Nonlinear Ringdown and Mode Coupling

Nonlinear ringdown can generate damped terms at sums and differences of linear QNM frequencies, but a fitted combination frequency is not sufficient identification. Direct second-order sourcing, an evolving mass and spin, spherical–spheroidal leakage, unresolved linear modes, and fitting degeneracy can overlap in the same finite window. A nonlinear assignment is strongest when frequency combination, angular selection, perturbative amplitude scaling, phase relation, resolution convergence, and window stability agree while a matched linear-only control fails.

Helpful background. Einstein Equations supplies perturbative expansion and nonlinear sources; Waves, Green Functions, and Geometric Optics supplies forced response; Black-Hole Perturbation Theory supplies the linear spectrum and second-order response problem; Gravitational Radiation and Waveforms supplies asymptotic strain and harmonic conventions; and Black-Hole Dynamics and Collapse supplies remnant relaxation and absorption.

From first-order parents to quadratic children

Use geometrized units $G=c=1$ for the remnant calculation: $M_f$ is the final mass as a time or length, while $M$ in quoted simulation intervals is the initial total mass. A duration $bM_f$ restores to $bGM_{f,\rm phys}/c^3$; cosmological propagation then multiplies that duration by $1+z$.

In a remnant-aligned inertial Bondi frame, use

$$H=r(h_+-ih_\times) =\sum_{\ell m}H_{\ell m}\,{}_{-2}Y_{\ell m},$$

with unit-normalized harmonics and remnant units $M_f\omega$. The time convention is $e^{-i\omega u}$, so every stable parent has $\operatorname{Im}\omega\lt 0$.

Expand a field schematically about a stationary Kerr background,

$$g=g^{(0)}+\varepsilon h^{(1)}+\varepsilon^2h^{(2)}+O(\varepsilon^3).$$

The perturbation equations have the structure

$$\mathcal L[h^{(1)}]=0, \qquad \mathcal L[h^{(2)}]=\mathcal S^{(2)}[h^{(1)},h^{(1)}],$$

where the same linear response operator acts at second order but is driven by a quadratic source. If two parent terms contain

$$h_a^{(1)}\sim A_a e^{-i\omega_a u+im_a\phi}, \qquad h_b^{(1)}\sim A_b e^{-i\omega_b u+im_b\phi},$$

their products drive

$$\omega_{a+b}=\omega_a+\omega_b, \quad m_{a+b}=m_a+m_b,$$

and products involving the complex conjugate drive

$$\omega_{a-b}=\omega_a-\omega_b^*, \quad m_{a-b}=m_a-m_b.$$

In both cases the imaginary part is $\operatorname{Im}\omega_a+\operatorname{Im}\omega_b\lt 0$, so the child decays. At leading order the two coefficient structures are

$$A_{a+b}=\kappa_{a+b}A_aA_b, \qquad A_{a-b}=\kappa_{a-b}A_aA_b^*,$$

with a coupling coefficient $\kappa$ that depends on background parameters, field and gauge conventions, angular projection, Green-function normalization, and source history. Azimuthal selection follows immediately; further $\ell$ and parity rules must be obtained by projecting the actual second-order source, not by multiplying two strain harmonics naively. The second-order source and sum-frequency construction are set out in Mitman et al. 2023, eqs. (1)–(5), pp. 1–3, PDF.

The driven solution can also contain homogeneous linear-QNM poles and prompt or tail response. A sum frequency is a frequency of the forcing, not a new eigenvalue of $\mathcal L$. If a forcing frequency coincides with a response pole, the usual nonresonant coefficient formula is replaced by a term such as $u e^{-i\omega u}$; its secular factor has to be included explicitly. For dimensional strain coefficients $H$ of length, $\kappa$ has inverse-length units; it becomes dimensionless only after all amplitudes are normalized by the same mass scale. These qualifications prevent an apparent quadratic relation caused merely by inconsistent amplitude units.

What would identify nonlinear content

A credible quadratic assignment needs mutually reinforcing tests.

• Frequency: a free-frequency fit approaches a predicted sum or difference, including the summed damping rate, over more than one window.
• Angular structure: the observed $m$ and allowed projected multipoles match the source selection rules in the declared frame.
• Amplitude and phase: across a controlled family, $|A_{a\circ b}|$ scales as $|A_aA_b|$, and its phase tracks $\arg A_a+\arg A_b+\arg\kappa$ or the corresponding difference.
• Numerical control: the coefficient and residual advantage converge with resolution and extraction, remain above the recovery floor, and survive modest start/end changes.
• Alternative model: a linear-only model with the same number of free parameters performs worse in held-out time or waveforms, not only on the selection interval.

No single item is decisive. Spherical–spheroidal mixing can place a linear frequency from one spheroidal $L$ into another spherical $\ell$ but does not create a sum frequency. Conversely, correlated linear overtones or prompt terms can approximate a sum-frequency contribution on a short interval. A combination frequency with the wrong amplitude scaling or unstable phase is therefore weak evidence.

Direct coupling versus a changing background

Direct quadratic modes are not the only nonlinear mechanism. Horizon absorption changes the background mass and spin, making the instantaneous linear spectrum evolve. A field that was a QNM of the earlier background is not a pure QNM of the later one and must be re-expanded. Sberna and collaborators named the resulting mechanism absorption-induced mode excitation (AIME). Their fully nonlinear demonstration used a spherically symmetric scalar field on a Schwarzschild–anti-de Sitter black hole; they then estimated asymptotically flat Schwarzschild gravitational excitation, while leaving a detailed Kerr calculation open Sberna et al. 2022, §§ III–VI, PDF.

This distinction matters. Direct second-order coupling produces source frequencies at quadratic combinations. AIME is a higher-order, potentially secular effect caused by background evolution and mode reprojection. Over a short window, either can appear as changing amplitudes or extra damped terms. A fixed Kerr sum can still be a useful representation, but usefulness does not by itself determine the mechanism.

Evidence status as of 2026-08-18

Controlled simulations support some quadratic content, while its complete decomposition, generic astrophysical importance, and uniqueness remain unresolved. No universal verdict follows from the current sample.

Evidence or result	Support	Qualification, scope, and limitation	Tension or failure mode
Second-order response structure	Quadratic sources predict sum/difference frequencies, azimuthal combinations, and coefficients proportional to products of parent amplitudes.	The observable second-order field depends on gauge, field reconstruction, regularization, angular projection, and QNM normalization; the linear pole spectrum itself is only part of the retarded response Berti, Cardoso, and Starinets 2009, § 3.2, pp. 18–20, PDF.	Selection rules establish allowed content, not its excitation strength or a complete finite-window expansion.
Quadratic $(4,4)$ evidence in 2023	Across 17 nonprecessing, quasicircular SXS simulations with mass ratios $1$–$8$ and remnant-spin groups near $0.5$ and $0.7$, the fitted $2\omega_{220}$ term scaled approximately as $A_{220}^2$; an equal-parameter linear-overtone control did not show that scaling Mitman et al. 2023, eqs. (2)–(8) and Figs. 1–3, pp. 2–5, PDF.	CCE strain was mapped to a remnant superrest frame; fixed frequencies, $u_1=100M$, start times mainly $15M$–$30M$, and selected $(2,2)$ and $(4,4)$ models were used.	The sample excluded precession and eccentricity, and fixed-frequency fits do not prove a universal nonlinear basis or eliminate every evolving-background alternative.
Multiple combinations in 2025	Two high-accuracy equal-mass aligned-spin simulations showed stable sum and difference candidates in the spherical $(2,2)$ channel alongside linear modes Giesler et al. 2025, §§ IV–V, PDF.	The analysis used CCE, remnant superrest-frame fixing, selected news content, variable projection, mode dropping, two resolutions, and a noise floor near $10^{-8}$.	The two-spin sample is narrow; lower-amplitude terms were noise-sensitive, and unstable early content before roughly $4M$ or $8M$ after peak was not uniquely decomposed.
Absorption-induced excitation in 2022	Fully nonlinear toy-model evolutions demonstrated excitation caused by absorbed flux changing the background; a sudden-change estimate agreed in the nonadiabatic regime Sberna et al. 2022, §§ III–IV, PDF.	The direct demonstration used a scalar field and asymptotically anti-de Sitter boundary conditions; the asymptotically flat gravitational calculation was an estimate for Schwarzschild.	Transfer to generic Kerr mergers, relative importance versus direct second-order coupling, and onset delays remain open.
Linear and fitting alternatives	Linear spheroidal leakage, overtones, prompt response, and a drifting Kerr background can all add structure to a spherical waveform channel.	Their frequencies and scalings differ in principle, but finite windows and correlated columns weaken separation.	A lower residual after adding one nonlinear term can reflect flexibility unless matched controls, stability, and held-out tests are supplied.

One position emphasizes that the 2023 scaling and 2025 stable combination-frequency results constitute positive evidence for quadratic modes in controlled numerical mergers. A qualifying position accepts that evidence while withholding claims of completeness, generic detectability, or unique early-time decomposition; it also keeps AIME and background evolution distinct from direct second-order coupling. These positions differ in evidential ceiling, not in the perturbative algebra.

Reproducibility, selection, and failure cases

A repeatable nonlinear-mode study should publish the following chain.

1. State waveform IDs, binary parameters, resolutions, extraction, BMS frame, fitted radiative field, peak definition, mass scaling, angular normalization, and every preprocessing subtraction.
2. Predict candidate frequencies and $m$ values before looking at their fitted amplitudes. Freeze a selection set or reserve simulations and time intervals for validation.
3. Across a source family, regress $\log|A_{a\circ b}|$ on $\log|A_aA_b|$ and report the slope, uncertainty, phase residual, spin dependence, and points below the recovery floor. The quadratic expectation is slope one in this product variable, or slope two when one parent is repeated.
4. Compare against a linear-only model with the same number of complex coefficients, free-frequency nuisance terms, and spherical–spheroidal leakage. Evaluate on held-out times as well as the fitted window.
5. Repeat at two or more resolutions and over a grid of start and end times. Require coefficient, phase, and frequency stability rather than residual reduction alone.
6. Mark failures explicitly: a scaling exponent inconsistent with the perturbative prediction, forbidden $m$, nonconvergent amplitude, frequency drift, optimizer dependence, disappearance under a frame correction, or reversal under a matched linear control.

This page does not reproduce production numerical evolutions; those algorithms and datasets belong to Computational Relativity. Detector-specific identifiability and empirical nonlinear-ringdown claims belong to Gravitational-Wave Astronomy.

Exercise: check a candidate child mode

Two parent modes have

$$M_f\omega_a=0.50-0.08i, \qquad M_f\omega_b=0.70-0.10i,$$

with $m_a=m_b=2$. Find the sum and conjugate-difference frequencies, their $m$ values, and their decay times. If both parent amplitudes are multiplied by $\lambda$, how should each quadratic child amplitude change at leading order?

Solution

The sum child has

$$M_f\omega_{a+b}=1.20-0.18i, \qquad m_{a+b}=4.$$

The conjugate-difference child has

$$M_f\omega_{a-b} =(0.50-0.08i)-(0.70+0.10i) =-0.20-0.18i, \qquad m_{a-b}=0.$$

Both decay because their imaginary part is negative, and both have $\tau/M_f=1/0.18\simeq5.56$. Since the sum child scales as $A_aA_b$ and the difference child as $A_aA_b^*$, multiplying both parent amplitudes by a real factor $\lambda$ multiplies either child by $\lambda^2$; their phases follow the corresponding sum or difference. Passing the sign, $m$, and scaling checks makes the assignment possible, not conclusive; convergence, window stability, angular projection, and a linear-only control are still required.

What could change the conclusion

The interpretation could change after a new second-order Kerr calculation fixes relevant normalization or selection rules, public numerical benchmarks broaden the binary sample, a credible disconfirmation changes the scaling evidence, or a cited result is corrected. A new comparison must distinguish direct coupling, AIME, and linear alternatives, and identify the date and inputs of each result.

References

• Berti, Emanuele, Vitor Cardoso, and Andrei O. Starinets, “Quasinormal Modes of Black Holes and Black Branes,” Classical and Quantum Gravity 26 (2009) 163001. DOI. Open PDF.
• Giesler, Matthew, Sizheng Ma, Keefe Mitman, Naritaka Oshita, Saul A. Teukolsky, Michael Boyle, Nils Deppe, Lawrence E. Kidder, Jordan Moxon, Kyle C. Nelli, Harald P. Pfeiffer, Mark A. Scheel, William Throwe, and Nils L. Vu, “Overtones and Nonlinearities in Binary Black Hole Ringdowns,” arXiv:2411.11269v2 (2025), dated April 25, 2025. arXiv. Open PDF.
• Mitman, Keefe, Macarena Lagos, Leo C. Stein, Sizheng Ma, Lam Hui, Yanbei Chen, Nils Deppe, François Hébert, Lawrence E. Kidder, Jordan Moxon, Mark A. Scheel, Saul A. Teukolsky, William Throwe, and Nils L. Vu, “Nonlinearities in Black Hole Ringdowns,” Physical Review Letters 130 (2023) 081402. DOI. Open PDF.
• Sberna, Laura, Pablo Bosch, William E. East, Stephen R. Green, and Luis Lehner, “Nonlinear Effects in the Black Hole Ringdown: Absorption-Induced Mode Excitation,” Physical Review D 105 (2022) 064046. DOI. Open PDF.