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Quantifying excitations of quasinormal mode systems

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Hans-Peter Nollert (Universitaet Tuebingen), Richard H. Price (University of Utah)

General Relativity and Quantum Cosmology · Submitted 22 Oct 1998 · 14 references

Pith's verdict: An honest negative result — the excitation coefficient really does fail — wrapped around a completeness proof whose rigor is conditional on heuristic root asymptotics.

T0 · AI review

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arXiv IDgr-qc/9810074 (version 1)
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CitationHans-Peter Nollert (Universitaet Tuebingen), Richard H. Price (University of Utah). "Quantifying excitations of quasinormal mode systems." arXiv:gr-qc/9810074 (1998). https://pith.science/paper/IYCC6KJK
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The paper

The paper claims

A candidate measure of quasinormal-mode content in black hole waveforms is constructed, tested on a solvable model, and found to give negative and over-100% values, motivating a conjecture that no algebraically natural measure can work.

The reading

When a black hole is disturbed, it rings like a struck bell, but the ringing is damped: the oscillation dies away. The frequencies of this ringdown are fixed by the black hole's mass and spin, and they dominate the gravitational waves that numerical simulations produce. A natural question is: how much of a given waveform is ringdown at a particular frequency? For ordinary, undamped oscillators (normal modes) this question has a clean answer, because the modes are orthogonal and each carries a definite share of the energy. Black hole ringdown modes are not orthogonal, and they decay, so the usual machinery does not apply directly. The authors set out to build the machinery anyway. They define what it would mean for a set of ringdown modes to be complete, define an inner product on waveforms, and construct a quantity they call the excitation coefficient that is invariant under the obvious ambiguity (shifting the initial data in space shifts the waveform in time, which rescales the naive coefficients). They then show that a simple model potential with one ringdown frequency can be modified, by adding a tiny spike, into a system with infinitely many ringdown frequencies that is provably complete, while the waveform barely changes. Finally they compute the excitation coefficient for this model. It comes out negative, and larger in magnitude than the total norm of the waveform, which is nonsense as a fraction. They conclude that this measure, and likely any measure built from the same algebra, cannot serve as a useful index of ringdown content.
Extended reading notes

Core claim

The excitation coefficient A_k = (a_k)*ā_k + (ã_k)*ā̃_k, constructed to be invariant under time shifts of the waveform, is of limited utility as a measure of quasinormal content. Table I makes this concrete: for the spiked truncated dipole potential with V_δ = 10^-6, A_1 = -57,933 while the waveform norm is 19,246, so the coefficient is negative and roughly three times the norm in magnitude. By the sum rule of Eq. (14), the remaining modes must then contribute about four times the norm. The paper states the conclusion directly: 'the excitation coefficient is of limited utility' (abstract:s5).

Load-bearing premise

The 'rigorous proof of completeness' in Appendix II, Lemma 1, uses the asymptotic root locations ℑ(s_n) = nπ/L + const and ℜ(s_n) = -(3/2L)ln(n) + const (Eqs. 87-88). Those formulas come from the iterative approximation of Appendix IC, which is a heuristic asymptotic expansion, not a proven bound on the roots of the Wronskian. Lemma 1's bound |I_Hn| ≤ const × n^(2-p-3d2/2L) therefore rests on an unproven input. The proof is rigorous conditional on those root asymptotics, but the paper presents it as unconditional.

Original abstract

Computations of the strong field generation of gravitational waves by black hole processes produce waveforms that are dominated by quasinormal (QN) ringing, a damped oscillation characteristic of the black hole. We describe here the mathematical problem of quantifying the QN content of the waveforms generated. This is done in several steps: (i) We develop the mathematics of QN systems that are complete (in a sense to be defined) and show that there is a quantity, the ``excitation coefficient,'' that appears to have the properties needed to quantify QN content. (ii) We show that incomplete systems can (at least sometimes) be converted to physically equivalent complete systems. Most notably, we give a rigorous proof of completeness for a specific modified model problem. (iii) We evaluate the excitation coefficient for the model problem, and demonstrate that the excitation coefficient is of limited utility. We finish by discussing the general question of quantification of QN excitations, and offer a few speculations about unavoidable differences between normal mode and QN systems.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,615 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.95 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.10 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Computations of the strong field generation of gravitational waves by black hole processes produce waveforms that are dominated by quasinormal (QN) ringing, a damped oscillation characteristic of the black hole. We describe here the mathematical problem of quantifying the QN content of the waveforms generated. This is done in several steps: (i) We develop the mathematics of QN systems that are complete (in a sense to be defined) and show that there is a quantity, the ``excitation coefficient,'' that appears to have the properties needed to quantify QN content. (ii) We show that incomplete systems can (at least sometimes) be converted to physically equivalent complete systems. Most notably, we give a rigorous proof of completeness for a specific modified model problem. (iii) We evaluate the excitation coefficient for the model problem, and demonstrate that the excitation coefficient is of limited utility. We finish by discussing the general question of quantification of QN excitations, and offer a few speculations about unavoidable differences between normal mode and QN systems.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Computations of the strong field generation of gravitational waves by black hole processes produce waveforms that are dominated by quasinormal (QN) ringing, a damped oscillation characteristic of the black hole. background 1.00 0.11 0.03 asserted 1.00
s1 We describe here the mathematical problem of quantifying the QN content of the waveforms generated. contribution 0.70 0.67 0.04 asserted 0.74
s2 This is done in several steps: (i) We develop the mathematics of QN systems that are complete (in a sense to be defined) and show that there is a quantity, the ``excitation coefficient,'' that appears to have the properties needed to quantify QN content. method 0.54 0.94 0.11 asserted 0.61
s3 (ii) We show that incomplete systems can (at least sometimes) be converted to physically equivalent complete systems. contribution 0.69 0.92 0.06 hedged possibility 0.74
s4 Most notably, we give a rigorous proof of completeness for a specific modified model problem. contribution 0.68 0.96 0.10 asserted 1.00
s5 (iii) We evaluate the excitation coefficient for the model problem, and demonstrate that the excitation coefficient is of limited utility. result 0.68 0.96 0.47 asserted 1.00
s6 We finish by discussing the general question of quantification of QN excitations, and offer a few speculations about unavoidable differences between normal mode and QN systems. future work 0.68 0.61 0.18 speculative 0.93

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1. Propagation of initial data to the observation locat
Figure 2. FIG. 2. Initial data shifted in location produces identical
Figure 3. FIG. 3. Oscillations produced by shifting and scaling initi
Figure 4. FIG. 4. Time evolution for initial data incident on the TDP an
Figure 5. FIG. 5. Quasinormal frequencies for the TDP with an ad
Figure 7. FIG. 7. Closing the contour for integration in the complex
Figure 8. FIG. 8. Values of the mode sum for different number of
Figure 10. FIG. 10. Logarithmic difference ∆
Figure 11. for two cases: the spiked TDP and the QN spectrum of Schwarzschild black holes (the QN modes of the Zerilli
Figure 12. FIG. 12. Cosine of angles between several contravariant
Figure 14. FIG. 14. Contour for proving convergence for the spiked TDP o

Showing 11 of the paper's 14 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:gr-qc/9810074 by the authors.

References
How later papers used it

81 later papers cite it

Evidence: some. 81 of the citing papers share no author with it, and 5 of those carry a confident label from Pith's citation classifier.

Pith read 7 of the 123 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 1 of 4 papers with this label
How the literature uses it

73 arXiv papers cite this in 100 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 69 sentences without a confident label

Uses its method or tool 4 papers
  • Here, we attack this problem by utilizing the excitation factors [ref] , which are the residue of a pole (the QNM) of the Green's function, and quantify the excitability of each QNM.

    Reconstruction of ringdown with excitation factors · Introduction · 0.92

  • Based on several works from the numerical GR community in flat space [ref] , we are able to numerically compute the residue of the QNMs of the system by just solving the ODEs that result from Laplace transforming the PDEs.

    Holographic quenches and anomalous transport · Introduction · 0.98

  • The mismatch between signals $h_1$ and $h_2$ is defined as [equation] where we use the following complex inner product [ref] [equation] We integrate from the ringdown start time, $t_0$, to an upper limit $T$ chosen such that the whole ringdown is captured (we use $T = t_0 + 100M$).

    Modelling the Ringdown from Precessing Black Hole Binaries · Aligned-spin Systems · 0.89

Cites it as background 25 papers

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