Ringdown and Black-Hole Spectroscopy as Waveform Theory

Ringdown waveform theory turns the retarded response of a settling black hole into a finite-window model of radiation at future null infinity. The central object is not a bare list of complex frequencies: it is a convention-explicit waveform with source-dependent amplitudes, angular mixing, a chosen onset and endpoint, prompt and tail limitations, and tests for nonlinear or rapidly damped content. This chapter develops that model while keeping current detector likelihoods and observational verdicts outside its scope.

Helpful background. Einstein Equations supplies perturbation theory about Kerr; Waves, Green Functions, and Geometric Optics supplies retarded poles, residues, and branch cuts; Black-Hole Perturbation Theory supplies QNM spectra and Green functions; Gravitational Radiation and Waveforms supplies the asymptotic radiation dictionary; and Black-Hole Dynamics and Collapse supplies remnant formation and relaxation.

Enter the finite-window problem

For the sustained remnant formulas in this chapter, use geometrized units $G=c=1$. Initial total mass $M$ and final mass $M_f$ are expressed as times or lengths. Restore a duration $bM_f$ as $bGM_{f,\mathrm{phys}}/c^3$ in the source frame; multiply by $1+z$ for detector time. This explicitly declared unit switch is the reason that $M_f\omega$ below is dimensionless.

The spacetime signature is $(-,+,+,+)$. In a declared inertial Bondi frame, normally aligned with the final remnant spin, write

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

with unit-normalized spin-$-2$ harmonics. Retarded time is $u$, the remnant has mass $M_f$ and spin $\chi_f$, and time dependence is $e^{-i\omega u}$. Hence a stable QNM satisfies $\operatorname{Im}\omega\lt 0$ and has damping time $\tau=-1/\operatorname{Im}\omega$. At the future horizon the retarded boundary condition is ingoing in advanced time; at future null infinity it is outgoing in retarded time.

When a curvature waveform is mentioned, use the outgoing Newman–Penrose tetrad $(\ell,n,m,\bar m)$ with $\ell\!\cdot n=-1$, $m\!\cdot\bar m=1$, and $\Psi_4=-C(n,\bar m,n,\bar m)$. Its leading asymptotic coefficient obeys $\Psi_4^0=\partial_u^2H$. The chapter models $H$ unless another field is explicitly named; it never identifies strain, news, and curvature without the required derivatives and integration data.

The labeled poles $\omega_{Lmn}(M_f,\chi_f)$ and their Green-function residues are calculated in Black-Hole Perturbation Theory. This chapter uses those results to build

$$H_{\rm mod}(u)= \sum_{q\in\mathcal Q} A_q(u_0)e^{-i\omega_q(u-u_0)}S_q +P(u)+T(u)+N^{(2)}(u), \qquad u_0\le u\le u_1.$$

Here the selected pole terms coexist with an explicitly modeled or bounded prompt part $P$, tail part $T$, and nonlinear contribution $N^{(2)}$. These symbols are not automatically separate fit columns; they mark physical content that a truncated QNM sum may omit or effectively absorb. The prompt–pole–branch-cut anatomy follows the retarded contour construction in Berti, Cardoso, and Starinets 2009, § 3.2, pp. 18–20, PDF.

The six-page route

The pages are ordered so that every later controversy rests on an explicit response and amplitude dictionary.

Page	Central question	Result to carry forward
Quasinormal-Mode Waveform Expansion	When may a late waveform be represented by a discrete damped sum?	A QNM sum is a finite-window pole model, not an ordinary complete basis; the boundary and damping signs and spherical–spheroidal overlaps are explicit.
Mode Excitation and Amplitudes	Why are universal frequencies compatible with nonuniversal amplitudes?	Pole position, residue factor, source projection, normalization, reference time, and fitted coefficient are distinct.
Ringdown Start Time	What evidence makes a linear model adequate after a chosen time?	Start time is observable-, frame-, norm-, window-, and purpose-dependent; near-peak usefulness and later amplitude stability answer different questions.
Overtones and Rapidly Damped Modes	When does an extra damped term become an identified overtone?	Fit improvement is weaker than stable amplitude, phase, frequency, injection recovery, and resolution evidence.
Nonlinear Ringdown and Mode Coupling	What distinguishes quadratic content from linear mixing or degeneracy?	Sum/difference frequency, angular selection, product-amplitude scaling, convergence, window stability, and a matched linear control must agree.
Requirements for a Spectroscopy Waveform Model	What must be established before interpreting several frequencies as one black-hole spectrum?	Resolvability, physical identification, waveform adequacy, and detector inference are separate gates; all labeled complex frequencies must fit one remnant.

Evidence levels and unresolved boundaries

Some statements are structural. The $e^{-i\omega u}$ damping sign, ingoing/outgoing boundary conditions, reciprocal normalization of mode coefficients, and $2K-2$ common-background restrictions for $K$ labeled complex frequencies follow from declared equations and parameter counting.

Other statements are conditional on a waveform and fitting protocol. Fixed-frequency overtone fits to SXS:BBH:0305 supported accurate near-peak remnant recovery in the documented 2019 setup Giesler et al. 2019, §§ III–IV, PDF. High-accuracy CCE waveforms, BMS-frame fixing, variable projection, synthetic-noise studies, and decay-compensated amplitude tests supplied stronger mode-specific evidence in two aligned-spin systems, while leaving early unmodeled content Giesler et al. 2025, §§ IV–V, PDF.

The start-time, overtone, and nonlinear pages therefore use evidence matrices with a cutoff of 2026-08-18. Their credible positions are preserved rather than compressed into consensus: useful near-peak representation need not imply a unique pole decomposition; stable overtones in controlled simulations need not establish completeness across binary parameter space; and supported quadratic modes need not fix the importance of every nonlinear mechanism. The 2023 quadratic-amplitude scaling result is positive primary evidence under its nonprecessing numerical selection, not a universal merger expansion Mitman et al. 2023, eqs. (2)–(8) and Figs. 1–3, pp. 2–5, PDF.

Constructing and testing a ringdown model

1. Fix the waveform object. State strain, news, or curvature; tetrad if applicable; Bondi/BMS frame; angular basis; mass and time units; peak definition; and Fourier sign.
2. Choose the response hypothesis. Select labeled poles from one remnant and state whether prompt, tail, mixing, overtones, nonlinear terms, or background drift are modeled or bounded.
3. Expose normalization. Record spheroidal and spherical conventions, overlap coefficients, amplitude reference time, and the map from source projection to fitted coefficients.
4. Scan the finite window. Vary both endpoints, model order, fixed versus free frequencies, and plausible nuisance terms. Transport amplitudes to a common time.
5. Compare with error. Use multiple numerical resolutions, extraction and frame variants, injection recovery, singular values, and held-out windows or waveforms. Treat a mode below its recovery floor as unresolved even when the global residual is small.
6. Apply the interface. Production evolution algorithms and convergence infrastructure are covered in Computational Relativity. Calibration, noise likelihoods, selection, empirical inference, and dated spectroscopy claims are covered in Gravitational-Wave Astronomy.

The workflow makes failure informative. A coefficient that moves reciprocally under a harmless mode renormalization is expected; a decay-compensated coefficient that wanders with start time is not. Spherical leakage preserves the parent linear frequency; a genuine quadratic source follows a sum or conjugate-difference rule. A low residual may establish approximation on one interval, but never completeness by itself.

Review the chapter

A free-frequency fit to a spherical $H_{22}$ waveform on $[u_{\rm peak},u_{\rm peak}+30M_f]$ returns

$$M_f\omega_0=0.52-0.08i, \quad M_f\omega_1=0.88-0.25i, \quad M_f\omega_2=1.16+0.03i,$$

with residual $10^{-6}$. Only one numerical resolution and one start time were used, and no BMS-frame correction or alternative prompt term was tested. What can be concluded? How should the coefficient of the first term transform if its reference time moves forward by $4M_f$?

Solution

The first two frequencies have the correct damping sign, but the third grows because $\operatorname{Im}\omega_2\gt 0$; it cannot be a stable retarded QNM under the chapter convention. The small residual supports only a three-column approximation on the stated window. It does not identify the first two terms as QNMs, because frame error, prompt response, spherical–spheroidal leakage, nonlinear content, numerical error, conditioning, and start-time stability were not tested. It also cannot support spectroscopy: labels and a single common $(M_f,\chi_f)$ have not been established, and there is no downstream noise inference.

For $u_0'=u_0+4M_f$,

$$A_0(u_0')=A_0(u_0)e^{-i\omega_0(4M_f)} =A_0(u_0)e^{-i2.08}e^{-0.32}.$$

Its magnitude is multiplied by $e^{-0.32}\simeq0.726$ and its phase changes by $-2.08$ radians. Transporting both coefficients to one common reference time should remove exactly this known change. Remaining drift would diagnose model or data limitations rather than harmless bookkeeping.

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, Maximiliano Isi, Mark A. Scheel, and Saul A. Teukolsky, “Black Hole Ringdown: The Importance of Overtones,” Physical Review X 9 (2019) 041060. 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.