Phase, Time, and Sign Conventions

Two waveform files can describe the same radiation yet appear complex conjugated, time reversed, phase shifted, or relabeled in $m$. Such disagreements usually arise because one convention was translated in isolation. The safe object is a complete convention set: complex polarization, dyad phase, harmonic phase, retarded time, Fourier kernel, curvature sign, and active/passive frame action. This page declares the set used throughout the volume and gives a crosswalk for common alternatives. It does not choose an orbital model or a detector phase convention.

Required background. Spacetime Symmetries and Killing Fields supplies rotations, azimuthal generators, and the meaning of the $m$ label.

Helpful background. Einstein Equations supplies curvature signs and the Weyl tensor; Waves, Green Functions, and Geometric Optics supplies retarded time and Fourier propagation; and Gravitational Radiation and Waveforms supplies the polarization and asymptotic-strain definitions.

The volume convention set

The complex asymptotic strain is

$$H=r(h_+-ih_\times)+O(r^{-1}),$$

with spin weight $-2$. A dyad rotation $m^A\mapsto e^{i\chi}m^A$ gives $H\mapsto e^{-2i\chi}H$. The unit-normalized harmonics use $e^{+im\phi}$ and the declared Condon–Shortley phase,

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

Retarded Bondi time labels outgoing null cuts. It is $u=t-r$ in the flat reference problem; a curved exterior may require a tortoise coordinate or an asymptotic logarithmic correction before the extracted time agrees with Bondi time. A positive-frequency outgoing wave has $e^{-i\omega u}$ with $\omega\gt0$. The Fourier pair is

$$\widetilde f(\omega)=\int f(u)e^{+i\omega u}du, \qquad f(u)=\int\frac{d\omega}{2\pi}\widetilde f(\omega)e^{-i\omega u}.$$

Therefore $\partial_u\leftrightarrow-i\omega$. Stable retarded quasinormal frequencies lie in the lower half of the complex $\omega$ plane.

The tetrad is ordered $(\ell,n,m,\bar m)$ with $\ell\!\cdot n=-1$ and $m\!\cdot\bar m=1$. Its asymptotic boost normalization is $\ell=(e_t+e_r)/\sqrt2$, $n=(e_t-e_r)/\sqrt2$ in the declared orthonormal Bondi-frame tetrad. Define

$$\Psi_4=-C_{abcd}n^a\bar m^bn^c\bar m^d.$$

With $\Psi_4^0=\lim r\Psi_4$ and $\mathcal N=\partial_uH$,

$$\Psi_4^0=\partial_u^2H, \qquad \widetilde\Psi_4^0=-\omega^2\widetilde H.$$

Bishop and Rezzolla give this extraction tetrad and the strain–curvature relation in Bishop and Rezzolla 2016, pp. 22–24, eqs. (95)–(105), PDF.

A complete crosswalk

Changed convention	Apparent change	Required translation
$h_++ih_\times$ instead of $h_+-ih_\times$	Complex conjugation and spin $+2$ rather than $-2$	Conjugate the field and use the conjugate spin basis
$e^{-im\phi}$ harmonics	Positive and negative azimuthal labels appear exchanged	Relabel $m$ and translate every conjugation and Wigner-$D$ phase
Inverse Fourier kernel in the forward transform	Derivatives carry $+i\omega$	Reverse the frequency label or translate both transform equations
$e^{+i\omega t}$ modal time dependence	Damped-mode half-plane appears reversed	Replace $\omega$ by $-\omega$ consistently, including boundary conditions
Advanced time $v=t+r$	Outgoing phase appears to propagate inward	Restore retarded time $u=t-r$ for future-null-infinity data
Opposite Riemann or $\Psi_4$ sign	Curvature differs by a minus sign	Translate the full curvature and tetrad definition
Passive rather than active rotation	Wigner phases and index order are inverted	Replace $D(R)$ by $D(R^{-1})=D(R)^\dagger$
Different Bondi cut or origin	Direction-dependent time and mode shifts	Apply the full BMS transformation, not a constant phase fit

Goldberg and collaborators fix how conjugation, $e^{im\phi}$, and spin weight fit together Goldberg et al. 1967, pp. 2155–2161. Boyle derives the corresponding rotations, boosts, translations, and supertranslations of asymptotic waveform data Boyle 2016, §§II–VI, pp. 3–19, PDF.

Orbital phase is not waveform phase by definition

In a suitably oriented nonprecessing source frame, a dominant mode often has

$$H_{\ell m}\sim A_{\ell m}e^{-im\Phi(u)}.$$

This relation is a physical approximation plus a convention, not the definition of $\Phi$. Tail phases, mode-dependent offsets, eccentricity, precession, and a different source-axis convention modify it. Defining a waveform phase as $-\arg H_{\ell m}/m$ is useful only after phase unwrapping, amplitude zeros, the $m$ sign, and the chosen mode have been stated.

An active delay, $H_2(u)=H_1(u-\Delta u)$, multiplies a monochromatic coefficient by $e^{+i\omega\Delta u}$ when both fields are expressed as functions of the same $u$. By contrast, under the passive relabeling $u'=u-\Delta u$ a scalar satisfies $H'(u')=H(u'+\Delta u)$ and its coefficient gains $e^{-i\omega\Delta u}$. A polarization-dyad rotation changes the coefficient by $e^{-2i\chi}$. These are independent operations: fitting one arbitrary phase cannot generally stand in for both across several $m$ values and frequencies.

Consistency checks

Use one monochromatic outgoing mode to test a convention converter. It should satisfy all of the following simultaneously:

• constant phase on $u=t-r$ propagates toward increasing $r$;
• $\dot H=-i\omega H$ and $\Psi_4^0=-\omega^2H$;
• a dyad rotation gives the spin-$-2$ phase;
• an active $z$ rotation gives $H_{\ell m}\mapsto e^{-im\alpha}H_{\ell m}$;
• the news-squared energy flux is unchanged by either constant rotation.

Passing only one check is insufficient because two sign mistakes can cancel in a single formula.

Exercise

A source uses $g=h_++ih_\times$ in a dyad $m'=e^{i\chi}m$ and reports the rescaled mode

$$G(u)=B e^{+i\omega(u-u_0)}, \qquad \omega>0.$$

Convert it to this volume’s $H=r(h_+-ih_\times)$ in the $m$ dyad, and find $\Psi_4^0$.

Solution

In the primed dyad, $G=\bar H'$. Since $H'=e^{-2i\chi}H$, one has $G=e^{+2i\chi}\bar H$ and therefore

$$H=e^{+2i\chi}\bar G =e^{+2i\chi}\bar B\,e^{-i\omega(u-u_0)}.$$

This now has the declared positive-frequency dependence. Differentiating twice gives

$$\Psi_4^0=\partial_u^2H=-\omega^2H.$$

The conjugation, dyad phase, time-origin phase, and curvature sign all had to be translated together.

References

• 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. “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.