Quasi-Normal Modes of Stars and Black Holes

Quasi-Normal Modes of Stars and Black Holes Kostas D. Kokkotas Bernd G. Schmidt

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Quasi-Normal Modes of Stars and Black Holes 1999 Living Reviews in Relativity 10.12942/lrr-1999-2 kokkotas1999-quasi-normal-modes-of-stars-and-bl kokkotas1999-quasi-normal-modes-of-stars-and-bl.txt, the project's staged copy 754edf18d75283265afcf721f24dadf902f78b96 cc-by physics

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0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them the complex characteristic frequencies at which a perturbed black hole or relativistic star rings down, and what a detected ringdown could be made to say about the object that produced it Quotation withheld because this record is not cleared for redistribution. which families of quasi-normal modes exist for black holes and for stars, whether they can be excited and detected, and what the perturbative approach can and cannot establish Quotation withheld because this record is not cleared for redistribution. a review of the theory of quasi-normal modes from the mathematical and the astrophysical side at once - the definition via poles of an analytically continued Green function, the families of black hole and stellar modes, the numerical techniques and, unusually, what each of them cannot see Quotation withheld because this record is not cleared for redistribution. critique Quotation withheld because this record is not cleared for redistribution. ringdown - a compact object losing energy to infinity oscillates at complex frequencies fixed by its own parameters and not by whatever disturbed it Quotation withheld because this record is not cleared for redistribution. review review Quotation withheld because this record is not cleared for redistribution. 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 a barrier that erases the initial data Quotation withheld because this record is not cleared for redistribution. 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 poles of a continuation, not solutions of the problem Quotation withheld because this record is not cleared for redistribution. 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 method that can only hear what it does not damp away Quotation withheld because this record is not cleared for redistribution. 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 an approximation with no internal error estimate Quotation withheld because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. no spectral theorem applies, so the completeness that normal modes enjoy cannot be expected for QNMs Quotation withheld because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. the Kerr-Newman spectrum has not been calculated at all, because the perturbation equations do not separate Quotation withheld because this record is not cleared for redistribution. time-domain evolution cannot return the complete spectrum - at most the two slowest damped frequencies survive into the signal Quotation withheld because this record is not cleared for redistribution. WKB is accurate for low-lying modes and fails for higher-order ones Quotation withheld because this record is not cleared for redistribution. the continued-fraction method reaches very high mode numbers and, being purely numerical, gives no intuition about the spectrum's properties Quotation withheld because this record is not cleared for redistribution. direct numerical integration for stars cannot reach the w-modes Quotation withheld because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. the energy released as gravitational radiation in stellar collapse is basically unknown, as is its distribution among the modes Quotation withheld because this record is not cleared for redistribution. linearised perturbation theory carries no built-in indication of its own accuracy Quotation withheld because this record is not cleared for redistribution. trapped modes require a star more compact than any realistic equation of state produces, so that family is of no observational relevance Quotation withheld because this record is not cleared for redistribution. whether the algebraically special purely imaginary frequencies are QNMs at all is still questioned Quotation withheld because this record is not cleared for redistribution. 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 Quotation withheld because this record is not cleared for redistribution. 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 Quotation withheld because this record is not cleared for redistribution. 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 eliminates: 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 · outcome: held Quotation withheld because this record is not cleared for redistribution. 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 eliminates: 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 · outcome: partial Quotation withheld because this record is not cleared for redistribution. 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 eliminates: 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 · outcome: held Quotation withheld because this record is not cleared for redistribution. 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 eliminates: the asymptotic mode distribution being an extrapolation from the few modes the other methods reach · outcome: held Quotation withheld because this record is not cleared for redistribution. the potential barrier at r = 3M treated as a filter on initial data, which is why different initial conditions give the same observed ringing eliminates: 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 · outcome (stated by the tagger, no quotation): absent-declared-unreported Quotation withheld because this record is not cleared for redistribution. 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 eliminates: 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 · outcome (stated by the tagger, no quotation): absent-declared-unreported Quotation withheld because this record is not cleared for redistribution. quasi-normal modes quasi-normal modes Quotation withheld because this record is not cleared for redistribution. normal modes normal modes Quotation withheld because this record is not cleared for redistribution. w-modes w-modes Quotation withheld because this record is not cleared for redistribution. f-mode f-mode Quotation withheld because this record is not cleared for redistribution. trapped modes trapped modes Quotation withheld because this record is not cleared for redistribution. r-modes r-modes Quotation withheld because this record is not cleared for redistribution. Kerr Kerr Quotation withheld because this record is not cleared for redistribution. completeness completeness Quotation withheld because this record is not cleared for redistribution. potential barrier potential barrier Quotation withheld because this record is not cleared for redistribution. evolution schemes evolution schemes Quotation withheld because this record is not cleared for redistribution. quadrupole formula quadrupole formula Quotation withheld because this record is not cleared for redistribution. radiative tail radiative tail Quotation withheld because this record is not cleared for redistribution. the QNM spectrum of a potential - FOR ANY POSITIVE MULTIPLE OF THE REGGE-WHEELER POTENTIAL: an infinite number of modes Quotation withheld because this record is not cleared for redistribution. 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 Quotation withheld because this record is not cleared for redistribution. the spectrum you obtain - BY EVOLVING THE TIME-DEPENDENT EQUATIONS: at most two frequencies, the slowest damped, whatever the initial data Quotation withheld because this record is not cleared for redistribution. 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 Quotation withheld because this record is not cleared for redistribution. 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 Quotation withheld because this record is not cleared for redistribution. 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 Quotation withheld because this record is not cleared for redistribution. what QNM properties settle about stability - AGAINST INSTABILITY: nothing. They cannot decide the issue Quotation withheld because this record is not cleared for redistribution. what QNM properties settle about stability - FOR INSTABILITY: everything. The appearance of a normal mode proves it Quotation withheld because this record is not cleared for redistribution. 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 Quotation withheld because this record is not cleared for redistribution. 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 Quotation withheld because this record is not cleared for redistribution. qualifies → the perturbative approach itself - the agreement with nonlinear collision calculations proves its power and indicates its limits in the same sentence Quotation withheld because this record is not cleared for redistribution. 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 Quotation withheld because this record is not cleared for redistribution. 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 Quotation withheld because this record is not cleared for redistribution. quasi-normal modes —distinct-from→ completeness quasi-normal modes completeness differs in: whether a general solution can be written as a sum of the modes · the relation itself is separately evidenced · strength of the relation: 0.95 · differs in (stated by the tagger, no quotation): whether a general solution can be written as a sum of the modes · the relation stated: the spectral theorem gives normal modes a convergent expansion and no similar result can be expected here 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 because this record is not cleared for redistribution. multiplying the Regge-Wheeler potential by any positive factor gives infinitely many QNMs, and at factor zero there are none - the count is discontinuous at zero multiplying the Regge-Wheeler potential by any positive factor gives infinitely many QNMs, and at factor zero there are none - the count is discontinuous at zero Quotation withheld because this record is not cleared for redistribution. quasi-normal mode properties cannot decide stability, while the appearance of a normal mode proves instability - the test is one-sided quasi-normal mode properties cannot decide stability, while the appearance of a normal mode proves instability - the test is one-sided Quotation withheld because this record is not cleared for redistribution. A black-hole evolution signal shows at most the two slowest-damped frequencies, so the rest of the spectrum is missing. A black-hole evolution signal shows at most the two slowest-damped frequencies, so the rest of the spectrum is missing. Quotation withheld because this record is not cleared for redistribution. the interferometers cover the f-mode band while the p- and w-modes carrying the rest of the stellar information lie above it, and the authors ask for detectors there the interferometers cover the f-mode band while the p- and w-modes carrying the rest of the stellar information lie above it, and the authors ask for detectors there Quotation withheld because this record is not cleared for redistribution. The potential barrier near r = 3M filters any initial data inside it, so an outside observer sees only the QNM ringing. The potential barrier near r = 3M filters any initial data inside it, so an outside observer sees only the QNM ringing. Quotation withheld because this record is not cleared for redistribution. the quasi eigenfunctions grow exponentially at both ends and are not physically relevant solutions, unlike normal modes the quasi eigenfunctions grow exponentially at both ends and are not physically relevant solutions, unlike normal modes Quotation withheld because this record is not cleared for redistribution. Linearized perturbation theory has no built-in indication of its accuracy; the difference from a second-order result quantifies the error. Linearized perturbation theory has no built-in indication of its accuracy; the difference from a second-order result quantifies the error. Quotation withheld because this record is not cleared for redistribution. massive scalar perturbations of Kerr are unstable, with growth times comparable to the age of the universe massive scalar perturbations of Kerr are unstable, with growth times comparable to the age of the universe Quotation withheld because this record is not cleared for redistribution. the r-mode instability slows a rapidly rotating neutron star within a year, which explains why only slowly rotating pulsars are associated with supernova remnants the r-mode instability slows a rapidly rotating neutron star within a year, which explains why only slowly rotating pulsars are associated with supernova remnants Quotation withheld because this record is not cleared for redistribution. perturbations of a stellar or black hole spacetime whose frequencies are complex because energy is radiated away - real part the oscillation frequency, imaginary part the damping quasi-normal modes Quotation withheld because this record is not cleared for redistribution. spacetime modes of a relativistic star: high frequency, very rapidly damped, hardly exciting any fluid motion, and closer to black hole QNMs than to the star's own fluid modes w-modes Quotation withheld because this record is not cleared for redistribution. the first w-mode frequency for a typical neutron star - above every detector band named in the same paragraph 5 – 12 kHz Quotation withheld because this record is not cleared for redistribution. the interferometers' sensitive band, which covers the f-mode and not the p- or w-modes 10 – 2000 Hz Quotation withheld because this record is not cleared for redistribution. the error with which the empirical relations recover a polytrope's mass and radius from its f-mode and first w-mode 10% Quotation withheld because this record is not cleared for redistribution. quasi-normal modes quasi-normal modes Quotation withheld because this record is not cleared for redistribution. completeness completeness Quotation withheld because this record is not cleared for redistribution.

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