Completeness, Excitation, and Ringdown Content

Quasinormal poles are indispensable to black-hole ringdown, but a pole sum is not automatically a complete expansion of arbitrary initial data. In an inverse Laplace transform, the retarded field generally contains a prompt high-frequency contribution, residues of resonant poles, and a branch-cut or threshold contribution. Their relative sizes depend on the source, observation region, time window, and norm. A finite sum of damped sinusoids may be an excellent waveform model without its fitted amplitudes being unique projections onto independently excited physical modes.

Required background. PDEs and Green Functions for Gravitation supplies inverse Laplace transforms, residues, branch cuts, and semigroup domains.

Helpful background. Black-Hole Geometry and Mechanics supplies asymptotic regions; Formulations of General Relativity supplies perturbation variables and gauge; Computational Relativity supplies evolution and fitting tests; Black-Hole Perturbation Theory supplies the master Green functions.

Three pieces of a retarded response

Fix one canonically rescaled Schwarzschild Regge–Wheeler or Zerilli field with radiative multipole $\ell\geq2$, and write $x=r_*$. Its source-free equation and compact Cauchy data are

$$(-\partial_t^2+\partial_x^2-V_\ell)X=0, \qquad X(0,x)=f(x),\quad \partial_tX(0,x)=p(x).$$

The source-normalized mode kernel obeys $(\partial_x^2+\omega^2-V_\ell)g_\ell=\delta(x-y)$. Integrating the one-sided transform by parts gives

$$\widehat{\partial_t^2X}=-p+i\omega f-\omega^2\widehat X, \qquad \widehat I(\omega,y)=i\omega f(y)-p(y).$$

Thus $\widehat X(\omega,x)=\int g_\ell(\omega;x,y)\widehat I(\omega,y)\,dy$. The minus sign multiplying $p$ is tied to the positive delta source of this kernel. The inversion line lies above the growth bound; for these stable Schwarzschild sectors we may choose $\gamma\gt0$:

$$X(t,x)=\frac{1}{2\pi} \int_{\mathbb R+i\gamma}e^{-i\omega t} \left[\int_{\mathbb R}g_\ell(\omega;x,y) \widehat I(\omega,y)\,dy\right]d\omega.$$

After justified continuation and contour deformation, one schematically obtains

$$X=X_{\rm prompt}+X_{\rm QNM}+X_{\rm cut},$$

with

$$X_{\rm QNM}(t,x) =-i\sum_n e^{-i\omega_nt} \operatorname*{Res}_{\omega=\omega_n} \int_{\mathbb R}g_\ell(\omega;x,y)\widehat I(\omega,y)\,dy.$$

The factor $-i$ follows from closing the left-to-right frequency line clockwise below. The sum runs over the enclosed simple poles, including the required frequency partners; higher-order poles require additional polynomials in time. A finite contour has a remainder, and an infinite sum requires a convergence argument.

$X_{\rm prompt}$ denotes retained large-frequency propagation terms in this representation. $X_{\rm cut}$ includes the zero-frequency branch structure that produces asymptotically flat curvature tails. Individual contour pieces need not be causal separately, and their separation depends on the representation. Leaver retains all three pieces in the Schwarzschild construction Leaver 1986, author-version § II B, pp. 7–8, Eqs. (20)–(24), PDF. The contour diagram and large-arc estimate explain why positive time alone does not license dropping the prompt term.

There need not be a pole-dominated interval for every source and observable. For generic asymptotically flat data with a nonzero tail coefficient, a power-law tail eventually outlives each fixed damped pole. This is not a proof that every possible infinite resummation fails: changing a function space or including a discretized cut changes the question. Nor does causality say that a residue expression vanishes before arrival; other pieces may have to cancel its premature signal.

Excitation factors are not excitation coefficients

Let $u_{\rm in}$ and $u_{\rm up}$ have unit ingoing horizon and outgoing infinity amplitudes, respectively, and define

$$W=u_{\rm in}\partial_xu_{\rm up} -u_{\rm up}\partial_xu_{\rm in}.$$

Then $g=u_{\rm in}(x_\lt)u_{\rm up}(x_\gt)/W$, where $x_\lt=\min(x,y)$ and $x_\gt=\max(x,y)$. Assume the continued endpoint solutions are analytic near a simple zero $\omega_n$ of $W$ and the kernel has a genuine pole there. At that frequency write

$$u_{\rm in}(\omega_n,x)=b_n u_n(x), \qquad u_n(x)=u_{\rm up}(\omega_n,x).$$

The proportionality amplitude $b_n$ cannot generally be set to one while keeping both endpoint normalizations. The residue and field coefficient are

$$\begin{aligned} \operatorname*{Res}_{\omega_n}g(\omega;x,y) &=\frac{b_n u_n(x)u_n(y)}{W'_n},\\ X_n^{\rm pole}(t,x)&=C_n u_n(x)e^{-i\omega_nt},\\ C_n&=-i\frac{b_n}{W'_n} \int_{\mathbb R}u_n(y)[i\omega_nf(y)-p(y)]\,dy, \qquad W'_n=\partial_\omega W(\omega_n). \end{aligned}$$

The background-dependent prefactor, together with the specified endpoint normalization and Fourier convention, plays the role of an excitation factor. Literature definitions can move amplitudes or frequency factors between terms. The excitation coefficient $C_n$ additionally depends on the data. The observable pole contribution includes $u_n(x)$ as well.

An explicit invariance check prevents a normalization mistake. For analytic nonzero functions $a(\omega),d(\omega)$, change the endpoint bases by

$$\begin{aligned} u_{\rm in}&\longmapsto a\,u_{\rm in}, &u_{\rm up}&\longmapsto d\,u_{\rm up},\\ W'_n&\longmapsto a_nd_nW'_n, &b_n&\longmapsto (a_n/d_n)b_n. \end{aligned}$$

The derivative terms from $\partial_\omega(ad)$ vanish because $W(\omega_n)=0$. Both the residue kernel and $C_n u_n(x)$ are unchanged; $1/W'_n$ alone is not. Compact support makes the displayed data integral finite. For noncompact data, spatial growth of continued QNMs can make it diverge; a specified analytic continuation or regularization is then required. Leaver separates these cases Leaver 1986, author-version § III, pp. 16–17, Eqs. (47)–(53), PDF.

For Kerr, a regularized symmetric complex bilinear form gives an analogous projection for the adjoint Teukolsky field with compact Cauchy support. Its contour prescription is part of the construction; it is not positive definite, and orthogonality does not establish a complete basis Green et al. 2023, arXiv v3, § VI, pp. 11–14, PDF. This is a well-defined linear-field result in its subextremal domain, not global uniqueness of nonlinear metric ringdown content.

What “complete” can mean

At least four claims are often compressed into one word:

1. Restricted contour completeness: inside a frequency region where a meromorphic resolvent expansion is controlled, pole residues plus a holomorphic remainder reproduce the resolvent acting on specified data.
2. Asymptotic dominance: after prompt propagation but before or alongside a tail, one or a few least-damped poles dominate a selected observable.
3. Finite-window usefulness: a chosen sum of damped exponentials fits a waveform within a declared norm and time interval.
4. Many independently excited modes: fitted coefficients are stable physical projections associated one-to-one with distinct QNMs.

These claims are not a hierarchy of increasingly good fits. A theorem about a resolvent need not identify fitted amplitudes, and a small fit residual need not prove a theorem. Asymptotic dominance can itself be rigorous when remainder bounds establish it; in a numerical waveform it remains a tested approximation. Physical interpretation additionally needs a specified field, source, normalization, and observation map.

What a local Keldysh expansion supplies

Let $F(\omega):\mathcal X\to\mathcal Y$ be a holomorphic family of bounded Fredholm operators of index zero between fixed complex Banach spaces, invertible somewhere in a connected domain $\Omega$. An unbounded wave operator needs a fixed, appropriately normed domain before this statement can be applied. In a relatively compact subdomain with no boundary poles, assume every enclosed characteristic value $\omega_n$ is simple. For right null vectors $v_n$ and dual left null functionals $\alpha_n$, define

$$F(\omega_n)v_n=0,\qquad \alpha_n F(\omega_n)=0, \qquad d_n=\alpha_n[F'(\omega_n)v_n]\ne0.$$

Subtracting these finite-rank principal parts leaves a holomorphic operator $H$:

$$F(\omega)^{-1} =\sum_n\frac{v_n\,\alpha_n[\,\cdot\,]}{d_n(\omega-\omega_n)} +H(\omega).$$

This is a local resolvent identity, not permission to discard $H$ during inverse transformation. Global time estimates additionally require contour growth bounds and control of the frequency-region limit. The residue uses dual pairing, not a chosen positive scalar product Besson and Jaramillo 2025, arXiv v2, § 2.1, pp. 9–10, Eqs. (11)–(21), PDF. Rescaling $v_n$ or $\alpha_n$ cancels between numerator and denominator. Choosing unit-norm bulk modes changes their numerical coefficients, not a fixed source-to-observer residue. Changing the operator domain or regularity space is a different operation and can change the spectral problem itself.

Numerical hyperboloidal expansions can accurately reconstruct early response and tails for selected data. A tail reconstructed using discretized branch-cut modes is not a proof of a continuum QNM-only expansion. The continuation sheet, retained spectrum, regularity, observation region, and limiting procedure must accompany that claim.

Comparing completeness, excitation, and waveform results

The literature status in this comparison is assessed through 2026-09-05. There is no single completeness question until the background, field, domain, data class, time region, and convergence sense are fixed. The sources below do not establish a universal, all-time, QNM-only expansion for arbitrary black-hole perturbations.

Evidence type	Background and source support	Time / region / norm	What it establishes	What it does not establish
Contour decomposition (Leaver 1986)	Schwarzschild Regge–Wheeler/Zerilli; compact and suitably continued extended radial data	Specified observer; pole, cut, and large-frequency pieces	Source-dependent excitation and tail contributions	An unconditional all-time pole-only theorem or generic Kerr completeness
Kerr bilinear projection (Green et al. 2023)	Subextremal Kerr Teukolsky field; compact-support Cauchy data, complex-$r$ contour	Formal pole contribution on a Cauchy slice; symmetric bilinear form, not a positive Hilbert norm	Orthogonality in the regulated form and equivalence of projection and Wronskian residue	Positive-definite normal-mode theory, obvious basis completeness, prompt/cut reconstruction
Hyperboloidal Keldysh numerics (Besson–Jaramillo 2025)	Solvable barrier and spherical black holes; selected Gaussian-like data	Boundary time series and fixed-time energy or $H^p$ diagnostics	Accurate finite reconstructions, regularity tests, and discretized cut contributions	A universal theorem, or a cut reconstructed from continuum QNMs alone
Scattering reconstruction (Besson–Jaramillo 2026)	Gaussian Schwarzschild scattering, with de Sitter/AdS, weakly charged and slowly rotating examples	Mainly boundary waveforms; separate QNM and discretized cut terms	Detailed numerical response reconstruction and a proposed algebraically special/continuum mechanism	Proof of the proposed mechanism, nonlinear merger onset, or convergent continuum cut eigenvalues
Finite-window overtone fit (Giesler et al. 2019)	Nonlinear binary-black-hole numerical-relativity waveform, dominant harmonic	Fit beginning near peak strain; waveform mismatch	More fixed Kerr frequencies improve early-window fits and remnant recovery in the reported tests	Unique physical projection for every fitted coefficient or completeness
Controlled linear scattering fits (Zhu et al. 2024)	Hyperboloidal Teukolsky evolution of ingoing Gaussian pulses on Kerr	Spatial/time fitting with varied windows	Early transients obscure high overtones even with spatial information	The same quantitative limit for mergers or detector data

The 2025 numerical construction is described in Besson and Jaramillo 2025, arXiv v2, § 3.2 and § 5.1, PDF. The 2026 preprint explicitly treats nonconvergent finite discretizations of the cut and labels its proposed mechanism as a conjecture Besson and Jaramillo 2026, v1, § 4, p. 6, and §§ 5–6, pp. 8–10, PDF. Its first version was submitted on August 12; it is a newly included source here, not a claimed post-August-12 discovery.

The successful merger fits use a different source and problem from the controlled scattering tests; the two results therefore need not contradict one another. Compare their precise tests in Giesler et al. 2019, arXiv v2, § IV, pp. 3–6, PDF and Zhu et al. 2024, arXiv v3, § IV C and § V A, pp. 7–10, PDF.

Nonlinear simulations add parameter drift, harmonic mixing, extraction effects, and genuinely quadratic responses. A good linear multi-exponential fit does not identify all of these as linear Kerr overtones. Conversely, contamination does not erase the linear poles. Waveform construction belongs to Gravitational Radiation, and detector likelihoods and spectroscopy verdicts belong to Gravitational-Wave Astronomy.

Source support and start time are physical diagnostics

Take two finite-energy initial data sets with the same total norm: one localized outside the potential peak, the other near the horizon. The poles $\omega_n$ are properties of the same background operator and do not change. The overlaps $C_n$, prompt arrival, and relative cut contribution can change. A claim of “the overtone content of the black hole” that omits the source therefore confuses spectrum with excitation.

A fit with fixed distinct frequencies has the form

$$h_N(t)=\sum_{n=0}^{N-1}A_n(t_0)e^{-i\omega_n(t-t_0)}.$$

Even for exact uncontaminated data, its amplitudes change when the reference time changes:

$$A_n(t_1)=A_n(t_0)e^{-i\omega_n(t_1-t_0)}, \qquad c_n=A_n(t_0)e^{i\omega_nt_0}.$$

Compare the coefficients $c_n$ at a common reference time, not the raw $A_n$. Otherwise ordinary decay is mistaken for unstable excitation.

Now set $s=t-t_0$ and minimize a weighted squared residual on $0\leq s\leq T$, with fixed positive weight $w(s)$. Put $\phi_n(s)=e^{-i\omega_ns}$ and write the true signal on that interval as $\sum A_n^{\rm true}\phi_n+q$. When the Gram matrix is nonsingular,

$$\begin{aligned} K_{jk}&=\int_0^T w(s)\overline{\phi_j(s)}\phi_k(s)\,ds,\\ b_j&=\int_0^T w(s)\overline{\phi_j(s)}q(t_0+s)\,ds,\\ \widehat{\boldsymbol A}-\boldsymbol A^{\rm true}&=K^{-1}\boldsymbol b. \end{aligned}$$

This follows by differentiating the quadratic objective with respect to the complex amplitudes. It separates three effects: the omitted signal $q$, its projection in the chosen window and weight, and amplification by a poorly conditioned $K$. A small omitted component can produce a large amplitude error. Merely comparing envelopes does not determine that error.

On the same samples, window, weight, and fixed frequency set, adding free amplitudes makes the minimum training residual non-increasing because the old model remains available. It need not decrease strictly or improve held-out predictions. Changing the window or normalizing the mismatch differently invalidates that simple comparison.

The minimum diagnostic varies all of the following independently:

• source support and width while holding the background fixed;
• fit start and end times;
• time-domain, energy, and—when regularity is at issue—specified $H^p$ norms;
• number of poles and whether prompt/cut templates are included;
• observation radius or spatial region.

Instability of the reference-corrected coefficients is a warning against unique many-mode interpretation. It is not, by itself, a proof that the underlying poles do not exist. Compare fixed physical observables as well: changing a modal normalization must not be confused with changing the data.

Two equal-energy sources in the same potential

An exact evolution makes the source dependence concrete. Use dimensionless $q=t/L$ and $\xi=x/L$, where $L$ is a fixed length, and the whole-line model

$$(-\partial_q^2+\partial_\xi^2-2\delta(\xi))X=0.$$

This is a repulsive point barrier, not a black-hole potential. Take zero initial displacement and velocities $p_A=\mathbf1_{[1,2]}$ and $p_B=\mathbf1_{[2,3]}$. These are finite-energy weak Cauchy data; smoothing their edges gives smooth approximations. Both have dimensionless energy

$$E(0)=\frac12\int_{\mathbb R} \left[(\partial_qX)^2+(\partial_\xi X)^2\right]d\xi+X(0,0)^2 =\frac12.$$

At the observer $\xi=4$, the exact delta-barrier kernel gives, for $[a,b]\subset(0,4)$,

$$X_{[a,b]}(q,4)=\frac12\int_a^b \left\{\Theta(q-4+y)-\Theta(q-4-y) [1-e^{-(q-4-y)}]\right\}dy.$$

The first term propagates directly from each source point; the second arrives after visiting the barrier. The overall sign follows from $\widehat I=-p$, so the initial-velocity response is $-G_{\rm ret}p$. Before $q=4-b$ there is no signal. After every barrier path has arrived, $q\gt4+b$, the two step-function integrals cancel and leave

$$X_{[a,b]}(q,4)=C_{[a,b]}e^{-q}, \qquad C_{[a,b]}=\frac{e^4}{2}(e^b-e^a).$$

Initial velocity support	First arrival	Pure-pole expression valid	Late coefficient
$[1,2]$	$q=2$	$q\gt6$	$C_A=e^4(e^2-e)/2$
$[2,3]$	$q=1$	$q\gt7$	$C_B=eC_A$

The same pole $\omega L=-i$ has different excitation and causal onset for equal-energy data at a fixed global time reference. The factor $e$ is not energy amplification: measured from their respective reflected-completion times, the two late exponential amplitudes agree. Large values of $C_A$ at the extrapolated reference time $q=0$ do not imply a large early signal: the pole expression is not valid there. This meromorphic model has no branch cut. It demonstrates source dependence and prompt cancellation, while Schwarzschild’s nonzero cut and late tail remain a separate physical effect. The existing single-impulse figure illustrates the same arrival mechanism before source integration.

What would change this assessment

These conclusions could change with a completeness theorem specifying the background, domain, regularity space, and remainder; independently reproduced source-resolved excitation; a nonlinear benchmark separating linear poles, quadratic modes, prompt response, and parameter drift; a controlled detector analysis; or a correction or withdrawal. Results after 2026-09-05 fall outside this page’s cutoff. Neither numerical success nor numerical failure in one finite realization settles all of these distinct questions.

Exercises

Move the reference time before comparing amplitudes

Use dimensionless time. For an exact mode with $\omega=2-i/2$, suppose its amplitude at $t_0=0$ is $A(0)=3$. Find its amplitude when the same signal is expressed relative to $t_1=2$. Has its physical excitation changed?

Solution

The new amplitude is $A(2)=3e^{-1-4i}$. Its smaller modulus is ordinary decay over two time units. Transporting back gives $A(2)e^{2i\omega}=3$, so the common-reference coefficient is unchanged. No refitting error or additional source is needed to produce this raw amplitude change.

Cancel the residue without removing the pole

In the delta-barrier example, choose initial velocity $p_C=p_B-e p_A$. Show that its observer signal is nonzero at $q=2$ but vanishes for $q\gt7$. Does this remove the resonance from the operator?

Solution

Linearity gives $X_C=X_B-eX_A$. At $q=2$ the $B$ interval has contributed its full direct integral, while $A$ has only reached its first endpoint, which has zero integration measure. No reflected contribution has arrived, so $X_C(2,4)=1/2$. At $q\gt7$ both signals are pure poles and $C_B-eC_A=0$. The nonzero initial data have energy $(1+e^2)/2$, but zero overlap with this pole. The resolvent and its pole at $\omega L=-i$ are unchanged; only this source-to-observer residue cancels.

Omitted oscillation and a misleading overtone fit

Use dimensionless time and the signal

$$h(t)=e^{-t}+0.2e^{-3t}+0.05e^{-t}\cos(8t), \qquad t\ge0.$$

Fit only the exponentials with decay rates $1$ and $3$. The cosine is an omitted oscillatory component, not a model of the prompt contour arc: its one-sided transform has its own poles at $\omega=\pm8-i$. First compare its envelope with the faster exponential. Then calculate the unweighted least-squares bias on $[t_0,\infty)$ and transport it to the common reference time zero. Does the envelope crossing determine the bias?

Solution

The envelope ratio is

$$R(t)=\frac{0.05e^{-t}}{0.2e^{-3t}} =0.25e^{2t}.$$

$R=1$ at

$$t=\frac12\log4=\log2\simeq0.693.$$

To calculate the actual fit, write $\theta=8t_0$ and $s=t-t_0$. The two basis functions are $e^{-s},e^{-3s}$, so

$$K=\begin{pmatrix}1/2&1/4\\1/4&1/6\end{pmatrix}, \qquad K^{-1}=\begin{pmatrix}8&-12\\-12&24\end{pmatrix}.$$

Use the elementary integral

$$\int_0^\infty e^{-a s}\cos(\theta+8s)\,ds =\frac{a\cos\theta-8\sin\theta}{a^2+64},\qquad a\gt0.$$

With $\epsilon=0.05$, the two projections of the omitted term are

$$\boldsymbol b=\epsilon e^{-t_0} \begin{pmatrix} (2\cos\theta-8\sin\theta)/68\\ (4\cos\theta-8\sin\theta)/80 \end{pmatrix}.$$

Multiplying by $K^{-1}$ and then by $e^{3t_0}$ for the faster coefficient gives

$$\widehat c_2-0.2 =\frac{\epsilon e^{2t_0}}{85} [72\cos(8t_0)-84\sin(8t_0)].$$

The bias oscillates, changes sign, and can vanish when the envelope is large. The crossing at $\log2$ is therefore only an amplitude-scale comparison, not a threshold for a definite fitting error.

The dependence on the norm is also concrete. Repeating the same normal equations on the finite interval $0\leq s\leq1$ gives the following reference-corrected faster coefficient (rounded):

Start time	Weight $w=1$	Weight $w=e^{-s}$
$0$	$0.248222$	$0.275783$
$\log2$	$0.510039$	$0.584830$
$1$	$-0.306392$	$-0.317641$

These values follow from integrating the two-by-two Gram matrix and the omitted cosine, not from changing the true coefficient $0.2$. This exact signal-model diagnostic is not a black-hole evolution or a detector fit.

References

• Besson, J., and Jaramillo, J. L. (2025). “Quasi-normal mode expansions of black hole perturbations: a hyperboloidal Keldysh’s approach.” General Relativity and Gravitation 57, 110. DOI. Open PDF, arXiv v2, 2025-06-08.
• Besson, J., and Jaramillo, J. L. (2026). “Vishveshwara’s waveform revisited: insights from a Keldysh quasinormal mode expansion.” arXiv:2608.11823v1, submitted 2026-08-12; preprint. Record. Open PDF.
• Giesler, M., Isi, M., Scheel, M. A., and Teukolsky, S. A. (2019). “Black hole ringdown: the importance of overtones.” Physical Review X 9, 041060. DOI.
• Green, S. R., Hollands, S., Sberna, L., Toomani, V., and Zimmerman, P. (2023). “Conserved currents for a Kerr black hole and orthogonality of quasinormal modes.” Physical Review D 107, 064030. DOI. Open PDF, arXiv v3, 2024-02-19.
• Leaver, E. W. (1986). “Spectral decomposition of the perturbation response of the Schwarzschild geometry.” Physical Review D 34, 384–408. DOI.
• Zhu, H., Ripley, J. L., Cárdenas-Avendaño, A., and Pretorius, F. (2024). “Challenges in quasinormal mode extraction: Perspectives from numerical solutions to the Teukolsky equation.” Physical Review D 109, 044010. DOI. Open PDF, arXiv v3, 2024-02-08.