony on a window that isolates the echo oscillations), and state a quantitative criterion for classifying a waveform as containing 'echoes' rather than merely extended ringdown.
minor comments (5)
  1. [Sec. II, Eq. (2) and the paragraph after Eq. (3)] The text says the regularization parameter is chosen as r_b = r_m, where r_m is the qOS minimal radius set by α, and then says 'In principle, we treat α and r_b as free parameters.' This is confusing: if r_b=r_m, then r_b and α are not independent. Please clarify whether the present scan treats r_b independently from α and whether all scanned configurations belong to the original LQBB construction or to a generalized family.
  2. [Eq. (A5)] The Gaussian initial-data parameters a and b are never specified. Even if the ringdown is insensitive to them, the echo amplitude and excitation can depend on the initial data; please state the values used and, if possible, show that the echo conclusions are unchanged for different a and b.
  3. [Tables I-III] The claim that Prony and DIM agree with only 'tiny numerical discrepancies' is somewhat overstated. Several rows differ by ~2-4% in Im ω (e.g., Table I, α=1.0, r_b=1.3: -0.0808969 vs -0.0825820; Table II, α=1.0, r_b=8.0: -0.0431197 vs -0.0450417). Please quantify the expected numerical error and state the agreement criterion used.
  4. [Sec. IV] There is no quantitative definition of an 'echo' or of when an echo is 'clear' versus 'not clearly distinguishable.' A simple criterion, such as a minimum amplitude ratio relative to the initial ringdown or a characteristic periodicity matching the well width, would make the classification reproducible.
  5. [Figures 5-7] The figure captions do not state the grid resolution, extraction radius, or the time window after which boundary effects may appear. Adding a vertical marker at the expected cavity round-trip time would help the reader connect the echo spacing to the potential-well width.

Circularity Check

0 steps flagged

No circularity: QNFs and echoes are computed from the LQBB metric without fitting to target data.

full rationale

The derivation chain is self-contained: the LQBB metric is taken from an external reference (Muniz et al.), the effective potential is derived from it, and the time-domain waveforms and QNFs are obtained by solving the wave equation numerically (FDM, Prony, DIM). No parameter is fitted to the target QNFs or echo signals; α and r_b are spacetime parameters from the model. The presence or absence of echoes is read off the computed waveforms and then correlated with the potential shape, not defined into existence. The Prony/DIM agreement is an internal cross-check; the 'echo' entries in Tables II and III merely indicate that Prony did not extract a damped sinusoid from echo-dominated tails, while DIM reports long-lived modes—a methodological limitation, not circularity. Self-citations (e.g., [18], [32], [38], [55]) are background or methodological references and are supported by external citations; none carries the central claim. Missing numerical convergence details (Δt, Δr_*, boundaries, Gaussian parameters) are a reproducibility/robustness concern, not circularity. Therefore no circular step is present.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

No data fitting is used; α and r_b are free model parameters from prior literature. The main hidden free choices are numerical: grid spacings and Gaussian initial-data parameters, which are never quantified. No new physical entities are introduced.

free parameters (4)
  • α (LQG quantum parameter) = not fitted; sampled values with M=1 (e.g., 0–4)
    Model parameter from Ref [25]; central to RBH vs wormhole classification and QNF trends.
  • r_b (bounce parameter) = not fitted; sampled values (e.g., 0–8) with M=1
    Model parameter from Simpson-Visser construction; controls horizon/throat structure and potential shape.
  • Gaussian initial-data parameters a and b = unspecified
    Initial wave packet in Eq. (A5); never assigned numerical values, so exact echo waveforms cannot be reproduced.
  • FDM grid spacings Δt and Δr_* = unspecified; only ratio Δt/Δr_*=1/2
    Numerical resolution is a free choice; without it, convergence of echoes/QNFs cannot be assessed.
axioms (4)
  • domain assumption The LQBB metric (Eqs. 1–2) is a valid spacetime solution supported by nonlinear electrodynamics and a scalar field.
    The paper inherits this from Ref [25] and does not rederive the matter content; if the solution is not physical, all subsequent signals are unphysical.
  • standard math Spherical-harmonic separation of the massless scalar field yields the Schrödinger-like equation with the stated effective potential V(r) (Eq. 7).
    Assumed standard decomposition; paper states result without derivation.
  • domain assumption For wormholes, perturbations decompose into even/odd parity sectors about the symmetric throat, and the even-parity sector is sufficient (Eqs. B5–B6).
    The paper studies only even parity; if odd modes dominated or the symmetry condition is wrong, the QNFs would be incomplete.
  • ad hoc to paper FDM/Prony/DIM numerical schemes converge with unspecified grid spacing and initial Gaussian width.
    The paper specifies only Δt/Δr_*=1/2 and no convergence study, so convergence is an untested assumption.

pith-pipeline@v1.3.0-alltime-deepseek · 14229 in / 16060 out tokens · 154608 ms · 2026-08-01T01:35:44.689030+00:00 · methodology

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read the original abstract

We investigate scalar perturbations of the static loop quantum black bounce (LQBB) spacetime with multipole index $l=1$, focusing on time-domain signals and fundamental quasinormal frequencies (QNFs). The LQBB model provides a unified description of regular black holes (RBHs) and traversable wormholes, governed by the quantum parameter $\alpha$ and the bounce parameter $r_b$. Using the finite difference method, we find no echoes for the displayed RBH configurations with a single-barrier effective potential, whereas clear echoes are produced by the potential well structure in selected traversable wormhole configurations. The QNFs obtained from the Prony method and the direct integration method are in good agreement. In the RBH case, increasing $r_b$ or $\alpha$ leads to a slower decay. In the wormhole case, the QNFs depend non-monotonically on the model parameters, and the emergence of echoes is closely tied to the effective potential profile. These results show that the LQBB spacetime provides a useful framework for studying wave dynamics in RBHs and traversable wormholes, and for clarifying how horizon and throat structures affect ringdown and echoes.

Figures

Figures reproduced from arXiv: 2607.25738 by Guoyang Fu, Huajie Gong, Jian-Pin Wu, Qin Tan, Qiyuan Pan, Shulan Li.

Figure 1
Figure 1. Figure 1: FIG. 1: Parameter space of the LQBB spacetime for [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: , we fix α = 1 and vary rb. The peak of the effective potential gradually decreases as 7 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: , we fix α = 1 (left panel) and α = 1.5 (right panel) to analyze the influence of rb. The choice of these two values of α is motivated by the different ranges of rb required for a wormhole spacetime: in the left panel, rb > 1.83929 is necessary for α = 1, whereas in the right panel, rb > 0 suffices for α = 1.5. This choice ensures the generality of our analysis. In both panels, as rb increases, the effecti… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The effective potential [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The time evolution of RBH for perturbations of the scalar field is shown for different [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The time evolution of the traversable wormhole for perturbations of the scalar field is shown [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The time evolution of the traversable wormhole for perturbations of the scalar field is [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗

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Reference graph

Works this paper leans on

69 extracted references · 54 linked inside Pith

  1. [1]

    Its metric isds 2 =−(1−2M/r+α 2M 2/r4)dt2 +(1−2M/r+α 2M 2/r4)−1dr2 + r2dΩ2, where the parameterαencodes the quantum effects from LQG

    By applying the quantization techniques of LQG to a spherically symmetric spacetime, one obtains a static, spherically symmetric BH solution incorporating loop quantum correc- tions [14, 15]. Its metric isds 2 =−(1−2M/r+α 2M 2/r4)dt2 +(1−2M/r+α 2M 2/r4)−1dr2 + r2dΩ2, where the parameterαencodes the quantum effects from LQG. Although this so- lution is val...

  2. [2]

    The LQBB metric is subsequently derived when the regularization parameter is chosen asr b =r m

    To globally remove the singularity, the model incorporates the Simpson-Visser pre- scription [20] by implementing the substitutionr→ p r2 +r 2 b , with a nonzero regularization parameterr b (also known as the bounce parameter), thereby extending the radial coordinate to the entire real domain. The LQBB metric is subsequently derived when the regularizatio...

  3. [3]

    They are clearly visible in the left panels of Figs

    Echoes are observed for selected configurations in the traversable wormhole regime of this model. They are clearly visible in the left panels of Figs. 6 and 7, consistent with the analysis in Sec. III. 10 α=1.5 α=2.0 α=3.0 α=4.0 0 100 200 300 400 500 600 10-9 10-7 10-5 0.001 0.100 10 t Log|Φ| l=1,r b=1.5 α=0 α=1.0 α=2.0 α=3.0 α=4.0 0 100 200 300 400 500 1...

  4. [4]

    6, asr b increases with fixedα= 1, the time interval between echo signals becomes shorter, and the echo signals gradually weaken and eventually disappear

    In the left panel of Fig. 6, asr b increases with fixedα= 1, the time interval between echo signals becomes shorter, and the echo signals gradually weaken and eventually disappear. This is because the width of the potential well in Fig. 3 becomes smaller, finally forming a single-peaked potential barrier. This conclusion is consistent with that in Ref. [2...

  5. [5]

    12 TABLE I: Fundamental QNFs of the RBH case of the LQBB spacetime forl= 1 andM= 1

    The QNFs from the Prony and DIM methods agree well, with only tiny numerical discrepancies, confirming the reliability and self-consistency of our computations. 12 TABLE I: Fundamental QNFs of the RBH case of the LQBB spacetime forl= 1 andM= 1. Prony DIM α= 1.0 rb = 0 0.299179 - 0.0922678i 0.299179 - 0.0922571i rb = 1.0 0.298410 - 0.0866014i 0.298426 - 0....

  6. [6]

    This indicates that an increase inrb reduces the oscillation frequency and slows the decay of the BH perturbation, leading to a longer-lived ringdown

    For the RBH case (see Table I), with fixed quantum parameterα= 1.0, as the bounce parameterr b increases, the real part of the QNF decreases slightly, while the absolute value of the imaginary part decreases significantly. This indicates that an increase inrb reduces the oscillation frequency and slows the decay of the BH perturbation, leading to a longer...

  7. [7]

    Echoes are identified atr b = 1.9 and 2.0 forα= 1.0, and atr b = 1.5 forα= 1.5

    Tables II and III show that bothr b andαaffect the wormhole QNFs and the appear- ance of echoes. Echoes are identified atr b = 1.9 and 2.0 forα= 1.0, and atr b = 1.5 forα= 1.5. At fixedα, both the real part and the magnitude of the imaginary part exhibit a non-monotonic dependence onr b within the sampled data. For fixed bounce parameterr b, the real part...

  8. [8]

    For the RBH case at fixedr b = 0.5 in Table I, the magnitude of the imaginary part decreases asαincreases, corresponding to a slower decay

    A comparison of Tables I and III reveals distinct decay behaviors with respect toα between the RBH and wormhole cases. For the RBH case at fixedr b = 0.5 in Table I, the magnitude of the imaginary part decreases asαincreases, corresponding to a slower decay. For the wormhole case at fixedr b in Table III, it generally increases withα, corresponding to a f...

    Pith/arXiv arXiv 2016

  9. [9]

    Penrose,Gravitational collapse and space-time singularities,Phys

    R. Penrose,Gravitational collapse and space-time singularities,Phys. Rev. Lett.14(Jan,

  10. [10]

    S. W. Hawking and R. Penrose,The singularities of gravitational collapse and cosmology, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences314 (1970), no. 1519 529–548

    1970

  11. [11]

    Rovelli,Quantum Gravity,Cambridge University Press, Cambridge, UK(2004)

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

    2004

  12. [12]

    Thiemann,Modern canonical quantum general relativity,gr-qc/0110034

    T. Thiemann,Modern canonical quantum general relativity,gr-qc/0110034

    Pith/arXiv arXiv

  13. [13]

    Ashtekar and J

    A. Ashtekar and J. Lewandowski,Background independent quantum gravity: A Status report, Class. Quant. Grav.21(2004) R53, [gr-qc/0404018]

    Pith/arXiv arXiv 2004

  14. [14]

    M. Han, W. Huang, and Y. Ma,Fundamental structure of loop quantum gravity,Int. J. Mod. Phys. D16(2007) 1397–1474, [gr-qc/0509064]

    Pith/arXiv arXiv 2007

  15. [15]

    J. Yang, C. Zhang, and X. Zhang,Alternative k=-1 loop quantum cosmology,Phys. Rev. D 107(2023), no. 4 046012, [arXiv:2212.05748]

    Pith/arXiv arXiv 2023

  16. [16]

    M. Han, H. Liu, D. Qu, F. Vidotto, and C. Zhang,Cosmological dynamics from covariant loop quantum gravity with scalar matter,Phys. Rev. D111(2025), no. 8 086012, [arXiv:2402.07984]

    Pith/arXiv arXiv 2025

  17. [17]

    Zhang, J

    C. Zhang, J. Lewandowski, Y. Ma, and J. Yang,Black holes and covariance in effective quantum gravity: A solution without Cauchy horizons,Phys. Rev. D112(2025), no. 4 044054, [arXiv:2412.02487]

    arXiv 2025

  18. [18]

    J. G. Kelly, R. Santacruz, and E. Wilson-Ewing,Effective loop quantum gravity framework for vacuum spherically symmetric spacetimes,Phys. Rev. D102(2020), no. 10 106024, [arXiv:2006.09302]

    Pith/arXiv arXiv 2020

  19. [19]

    Lewandowski, Y

    J. Lewandowski, Y. Ma, J. Yang, and C. Zhang,Quantum Oppenheimer-Snyder and Swiss Cheese Models,Phys. Rev. Lett.130(2023), no. 10 101501, [arXiv:2210.02253]

    Pith/arXiv arXiv 2023

  20. [20]

    J. Yang, C. Zhang, and Y. Ma,Shadow and stability of quantum-corrected black holes,Eur. 19 Phys. J. C83(2023), no. 7 619, [arXiv:2211.04263]

    Pith/arXiv arXiv 2023

  21. [21]

    Zhang, Y

    C. Zhang, Y. Ma, and J. Yang,Black hole image encoding quantum gravity information, arXiv:2302.02800

    Pith/arXiv arXiv

  22. [22]

    H. Gong, S. Li, D. Zhang, G. Fu, and J.-P. Wu,Quasinormal modes of quantum-corrected black holes,arXiv:2312.17639

    Pith/arXiv arXiv

  23. [23]

    Yang, Y.-P

    S. Yang, Y.-P. Zhang, T. Zhu, L. Zhao, and Y.-X. Liu,Gravitational waveforms from periodic orbits around a quantum-corrected black hole,JCAP01(2025) 091, [arXiv:2407.00283]

    Pith/arXiv arXiv 2025

  24. [24]

    Simpson and M

    A. Simpson and M. Visser,Black-bounce to traversable wormhole,JCAP02(2019) 042, [arXiv:1812.07114]

    Pith/arXiv arXiv 2019

  25. [25]

    M. S. Churilova and Z. Stuchlik,Ringing of the regular black-hole/wormhole transition, Class. Quant. Grav.37(2020), no. 7 075014, [arXiv:1911.11823]

    Pith/arXiv arXiv 2020

  26. [26]

    Y. Yang, D. Liu, Z. Xu, Y. Xing, S. Wu, and Z.-W. Long,Echoes of novel black-bounce spacetimes,Phys. Rev. D104(2021), no. 10 104021, [arXiv:2107.06554]

    Pith/arXiv arXiv 2021

  27. [27]

    S. R. Wu, B. Q. Wang, D. Liu, and Z. W. Long,Echoes of charged black-bounce spacetimes, Eur. Phys. J. C82(2022), no. 11 998, [arXiv:2201.08415]

    Pith/arXiv arXiv 2022

  28. [28]

    Y. Yang, D. Liu, Z. Xu, and Z.-W. Long,Ringing and echoes from black bounces surrounded by the string cloud,Eur. Phys. J. C83(2023), no. 3 217, [arXiv:2210.12641]

    Pith/arXiv arXiv 2023

  29. [29]

    C. R. Muniz, G. Alencar, M. S. Cunha, and G. J. Olmo,Static and stationary black bounces inspired by loop quantum gravity,Phys. Rev. D112(2025), no. 2 024018, [arXiv:2408.08542]

    Pith/arXiv arXiv 2025

  30. [30]

    K.-J. He, H. Ye, X.-X. Zeng, L.-F. Li, and P. Xu,Shadow and accretion disk images of the rotation loop quantum black bounce*,Chin. Phys.49(2025), no. 12 125103, [arXiv:2502.08388]. [27]LIGO Scientific, VIRGO, KAGRACollaboration, R. Abbott et al.,Tests of General Relativity with GWTC-3,arXiv:2112.06861

    Pith/arXiv arXiv 2025

  31. [31]

    Berti et al.,Testing General Relativity with Present and Future Astrophysical Observations,Class

    E. Berti et al.,Testing General Relativity with Present and Future Astrophysical Observations,Class. Quant. Grav.32(2015) 243001, [arXiv:1501.07274]

    Pith/arXiv arXiv 2015

  32. [32]

    Berti, K

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

    Pith/arXiv arXiv 2018

  33. [33]

    Cardoso and P

    V. Cardoso and P. Pani,Testing the nature of dark compact objects: a status report,Living 20 Rev. Rel.22(2019), no. 1 4, [arXiv:1904.05363]

    Pith/arXiv arXiv 2019

  34. [34]

    G. Fu, D. Zhang, P. Liu, X.-M. Kuang, Q. Pan, and J.-P. Wu,Quasinormal modes and Hawking radiation of a charged Weyl black hole,Phys. Rev. D107(2023), no. 4 044049, [arXiv:2207.12927]

    Pith/arXiv arXiv 2023

  35. [35]

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

    Pith/arXiv arXiv 2024

  36. [36]

    Z.-W. Xia, H. Yang, and Y.-G. Miao,Scalar fields around a rotating loop quantum gravity black hole: waveform, quasi-normal modes and superradiance,Class. Quant. Grav.41 (2024), no. 16 165010, [arXiv:2310.00253]

    Pith/arXiv arXiv 2024

  37. [37]

    W.-D. Guo, Q. Tan, and Y.-X. Liu,Quasinormal modes and greybody factor of a Lorentz-violating black hole,JCAP07(2024) 008, [arXiv:2312.16605]

    Pith/arXiv arXiv 2024

  38. [38]

    Zhang, H

    D. Zhang, H. Gong, G. Fu, J.-P. Wu, and Q. Pan,Quasinormal modes of a regular black hole with sub-Planckian curvature,Eur. Phys. J. C84(2024), no. 6 564, [arXiv:2402.15085]

    Pith/arXiv arXiv 2024

  39. [39]

    Z. Song, H. Gong, H.-L. Li, G. Fu, L.-G. Zhu, and J.-P. Wu,Quasinormal modes and ringdown waveform of the Frolov black hole,arXiv:2406.04787

    Pith/arXiv arXiv

  40. [40]

    Yang, Z.-W

    H. Yang, Z.-W. Xia, and Y.-G. Miao,Echoes and quasi-normal modes of perturbations around Schwarzschild traversable wormholes,Eur. Phys. J. C85(2025), no. 7 742, [arXiv:2406.00377]

    Pith/arXiv arXiv 2025

  41. [41]

    Q. Tan, S. Long, W. Deng, and J. Jing,Quasinormal modes and echoes of a double braneworld,JHEP02(2025) 055, [arXiv:2410.06945]

    arXiv 2025

  42. [42]

    Z. Dong, D. Zhang, G. Fu, and J.-P. Wu,Quasinormal modes of a d-dimensional regular black hole featuring an integrable singularity,Eur. Phys. J. C85(2025), no. 2 215, [arXiv:2412.20457]

    Pith/arXiv arXiv 2025

  43. [43]

    W. Deng, S. Long, Q. Tan, Z.-C. Chen, and J. Jing,Scalar-gravitational quasinormal modes and echoes in a five dimensional thick brane,JHEP01(2026) 066, [arXiv:2508.20937]

    Pith/arXiv arXiv 2026

  44. [44]

    Cardoso, S

    V. Cardoso, S. Hopper, C. F. B. Macedo, C. Palenzuela, and P. Pani,Gravitational-wave signatures of exotic compact objects and of quantum corrections at the horizon scale,Phys. Rev. D94(2016), no. 8 084031, [arXiv:1608.08637]

    Pith/arXiv arXiv 2016

  45. [45]

    S. Hui, B. Mu, and P. Wang,Echoes from charged black holes influenced by quintessence, Phys. Dark Univ.43(2024) 101396, [arXiv:2305.11200]. 21

    Pith/arXiv arXiv 2024

  46. [46]

    Zhang, Y

    C. Zhang, Y. Gao, C.-J. Xia, and R. Xu,Rescaling strange-cluster stars and its implications on gravitational-wave echoes,Phys. Rev. D108(2023), no. 6 063002, [arXiv:2305.13323]

    Pith/arXiv arXiv 2023

  47. [47]

    Liu, W.-L

    H. Liu, W.-L. Qian, Y. Liu, J.-P. Wu, B. Wang, and R.-H. Yue,Alternative mechanism for black hole echoes,Phys. Rev. D104(2021), no. 4 044012, [arXiv:2104.11912]

    Pith/arXiv arXiv 2021

  48. [48]

    Abedi, H

    J. Abedi, H. Dykaar, and N. Afshordi,Echoes from the Abyss: Tentative evidence for Planck-scale structure at black hole horizons,Phys. Rev. D96(2017), no. 8 082004, [arXiv:1612.00266]

    Pith/arXiv arXiv 2017

  49. [49]

    R. S. Conklin, B. Holdom, and J. Ren,Gravitational wave echoes through new windows, Phys. Rev. D98(2018), no. 4 044021, [arXiv:1712.06517]

    Pith/arXiv arXiv 2018

  50. [50]

    Cardoso and P

    V. Cardoso and P. Pani,Tests for the existence of black holes through gravitational wave echoes,Nature Astron.1(2017), no. 9 586–591, [arXiv:1709.01525]

    Pith/arXiv arXiv 2017

  51. [51]

    Z. Mark, A. Zimmerman, S. M. Du, and Y. Chen,A recipe for echoes from exotic compact objects,Phys. Rev. D96(2017), no. 8 084002, [arXiv:1706.06155]

    Pith/arXiv arXiv 2017

  52. [52]

    R. A. Konoplya, Z. Stuchl ´ ık, and A. Zhidenko,Echoes of compact objects: new physics near the surface and matter at a distance,Phys. Rev. D99(2019), no. 2 024007, [arXiv:1810.01295]

    Pith/arXiv arXiv 2019

  53. [53]

    L. F. Longo Micchi and C. Chirenti,Spicing up the recipe for echoes from exotic compact objects: orbital differences and corrections in rotating backgrounds,Phys. Rev. D101(2020), no. 8 084010, [arXiv:1912.05419]

    Pith/arXiv arXiv 2020

  54. [54]

    K. A. Bronnikov and R. A. Konoplya,Echoes in brane worlds: ringing at a black hole–wormhole transition,Phys. Rev. D101(2020), no. 6 064004, [arXiv:1912.05315]

    Pith/arXiv arXiv 2020

  55. [55]

    Dutta Roy, S

    P. Dutta Roy, S. Aneesh, and S. Kar,Revisiting a family of wormholes: geometry, matter, scalar quasinormal modes and echoes,Eur. Phys. J. C80(2020), no. 9 850, [arXiv:1910.08746]

    Pith/arXiv arXiv 2020

  56. [56]

    Chowdhury and N

    A. Chowdhury and N. Banerjee,Echoes from a singularity,Phys. Rev. D102(2020), no. 12 124051, [arXiv:2006.16522]

    Pith/arXiv arXiv 2020

  57. [57]

    Chowdhury, S

    A. Chowdhury, S. Devi, and S. Chakrabarti,Naked singularity in 4D Einstein-Gauss-Bonnet novel gravity: Echoes and instability,Phys. Rev. D106(2022), no. 2 024023, [arXiv:2202.13698]

    Pith/arXiv arXiv 2022

  58. [58]

    Zhang, Q

    D. Zhang, Q. Tan, G. Fu, H. Gong, J.-P. Wu, and Q. Pan,Echoes from the Minkowski-core spacetime,Sci. China Phys. Mech. Astron.69(2026), no. 5 250412, [arXiv:2509.23215]. 22

    arXiv 2026

  59. [59]

    Abdalla, C

    E. Abdalla, C. E. Pellicer, J. de Oliveira, and A. B. Pavan,Phase transitions and regions of stability in Reissner-Nordstr¨ om holographic superconductors,Phys. Rev. D82(2010) 124033, [arXiv:1010.2806]

    Pith/arXiv arXiv 2010

  60. [60]

    Zhu, S.-J

    Z. Zhu, S.-J. Zhang, C. E. Pellicer, B. Wang, and E. Abdalla,Stability of Reissner-Nordstr¨ om black hole in de Sitter background under charged scalar perturbation,Phys. Rev. D90 (2014), no. 4 044042, [arXiv:1405.4931]. [Addendum: Phys. Rev. D 90 (2014) 049904 ]

    Pith/arXiv arXiv 2014

  61. [61]

    Lin and W.-L

    K. Lin and W.-L. Qian,Echoes of axial gravitational perturbations in stars of uniform density,Chin. Phys. C47(2023), no. 8 085101, [arXiv:2204.09531]

    Pith/arXiv arXiv 2023

  62. [62]

    Berti, V

    E. Berti, V. Cardoso, J. A. Gonzalez, and U. Sperhake,Mining information from binary black hole mergers: A Comparison of estimation methods for complex exponentials in noise, Phys. Rev. D75(2007) 124017, [gr-qc/0701086]

    Pith/arXiv arXiv 2007

  63. [63]

    R. A. Konoplya and A. Zhidenko,Quasinormal modes of black holes: From astrophysics to string theory,Rev. Mod. Phys.83(2011) 793–836, [arXiv:1102.4014]

    Pith/arXiv arXiv 2011

  64. [64]

    Ferrari, L

    V. Ferrari, L. Gualtieri, and S. Marassi,A New approach to the study of quasi-normal modes of rotating stars,Phys. Rev. D76(2007) 104033, [arXiv:0709.2925]

    Pith/arXiv arXiv 2007

  65. [65]

    P. Pani, V. Cardoso, L. Gualtieri, E. Berti, and A. Ishibashi,Perturbations of slowly rotating black holes: massive vector fields in the Kerr metric,Phys. Rev. D86(2012) 104017, [arXiv:1209.0773]

    Pith/arXiv arXiv 2012

  66. [66]

    Pierini,Quasinormal modes of black holes in Einstein-dilaton Gauss-Bonnet gravity, Rome U.(2023)

    L. Pierini,Quasinormal modes of black holes in Einstein-dilaton Gauss-Bonnet gravity, Rome U.(2023)

    2023

  67. [67]

    Moderski and M

    R. Moderski and M. Rogatko,Late time evolution of a selfinteracting scalar field in the space-time of dilaton black hole,Phys. Rev. D64(2001) 044024, [gr-qc/0105056]

    Pith/arXiv arXiv 2001

  68. [68]

    Moderski and M

    R. Moderski and M. Rogatko,Late time evolution of a charged massless scalar field in the space-time of a dilaton black hole,Phys. Rev. D63(2001) 084014

    2001

  69. [69]

    Moderski and M

    R. Moderski and M. Rogatko,Evolution of a self-interacting scalar field in the spacetime of a higher dimensional black hole,Phys. Rev. D72(2005) 044027, [hep-th/0508175]. 23

    Pith/arXiv arXiv 2005