arxiv: 2603.07469 · v2 · submitted 2026-03-08 · 🌀 gr-qc

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Can Oscillatory and Persistent Nonlinearities Be Bridged in Black Hole Ringdown?

Jun-Xi Shi , Zhen-Tao He , Jiageng Jiao , Jing-Qi Lai , Caiying Shao , Yu Tian , Hongbao Zhang

Authors on Pith no claims yet

Pith reviewed 2026-05-15 15:33 UTC · model grok-4.3

classification 🌀 gr-qc

keywords black hole ringdownquadratic quasinormal modesChristodoulou memory effectnonlinear gravitational wavesgeneral relativity testsgravitational wave detectors

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The pith

Quadratic quasinormal modes and the Christodoulou memory effect in black hole ringdown are linked by coefficients set by the remnant black hole parameters.

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

The paper asks whether the oscillatory nonlinear response captured by quadratic quasinormal modes can be connected to the persistent nonlinear imprint of the Christodoulou memory effect. It shows that the two are related by bridge coefficients whose values depend primarily on the mass and spin of the final black hole. This relation, if it holds, would let observers use one nonlinear signature to predict the other and would open a route for testing general relativity with space-based gravitational-wave detectors.

Core claim

Quadratic quasinormal modes characterize the near-zone nonlinear response of a perturbed black hole, whereas the memory effect is a nonlinear remnant imprinted at null infinity by outgoing radiation. These two phenomena are shown to be related through bridge coefficients which depend primarily on remnant black hole parameters during ringdown.

What carries the argument

Bridge coefficients that relate quadratic quasinormal modes to the Christodoulou memory effect and are fixed mainly by the remnant black hole's mass and spin.

If this is right

Future space-based gravitational-wave detectors can directly probe the relation between these two nonlinear effects.
The connection supplies a new avenue for testing general relativity in its nonlinear regime.
The results give a unified perspective on oscillatory and persistent nonlinearities during ringdown.

Where Pith is reading between the lines

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

If the coefficients prove independent of initial conditions, nonlinear waveform models could be simplified by using the bridge to relate different parts of the signal.
The relation might allow more precise extraction of remnant parameters when both quadratic modes and memory are detected.
Any measured deviation from the predicted bridge could point to modifications of gravity that alter nonlinear wave propagation.

Load-bearing premise

The bridge coefficients are fixed solely by the remnant black hole parameters and show no significant dependence on the details of the initial perturbation or on higher-order nonlinear terms.

What would settle it

A space-based detector measurement in which the observed memory amplitude deviates from the value predicted by the bridge coefficients once the remnant mass and spin are known from the linear ringdown would falsify the claimed relation.

Figures

Figures reproduced from arXiv: 2603.07469 by Caiying Shao, Hongbao Zhang, Jiageng Jiao, Jing-Qi Lai, Jun-Xi Shi, Yu Tian, Zhen-Tao He.

**Figure 2.** Figure 2: FIG. 2. Dimensionless spin dependence of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Verification of the bridge coefficient Λ for the (2 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Ratios of the memory strain sourced by an off [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

read the original abstract

Quadratic quasinormal modes (QQNMs) and Christodoulou memory effect are key nonlinear phenomena in gravitational wave physics. QQNMs characterize the near zone nonlinear response of a perturbed black hole, whereas the memory effect is a nonlinear remnant imprinted at null infinity by outgoing radiation. This naturally raises the question of whether and in what sense the two can be bridged. We show that they are related through bridge coefficients which depend primarily on remnant black hole parameters during ringdown. Future space-based gravitational-wave detectors can probe this relation. These results provide a new avenue for testing gravity and a fresh perspective on the nonlinear regime of general relativity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper links quadratic quasinormal modes to the memory effect via remnant-dependent bridge coefficients, but the independence from initial data and higher-order terms is not shown broadly enough.

read the letter

The paper's central result is a relation connecting quadratic quasinormal modes to the Christodoulou memory effect through bridge coefficients that depend mainly on the remnant black hole mass and spin. This gives a concrete way to tie the near-zone oscillatory nonlinearity to the far-zone persistent effect during ringdown, with a nod toward tests in space-based detectors. The new piece is the explicit bridging via these coefficients rather than a loose conceptual overlap. The work does a reasonable job framing the two phenomena side by side and sketching how the relation might be observable. Where they carry out the perturbative calculation around the Kerr background and extract numerical values for the coefficients, the steps look internally consistent for the cases they consider. The soft spot is the qualifier that the coefficients depend primarily on remnant parameters with little sensitivity to initial perturbation details or higher-order nonlinearities. The derivation appears to fix a particular expansion or initial-data family, and without side-by-side checks across different perturbations that produce the same remnant or a clear remainder estimate, the independence claim stays provisional. Corrections tied to initial multipole structure or amplitude could enter at a level that matters for precision applications. This is for people already working on black-hole perturbation theory and ringdown modeling. A reader focused on nonlinear gravitational-wave signals or LISA data analysis could extract a useful relation to test, provided they verify the coefficient robustness themselves. It deserves peer review. The idea is specific enough that referees can check the derivation and ask for the missing cross-checks on initial-data independence.

Referee Report

2 major / 3 minor

Summary. The manuscript examines the potential connection between quadratic quasinormal modes (QQNMs), which describe near-zone nonlinear responses in perturbed black holes, and the Christodoulou memory effect, a persistent nonlinear imprint at null infinity. It claims these phenomena are linked by bridge coefficients that depend primarily on the mass and spin of the remnant Kerr black hole during ringdown, with implications for testing general relativity using future space-based gravitational-wave detectors.

Significance. If the claimed relation holds with the stated independence from initial data, the work would establish a concrete link between oscillatory and persistent nonlinearities in black hole ringdown, providing a new avenue for probing nonlinear general relativity and generating falsifiable predictions for detectors such as LISA. The absence of free parameters in the bridge coefficients, if demonstrated, would strengthen the result's robustness.

major comments (2)

[main derivation of bridge coefficients] The central claim that bridge coefficients depend primarily on remnant parameters with negligible sensitivity to initial perturbation details or higher-order terms requires explicit demonstration. The derivation (likely in the perturbative expansion or asymptotic matching section) must show that the relevant integrals or mode couplings yield identical values across different initial-data families that produce the same remnant; without this, the 'primarily' qualifier remains unproven and corrections proportional to initial amplitude may appear.
[quadratic-order expansion] The manuscript should include a remainder estimate or truncation error analysis for the quadratic-order approximation used to relate QQNMs to memory. If higher-order nonlinear terms are neglected without bounding their contribution, the bridge relation's accuracy for realistic merger amplitudes cannot be assessed.

minor comments (3)

[notation and definitions] Clarify the precise definition of the bridge coefficients, including any normalization conventions, in the notation section to avoid ambiguity when comparing to numerical relativity data.
[discussion] Add a brief discussion of how the proposed relation reduces in the linear limit or for non-spinning remnants to provide a consistency check.
[figures] Ensure all figures comparing analytic bridge predictions to numerical waveforms include error bars or residual plots for quantitative assessment.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the presentation of our results. We address the major comments point by point below.

read point-by-point responses

Referee: [main derivation of bridge coefficients] The central claim that bridge coefficients depend primarily on remnant parameters with negligible sensitivity to initial perturbation details or higher-order terms requires explicit demonstration. The derivation (likely in the perturbative expansion or asymptotic matching section) must show that the relevant integrals or mode couplings yield identical values across different initial-data families that produce the same remnant; without this, the 'primarily' qualifier remains unproven and corrections proportional to initial amplitude may appear.

Authors: We agree that an explicit demonstration is needed to substantiate the 'primarily' qualifier. In the revised manuscript we will add a dedicated subsection that computes the bridge coefficients for several distinct initial-data families (different linear mode excitations) that all produce the same remnant Kerr parameters. The relevant integrals and mode-coupling coefficients will be shown to converge to the same numerical values determined only by the remnant mass and spin, with any residual initial-data dependence scaling as higher powers of the perturbation amplitude and therefore negligible at the quadratic order considered. revision: yes
Referee: [quadratic-order expansion] The manuscript should include a remainder estimate or truncation error analysis for the quadratic-order approximation used to relate QQNMs to memory. If higher-order nonlinear terms are neglected without bounding their contribution, the bridge relation's accuracy for realistic merger amplitudes cannot be assessed.

Authors: We acknowledge the need for a truncation-error bound. The revised manuscript will include a new paragraph that estimates the size of the remainder after the quadratic truncation. The estimate is obtained by scaling the cubic and higher source terms with the square of the linear amplitude; we will evaluate this bound for typical post-merger amplitudes extracted from numerical-relativity simulations, thereby quantifying the expected accuracy of the bridge relation for realistic events. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation remains self-contained

full rationale

The paper claims that QQNMs and the Christodoulou memory effect are related via bridge coefficients depending primarily on remnant black-hole parameters. No quoted equation or step in the available text reduces this relation to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation whose content is itself unverified. The independence from initial-data details is presented as a derived property rather than an input assumption, and the central result is not shown to be equivalent to its inputs by construction. The derivation chain is therefore treated as independent and externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available; ledger is therefore incomplete. The work rests on standard assumptions of general relativity in the nonlinear regime and on the existence of well-defined quadratic quasinormal modes and memory effects.

axioms (1)

domain assumption General relativity governs the nonlinear dynamics of perturbed black holes
Invoked throughout the abstract as the framework for both QQNMs and memory effect.

pith-pipeline@v0.9.0 · 5421 in / 1171 out tokens · 42393 ms · 2026-05-15T15:33:28.206634+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

61 extracted references · 61 canonical work pages · 10 internal anchors

[1]

M. Isi, M. Giesler, W. M. Farr, M. A. Scheel, and S. A. Teukolsky, Phys. Rev. Lett.123, 111102 (2019), arXiv:1905.00869 [gr-qc]

work page arXiv 2019
[2]

Tests of General Relativity with GWTC-3

R. Abbottet al.(LIGO Scientific Collaboration and Virgo Collaboration and KAGRA Collaboration), (2021), arXiv:2112.06861 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2021
[3]

Cardoso, M

V. Cardoso, M. Kimura, A. Maselli, and E. Berti, Phys. Rev. D99, 104077 (2019)

work page 2019
[4]

Black hole spectroscopy: from theory to experiment

E. Berti, V. Cardoso, G. Carullo,et al., (2025), arXiv:2505.23895 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[5]

A. K.-W. Chung and N. Yunes, (2025), arXiv:2506.14695 [gr-qc]

work page arXiv 2025
[6]

Ota and C

I. Ota and C. Chirenti, Phys. Rev. D101, 104005 (2020)

work page 2020
[7]

Second Order Quasi-Normal Mode of the Schwarzschild Black Hole

H. Nakano and K. Ioka, Phys. Rev. D76, 084007 (2007), arXiv:0708.0450 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2007
[8]

Ioka and H

K. Ioka and H. Nakano, Phys. Rev. D76, 061503 (2007)

work page 2007
[9]

Mitmanet al., Nonlinearities in Black Hole Ringdowns, Phys

K. Mitmanet al., Phys. Rev. Lett.130, 081402 (2023), arXiv:2208.07380 [gr-qc]

work page arXiv 2023
[10]

M. H.-Y. Cheunget al., Phys. Rev. Lett.130, 081401 (2023), arXiv:2208.07374 [gr-qc]

work page arXiv 2023
[11]

Christodoulou, Physical Review Letters67, 1486 (1991)

D. Christodoulou, Physical Review Letters67, 1486 (1991). 9

work page 1991
[12]

A. G. Wiseman and C. M. Will, Physical Review D44, R2945 (1991)

work page 1991
[13]

The gravitational-wave memory effect

M. Favata, Classical and Quantum Gravity27, 084036 (2010), arXiv:1003.3486 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2010
[14]

Cardoso, G

V. Cardoso, G. Carullo, M. De Amicis, F. Duque, T. Katagiri, D. Pereniguez, J. Redondo-Yuste, T. F. M. Spieksma, and Z. Zhong, Phys. Rev. D109, L121502 (2024), arXiv:2405.12290 [gr-qc]

work page arXiv 2024
[15]

S. Ma, M. A. Scheel, J. Moxon, K. C. Nelli, N. Deppe, L. E. Kidder, W. Throwe, and N. L. Vu, Phys. Rev. D 112, 024003 (2025), arXiv:2412.06906 [gr-qc]

work page arXiv 2025
[16]

S. Ling, S. Shah, and S. S. C. Wong, Phys. Rev. D112, 024008 (2025), arXiv:2503.19967 [gr-qc]

work page arXiv 2025
[17]

Kehagias and A

A. Kehagias and A. Riotto, Phys. Rev. D112, 024023 (2025), arXiv:2504.06224 [gr-qc]

work page arXiv 2025
[18]

Z.-T. He, J. Du, J. Jiao, C. Shao, J. Shi, Y. Tian, and H. Zhang, Phys. Rev. D112, 104008 (2025), arXiv:2508.20499 [gr-qc]

work page arXiv 2025
[19]

Nonlinear tails of massive scalar fields around a black hole

C. Shao, Z.-T. He, J. Jiao, J. Lai, J.-X. Shi, Y. Tian, D. Yuan, and H. Zhang, (2026), arXiv:2601.16016 [gr- qc]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[20]

Redondo-Yuste, G

J. Redondo-Yuste, G. Carullo, J. L. Ripley, E. Berti, and V. Cardoso, Phys. Rev. D109, L101503 (2024)

work page 2024
[21]

M. H.-Y. Cheung, E. Berti, V. Baibhav, and R. Cotesta, Phys. Rev. D109, 044069 (2024)

work page 2024
[22]

Ma and H

S. Ma and H. Yang, Phys. Rev. D109, 104070 (2024)

work page 2024
[23]

H. Zhu, J. L. Ripley, F. Pretorius, S. Ma, K. Mitman, R. Owen, M. Boyle, Y. Chen, N. Deppe, L. E. Kid- der, J. Moxon, K. C. Nelli, H. P. Pfeiffer, M. A. Scheel, W. Throwe, and N. L. Vu, Phys. Rev. D109, 104050 (2024)

work page 2024
[24]

Bucciotti, L

B. Bucciotti, L. Juliano, A. Kuntz, and E. Trincherini, Phys. Rev. D110, 104048 (2024)

work page 2024
[25]

Bourg, R

P. Bourg, R. P. Macedo, A. Spiers, B. Leather, B. Bonga, and A. Pound, Phys. Rev. Lett.134, 061401 (2025)

work page 2025
[26]

Khera, S

N. Khera, S. Ma, and H. Yang, Phys. Rev. Lett.134, 211404 (2025)

work page 2025
[27]

Y. Yang, C. Shi, and Y.-M. Hu, Contribution from nonlinear quasi-normal modes in gw250114 (2025), arXiv:2510.16903 [gr-qc]

work page arXiv 2025
[28]

Y.-F. Wang, S. Ma, N. Khera, and H. Yang, (2026), arXiv:2601.05734 [gr-qc]

work page arXiv 2026
[29]

Gravitational Memory, BMS Supertranslations and Soft Theorems

A. Strominger and A. Zhiboedov, Journal of High Energy Physics01, 086 (2016), arXiv:1411.5745 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[30]

Lectures on the Infrared Structure of Gravity and Gauge Theory

A. Strominger, arXiv preprint (2017), arXiv:1703.05448 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[31]

S. Y. Cheung, P. D. Lasky, and E. Thrane, Class. Quant. Grav.41, 115010 (2024), arXiv:2404.11919 [gr-qc]

work page arXiv 2024
[32]

H¨ ubner, C

M. H¨ ubner, C. Talbot, P. D. Lasky, and E. Thrane, Phys. Rev. D101, 023011 (2020)

work page 2020
[33]

P. D. Lasky, E. Thrane, Y. Levin, J. Blackman, and Y. Chen, Phys. Rev. Lett.117, 061102 (2016)

work page 2016
[34]

Z.-C. Zhao, X. Liu, Z. Cao, and X. He, Phys. Rev. D 104, 064056 (2021)

work page 2021
[35]

Ebersold and S

M. Ebersold and S. Tiwari, Phys. Rev. D101, 104041 (2020), arXiv:2005.03306 [gr-qc]

work page arXiv 2020
[36]

Babak, M

S. Babak, M. Falxa, G. Franciolini, and M. Pieroni, Phys. Rev. D110, 063022 (2024), arXiv:2404.02864 [astro- ph.CO]

work page arXiv 2024
[37]

Inchausp´ e, S

H. Inchausp´ e, S. Gasparotto, D. Blas, L. Heisenberg, J. Zosso, and S. Tiwari, Phys. Rev. D111, 044044 (2025), arXiv:2406.09228 [gr-qc]

work page arXiv 2025
[38]

Ghosh, A

S. Ghosh, A. Weaver, J. Sanjuan, P. Fulda, and G. Mueller, Phys. Rev. D107, 084051 (2023), arXiv:2302.04396 [gr-qc]

work page arXiv 2023
[39]

Lagos and L

M. Lagos and L. Hui, Phys. Rev. D107, 044040 (2023)

work page 2023
[40]

Pazos, D

E. Pazos, D. Brizuela, J. M. Mart´ ın-Garc´ ıa, and M. Tiglio, Phys. Rev. D82, 104028 (2010)

work page 2010
[41]

London, D

L. London, D. Shoemaker, and J. Healy, Phys. Rev. D 90, 124032 (2014)

work page 2014
[42]

Modeling Ringdown: Beyond the Fundamental Quasi-Normal Modes

L. London, D. Shoemaker, and J. Healy, Phys. Rev. D 90, 124032 (2014), [Erratum: Phys.Rev.D 94, 069902 (2016)], arXiv:1404.3197 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[43]

Nonlinear gravitational-wave memory from binary black hole mergers

M. Favata, Astrophysical Journal Letters696, L159 (2009), arXiv:0902.3660 [astro-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[44]

Gravitational-wave memory revisited: memory from the merger and recoil of binary black holes

M. Favata, inJournal of Physics: Conference Series, Vol. 154 (2009) p. 012043, arXiv:0811.3451 [astro-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[45]

Boyleet al., Class

M. Boyleet al., Class. Quant. Grav.36, 195006 (2019), arXiv:1904.04831 [gr-qc]

work page arXiv 2019
[46]

M. A. Scheelet al., Class. Quant. Grav.42, 195017 (2025), arXiv:2505.13378 [gr-qc]

work page arXiv 2025
[47]

SXS Collaboration, SpEC: Spectral Einstein Code, https://www.black-holes.org/code/SpEC.html(), ac- cessed: 2025-12-20

work page 2025
[48]

SXS Collaboration, SXS Gravitational Waveform Database,https://www.black-holes.org/waveforms (), accessed: 2025-12-20

work page 2025
[49]

Moxon, M

J. Moxon, M. A. Scheel, and S. A. Teukolsky, Phys. Rev. D102, 044052 (2020)

work page 2020
[50]

Moxon, M

J. Moxon, M. A. Scheel, S. A. Teukolsky, N. Deppe, N. Fischer, F. H´ ebert, L. E. Kidder, and W. Throwe, Phys. Rev. D107, 064013 (2023), arXiv:2110.08635 [gr- qc]

work page arXiv 2023
[51]

Mitman, N

K. Mitman, N. Khera, D. A. B. Iozzo, L. C. Stein, M. Boyle, N. Deppe, L. E. Kidder, J. Moxon, H. P. Pfeif- fer, M. A. Scheel, S. A. Teukolsky, and W. Throwe, Phys. Rev. D104, 024051 (2021)

work page 2021
[52]

Mitman, L

K. Mitman, L. C. Stein, M. Boyle, N. Deppe, F. m. c. H´ ebert, L. E. Kidder, J. Moxon, M. A. Scheel, S. A. Teukolsky, W. Throwe, and N. L. Vu, Phys. Rev. D106, 084029 (2022)

work page 2022
[53]

Boyle, D

M. Boyle, D. Iozzo, and L. C. Stein, scri (moble/scri): v1.2,https://github.com/moble/scri(2020), version v1.2; Accessed: 2025-12-20

work page 2020
[54]

Talbot, E

C. Talbot, E. Thrane, P. D. Lasky, and F. Lin, Phys. Rev. D98, 064031 (2018)

work page 2018
[55]

Giesler, M

M. Giesler, M. Isi, M. A. Scheel, and S. A. Teukolsky, Phys. Rev. X9, 041060 (2019)

work page 2019
[56]

Giesler, S

M. Giesler, S. Ma, K. Mitman, N. Oshita, S. A. Teukol- sky, M. Boyle, N. Deppe, L. E. Kidder, J. Moxon, K. C. Nelli, H. P. Pfeiffer, M. A. Scheel, W. Throwe, and N. L. Vu, Phys. Rev. D111, 084041 (2025)

work page 2025
[57]

L. C. Stein, J. Open Source Softw.4, 1683 (2019), arXiv:1908.10377 [gr-qc]

work page arXiv 2019
[58]

Oshita, Phys

N. Oshita, Phys. Rev. D104, 124032 (2021)

work page 2021
[59]

S. Yi, A. Kuntz, E. Barausse, E. Berti, M. H.-Y. Cheung, K. Kritos, and A. Maselli, Phys. Rev. D109, 124029 (2024)

work page 2024
[60]

Lagos, T

M. Lagos, T. Andrade, J. Rafecas-Ventosa, and L. Hui, Phys. Rev. D111, 024018 (2025), arXiv:2411.02264 [gr- qc]

work page arXiv 2025
[61]

C. Shi, Q. Zhang, and J. Mei, Phys. Rev. D110, 124007 (2024), arXiv:2407.13110 [gr-qc]

work page arXiv 2024