Quasinormal Frequencies, Tails, and Response

Black-hole response comprises poles, operator residues, source excitations, prompt and branch-cut terms, scattering and tidal coefficients, and detector fits. They are distinct. This card fixes the $e^{-i\omega t}$ convention and records each object’s defining choices.

One retarded problem, several quantities

For a fixed field, gauge-invariant variable, background, and angular sector, write the frequency-domain problem schematically as

$$\mathcal L(\omega)\widetilde\psi(\omega,x) =\widetilde S(\omega,x), \qquad \widetilde\psi=G_{\mathrm{ret}}(\omega)\widetilde S.$$

The pencil includes its mapping spaces, domain, and endpoint conditions. The usual asymptotically flat prescription is ingoing at the future horizon and outgoing at infinity, continued on a declared analytic sheet. With

$$\psi(t)=\int_{\Gamma}\frac{d\omega}{2\pi} \,\widetilde\psi(\omega)e^{-i\omega t},$$

decay requires $\operatorname{Im}\omega_n\lt0$.

A quasinormal frequency $\omega_n$ is a pole of this continued resolvent, not a waveform amplitude or detector reading. At a simple pole,

$$G_{\mathrm{ret}}(\omega) =\frac{\mathsf P_n}{\omega-\omega_n}+G_{\mathrm{reg}}(\omega), \qquad \mathsf P_n=\mathop{\mathrm{Res}}_{\omega_n}G_{\mathrm{ret}}.$$

With fixed mapping spaces and physical variables, $\mathsf P_n$ is a Green-operator residue. Analytic nonzero rescaling of an auxiliary endpoint solution changes raw Wronskians and scalar mode or kernel factors, but cannot move the pole or change $\mathsf P_n$. Components of $\mathsf P_n$ depend on bases and projections; a source-excitation coefficient also depends on specified data or forcing. For the operator construction, see Quasinormal Modes, Resonances, and Pseudospectra.

What each name denotes

Quantity	Definition or calculation	Choices that remain visible
Quasinormal pole or resonance	Pole $\omega_n$ of a continued retarded resolvent	Field variable, operator domain, horizon and outer conditions, Fourier sign, analytic sheet
Green-operator residue	Operator $\mathsf P_n=\operatorname{Res}_{\omega_n}G_{\mathrm{ret}}$ between fixed source and field spaces	Physical variables, spaces, domain, and boundary conditions; invariant under auxiliary basis rescaling
Scalar mode or kernel excitation factor	Component or factorization coefficient of $\mathsf P_n$ in chosen bases	Mode, endpoint, angular, kernel, and projection normalization
Source-excitation coefficient	Scalar coefficient after $\mathsf P_n$ acts on specified initial data or forcing and is projected	Source history, observable projection, reference time, modal convention
Prompt response	Direct high-frequency or causal-front contribution before a pole expansion dominates	Source and receiver, contour closure, geometry, time interval
Branch-cut tail	Integral along a declared cut of the continued Green function	Sheet, cut placement, asymptotics, source moments, observation region
Reflection, transmission, or absorption coefficient	Ratio of normalized asymptotic channel amplitudes or fluxes at real frequency	Channel phases, unit-flux normalization, horizon generator, frequency range
Evolution resolvent or pseudospectrum	For $i\partial_t\Psi=\mathsf A\Psi$, growth of $\lVert(zI-\mathsf A)^{-1}\rVert$ or spectra under bounded perturbations	Generator domain, energy norm, perturbation class, discretization
Tidal response or Love coefficient	Induced multipole divided by a declared applied field, in a stated low-frequency expansion	Gauge-invariant matching, source/response split, subtraction scheme, order of limits
Detector-fitted ringdown parameter	Frequency, damping time, phase, or amplitude inferred through a waveform and noise model	Source excitation, angular projection, detector transfer, fit interval, likelihood, priors

Poles, prompt response, and tails

Deforming the inverse-transform contour can produce the schematic response

$$\psi(t)=\psi_{\mathrm{prompt}}(t) +\sum_n A_n e^{-i\omega_n(t-t_0)} +\psi_{\mathrm{cut}}(t).$$

This is not a universal completeness theorem: relevant arcs, poles, and cuts depend on the analytic problem and time. Prompt response can dominate early, poles an intermediate interval, and a branch point the late-time tail. A tail is not a “very late quasinormal mode,” and a finite pole fit does not prove an exact pole-only solution.

For Schwarzschild perturbations, Leaver’s source-explicit inverse transform exhibits prompt, pole, and branch-cut pieces Leaver 1986, full printed pp. 384–408. Its residue gives an excitation only after source projection. Changing the source can suppress a pole without moving it.

Pseudospectra require a distinct first-order evolution problem,

$$i\partial_t\Psi=\mathsf A\Psi, \qquad \sigma_\epsilon(\mathsf A)=\sigma(\mathsf A)\cup \left\{z:\left\lVert(zI-\mathsf A)^{-1}\right\rVert \gt\epsilon^{-1}\right\}.$$

The generator $\mathsf A$ is not the frequency-domain pencil $\mathcal L(\omega)$; relating them requires a stated first-order reformulation. Its domain, energy norm, boundary behavior, and perturbation class are part of the result Jaramillo, Macedo, and Al Sheikh 2021, §II.D, Eqs. (17)–(25), pp. 7–9; §III and Appendices B–C, pp. 9–17 and 32–36. A large pseudospectrum can signal sensitivity or transient amplification, but does not by itself imply a large time-domain waveform change for a specified source.

Scattering and tidal response

At real frequency, scattering coefficients compare incoming, outgoing, or transmitted channels. Amplitude ratios depend on channel normalizations; flux ratios require unit-flux normalization and the correct horizon-frequency factor. Their continuations can share poles without equaling a residue or source-excited waveform.

Tidal response asks for an applied field and an induced multipole. In the static spherical problem, the growing and decaying asymptotic solutions must be separated by gauge-invariant matching before their ratio is called a Love number Binnington and Poisson 2009, §III, Eqs. (3.9)–(3.13) and (3.19)–(3.22), pp. 6–10. At finite frequency the response is complex: its conservative and dissipative parts require a Fourier convention, a near-zone expansion, horizon condition, and matching prescription Chia 2021, §§III–VII, Eqs. (5)–(16), pp. 2–6.

Vanishing static Schwarzschild Love numbers therefore do not imply vanishing absorption or dynamical response. A resonance residue cannot be relabeled a Love coefficient without the source/induced split, gauge, subtraction, and order of limits. The full treatment is Frequency-Dependent Response and Dissipation.

From response to a detector fit

Source data, angular projection, propagation, and the observable set mode amplitudes and phases; see Ringdown and Black-Hole Spectroscopy as Waveform Theory.

A detector analysis adds calibration, sampling, noise likelihood, priors, a fit interval, and start/reference times. Isi and Farr make these dependencies explicit Isi and Farr 2021, §§II–III, arXiv v2 PDF pp. 4–17.

Thus a damped-sinusoid posterior is detector- and model-conditioned, not a direct observation of an operator pole. These observational tests are developed in Ringdown Tests and Strong-Field Spectroscopy. To assess a particular event or detection claim, use its dated analysis and reproducible spectrum calculation alongside these definitions.

Common pitfalls

Pole equals amplitude. A pole can have negligible projection for a particular source.

A pole sum is automatically complete. Prompt arcs and branch cuts also depend on time and the source-observer configuration.

Operator residue equals scalar excitation factor. The former is an operator Laurent coefficient between fixed spaces; the latter is a representation-dependent component or factorization of it.

References

• Binnington, Taylor, and Eric Poisson. “Relativistic Theory of Tidal Love Numbers.” Physical Review D 80 (2009): 084018. DOI. Open preprint.
• Chia, Horng Sheng. “Tidal Deformation and Dissipation of Rotating Black Holes.” Physical Review D 104 (2021): 024013. DOI. Open preprint.
• Isi, Maximiliano, and Will M. Farr. “Analyzing Black-Hole Ringdowns.” arXiv:2107.05609v2, 2021. Open preprint.
• Jaramillo, José Luis, Rodrigo Panosso Macedo, and Lamis Al Sheikh. “Pseudospectrum and Black Hole Quasi-Normal Mode (In)stability.” Physical Review X 11 (2021): 031003. DOI. Open preprint.
• Leaver, Edward W. “Spectral Decomposition of the Perturbation Response of the Schwarzschild Geometry.” Physical Review D 34 (1986): 384–408. DOI.