Modes, Harmonics, and Waveform Conventions

Waveform modes become interoperable only when their angular basis, frame, phase, time, and curvature conventions are specified together. This chapter fixes one coherent dictionary for the spin-$-2$ strain $h_+-ih_\times$, unit-normalized spherical harmonics, strain–news–curvature derivatives, Wigner rotations, precessing frames, BMS transformations, and spherical–spheroidal overlaps. It explains how representations change without treating a basis coefficient as an invariant observable. Source-specific dynamical models and detector calibration remain outside the chapter.

Helpful background. Spacetime Symmetries and Killing Fields supplies rotations and mode generators; Einstein Equations supplies radiative curvature; Waves, Green Functions, and Geometric Optics supplies outgoing modes and Fourier analysis; and Gravitational Radiation and Waveforms supplies strain, news, and asymptotic frames.

Enter the modal dictionary

The pages proceed from the angular basis to increasingly structured frame changes:

1. Spin-Weighted Harmonics for Waveforms fixes spin weight, Condon–Shortley and dyad phases, orthogonality, conjugation, and eth ladders.
2. Strain and Curvature Mode Decompositions projects strain, news, and $\Psi_4$ onto one basis and derives their time-derivative relations.
3. Source, Inertial, and Radiation Frames distinguishes Bondi, inertial, source-attached, radiation-axis, coprecessing, corotating, and screen frames.
4. Phase, Time, and Sign Conventions gives a complete crosswalk for complex strain, Fourier kernels, retarded time, harmonic phases, and curvature signs.
5. Precessing Frames and Mode Mixing separates unitary rotational mixing from intrinsic waveform evolution and imposes minimal rotation.
6. Spherical and Spheroidal Ringdown Modes explains why a spherical mode can contain several spheroidal quasinormal frequencies.

Readers comparing two waveform files should read pages 1–4 before attempting an alignment. Ringdown comparisons then add page 6; time-dependent source or radiation axes require page 5.

The convention card

Choose a complex unit-sphere dyad with $m^A\bar m_A=1$. The rescaled strain is

$$H=C_{AB}\bar m^A\bar m^B =\lim_{r\to\infty}r(h_+-ih_\times).$$

Under $m^A\mapsto e^{i\chi}m^A$, one has $H\mapsto e^{-2i\chi}H$, so

$$H=\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell} H_{\ell m}\,{}_{-2}Y_{\ell m}.$$

The harmonics use $e^{+im\phi}$, the declared Condon–Shortley phase, and unit-sphere normalization. Their definitions and rotation identities follow Goldberg et al. 1967, pp. 2155–2161.

Using Bondi retarded time $u$ ($u=t-r$ in flat spacetime), the Fourier pair uses $e^{+i\omega u}$ forward and $e^{-i\omega u}$ inverse. Thus positive-frequency outgoing waves have $e^{-i\omega u}$ and

$$\mathcal N=\partial_uH, \qquad \Psi_4^0=\partial_u^2H,$$ $$\widetilde{\mathcal N}=-i\omega\widetilde H, \qquad \widetilde\Psi_4^0=-\omega^2\widetilde H.$$

Here the tetrad order is $(\ell,n,m,\bar m)$, with asymptotic normalization $\ell=(e_t+e_r)/\sqrt2$, $n=(e_t-e_r)/\sqrt2$, and $m=(e_\theta+i e_\phi)/\sqrt2$ in an orthonormal frame. Thus $\ell\!\cdot n=-1$ and $\Psi_4=-C_{abcd}n^a\bar m^bn^c\bar m^d$. The symmetric normalization matters: a tetrad boost preserving $\ell\!\cdot n$ would still rescale $\Psi_4$. Bishop and Rezzolla give this extraction dictionary in Bishop and Rezzolla 2016, pp. 22–24, eqs. (95)–(105), PDF.

Recognize the transformation by its mixing

Transformation	Angular action	Essential extra information
Rigid active rotation $R$	Mixes $m$ at fixed $\ell$ through $D^\ell_{m m'}(R)$	Active direction, Wigner index order, dyad phase
Passive axes rotation	Uses $D^\ell(R^{-1})=D^\ell(R)^\dagger$	Which axes moved and relative to what
Time-dependent rotation	Same instantaneous mixing, but news includes $\dot D H$	Angular velocity and residual axial gauge
Translation, boost, or supertranslation	Can mix $\ell$, angles, retarded time, and spin/conformal weights	Full Bondi frame and BMS map
Spherical–spheroidal change	Mixes spherical $\ell$ at fixed $m$ through $\mu^m_{\ell L}(a\omega)$	Spin, complex frequency, remnant $a$, normalization

Boyle derives the full BMS waveform action and distinguishes it from a simple rotation Boyle 2016, §§II–VI, pp. 3–19, PDF. Boyle and collaborators develop rotating radiation frames and show which precessing asymmetries survive them Boyle et al. 2014, §§II–VI, PDF. The spheroidal angular equation and its spherical limit are reviewed in Berti, Cardoso, and Starinets 2009, p. 16, eq. (25), PDF.

Frame and derivative checks

A valid rigid rotation preserves

$$\sum_m\lvert H^R_{\ell m}\rvert^2 =\sum_m\lvert H_{\ell m}\rvert^2.$$

For a time-dependent rotation, the inertial news is

$$N^I=\dot D\,H^C+D\,\dot H^C,$$

not merely $D\dot H^C$. A minimal-rotation coprecessing frame removes angular velocity about its tracked radiation axis; a corotating frame deliberately adds such an axial rotation to follow waveform phase.

Curvature integration requires two constants per mode. A constant strain offset changes neither news nor curvature; a linear strain drift changes news but not curvature; physical memory is an endpoint difference in strain. At zero frequency, division by $\omega^2$ cannot distinguish these pieces without boundary information.

For spheroidal modes, the decisive check is

$$\mu^m_{\ell L}(0)=\delta_{\ell L}.$$

When quasinormal $\omega$ is complex, the angular operator is non-Hermitian and ordinary right-eigenfunction orthogonality must not be assumed. Use declared left/right normalization or retain the spherical overlap matrix directly.

Review the chapter

In a coprecessing frame, let $H^C_{22}=Ae^{-i\omega u}$ be the only mode. The inertial frame is related by the active rotation $R_z(\alpha(u))$. Find the rotated strain and news. In a separate example in a fixed remnant frame, a single spheroidal mode has time dependence $Ae^{-i\omega u}$ and angular expansion $S_{22}=\sqrt{1-\varepsilon^2}Y_{22}+\varepsilon Y_{32}$ for real $\varepsilon$. State where that frequency appears after spherical projection. All harmonics here have spin weight $-2$.

Solution

The active Wigner phase gives

$$H^I_{22}=Ae^{-i\omega u-2i\alpha(u)}, \qquad N^I_{22}=-i(\omega+2\dot\alpha)H^I_{22}.$$

The $2\dot\alpha$ term is frame motion; it vanishes for a rigid orientation. Spheroidal projection places the same physical frequency in two spherical coefficients:

$$H_{22}=\sqrt{1-\varepsilon^2}\,Ae^{-i\omega u}, \qquad H_{32}=\varepsilon Ae^{-i\omega u}.$$

The axial rotation changed only an $m$-dependent phase within fixed $\ell$; a general rotation also mixes $m$. The spherical–spheroidal change mixed $\ell$ at fixed $m$ and left the time dependence unchanged. A time-dependent frame can shift the apparent instantaneous frequency by $2\dot\alpha$, but it has not introduced a new quasinormal mode of the remnant.

References

• Berti, Emanuele, Vitor Cardoso, and Andrei O. Starinets. “Quasinormal Modes of Black Holes and Black Branes.” Classical and Quantum Gravity 26 (2009): 163001. DOI. Open PDF.
• Bishop, Nigel T., and Luciano Rezzolla. “Extraction of Gravitational Waves in Numerical Relativity.” Living Reviews in Relativity 19 (2016): 2. DOI. Open PDF.
• Boyle, Michael, Lawrence E. Kidder, Serguei Ossokine, and Harald P. Pfeiffer. “Gravitational-Wave Modes from Precessing Black-Hole Binaries.” Physical Review D 90 (2014): 124010. DOI. Open PDF.
• Boyle, Michael. “Transformations of Asymptotic Gravitational-Wave Data.” Physical Review D 93 (2016): 084031. DOI. Open PDF.
• Goldberg, Joshua N., Alan J. Macfarlane, Ezra T. Newman, Fritz Rohrlich, and E. C. G. Sudarshan. “Spin-s Spherical Harmonics and ð.” Journal of Mathematical Physics 8 (1967): 2155–2161. DOI.