Quasinormal-Mode Waveform Expansion

A quasinormal-mode (QNM) waveform is the pole part of a retarded black-hole response, expressed over a finite interval in which that part is useful. It is not an ordinary complete basis expansion of an arbitrary merger waveform. This page derives the damped-sinusoid representation, checks its signs and boundary conditions, and shows how spherical–spheroidal mixing and the fitting window limit its interpretation. Black-Hole Perturbation Theory supplies the spectrum and Green-function poles; here the task is to turn those inputs into a convention-controlled waveform model.

Helpful background. Einstein Equations supplies the linearization about a stationary remnant; Waves, Green Functions, and Geometric Optics supplies retarded response and contour deformation; Black-Hole Perturbation Theory supplies QNM spectra and radial boundary-value problems; Gravitational Radiation and Waveforms supplies the asymptotic strain; and Black-Hole Dynamics and Collapse supplies the formation and relaxation of the remnant.

The radiative field and remnant frame

For the sustained remnant calculation on this page, use geometrized units $G=c=1$: $M_f$ denotes the remnant mass as a length or time. To restore dimensions, a time $bM_f$ means $bGM_{f,\mathrm{phys}}/c^3$ and a frequency quoted as $M_f\omega=\widehat\omega$ means $\omega=\widehat\omega c^3/(GM_{f,\mathrm{phys}})$. The reference background is an asymptotically flat, uncharged, subextremal Kerr black hole, $0\le\chi_f\lt1$.

At future null infinity use the complex, radius-rescaled strain

$$H(u,\theta,\phi)=r\bigl(h_+-i h_\times\bigr),$$

in an inertial Bondi frame whose spatial axes are aligned with the late-time remnant spin. Retarded time is $u$, and the stationary reference black hole has mass $M_f$, dimensionless spin $\chi_f$, and Kerr parameter $a_f=\chi_f M_f$. The time dependence is $e^{-i\omega u}$; therefore a stable mode has $\operatorname{Im}\omega\lt 0$. Frequencies will usually be quoted as the dimensionless products $M_f\omega$.

Let ${}_{-2}S_{Lm}(\theta,\phi;a_f\omega)$ denote a right spin-$-2$ spheroidal eigenfunction. Fix its otherwise arbitrary phase and impose

$$\int_{S^2}\left|{}_{-2}S_{Lm}\right|^2d\Omega=1.$$

This normalization is part of the amplitude convention. Because $a_f\omega$ is complex, the angular problem is non-Hermitian; unit norm does not justify treating distinct right eigenfunctions as an orthogonal basis.

From a response pole to a damped mode

For a separated retarded Green function, suppose the frequency-domain response near a simple QNM pole has the local form

$$\widetilde G(\omega;x,x')= \frac{\mathcal R_q(x,x')}{\omega-\omega_q} +\text{regular terms}.$$

Deforming the inverse-transform contour into the lower half-plane for positive elapsed time picks up the residue. Numerical constants and the source integral can be absorbed into a complex amplitude $A_q$, leaving $A_qe^{-i\omega_q(u-u_0)}$. Summing a selected set $\mathcal Q$ gives the finite pole model

$$H_{\rm QNM}(u,\theta,\phi;u_0) = \sum_{q=(L,m,n)\in\mathcal Q} A_q(u_0)e^{-i\omega_q(u-u_0)} {}_{-2}S_{Lm}(\theta,\phi;a_f\omega_q), \qquad u\ge u_0.$$

Here $n=0$ labels the least-damped member of a fixed angular family and $n\gt 0$ its more rapidly damped overtones. The pole position depends only on the chosen stationary background and boundary conditions; $A_q$ also depends on source history, initial data, field normalization, frame, and reference time. The Green-function construction and this separation between poles and excitation are developed in Berti, Cardoso, and Starinets 2009, § 3.2, pp. 18–20, PDF.

The chosen set must also state which frequency branches it contains. For Kerr, the symmetry maps a mode at $(m,\omega)$ to a mirror solution at $(-m,-\omega^*)$; both have negative imaginary part. A waveform may excite both branches, and generic precessing data do not justify fixing one amplitude by conjugating another. The displayed $(L,m,n)$ notation suppresses a branch label; when both branches are retained, include that label in $q$ and sum them independently. The spin-weighted complex field $H$ itself is not a real scalar that could supply such an extra reality constraint.

Changing the reference time does not create a new physical amplitude. The same single mode written about $u_0'$ has

$$A_q(u_0')=A_q(u_0)e^{-i\omega_q(u_0'-u_0)}.$$

Consequently, start-time comparisons must either transport amplitudes to one common reference time or explicitly divide out the known damping and phase rotation.

Boundary and damping sign check

First consider a Schwarzschild master field, with $r_*$ increasing from the future horizon to infinity. With the inherited $e^{-i\omega t}$ convention, its radiative phase obeys

$$\Psi\sim \begin{cases} e^{-i\omega(t+r_*)}=e^{-i\omega v}, & r_*\to-\infty,\\[3pt] e^{-i\omega(t-r_*)}=e^{-i\omega u}, & r_*\to+\infty. \end{cases}$$

The first line is ingoing in advanced time $v=t+r_*$; the second is outgoing in retarded time $u=t-r_*$. Kerr requires the corresponding horizon rotation. A separated mode $e^{-i\omega t+im\phi}R(r)$ has ingoing radial phase

$$R\propto e^{-i(\omega-m\Omega_H)r_*}, \qquad \Omega_H=\frac{a_f}{r_+^2+a_f^2}, \qquad r_+=M_f+\sqrt{M_f^2-a_f^2}.$$

In regular ingoing Kerr coordinates, $v=t+r_*$ and $\widetilde\phi=\phi+\int a_f\,dr/\Delta$ with $\Delta=r^2-2M_fr+a_f^2$, the full phase becomes $e^{-i\omega v+im\widetilde\phi}$ at the horizon. Along its generators the frequency is $\omega-m\Omega_H$. Teukolsky radial functions additionally have spin-dependent powers (for spin $s$, an ingoing factor $\Delta^{-s}$), which must be combined with a regular tetrad before testing field regularity. These factors do not change which phase is ingoing; the separated Kerr operator is given in Berti, Cardoso, and Starinets 2009, §2.3, eqs. (24)–(26), p. 16, PDF.

At infinity, the time factor remains $e^{-i\omega u}$. Writing $\omega=\omega_R+i\omega_I$ gives

$$e^{-i\omega\Delta u} =e^{-i\omega_R\Delta u}e^{\omega_I\Delta u}.$$

Thus temporal damping for $\Delta u\gt 0$ requires $\omega_I\lt 0$, with decay time $\tau=-1/\omega_I$. These boundary conditions and the lower-half-plane sign are stated explicitly in Berti, Cardoso, and Starinets 2009, § 3.1, pp. 17–18, PDF. They also explain a subtle limitation: after separating off $e^{-i\omega t}$, the radial factor grows exponentially at both asymptotic ends for a damped QNM. QNMs are resonances, not square-integrable normal modes.

What a spherical waveform mode contains

Numerical and analytical waveforms are often stored in unit-normalized spin-weighted spherical harmonics,

$$H=\sum_{\ell m}H_{\ell m}(u)\,{}_{-2}Y_{\ell m}(\theta,\phi).$$

Expanding each spheroidal function as

$${}_{-2}S_{Lm}(a_f\omega_q) =\sum_{\ell}\mu^m_{\ell L}(a_f\omega_q)\,{}_{-2}Y_{\ell m}$$

yields

$$H_{\ell m}^{\rm QNM}(u) =\sum_{L,n}\mu^m_{\ell L}(a_f\omega_{Lmn}) A_{Lmn}(u_0)e^{-i\omega_{Lmn}(u-u_0)}.$$

A frequency found in a spherical $(\ell,m)$ channel therefore need not carry spheroidal label $L=\ell$. This is angular-basis leakage, not nonlinear frequency generation. The normalization and overlap convention must accompany every reported amplitude; see Spherical–Spheroidal Mode Mixing for the full dictionary.

Finite windows and the completeness limit

A causal black-hole response is schematically

$$H=H_{\rm prompt}+H_{\rm poles}+H_{\rm tail}.$$

The prompt term comes from the high-frequency part of the contour, QNMs from isolated poles, and late-time tails from branch-cut structure. This contour anatomy is reviewed in Berti, Cardoso, and Starinets 2009, § 3.2, pp. 18–20, PDF. A truncated QNM sum can approximate their combined waveform over some interval without proving that the full response equals a discrete sum.

For a detector-independent diagnostic, define the unweighted finite-window residual

$$\epsilon^2(u_0,u_1)= \frac{\displaystyle\int_{u_0}^{u_1}du\int d\Omega\, \left|H-H_{\rm QNM}\right|^2} {\displaystyle\int_{u_0}^{u_1}du\int d\Omega\,|H|^2}.$$

Vary both endpoints, the mode set, and the numerical resolution. Early-window drift can reveal prompt response, merger dynamics, or an evolving background; late-window drift can reveal tails or a numerical noise floor. A smaller residual alone does not identify each fitted term with a unique pole. Detector likelihoods and empirical spectroscopy claims lie outside this page and belong to Gravitational-Wave Astronomy.

Exercise: diagnose one damped term

A fitted mode has $M_f\omega=0.50-0.08i$. Find its damping time, its amplitude and phase change over $\Delta u=10M_f$, and the coefficient transformation when the reference time is shifted forward by $5M_f$. Then state why the result does not establish completeness.

Solution

The decay time is $\tau/M_f=1/0.08=12.5$. Over $10M_f$,

$$e^{-i\omega\Delta u}=e^{-i5}e^{-0.8},$$

so the magnitude is multiplied by $e^{-0.8}\simeq0.449$ and the phase changes by $-5$ radians. With $u_0'=u_0+5M_f$,

$$A(u_0')=A(u_0)e^{-i(0.50-0.08i)5},$$

whose magnitude is $e^{-0.4}|A(u_0)|\simeq0.670|A(u_0)|$ and whose phase is shifted by $-2.5$ radians. Transporting this later coefficient back to $u_0$ recovers the original coefficient exactly. The calculation checks the lower-half-plane damping sign and reference-time bookkeeping, but it says nothing about prompt or branch-cut contributions; therefore it cannot prove that a discrete mode set is complete.

References

• Emanuele Berti, Vitor Cardoso, and Andrei O. Starinets, “Quasinormal Modes of Black Holes and Black Branes,” Classical and Quantum Gravity 26 (2009) 163001. DOI. Open PDF.