Pith. sign in

REVIEW 3 major objections 4 minor 6 cited by

Plane-wave perturbations yield selection rules for quadratic quasinormal modes.

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 →

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

Second-order gravitational perturbations on plane waves are solved with a GHP master equation and tensor harmonics, yielding quadratic quasinormal mode ratios and selection rules.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection Solid plane-wave perturbative framework with new selection rules; the Kerr lightring interpretation is still a conjecture. the 3 major comments →

arxiv 2509.03598 v1 pith:YUHESTKA submitted 2025-09-03 gr-qc hep-th

Perturbations of Plane Waves and Quadratic Quasinormal Modes on the Lightring

Kwinten Fransen , David Pere\~niguez , Jaime Redondo-Yuste This is my paper
classification gr-qc hep-th MSC 83C3583C5783C60 PACS 04.30.-w04.70.-s
keywords plane wavesPenrose limitquadratic quasinormal modesblack hole lightringsTeukolsky equationGeroch-Held-Penrose formalismsecond-order perturbation theoryKerr ringdown
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 reading

The paper claims that second-order gravitational perturbations around vacuum plane waves are governed by a single sourced master equation, with the source written entirely in terms of a first-order Hertz potential. On homogeneous plane waves—the Penrose limits of equatorial lightrings in Kerr—the mode solutions are eigenfunctions of simple and inverted harmonic oscillators, so the sourced equation reduces to algebra. The authors use this to compute quadratic quasinormal mode (QQNM) excitation ratios for arbitrary parent-mode combinations and uncover emergent selection rules: the daughter overtone numbers are bounded by the sums of the parent overtone numbers in finite excitation channels, with parity preserved. Because the linear plane-wave modes map to eikonal Kerr quasinormal modes, these ratios are proposed as the high-frequency limit of black hole QQNM excitation. The paper also introduces a geodesic, parallel, and transverse (GPT) gauge that makes the ratios invariant under residual gauge freedom and therefore physically meaningful.

Core claim

The paper's central claim is that the second-order Teukolsky equation (3.40), with the explicit source terms (3.41)-(3.42), describes quadratic curvature fluctuations around vacuum plane waves, and that on homogeneous plane waves the resulting QQNM excitation ratios satisfy the selection rules n'_y <= n_a_y + n_b_y in all channels and n'_x <= n_a_x + n_b_x in finite excitation channels, together with the parity rule n'_i mod 2 = n_a_i + n_b_i mod 2. The authors construct the first-order solution from a Hertz potential, reconstruct the metric and frame perturbations in the GPT gauge, and express the second-order source purely in terms of that potential and its derivatives. On homogeneous plan

What carries the argument

The load-bearing object is the Hertz potential Ψ_H, a GHP scalar of weight (-4,0) satisfying the massless wave equation; formulas (3.31)-(3.35) reconstruct from it the full first-order metric, frame, spin-coefficient, connection, and Weyl-scalar perturbations. The second-order source terms S± are quadratic in Ψ_H, and on homogeneous plane waves the modes are ladder representations of the Heisenberg algebra of the simple and inverted harmonic oscillators, so the master equation becomes algebraic frequency-space inversion. The GPT gauge (geodesic, parallel, transverse) fixes the linear frame so that the second-order Weyl scalar ¨Ψ0 and the derived ratios are invariant under residual gauge free

Load-bearing premise

The linear-order Penrose-limit dictionary between plane-wave modes and Kerr lightring quasinormal modes is assumed to extend to second order, so the computed ratios represent black hole QQNMs; the paper explicitly leaves matching Kerr gauge and frame to the GPT gauge and translating the ratio from the lightring to future null infinity to future work.

What would settle it

A direct second-order Kerr perturbation calculation in the eikonal limit, with the frame matched to the GPT gauge and amplitudes transported to future null infinity, should reproduce the plane-wave ratio for fundamental-mode self-coupling, whose leading term is 16i/p_v^2 in the ++ channel; disagreement at that order would falsify the extension. Equivalently, numerical-relativity ringdowns with high azimuthal number could search for excited overtone combinations that violate the selection rule n'_y <= n_a_y + n_b_y.

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

If this is right

  • QQNM excitation ratios can now be computed for arbitrary combinations of linear modes, not just the fundamental-mode special cases, and the accompanying code supplies them.
  • The selection rules imply, for example, that two fundamental modes in a finite excitation channel couple only to a fundamental daughter mode, with no overtone excitation; overtones are generated only when parent modes carry overtones.
  • In the high-frequency limit the leading ratios fall off as powers of the parent frequency, but if one parent frequency stays finite while the other grows, the ratios grow linearly with the large frequency.
  • The GPT gauge and Hertz-potential reconstruction extend Teukolsky-Starobinsky identities to generic plane waves, giving a template for higher-order perturbation theory on algebraically special backgrounds.
  • The frequencies of second-order modes are sums and differences of linear frequencies, and the system is non-resonant at second order; zero-frequency modes are set aside as a separate, memory-like sector.

Where Pith is reading between the lines

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

  • If the Penrose-limit dictionary survives at second order, the selection rules are a testable prediction for Kerr ringdown in the eikonal regime: full second-order Kerr calculations or numerical relativity should show suppressed excitation of overtone combinations that violate n'_y <= n_a_y + n_b_y.
  • The zero-frequency sector likely belongs to nonlinear memory or background renormalization rather than to quasinormal modes; the paper notes these contributions may renormalize the background within the class of spacetimes retaining ∂_v as an isometry.
  • The metric-harmonic method, worked out for symmetric plane waves, is expected to generalize to all homogeneous plane waves, which would make algebraic perturbation theory available in broader plane-wave backgrounds.
  • Third-order perturbation theory may develop driven quasi-resonances because the zero-frequency second-order modes source slow dynamics; the paper hints at this, and it could connect to nonlinear instability or turbulence questions.
Share X Bluesky LinkedIn Reddit HN

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

3 major / 4 minor

Summary. This paper develops second-order gravitational perturbation theory around vacuum plane-wave spacetimes. In the GHP formalism it introduces a geodesic-parallel-transverse (GPT) gauge, derives a sourced second-order Teukolsky-type master equation for Ψ₀ with explicit source terms constructed from a first-order Hertz potential, and establishes Teukolsky–Starobinsky-type identities. For homogeneous plane waves it constructs scalar, vector, and tensor oscillator harmonics, showing that the linearized Einstein equations become algebraic. It then defines quadratic quasinormal-mode (QQNM) excitation ratios for the Weyl scalar Ψ₀, computes them for several mode couplings, extracts selection rules for overtone numbers and parities, and provides a Mathematica code. The stated motivation is that the plane waves arise as Penrose limits of Kerr equatorial lightrings, so that the computed ratios represent the high-frequency limit of black-hole QQNM amplitudes. The paper explicitly acknowledges, however, that the matching of Kerr gauge/frame to the GPT gauge and the translation of ratios from the lightring to null infinity are left to future work.

Significance. If the plane-wave results are taken on their own terms, this is a substantial technical contribution: it provides a complete GHP-based second-order formalism for type-N plane waves, a novel gauge fixing, two independent approaches that cross-check the selection rules, and analytical formulas for excitation ratios together with public code. The derivation of the source term is self-contained and its coordinate form is stated to reduce to earlier literature. The paper also makes a set of concrete, checkable predictions—the selection rules and the scaling of the ratios—which is a strength. The main limitation is that the advertised bridge to Kerr lightring QQNMs is not established: the Penrose limit is taken at linear order, while the nonlinear dictionary is assumed. The paper itself flags the missing steps. Because the plane-wave formalism is sound but the central black-hole interpretation is currently conjectural, the appropriate outcome is a major revision that either supplies the missing second-order dictionary or explicitly repositions the results as plane-wave statements with a tentative lightring application.

major comments (3)
  1. [§6; §5.2 after Eq. (5.18)] The advertised interpretation as high-frequency Kerr QQNM amplitudes is not proven. The Penrose limit is a λ→0 rescaling, and the second-order source (3.40)–(3.42) is derived on the exact type-N plane-wave background. The Kerr second-order source contains background-Weyl terms and products of first-order perturbations whose λ-weighting relative to (˙Ψ₀)² is not analyzed; the paper does not show that taking the Penrose limit commutes with second-order perturbation theory. The Conclusions explicitly defer matching the Kerr gauge/frame to GPT and translating the ratio from the lightring to null infinity. The p′ᵥ=0 case (Sec. 5.1, Eq. (5.12)) is a concrete mismatch: equal-m parents are a legitimate m′=0 QQNM channel in Kerr, but on the plane wave this channel is non-oscillatory and cannot be interpreted as a black-hole QQNM. Hence Eqs. (5.30)–(5.31) and (5.23)–(5.25) are plane-wave results;
  2. [§5.2, Eq. (5.17)] The gravitational source decomposition is postulated, not derived. All QQNM ratios in §5.4 and selection rules in §5.3 inherit this assumption. When the parent pᵥ have opposite signs (or for the +− channel with the same sign), the Gaussian weight is |p′ᵥ|+Δ/2 rather than |p′ᵥ|, so the expansion in n′ₓ is infinite; no completeness or uniqueness statement for the modes (4.6) is provided, and the I_y contour in Eq. (5.5) is not fully specified. Please supply a proof of the expansion, or state it as an explicit assumption and test the convergence of the infinite sums used in the code.
  3. [§4.1, Eq. (4.7)] The normalization formula is inconsistent with the stated normalization property. For nₓ=n_y=0 it gives N=1/π², while Φ(pᵤ,pᵥ,0,0)=N at x=y=0, which is not 1. For nₓ=1 or n_y=1, Γ(0) appears in a denominator, making N=0 rather than giving the natural first-derivative normalization described in the text. Since all ratios (5.18), (5.30)–(5.31), (5.34) are normalization-dependent, this must be corrected and the numerical code must be re-checked against the corrected convention.
minor comments (4)
  1. [References [150]] Reference [150] is given only as a URL placeholder. The paper should provide a stable identifier or arXiv/DOI link to the code repository.
  2. [Appendix A, §4.2.2, §5.3] There are typographical errors: 'oscilator' in Appendix A, 'Relavity' in §4.2.2, and 'anhilate' in §5.3. A careful proofread is needed.
  3. [§5.4.2, Eq. (5.35)] The crossing symmetry (5.35) between x and y is stated without derivation. A short argument, analogous to the one used for the selection rules, would improve transparency.
  4. [Figure 2] The captions of Figure 2 do not specify the normalization convention used for the plotted ratios. Since the ratios are normalization-dependent, each panel should state whether the normalization of Eq. (4.7) is used and how the plotted quantity is defined.

Circularity Check

0 steps flagged

No circular derivation: the second-order source and QQNM ratios are computed from the background and first-order Hertz potential without feeding back the target result. The only self-citation is the linear Penrose-limit/QNM dictionary, which is not load-bearing for the part of the claim presented as proven.

full rationale

The paper's main derivation is self-contained. The second-order Teukolsky equation (3.40) with explicit sources (3.41)-(3.42) is obtained by perturbing the exact curvature wave equation (3.2) on the plane-wave background and evaluating the resulting quadratic terms on the first-order Hertz-potential solution (3.31)-(3.35). The QQNM ratios (5.18) are then obtained by decomposing the source into scalar eigenmodes and inverting the wave operator; the selection rules (5.23)-(5.25) follow from the polynomial structure of the Hermite-mode products and the Gaussian factors. No target Kerr QQNM amplitude is used as input, and no fitted parameter is renamed as a prediction. The one potentially self-citational element is Ref. [81] (by one of the present authors), used to identify plane-wave mode labels with eikonal Kerr QNMs: 'we find that pu = ωℓmn becomes the subleading correction to the frequency of QNMs in the eikonal or high frequency regime [81].' This is an externally published, checkable linear dictionary, and the paper explicitly does not claim its second-order extension is proven: 'the main steps that should be taken to perform a precise match with the black hole quadratic quasinormal modes are: (i) matching the gauge and frame used in describing perturbations of Kerr to the GPT gauge presented here, and (ii) translating the ratio extracted at the lightring to the ratio extracted at future null infinity. We leave this for future work.' The p'_v=0 non-oscillatory channel is also explicitly flagged as not interpretable as a black-hole QQNM. These are open-assumption limitations, not circularity. Score 2 reflects only the presence of a minor, non-load-bearing self-citation in the motivational dictionary.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

No new physical entities are introduced. The GPT gauge is a gauge choice, and the tensor oscillator harmonics are derivatives of the known Heisenberg algebra generators. The central claims rest on standard GR spinor formalism, the Penrose limit construction, and the assumed completeness of the mode decomposition.

free parameters (1)
  • Mode normalization N(pv,nx,ny) = Equation (4.7)
    The normalization of the off-shell modes (4.6) is chosen by hand so that even modes are unity at the lightring. The raw QQNM ratios R depend on this choice (see Eq 5.10); invariant combinations such as (5.21) are constructed to be independent.
axioms (5)
  • standard math Weyl spinor wave equation (3.1) and its projections (3.2),(3.4) are exact on Ricci-flat spacetimes.
    Invoked in Section 3 as the starting point for deriving the first- and second-order Teukolsky equations; from Ref [134].
  • domain assumption The homogeneous plane wave (4.1) with Omega=Lambda is the Penrose limit of the equatorial lightring of Kerr.
    Used in Section 4 to identify Omega and Lambda via Eq (4.2). This is a result from Refs [81,130], not re-derived here.
  • domain assumption The dictionary p_v = m*Omega, n_x = l - |m|, n_y = n maps plane wave modes to eikonal Kerr QNMs.
    Used in Section 5 to interpret QQNM ratios as lightring quantities. This correspondence is established at linear order in Ref [81]; its validity at second order is assumed.
  • standard math A Hertz potential satisfying (3.30) generates the complete first-order solution via (3.31)-(3.35).
    Adaptation of Wald/Chrzanowski reconstruction [138] to type N; used in Section 3.2.
  • ad hoc to paper The second-order source can be uniquely decomposed into eigenfunctions of the wave operator (Eq 5.17).
    Explicitly postulated in Section 5.2 ('we postulate'). The scalar toy model and selection rules provide consistency checks but no proof of completeness is given.

reviewed 2026-08-05 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Perturbations of Plane Waves and Quadratic Quasinormal Modes on the Lightring." pith.science (2026). https://pith.science/paper/YUHESTKA

@misc{pith2026250903598,
  author       = {Pith},
  title        = {Pith review of: Perturbations of Plane Waves and Quadratic Quasinormal Modes on the Lightring},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YUHESTKA}},
  note         = {Machine review of arXiv:2509.03598}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

We study second order gravitational perturbations on plane wave spacetimes from both the metric and curvature perturbation points of view. For the former, we explicitly use the isometries of the background to introduce tensor oscillator harmonics, which render Einstein's equations algebraic around symmetric plane waves. For the latter, we formulate the first and second order Teukolsky equations in a Geroch-Held-Penrose covariant way. Both approaches are useful in their own right, and together with our discussion on gauge freedom, they provide a foundation for the study of higher-order gravitational dynamics around plane wave spacetimes. Taking the perspective that these plane wave spacetimes arise from Penrose limits, we subsequently use these results to explore the nonlinear gravitational dynamics close to black hole lightrings. Specifically, we define and discuss quadratic quasinormal mode ratios, observe that they satisfy emergent selection rules, and make publicly available a code to compute them.

discussion (0)

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

Forward citations

Cited by 6 Pith papers

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

  1. Unifying the Regge-Wheeler-Zerilli and Bardeen-Press-Teukolsky formalisms on spherical backgrounds

    gr-qc 2026-05 unverdicted novelty 7.0

    A self-dual curvature formulation unifies the Regge-Wheeler-Zerilli and Bardeen-Press-Teukolsky equations on spherical backgrounds as components of one tensorial curvature equation.

  2. Gravitational electric-magnetic duality at the light ring and quasinormal mode isospectrality in effective field theories

    gr-qc 2026-05 unverdicted novelty 7.0

    Gravitational electric-magnetic duality at the light ring organizes and preserves quasinormal mode isospectrality in GR and selects duality-invariant higher-derivative corrections in effective field theories.

  3. Quasinormal Modes of pp-Wave Spacetimes and Zero Temperature Dissipation

    gr-qc 2026-04 unverdicted novelty 7.0

    Scalar quasinormal modes on pp-wave spacetimes show zero-temperature dissipation for d >= 3 via an irregular singular point acting as absorber, with exact non-dissipative spectrum for d=2 and gapped modes proven by re...

  4. The Bondi--Sachs gauge, BMS frames, and memory in black hole perturbation theory

    gr-qc 2026-06 unverdicted novelty 6.0

    Introduces a gauge transformation framework for BMS frames in multiscale black hole perturbation theory on Kerr that incorporates memory effects and avoids infrared divergences.

  5. Black Hole Ringdown Nonlinearities in the Large-D Limit

    gr-qc 2026-06 unverdicted novelty 6.0

    In the large-D limit, analytic third-order nonlinear corrections to quasinormal modes improve ringdown modeling accuracy by several orders of magnitude for head-on black hole collisions.

  6. Computing nonlinearity ratios using second order black hole perturbation theory

    gr-qc 2025-11 conditional novelty 6.0

    For the (2,2)×(2,2)→(4,4) channel, the WKB/matched-asymptotics scheme yields nonlinearity ratio 0.164 at infinity and 0.055 at the horizon, matching numerical relativity within its spread.

Reference graph

Works this paper leans on

157 extracted references · 26 canonical work pages · cited by 6 Pith papers · 25 internal anchors

  1. [1]

    LIGO Scientific, Virgo collaboration, Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett.116 (2016) 061102 [1602.03837]

    Pith/arXiv arXiv 2016

  2. [2]

    TianQin collaboration, TianQin: a space-borne gravitational wave detector, Class. Quant. Grav. 33 (2016) 035010 [1512.02076]

    Pith/arXiv arXiv 2016

  3. [3]

    ET collaboration, Science Case for the Einstein Telescope, JCAP 03 (2020) 050 [1912.02622]

    Pith/arXiv arXiv 2020

  4. [4]

    Badurina et al.,AION: An Atom Interferometer Observatory and Network, JCAP 05 (2020) 011 [1911.11755]

    L. Badurina et al.,AION: An Atom Interferometer Observatory and Network, JCAP 05 (2020) 011 [1911.11755]

    Pith/arXiv arXiv 2020

  5. [5]

    LISA collaboration, Astrophysics with the Laser Interferometer Space Antenna, Living Rev. Rel. 26 (2023) 2 [2203.06016]

    Pith/arXiv arXiv 2023

  6. [6]

    Goldberger and I.Z

    W.D. Goldberger and I.Z. Rothstein,An Effective field theory of gravity for extended objects, Phys. Rev. D73 (2006) 104029 [hep-th/0409156]

    Pith/arXiv arXiv 2006

  7. [7]

    Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M.P. Solon and M. Zeng,Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order, Phys. Rev. Lett.122 (2019) 201603 [1901.04424]

    Pith/arXiv arXiv 2019

  8. [8]

    Porto,The effective field theorist’s approach to gravitational dynamics, Phys

    R.A. Porto,The effective field theorist’s approach to gravitational dynamics, Phys. Rept. 633 (2016) 1 [1601.04914]

    Pith/arXiv arXiv 2016

  9. [9]

    Blanchet,Post-Newtonian Theory for Gravitational Waves, Living Rev

    L. Blanchet,Post-Newtonian Theory for Gravitational Waves, Living Rev. Rel.17 (2014) 2 [1310.1528]

    Pith/arXiv arXiv 2014

  10. [10]

    Buonanno, G.B

    A. Buonanno, G.B. Cook and F. Pretorius,Inspiral, merger and ring-down of equal-mass black-hole binaries, Phys. Rev. D75 (2007) 124018 [gr-qc/0610122]

    Pith/arXiv arXiv 2007

  11. [11]

    Cardoso, A.S

    V. Cardoso, A.S. Miranda, E. Berti, H. Witek and V.T. Zanchin,Geodesic stability, Lyapunov exponents and quasinormal modes, Phys. Rev. D79 (2009) 064016 [0812.1806]

    Pith/arXiv arXiv 2009

  12. [12]

    Barack and A

    L. Barack and A. Pound,Self-force and radiation reaction in general relativity, Rept. Prog. Phys. 82 (2019) 016904 [1805.10385]. – 38 –

    Pith/arXiv arXiv 2019

  13. [13]

    Pound and B

    A. Pound and B. Wardell,Black hole perturbation theory and gravitational self-force, 2101.04592

    Pith/arXiv arXiv

  14. [14]

    Berti et al.,Black hole spectroscopy: from theory to experiment, 2505.23895

    E. Berti et al.,Black hole spectroscopy: from theory to experiment, 2505.23895

    Pith/arXiv arXiv

  15. [15]

    Baumgarte and S.L

    T.W. Baumgarte and S.L. Shapiro,Numerical relativity and compact binaries, Phys. Rept. 376 (2003) 41 [gr-qc/0211028]

    Pith/arXiv arXiv 2003

  16. [16]

    S. Husa, S. Khan, M. Hannam, M. Pürrer, F. Ohme, X. Jiménez Forteza et al., Frequency-domain gravitational waves from nonprecessing black-hole binaries. I. New numerical waveforms and anatomy of the signal, Phys. Rev. D93 (2016) 044006 [1508.07250]

    Pith/arXiv arXiv 2016

  17. [17]

    Blackman, S.E

    J. Blackman, S.E. Field, M.A. Scheel, C.R. Galley, C.D. Ott, M. Boyle et al.,Numerical relativity waveform surrogate model for generically precessing binary black hole mergers, Phys. Rev. D96 (2017) 024058 [1705.07089]

    Pith/arXiv arXiv 2017

  18. [18]

    Rashti, R

    A. Rashti, R. Gamba, K. Chandra, D. Radice, B. Daszuta, W. Cook et al.,Binary black hole waveforms from high-resolution gr-athena++ simulations, Phys. Rev. D111 (2025) 104078 [2411.11989]

    Pith/arXiv arXiv 2025

  19. [19]

    Elvang and Y.-t

    H. Elvang and Y.-t. Huang,Scattering Amplitudes in Gauge Theory and Gravity, Cambridge University Press (4, 2015)

    2015

  20. [20]

    Travaglini et al.,The SAGEX review on scattering amplitudes, J

    G. Travaglini et al.,The SAGEX review on scattering amplitudes, J. Phys. A55 (2022) 443001 [2203.13011]

    Pith/arXiv arXiv 2022

  21. [21]

    Liu,Scattering in anti-de Sitter space and operator product expansion, Phys

    H. Liu,Scattering in anti-de Sitter space and operator product expansion, Phys. Rev. D60 (1999) 106005 [hep-th/9811152]

    Pith/arXiv arXiv 1999

  22. [22]

    Aharony, L.F

    O. Aharony, L.F. Alday, A. Bissi and E. Perlmutter,Loops in AdS from Conformal Field Theory, JHEP 07 (2017) 036 [1612.03891]

    Pith/arXiv arXiv 2017

  23. [23]

    Rastelli and X

    L. Rastelli and X. Zhou,How to Succeed at Holographic Correlators Without Really Trying, JHEP 04 (2018) 014 [1710.05923]

    Pith/arXiv arXiv 2018

  24. [24]

    Giombi, C

    S. Giombi, C. Sleight and M. Taronna,Spinning AdS Loop Diagrams: Two Point Functions, JHEP 06 (2018) 030 [1708.08404]

    Pith/arXiv arXiv 2018

  25. [25]

    Carmi, L

    D. Carmi, L. Di Pietro and S. Komatsu,A Study of Quantum Field Theories in AdS at Finite Coupling, JHEP 01 (2019) 200 [1810.04185]

    Pith/arXiv arXiv 2019

  26. [26]

    Jepsen and S

    C.B. Jepsen and S. Parikh,Propagator identities, holographic conformal blocks, and higher-point AdS diagrams, JHEP 10 (2019) 268 [1906.08405]

    Pith/arXiv arXiv 2019

  27. [27]

    Eberhardt, S

    L. Eberhardt, S. Komatsu and S. Mizera,Scattering equations in AdS: scalar correlators in arbitrary dimensions, JHEP 11 (2020) 158 [2007.06574]

    Pith/arXiv arXiv 2020

  28. [28]

    Rostworowski,Towards a theory of nonlinear gravitational waves: A systematic approach to nonlinear gravitational perturbations in the vacuum, Phys

    A. Rostworowski,Towards a theory of nonlinear gravitational waves: A systematic approach to nonlinear gravitational perturbations in the vacuum, Phys. Rev. D96 (2017) 124026 [1705.02258]

    Pith/arXiv arXiv 2017

  29. [29]

    Rostworowski,Higher order perturbations of anti–de Sitter space and time-periodic solutions of vacuum Einstein equations, Phys

    A. Rostworowski,Higher order perturbations of anti–de Sitter space and time-periodic solutions of vacuum Einstein equations, Phys. Rev. D95 (2017) 124043 [1701.07804]

    Pith/arXiv arXiv 2017

  30. [30]

    Dias and J.E

    O.J.C. Dias and J.E. Santos,AdS nonlinear instability: moving beyond spherical symmetry, Class. Quant. Grav.33 (2016) 23LT01 [1602.03890]

    Pith/arXiv arXiv 2016

  31. [31]

    Dias and J.E

    Ó.J.C. Dias and J.E. Santos,AdS nonlinear instability: breaking spherical and axial symmetries, Class. Quant. Grav.35 (2018) 185006 [1705.03065]. – 39 –

    Pith/arXiv arXiv 2018

  32. [32]

    Choptuik, Ó.J.C

    M.W. Choptuik, Ó.J.C. Dias, J.E. Santos and B. Way,Collapse and Nonlinear Instability of AdS Space with Angular Momentum, Phys. Rev. Lett.119 (2017) 191104 [1706.06101]

    Pith/arXiv arXiv 2017

  33. [33]

    Arkani-Hamed and J

    N. Arkani-Hamed and J. Maldacena,Cosmological Collider Physics, 1503.08043

    Pith/arXiv arXiv

  34. [34]

    Benincasa,Amplitudes meet Cosmology: A (Scalar) Primer, 2203.15330

    P. Benincasa,Amplitudes meet Cosmology: A (Scalar) Primer, 2203.15330

    Pith/arXiv arXiv

  35. [35]

    Cahen and N

    M. Cahen and N. Wallach,Lorentzian symmetric spaces, Bulletin of the American Mathematical Society 76 (1970) 585

    1970

  36. [36]

    Gibbons,Quantized Fields Propagating in Plane Wave Space-Times, Commun

    G.W. Gibbons,Quantized Fields Propagating in Plane Wave Space-Times, Commun. Math. Phys. 45 (1975) 191

    1975

  37. [37]

    Mason,On ward’s integral formula for the wave equation in plane wave space-times, Twistor Newsletter 28 (1989) 17

    L. Mason,On ward’s integral formula for the wave equation in plane wave space-times, Twistor Newsletter 28 (1989) 17

    1989

  38. [38]

    Horowitz and A.R

    G.T. Horowitz and A.R. Steif,Strings in Strong Gravitational Fields, Phys. Rev. D42 (1990) 1950

    1990

  39. [39]

    Gleiser, C.O

    R.J. Gleiser, C.O. Nicasio, R.H. Price and J. Pullin,Second order perturbations of a Schwarzschild black hole, Class. Quant. Grav.13 (1996) L117 [gr-qc/9510049]

    Pith/arXiv arXiv 1996

  40. [40]

    Gleiser, C.O

    R.J. Gleiser, C.O. Nicasio, R.H. Price and J. Pullin,Gravitational radiation from Schwarzschild black holes: The Second order perturbation formalism, Phys. Rept. 325 (2000) 41 [gr-qc/9807077]

    Pith/arXiv arXiv 2000

  41. [41]

    Nicasio, R.J

    C.O. Nicasio, R.J. Gleiser, R.H. Price and J. Pullin,The Collision of boosted black holes: Second order close limit calculations, Phys. Rev. D59 (1999) 044024 [gr-qc/9802063]

    Pith/arXiv arXiv 1999

  42. [42]

    Campanelli and C.O

    M. Campanelli and C.O. Lousto,Second order gauge invariant gravitational perturbations of a Kerr black hole, Phys. Rev. D59 (1999) 124022 [gr-qc/9811019]

    Pith/arXiv arXiv 1999

  43. [43]

    Abramo and F

    L.R.W. Abramo and F. Finelli,Back reaction of gravitational radiation on the Schwarzschild black hole, Gen. Rel. Grav.33 (2001) 339 [gr-qc/9907102]

    Pith/arXiv arXiv 2001

  44. [44]

    Ioka and H

    K. Ioka and H. Nakano,Second and higher-order quasi-normal modes in binary black hole mergers, Phys. Rev. D76 (2007) 061503 [0704.3467]

    Pith/arXiv arXiv 2007

  45. [45]

    Nakano and K

    H. Nakano and K. Ioka,Second Order Quasi-Normal Mode of the Schwarzschild Black Hole, Phys. Rev. D76 (2007) 084007 [0708.0450]

    Pith/arXiv arXiv 2007

  46. [46]

    Okuzumi, K

    S. Okuzumi, K. Ioka and M.-a. Sakagami,Possible Discovery of Nonlinear Tail and Quasinormal Modes in Black Hole Ringdown, Phys. Rev. D77 (2008) 124018 [0803.0501]

    Pith/arXiv arXiv 2008

  47. [47]

    Brizuela, J.M

    D. Brizuela, J.M. Martin-Garcia and M. Tiglio,A Complete gauge-invariant formalism for arbitrary second-order perturbations of a Schwarzschild black hole, Phys. Rev. D80 (2009) 024021 [0903.1134]

    Pith/arXiv arXiv 2009

  48. [48]

    Pazos, D

    E. Pazos, D. Brizuela, J.M. Martin-Garcia and M. Tiglio,Mode coupling of Schwarzschild perturbations: Ringdown frequencies, Phys. Rev. D82 (2010) 104028 [1009.4665]

    Pith/arXiv arXiv 2010

  49. [49]

    Loutrel, J.L

    N. Loutrel, J.L. Ripley, E. Giorgi and F. Pretorius,Second Order Perturbations of Kerr Black Holes: Reconstruction of the Metric, Phys. Rev. D103 (2021) 104017 [2008.11770]

    Pith/arXiv arXiv 2021

  50. [50]

    Ripley, N

    J.L. Ripley, N. Loutrel, E. Giorgi and F. Pretorius,Numerical computation of second order vacuum perturbations of Kerr black holes, Phys. Rev. D103 (2021) 104018 [2010.00162]

    Pith/arXiv arXiv 2021

  51. [51]

    Wardell, A

    B. Wardell, A. Pound, N. Warburton, J. Miller, L. Durkan and A. Le Tiec,Gravitational Waveforms for Compact Binaries from Second-Order Self-Force Theory, Phys. Rev. Lett. 130 (2023) 241402 [2112.12265]. – 40 –

    Pith/arXiv arXiv 2023

  52. [52]

    Albertini, A

    A. Albertini, A. Nagar, A. Pound, N. Warburton, B. Wardell, L. Durkan et al.,Comparing second-order gravitational self-force, numerical relativity, and effective one body waveforms from inspiralling, quasicircular, and nonspinning black hole binaries, Phys. Rev. D106 (2022) 084061 [2208.01049]

    Pith/arXiv arXiv 2022

  53. [53]

    Spiers, A

    A. Spiers, A. Pound and J. Moxon,Second-order Teukolsky formalism in Kerr spacetime: Formulation and nonlinear source, Phys. Rev. D108 (2023) 064002 [2305.19332]

    Pith/arXiv arXiv 2023

  54. [54]

    Sberna, P

    L. Sberna, P. Bosch, W.E. East, S.R. Green and L. Lehner,Nonlinear effects in the black hole ringdown: Absorption-induced mode excitation, Phys. Rev. D105 (2022) 064046 [2112.11168]

    Pith/arXiv arXiv 2022

  55. [55]

    Redondo-Yuste, D

    J. Redondo-Yuste, D. Pereñiguez and V. Cardoso,Ringdown of a dynamical spacetime, Phys. Rev. D109 (2024) 044048 [2312.04633]

    Pith/arXiv arXiv 2024

  56. [56]

    T. May, S. Ma, J.L. Ripley and W.E. East,Nonlinear effect of absorption on the ringdown of a spinning black hole, Phys. Rev. D110 (2024) 084034 [2405.18303]

    Pith/arXiv arXiv 2024

  57. [57]

    Zhu et al.,Imprints of changing mass and spin on black hole ringdown, Phys

    H. Zhu et al.,Imprints of changing mass and spin on black hole ringdown, Phys. Rev. D 110 (2024) 124028 [2404.12424]

    Pith/arXiv arXiv 2024

  58. [58]

    Cheung et al.,Nonlinear Effects in Black Hole Ringdown, Phys

    M.H.-Y. Cheung et al.,Nonlinear Effects in Black Hole Ringdown, Phys. Rev. Lett.130 (2023) 081401 [2208.07374]

    Pith/arXiv arXiv 2023

  59. [59]

    Mitman et al.,Nonlinearities in Black Hole Ringdowns, Phys

    K. Mitman et al.,Nonlinearities in Black Hole Ringdowns, Phys. Rev. Lett.130 (2023) 081402 [2208.07380]

    Pith/arXiv arXiv 2023

  60. [60]

    Redondo-Yuste, G

    J. Redondo-Yuste, G. Carullo, J.L. Ripley, E. Berti and V. Cardoso,Spin dependence of black hole ringdown nonlinearities, Phys. Rev. D109 (2024) L101503 [2308.14796]

    Pith/arXiv arXiv 2024

  61. [61]

    Perrone, T

    D. Perrone, T. Barreira, A. Kehagias and A. Riotto,Non-linear black hole ringdowns: An analytical approach, Nucl. Phys. B 999 (2024) 116432 [2308.15886]

    Pith/arXiv arXiv 2024

  62. [62]

    Bucciotti, L

    B. Bucciotti, L. Juliano, A. Kuntz and E. Trincherini,Amplitudes and polarizations of quadratic quasi-normal modes for a Schwarzschild black hole, JHEP 09 (2024) 119 [2406.14611]

    Pith/arXiv arXiv 2024

  63. [63]

    Bucciotti, L

    B. Bucciotti, L. Juliano, A. Kuntz and E. Trincherini,Quadratic quasinormal modes of a Schwarzschild black hole, Phys. Rev. D110 (2024) 104048 [2405.06012]

    Pith/arXiv arXiv 2024

  64. [64]

    Bourg, R

    P. Bourg, R. Panosso Macedo, A. Spiers, B. Leather, B. Bonga and A. Pound,Quadratic Quasinormal Mode Dependence on Linear Mode Parity, Phys. Rev. Lett.134 (2025) 061401 [2405.10270]

    Pith/arXiv arXiv 2025

  65. [65]

    Bourg, R

    P. Bourg, R. Panosso Macedo, A. Spiers, B. Leather, B. Béatrice and A. Pound,Quadratic quasinormal modes at null infinity on a Schwarzschild spacetime, 2503.07432

    Pith/arXiv arXiv

  66. [66]

    Zhu et al.,Nonlinear effects in black hole ringdown from scattering experiments: Spin and initial data dependence of quadratic mode coupling, Phys

    H. Zhu et al.,Nonlinear effects in black hole ringdown from scattering experiments: Spin and initial data dependence of quadratic mode coupling, Phys. Rev. D109 (2024) 104050 [2401.00805]

    Pith/arXiv arXiv 2024

  67. [67]

    Bucciotti, V

    B. Bucciotti, V. Cardoso, A. Kuntz, D. Pereñiguez and J. Redondo-Yuste,Ringdown nonlinearities in the eikonal regime, Phys. Rev. D111 (2025) L081502 [2501.17950]

    Pith/arXiv arXiv 2025

  68. [68]

    Ma and H

    S. Ma and H. Yang,Excitation of quadratic quasinormal modes for Kerr black holes, Phys. Rev. D 109 (2024) 104070 [2401.15516]

    Pith/arXiv arXiv 2024

  69. [69]

    Khera, S

    N. Khera, S. Ma and H. Yang,Quadratic Mode Couplings in Rotating Black Holes and Their Detectability, Phys. Rev. Lett.134 (2025) 211404 [2410.14529]. – 41 –

    Pith/arXiv arXiv 2025

  70. [70]

    Press,Long Wave Trains of Gravitational Waves from a Vibrating Black Hole, Astrophys

    W.H. Press,Long Wave Trains of Gravitational Waves from a Vibrating Black Hole, Astrophys. J. Lett.170 (1971) L105

    1971

  71. [71]

    vibrations

    Goebel, C. J.,Comments on the “vibrations” of a Black Hole., Astrophys. J. Lett.172 (1972) L95

    1972

  72. [72]

    Mashhoon,Stability of charged rotating black holes in the eikonal approximation, Phys

    B. Mashhoon,Stability of charged rotating black holes in the eikonal approximation, Phys. Rev. D 31 (1985) 290

    1985

  73. [73]

    Schutz and C.M

    B.F. Schutz and C.M. Will,BLACK HOLE NORMAL MODES: A SEMIANALYTIC APPROACH, Astrophys. J. Lett.291 (1985) L33

    1985

  74. [74]

    Nollert,TOPICAL REVIEW: Quasinormal modes: the characteristic ‘sound’ of black holes and neutron stars, Class

    H.-P. Nollert,TOPICAL REVIEW: Quasinormal modes: the characteristic ‘sound’ of black holes and neutron stars, Class. Quant. Grav.16 (1999) R159

    1999

  75. [75]

    Event Horizon Telescope collaboration, First Sagittarius A* Event Horizon Telescope Results. VI. Testing the Black Hole Metric, Astrophys. J. Lett.930 (2022) L17 [2311.09484]

    Pith/arXiv arXiv 2022

  76. [76]

    Chael, M.D

    A. Chael, M.D. Johnson and A. Lupsasca,Observing the Inner Shadow of a Black Hole: A Direct View of the Event Horizon, Astrophys. J. 918 (2021) 6 [2106.00683]

    Pith/arXiv arXiv 2021

  77. [77]

    Johnson et al.,The Black Hole Explorer: motivation and vision, Proc

    M.D. Johnson et al.,The Black Hole Explorer: motivation and vision, Proc. SPIE Int. Soc. Opt. Eng. 13092 (2024) 130922D [2406.12917]

    Pith/arXiv arXiv 2024

  78. [78]

    Lupsasca, A

    A. Lupsasca, A. Cárdenas-Avendaño, D.C.M. Palumbo, M.D. Johnson, S.E. Gralla, D.P. Marrone et al.,The Black Hole Explorer: photon ring science, detection, and shape measurement, Proc. SPIE Int. Soc. Opt. Eng.13092 (2024) 130926Q [2406.09498]

    Pith/arXiv arXiv 2024

  79. [79]

    Davis, R

    M. Davis, R. Ruffini, W.H. Press and R.H. Price,Gravitational radiation from a particle falling radially into a schwarzschild black hole, Phys. Rev. Lett.27 (1971) 1466

    1971

  80. [80]

    Cardoso, F

    V. Cardoso, F. Duque and G. Khanna,Gravitational tuning forks and hierarchical triple systems, Phys. Rev. D103 (2021) L081501 [2101.01186]

    Pith/arXiv arXiv 2021

Showing first 80 references.

This paper was first reviewed by deepseek-v4-flash on August 5, 2026.