Source, Inertial, and Radiation Frames

A gravitational waveform is one spin-weight $-2$ field, but its mode coefficients depend on how cuts, angles, axes, and the polarization dyad are chosen. A rigid rotation can move amplitude among $m$ modes without changing the radiation. A time-dependent source-attached rotation can add apparent modulation, while a boost or supertranslation changes more than the orientation. This page separates Bondi, inertial, source, radiation-axis, and coprecessing frames and declares the active/passive Wigner-$D$ convention used to compare them. Detector calibration and source dynamics are not part of this frame dictionary.

Required background. Spacetime Symmetries and Killing Fields supplies rotations, active group actions, and Killing generators.

Helpful background. Einstein Equations supplies asymptotic tensor data; Waves, Green Functions, and Geometric Optics supplies angular mode bases; and Gravitational Radiation and Waveforms supplies $h=h_+-ih_\times$ and its spin weight.

Frames answer different questions

The word “frame” is overloaded. A reproducible waveform states which of the following structures it uses:

Frame	Defining structure	What changes when it changes
Bondi frame	Cuts $u=\text{constant}$, sphere coordinates, conformal scale, and dyad at $\mathscr I^+$	Strain, modes, retarded time, and charge aspects under the full BMS group
Asymptotically inertial frame	Time-independent axes and origin within a fixed asymptotic Poincaré frame	Mode labels under rigid rotations; momentum components under boosts
Source-attached frame	Axes built from source directions such as separation or angular momentum	Coefficients acquire the time dependence of those axes
Radiation-axis frame	An axis inferred from the angular structure of the radiation	Dominant power may concentrate in selected $m$ values; prescription dependent
Coprecessing frame	A time-dependent rotation carrying a radiation or source axis to $z$	Precession-induced mixing is reduced, with a residual axial gauge
Corotating frame	A coprecessing frame with an additional rotation following waveform phase	Much of the rapid phase can be removed from selected modes
Line-of-sight screen	Propagation direction plus two transverse polarization axes	$h_+$ and $h_\times$ mix while the tensor field is unchanged

A source frame need not be inertial. A radiation-axis frame need not coincide with an orbital-angular-momentum frame. A coprecessing frame is a data-dependent choice, not a new observer or a different spacetime.

Active and passive rotations

Expand the asymptotic strain in a fixed unit-normalized basis,

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

Define an active rotation $R$ by rotating the radiation pattern while holding the coordinate axes fixed. In this page its modes transform as

$$H^{R}_{\ell m} =\sum_{m'=-\ell}^{\ell} D^\ell_{m m'}(R)H_{\ell m'}.$$

The spin-frame phase is included in this Wigner action. For an active $z$ rotation through $\alpha$,

$$D^\ell_{m m'}(R_z(\alpha)) =e^{-im\alpha}\delta_{mm'}.$$

A passive rotation of the coordinate axes by the same geometric $R$ uses the inverse matrix,

$$H^{\rm new\ axes}_{\ell m} =\sum_{m'}D^\ell_{m m'}(R^{-1})H_{\ell m'} =\sum_{m'}D^{\ell *}_{m' m}(R)H_{\ell m'}.$$

Writing only “rotate by $R$” without active/passive direction and index order is incomplete. Boyle gives a convention-explicit treatment of rotations, spin phase, and the larger BMS action in Boyle 2016, §§II–IV, pp. 3–13, PDF.

What a rotation preserves

Each $D^\ell$ is unitary, so a rigid rotation preserves fixed-$\ell$ angular power:

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

It mixes only $m$ at fixed $\ell$. If an alleged rigid rotation changes this norm or couples different $\ell$ values, then the implementation has an index, normalization, interpolation, or frame error. The same statement applies to news only when the rotation is time independent. For $R=R(u)$, differentiating the inertial strain produces both $D\dot H$ and $\dot D H$.

A radiation-axis prescription can be checked by rotating a simple mode set into and out of its preferred axis and verifying norm preservation. It cannot prove that all remaining asymmetry is gauge. Boyle and collaborators show explicitly that generic precessing radiation retains physical asymmetries even in a well-chosen rotating frame Boyle et al. 2014, §§II–VI, PDF.

Beyond rotations: the Bondi frame

Rotations are only part of the BMS freedom. A time translation changes mode phases through the retarded-time origin. A spatial translation induces direction-dependent retarded-time shifts. A boost aberrates angles and changes conformal and spin factors. A proper supertranslation changes the cuts by an arbitrary allowed angle-dependent function and adds a time-independent electric-parity term to the shear.

These transformations can couple different $\ell$ values and, because they relabel $u$, can mix angular and temporal structure. They are not reproducible by a Wigner matrix acting separately at each time. Boyle derives the full asymptotic waveform transformation and illustrates translation, boost, and supertranslation mode coupling in Boyle 2016, §§III–VI, pp. 9–19, PDF.

A comparison should therefore record the Bondi cut convention, asymptotic rest frame, origin, inertial axes, polarization dyad, and any time-dependent source or radiation frame. Aligning only one time and phase is sufficient only after the remaining freedom has been controlled.

Exercise

In an inertial frame, suppose the only nonzero coefficient is $H_{22}=A$. Compare an active rotation $R_z(\alpha)$ with a passive rotation of the axes by the same $R_z(\alpha)$, starting from the original waveform in each case. Find the coefficient in each case and check the power. What happens if the two transformations are applied successively?

Solution

The active convention gives

$$H^R_{22}=e^{-2i\alpha}A.$$

The passive axes use $R^{-1}$, so

$$H^{\rm new\ axes}_{22}=e^{+2i\alpha}A.$$

In either case $\lvert H_{22}\rvert^2=\lvert A\rvert^2$. The opposite phases do not represent opposite physics; they record whether the pattern or the coordinate basis was rotated. Applying both transformations successively returns the original coefficient.

References

• 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.