phenomenon · claim-type · cross-domain-transfer · mechanism · claim-relation · replication-status · limitation — grounded in the full text| phenomenon | ringdown - a compact object losing energy to infinity oscillates at complex frequencies fixed by its own parameters and not by whatever disturbed it— “Although there are presently several indirect ways of identifying black holes in the universe, gravitational waves emitted by an oscillating black hole will carry a unique fingerprint which would lead to the direct identification of their existence. As we mentioned earlier, gravitational radiation from black hole oscillations exhibits certain… | |
| claim-type | critique— “Numerical investigations of the fully nonlinear equations of general relativity have provided results which agree with the results of perturbation calculations; in particular numerical studies of the head-on collision of two black holes (cf. Figure 1) and gravitational collapse to a Kerr hole. Recently, Price, Pullin and collaborators have… | |
| method — evidence | — “A major disadvantage of the evolution schemes is that one cannot get the complete spectrum of the QNMs neither for a star nor for a black hole. The reason is that although any perturbation is the sum of the harmonics involved, in practice only a few of them will be observed and in the best case one can succeed in getting a few extra modes by… | |
| object of study — evidence | — “In the framework of general relativity (GR) quasi-normal modes (QNM) arise, as perturbations (electromagnetic or gravitational) of stellar or black hole spacetimes. Due to the emission of gravitational waves there are no normal mode oscillations but instead the frequencies become “quasi-normal” (complex), with the real part representing the… | |
| research question — evidence | — “A critical issue related to the discussions of the previous sections is the excitation of the QNMs. The truth is that, although the QNMs are predicted by our perturbation equations, it is not always clear which ones will be excited and under what initial conditions. As we have already mentioned in the introduction there is an excellent… | |
| contribution — evidence | — “In the framework of general relativity (GR) quasi-normal modes (QNM) arise, as perturbations (electromagnetic or gravitational) of stellar or black hole spacetimes. Due to the emission of gravitational waves there are no normal mode oscillations but instead the frequencies become “quasi-normal” (complex), with the real part representing the… | |
| transfer | from quantum mechanics: resonance theory from quantum mechanics - alpha decay, poles of a continued Green function, the Bohr-Sommerfeld rule and WKB scattering off a barrier - carried into the calculation of black hole oscillation frequencies— “In general with this approach one can calculate quite accurately the low-lying (relatively small imaginary part) QNM modes, but it fails to give accurate results for higher-order modes.” | transfers[0] |
| transfer | from classical astrophysics: the asteroseismological programme of inferring stellar interiors from an oscillation spectrum, carried from classical astrophysics to neutron stars where the spectrum includes modes of the spacetime— “From tests performed using polytropic stars to provide data for the above relations it was seen that these equations predict the masses and the radii of the polytropes usually with an error less than 10%. There is work underway towards extracting the parameters of the star from a noisy signal.” | transfers[1] |
| mechanism | a barrier that erases the initial data the effective potential peaks near the unstable photon orbit at r = 3M; whatever is imposed inside that radius has to cross the peak to reach an observer, and the peak passes only the ringing. This is why the frequencies are a fingerprint of the object rather than of the disturbance, and why a crude close-limit approximation to a black hole collision predicts the waveform correctly— “That is, for any set of initial data that one can impose, the QNMs will critically depend on the shape of the potential barrier, and this is the reason that the close limit approximation of the two black-hole collision used by Price and Pullin was so successful, because whatever initial data you provide inside the r < 3M region (the peak… | mechanisms[0] |
| mechanism | poles of a continuation, not solutions of the problem the quasi-normal frequencies are the zeros of a Wronskian in the analytic continuation of the Green function to Re(s) < 0. The corresponding functions grow exponentially at both ends and are not physical states; what is physical is that they control the late-time decay in any bounded region— “First of all we should note that because of Re(s) < 0 the function f+ grows exponentially for small and large x! The corresponding spacetime solution is therefore not a physically relevant solution, unlike the normal modes.” | mechanisms[1] |
| mechanism | a method that can only hear what it does not damp away evolving the equations in time and Fourier-transforming the result recovers the modes still present in the signal - the slowest damped ones. Everything with a larger imaginary part has decayed before the window opens, so the method's reach is set by its own arithmetic and not by the spectrum— “A major disadvantage of the evolution schemes is that one cannot get the complete spectrum of the QNMs neither for a star nor for a black hole. The reason is that although any perturbation is the sum of the harmonics involved, in practice only a few of them will be observed and in the best case one can succeed in getting a few extra modes by… | mechanisms[2] |
| mechanism | an approximation with no internal error estimate linearised perturbation theory gives no built-in indication of how good it is; the only quantitative check is to compute the next order and take the difference. So the accuracy of a first-order result is not a property the calculation can report about itself— “A basic feature of linearized perturbation theory is that there is no “built-in” indication of how good the approximation is. But if one has results for a physical quantity to second order in a perturbation parameter, then the difference between the results of the first and second order theory is a quantitative indication of the error in the… | mechanisms[3] |
| critiques | the quadrupole formula as the route to radiated energy - it accounts only for fluid deformations and misses the emission carried by the spacetime modes— “Up to now the general way of calculating the energy emission has been the quadrupole formula. But this formula only accounts for the fluid deformations and motions, and as we have seen this accounts only for a part of the total energy.” | claim_relations[0] |
| critiques | fully relativistic simulation taken on its own - it gives numerical answers without indicating which principles govern the process, and can be simply incorrect or misleading— “Fully relativistic computer simulations give only numerical answers to problems; often these answers do not provide physical understanding of which principles are important, or even what principles govern a given process. Even more, in some cases, simulation results can be simply incorrect or misleading. By closely coupling various… | claim_relations[1] |
| qualifies | the perturbative approach itself - the agreement with nonlinear collision calculations proves its power and indicates its limits in the same sentence— “Numerical investigations of the fully nonlinear equations of general relativity have provided results which agree with the results of perturbation calculations; in particular numerical studies of the head-on collision of two black holes (cf. Figure 1) and gravitational collapse to a Kerr hole. Recently, Price, Pullin and collaborators have… | claim_relations[2] |
| qualifies | the prospect of neutron-star asteroseismology - the empirical relations recover mass and radius to about 10%, and require observing a mode outside the detectors' band— “For the QNM frequencies of stars, there is a lack of appropriate detectors. Laser interferometers will be sensitive enough in the frequency regime of the f-mode but it will be very hard to detect signals in the frequencies of the p- and w-modes. Nevertheless, from the discussion in section 5 it is apparent that there is a wealth of… | claim_relations[3] |
| extends | helioseismology and asteroseismology, whose programme of reading a star's interior from its oscillation spectrum is carried here to objects whose oscillations are of the spacetime as well as the matter— “From tests performed using polytropic stars to provide data for the above relations it was seen that these equations predict the masses and the radii of the polytropes usually with an error less than 10%. There is work underway towards extracting the parameters of the star from a noisy signal.” | claim_relations[4] |
| sense | the QNM spectrum of a potential - FOR ANY POSITIVE MULTIPLE OF THE REGGE-WHEELER POTENTIAL: an infinite number of modes— “However this is a delicate issue as we see if, for example, we multiply the Regge-Wheeler potential by a factor ∊: For any ∊ > 0 we obtain an infinite number of QNMs, for ∊ = 0, however there is no QNM!” | senses[0] |
| sense | the QNM spectrum of a potential - AT MULTIPLIER EXACTLY ZERO: no QNM at all. The count is discontinuous and no precision on the multiplier removes the jump— “However this is a delicate issue as we see if, for example, we multiply the Regge-Wheeler potential by a factor ∊: For any ∊ > 0 we obtain an infinite number of QNMs, for ∊ = 0, however there is no QNM!” | senses[1] |
| sense | the spectrum you obtain - BY EVOLVING THE TIME-DEPENDENT EQUATIONS: at most two frequencies, the slowest damped, whatever the initial data— “A major disadvantage of the evolution schemes is that one cannot get the complete spectrum of the QNMs neither for a star nor for a black hole. The reason is that although any perturbation is the sum of the harmonics involved, in practice only a few of them will be observed and in the best case one can succeed in getting a few extra modes by… | senses[2] |
| sense | the spectrum you obtain - BY CONTINUED FRACTIONS: mode numbers up to 100,000, and no intuition about the spectrum's properties, because the method is purely numerical— “As a final comment on this excellent method we should point out that it has a disadvantage compared to the WKB based methods in that it is a purely numerical method and it cannot provide much intuition about the properties of the QNM spectrum.” | senses[3] |
| sense | stellar QNM information - AS THE THEORY OFFERS IT: mass, radius and equation of state to about 10%, IF the f-mode and the first w-mode are both observed— “From tests performed using polytropic stars to provide data for the above relations it was seen that these equations predict the masses and the radii of the polytropes usually with an error less than 10%. There is work underway towards extracting the parameters of the star from a noisy signal.” | senses[4] |
| sense | stellar QNM information - AS THE DETECTORS CAN TAKE IT: the f-mode band only; the w-modes sit at 5-12 kHz where the interferometers are not sensitive— “An important factor for the detection of gravitational waves are the pulsation mode frequencies. Existing resonant gravitational wave detectors, as well as laser interferometric ones which are under construction, are only sensitive in a certain bandwidth. The spherical and bar detectors are typically tuned to 0.6 – 3 kHz, while the… | senses[5] |
| sense | what QNM properties settle about stability - AGAINST INSTABILITY: nothing. They cannot decide the issue— “If we define stability as boundedness in time for all solutions with data of compact support, properties of quasi-normal modes can not decide the stability issue. However, the appearance of a normal mode proves instability.” | senses[6] |
| sense | what QNM properties settle about stability - FOR INSTABILITY: everything. The appearance of a normal mode proves it— “If we define stability as boundedness in time for all solutions with data of compact support, properties of quasi-normal modes can not decide the stability issue. However, the appearance of a normal mode proves instability.” | senses[7] |
| control | perturbation-theory waveforms compared against fully nonlinear numerical relativity for the head-on collision of two black holes and for collapse to a Kerr hole — held— “Numerical investigations of the fully nonlinear equations of general relativity have provided results which agree with the results of perturbation calculations; in particular numerical studies of the head-on collision of two black holes (cf. Figure 1) and gravitational collapse to a Kerr hole. Recently, Price, Pullin and collaborators have… — rules out the linearised treatment being an artefact of linearisation - the comparison is against a calculation that makes none of its approximations, on the same physical problem— “Numerical investigations of the fully nonlinear equations of general relativity have provided results which agree with the results of perturbation calculations; in particular numerical studies of the head-on collision of two black holes (cf. Figure 1) and gravitational collapse to a Kerr hole. Recently, Price, Pullin and collaborators have… | controls[0] |
| control | second-order perturbation theory proposed as the quantitative error estimate for a first-order calculation, since the difference between the orders is the only available measure — partial— “A basic feature of linearized perturbation theory is that there is no “built-in” indication of how good the approximation is. But if one has results for a physical quantity to second order in a perturbation parameter, then the difference between the results of the first and second order theory is a quantitative indication of the error in the… — rules out a first-order result being reported without any indication of its accuracy - the method has no internal one and the authors say so before proposing this— “A basic feature of linearized perturbation theory is that there is no “built-in” indication of how good the approximation is. But if one has results for a physical quantity to second order in a perturbation parameter, then the difference between the results of the first and second order theory is a quantitative indication of the error in the… | controls[1] |
| control | five independent numerical routes to the same frequencies - time evolution, integration of the time-independent equation, WKB and phase-integral, continued fractions, and resonance searching - each reported with the part of the spectrum it cannot reach — held— “In general with this approach one can calculate quite accurately the low-lying (relatively small imaginary part) QNM modes, but it fails to give accurate results for higher-order modes.” — rules out any one method's blind spot being mistaken for a property of the spectrum; the modes WKB loses are the ones continued fractions find, and vice versa— “In general with this approach one can calculate quite accurately the low-lying (relatively small imaginary part) QNM modes, but it fails to give accurate results for higher-order modes.” | controls[2] |
| control | Nollert's continued-fraction calculation carried to mode number 100,000, on a method whose convergence for quasi-normal modes is not obvious because the relevant solution grows exponentially — held— “In the case of quasi-normal modes this is not so clear because f+ grows exponentially. Nevertheless, the method works very well numerically and was improved by Nollert such that he was able to calculate very high mode numbers (up to 100, 000!). In this way he obtained the asymptotic distributions of modes described in (31).” — rules out the asymptotic mode distribution being an extrapolation from the few modes the other methods reach— “In the case of quasi-normal modes this is not so clear because f+ grows exponentially. Nevertheless, the method works very well numerically and was improved by Nollert such that he was able to calculate very high mode numbers (up to 100, 000!). In this way he obtained the asymptotic distributions of modes described in (31).” | controls[3] |
| control | the potential barrier at r = 3M treated as a filter on initial data, which is why different initial conditions give the same observed ringing — absent-declared-unreported — rules out the initial data as an explanation of the ringing - the potential barrier at r = 3M filters whatever is imposed, which is why different initial conditions give the same modes— “That is, for any set of initial data that one can impose, the QNMs will critically depend on the shape of the potential barrier, and this is the reason that the close limit approximation of the two black-hole collision used by Price and Pullin was so successful, because whatever initial data you provide inside the r < 3M region (the peak… | controls[4] |
| control | the Cowling approximation, which freezes the spacetime to isolate the fluid modes, and the inverse Cowling approximation, which freezes the fluid to isolate the spacetime modes — absent-declared-unreported — rules out the coupling between fluid and spacetime - the Cowling approximation freezes the spacetime to leave the fluid modes alone, and the inverse freezes the fluid to leave the spacetime modes— “From the form of the above equations one can easily see two limiting cases. Let us first assume that the gravitational field is very weak. Then equation and can be omitted (actually S → 0 in the weak field limit) and we find that one equation is enough to describe (with acceptable accuracy) the oscillations of the fluid. This approach is… | controls[5] |
| weight | the quadrupole formula as the route to radiated energy - it accounts only for fluid deformations and misses the emission carried by the spacetime modes — 0.95— “Up to now the general way of calculating the energy emission has been the quadrupole formula. But this formula only accounts for the fluid deformations and motions, and as we have seen this accounts only for a… | claim_relations[0] |
| weight | fully relativistic simulation taken on its own - it gives numerical answers without indicating which principles govern the process, and can be simply incorrect or misleading — 0.95 | claim_relations[1] |
| weight | the perturbative approach itself - the agreement with nonlinear collision calculations proves its power and indicates its limits in the same sentence — 0.95 Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | claim_relations[2] |
| weight | the prospect of neutron-star asteroseismology - the empirical relations recover mass and radius to about 10%, and require observing a mode outside the detectors' band — 0.95 Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | claim_relations[3] |
| weight | helioseismology and asteroseismology, whose programme of reading a star's interior from its oscillation spectrum is carried here to objects whose oscillations are of the spacetime as well as the matter — 0.9 Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | claim_relations[4] |
| audit | quasi-normal modes —distinct-from→ completeness — Audit 2026-09-28: DOUBTFUL - read against its span and kept, but the span sets QNM against the completeness of normal modes; distinct-from is loose. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[2] |
| differs in (stated by the tagger, no quotation) | quasi-normal modes —distinct-from→ completeness — whether a general solution can be written as a sum of the modes | concept_edges[2] |
| relation | quasi-normal modes —distinct-from→ completeness — the spectral theorem gives normal modes a convergent expansion and no similar result can be expected here Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[2] |
| weight | quasi-normal modes —distinct-from→ completeness — 0.95 Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[2] |
| outcome | control — held Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | controls[0] |
| outcome | control — partial Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | controls[1] |
| outcome | control — held Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | controls[2] |
| outcome | control — held Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | controls[3] |
| outcome (stated by the tagger, no quotation) | control — absent-declared-unreported Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | controls[4] |
| outcome (stated by the tagger, no quotation) | control — absent-declared-unreported Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | controls[5] |
| limitation | whether the late-time solution can be expanded pointwise in quasi-normal modes is unresolved, and has been shown only for one particular potential Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[0] |
| limitation | no spectral theorem applies, so the completeness that normal modes enjoy cannot be expected for QNMs Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[1] |
| limitation | properties of quasi-normal modes cannot decide stability; only the appearance of a normal mode settles the question, and then only in the direction of instability Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[2] |
| limitation | the stability of Kerr is made very plausible but is not established as conclusively as for Schwarzschild, and it is not known whether all solutions stay bounded in time Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[3] |
| limitation | the Kerr-Newman spectrum has not been calculated at all, because the perturbation equations do not separate Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[4] |
| limitation | time-domain evolution cannot return the complete spectrum - at most the two slowest damped frequencies survive into the signal Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[5] |
| limitation | WKB is accurate for low-lying modes and fails for higher-order ones Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[6] |
| limitation | the continued-fraction method reaches very high mode numbers and, being purely numerical, gives no intuition about the spectrum's properties Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[7] |
| limitation | direct numerical integration for stars cannot reach the w-modes Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[8] |
| limitation | the detectors are sensitive in the f-mode band while the p- and w-modes carrying the rest of the information sit above it, and the authors call for detectors in that range Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[9] |
| limitation | which modes are excited, and by how much, is not determined by the theory - it depends on initial data that are themselves arbitrary until a full nonlinear calculation supplies them Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[10] |
| limitation | the energy released as gravitational radiation in stellar collapse is basically unknown, as is its distribution among the modes Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[11] |
| limitation | linearised perturbation theory carries no built-in indication of its own accuracy Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[12] |
| limitation | trapped modes require a star more compact than any realistic equation of state produces, so that family is of no observational relevance Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[13] |
| limitation | whether the algebraically special purely imaginary frequencies are QNMs at all is still questioned Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[14] |
finding · quantity · variable · definition · key-term — grounded in the full text| finding | Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[0] |
| finding | Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[1] |
| finding | Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[2] |
| finding | Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[3] |
| finding | Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[4] |
| finding | Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[5] |
| finding | Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[6] |
| finding | Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[7] |
| finding | Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[8] |
| quantity | the first w-mode frequency for a typical neutron star - above every detector band named in the same paragraph Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.quantities[0] |
| quantity | the interferometers' sensitive band, which covers the f-mode and not the p- or w-modes Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.quantities[1] |
| quantity | the error with which the empirical relations recover a polytrope's mass and radius from its f-mode and first w-mode Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.quantities[2] |
| variable | quasi-normal modes Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.variables[0] |
| variable | completeness Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.variables[1] |
| definition | quasi-normal modes Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[0] |
| definition | w-modes Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[1] |
| concept | quasi-normal modes — not defined in this paper's own words; complex characteristic frequencies of an open system losing energy to infinity - real part the oscillation, imaginary part the damping; poles of an analytically continued Green function Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[0] |
| concept | normal modes — not defined in this paper's own words; the real-frequency eigenmodes of a closed system, with a spectral theorem behind them and square-integrable eigenfunctions - the thing QNMs are named after and are not Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[1] |
| concept | w-modes — not defined in this paper's own words; spacetime modes of a relativistic star - rapidly damped, hardly exciting any fluid motion, and closer to black hole QNMs than to the star's own fluid modes Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[2] |
| concept | f-mode — not defined in this paper's own words; the fundamental fluid mode, 1.5-3 kHz for a typical neutron star and the only stellar QNM in the band the interferometers actually cover Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[3] |
| concept | trapped modes — not defined in this paper's own words; slowly damped modes existing only when the stellar surface lies inside the potential peak - and no realistic equation of state makes a star that compact Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[4] |
| concept | r-modes — not defined in this paper's own words; rotational modes, generically unstable to gravitational radiation by the Chandrasekhar-Friedman-Schutz mechanism - the instability that explains why supernova remnants hold slow pulsars Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[5] |
| concept | Kerr — not defined in this paper's own words; the rotating black hole solution; its perturbation operator is not elliptic in the ergosphere, and whether all its solutions stay bounded in time is stated here as unknown Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[6] |
| concept | completeness — not defined in this paper's own words; the property that lets a general solution be expanded in modes; guaranteed for normal modes by the spectral theorem and, the authors say, not to be expected for QNMs Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[7] |
| concept | potential barrier — not defined in this paper's own words; the peak of the effective potential near r = 3M, which filters whatever initial data are imposed inside it so that an outside observer sees only the ringing Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[8] |
| concept | evolution schemes — not defined in this paper's own words; time-domain integration of the perturbation equations - free of boundary-eigenvalue difficulties, and structurally unable to return more than the slowest-damped modes Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[9] |
| concept | quadrupole formula — not defined in this paper's own words; the standard route to radiated energy, which accounts only for fluid deformations and therefore misses the part of the emission carried by the spacetime modes Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[10] |
| concept | radiative tail — not defined in this paper's own words; the power-law decay that swamps the ringing at late times, produced by the slow fall-off of the potential rather than by the object Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[11] |
When we have changed our mind. Some entries in the table begin with a word in capitals and the reason for it. EVIDENCE CORRECTED — the finding still stands, but we had quoted the wrong sentence from the paper, and the quotation has been replaced with the right one. RETRACTED — the entry itself was wrong; it stays on the page, marked, with the reason, rather than being deleted. WITHDRAWN AND REPLACED — the entry was wrong, and a corrected version of it appears in the entries below. Each mark carries its own reason, in brackets, because the reasons differ from one entry to the next. An entry we have WITHDRAWN is removed from this page and kept in the record, with its reason. A line reading Audit and a date is a later re-reading of the entry against its whole sentence: DOUBTFUL means it was kept, and the reason says what is loose about it. Quotation marks inside the tagger's sentence. Where a tag is the tagger's own sentence, a phrase in quotation marks inside it is borrowed from the paper — the tagger's wording around it is not. The paper's own sentence is the evidence quote for that row; on this page it is withheld with the article, and the article is at its publisher. Nothing inside quotation marks anywhere on this page is ours: if it is quoted, it is the paper's. Whose words a concept carries. A concept entry records a term the paper itself uses, so the term, the sense given for it, and its evidence are the paper's own words, quoted where the paper fixes them. This is the opposite of a sense entry in the tag table, which is the tagger's sentence about how the paper uses a term. Where a concept entry could not be grounded in the paper's words, the entry says so on its own row.
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