Nonlinear Merger and Ringdown Onset

There is no known universal, invariant instant at which a merger spacetime becomes “linear.” Ringdown onset is instead a controlled-model question: choose an observable, a candidate Kerr background, a norm and tolerance, and a family of perturbative models; then ask when the residual and inferred remnant parameters become stable under changes of start time and model complexity. A waveform fit alone can be accurate without proving that the full strong-field spacetime is perturbatively close to Kerr.

Helpful background. Quasi-local remnant geometry is developed in Black-Hole Geometry and Mechanics; mode spectra and Green-function response belong to Black-Hole Perturbation Theory; gauge and extraction variables come from Formulations of General Relativity; convergence and uncertainty belong to Computational Relativity; and the preceding binary dynamics is organized in Relativistic Two-Body Dynamics.

What an onset criterion must specify

Let $d(t)$ denote a versioned waveform or horizon-data product and $m(t;\vartheta,t_0)$ a linear perturbative model applied for $t\ge t_0$. A useful criterion does not merely minimize a residual once. It requires a plateau in inferred remnant parameters, residuals consistent with the data error, and robustness under changing $t_0$, included angular modes, overtone count, and observable. Schematically,

$$R(t_0)=\frac{\lVert d-m(\hat\vartheta(t_0),t_0)\rVert_W} {\lVert d\rVert_W},$$

where the window, weighting $W$, alignment, and norm are part of the definition. A small $R$ is not decisive if extra damped sinusoids can absorb prompt nonlinear structure or if the fit parameters drift strongly with $t_0$.

Horizon diagnostics pose the parallel question locally. One may track shear, curvature invariants, or multipoles and compare their decay with perturbations of the independently measured remnant. Mourier and collaborators demonstrate that horizon multipoles can exhibit quasinormal behavior after merger, but the interpretation depends on the observable and fitting interval Mourier et al. 2021, §§ II–V. Waveform studies show that overtones can fit numerical strain from very early times Giesler et al. 2019, pp. 1–5 and Figs. 1–3; this is evidence for model adequacy in that data set, not a theorem about pointwise spacetime linearity.

Comparing ringdown-onset diagnostics

The evidence cutoff for this assessment is 2026-08-12. The following matrix separates claims that are often conflated.

Diagnostic	Controlled evidence	Main limitation	Stronger test
Fundamental-mode waveform fit	Late-time parameters stabilize and residuals approach numerical error	Start time, frame, extraction, and mode mixing affect the result	Repeat across resolutions, extraction methods, modes, and frames
Many-overtone fit	Can represent earlier post-peak waveforms in selected simulations	Flexible damped sums can fit nonmodal or nonlinear content	Out-of-sample prediction and stable amplitudes under model changes
Horizon multipole decay	Directly probes the remnant’s evolving geometry	Foliation and quasi-local definitions remain	Compare several invariants with a common remnant mass and spin
Curvature speciality	Measures algebraic approach toward type D Okounkova 2020, §§ II–III and Figs. 3–5	Algebraic speciality alone does not establish a global Kerr exterior, and numerical Kerrness measures have a resolution-dependent floor	Add multipole, flux, convergence, and decay estimates in a controlled region

Two credible positions remain. A pragmatic waveform position defines onset by stable predictive accuracy for a selected observable. A geometric position reserves “linear regime” for simultaneous smallness and decay of invariant perturbations throughout an appropriate exterior region. These are not contradictory definitions, but they answer different questions. No current result makes their timestamps universal across mass ratio, spin, eccentricity, gauge, and data topology.

Data and statistical discipline

A reproducible analysis records the public simulation identifier and data revision, extraction method, extrapolation order or characteristic-extraction product, mode convention, retarded-time and Bondi-frame construction, and horizon-data normalization. Selections—mass ratio, spins, eccentricity, resolution, modes, start-time grid, and data-quality cuts—must be fixed before interpreting onset. Calibration uncertainty includes extraction and frame alignment; numerical uncertainty includes truncation and finite-boundary effects.

The fitted model must state whether complex frequencies are fixed by independently inferred Kerr parameters or varied, how amplitudes and phases are regularized, which covariance or weighting defines the residual, and how many effective degrees of freedom are added. Robustness tests shift the fit window, remove modes, compare resolutions, and predict held-out times. Relevant systematics include missing angular modes, spherical–spheroidal mixing, nonlinear combination frequencies, tail contamination, parameter covariance, and the finite size of the simulation catalog.

The analysis should retain input identifiers, code or explicit equations, fit windows, parameter tables, and machine-readable residuals. Nothing on this page reports a fresh fit or rerun. It states the reproduction boundary needed for one.

Limits of ringdown-onset criteria

The unresolved issue is not whether linear models eventually work; sufficiently late, decaying perturbations make that expectation precise in controlled settings. The open question is how early a selected linear description is explanatory rather than merely flexible, and whether one diagnostic transfers to another. Conclusions depend on formulation, norm, tolerance, observable, parameter regime, and desired prediction.

A fixed criterion tested on independently converged simulations could establish agreement between onset times measured from different observables. A theorem controlling the nonlinear-to-linear transition for a broader binary data class would strengthen the interpretation; reproducible failures outside a criterion’s calibration region would delimit it.

Transfer diagnostic

A ten-overtone fit reaches a tiny in-sample residual at the strain peak, but its remnant mass shifts by $4\%$ when the start time moves by $2GM/c^3$ and its early amplitudes do not predict the next $10GM/c^3$ of data when refitted on a shorter window, where $M$ is the total mass scale. Has linear ringdown onset been established?

Solution

No. The low in-sample residual shows representational flexibility. Parameter drift and failed held-out prediction show that the inferred linear model is not stable under discriminating changes. One should compare numerical error, reduce or vary the model, test other modes and horizon diagnostics, and identify the earliest interval where remnant parameters and predictions remain stable. The answer may be a useful observable-dependent onset, not a universal spacetime time.

References

• Giesler, Matthew, Maximiliano Isi, Mark A. Scheel, and Saul A. Teukolsky. “Black Hole Ringdown: The Importance of Overtones.” Physical Review X 9 (2019): 041060. doi:10.1103/PhysRevX.9.041060.
• Mourier, Pierre, Xisco Jiménez Forteza, Daniel Pook-Kolb, Badri Krishnan, and Erik Schnetter. “Quasinormal Modes and Their Overtones at the Common Horizon in a Binary Black Hole Merger.” Physical Review Letters 126 (2021): 041101. doi:10.1103/PhysRevLett.126.041101.
• Okounkova, Maria. “Revisiting Nonlinearity in Binary Black Hole Mergers.” Physical Review D 102 (2020): 084046. doi:10.1103/PhysRevD.102.084046.