Gravitational-Wave Strain, News, Flux, and Ψ₄

Strain, Bondi news, and outgoing Newman–Penrose curvature are related only after the screen orientation, asymptotic frame, null tetrad, retarded time, radius, Fourier sign, and harmonic phases have been fixed together. This card states one complete package and marks the boundaries between local curvature, null-infinity data, finite-radius extraction, and a calibrated detector channel.

One strain–news–curvature package

Use retarded Bondi coordinates $(u,r,x^A)$ at future null infinity, with $r$ an areal radius and $q_{AB}$ the positively oriented unit-sphere metric. Choose a complex dyad satisfying $m^A\bar m_A=1$. The radiative angular metric and the site’s news tensor are

$$g_{AB}=r^2q_{AB}+rC_{AB}+O(1), \qquad q^{AB}C_{AB}=0, \qquad N_{AB}:=\partial_uC_{AB}.$$

With the oriented detector-screen convention

$$h:=h_+-ih_\times, \qquad H:=C_{AB}\bar m^A\bar m^B =\lim_{r\to\infty}r h, \qquad \mathcal N:=N_{AB}\bar m^A\bar m^B=\partial_uH.$$

A time-independent dyad rotation $m^A\mapsto e^{i\chi}m^A$ sends $H,\mathcal N\mapsto e^{-2i\chi}(H,\mathcal N)$, so both fields have spin weight $-2$. The underlying tensors do not change. The complete polarization construction is Plus and Cross Polarizations.

Mädler and Winicour define a tensor news smaller by a factor of two:

$$N_{AB}^{\rm MW}=\frac12N_{AB}.$$

That factor must be translated before their mass-loss equation is imported; otherwise a news-squared flux is wrong by a factor of four. Their Bondi expansion, constraint, and conformal construction appear in Mädler and Winicour 2018, §3, Eqs. (33)–(62), pp. 8–16 and §4, pp. 17–20. The site’s tensor and scalar normalizations are developed in Bondi Shear and News.

Near infinity, begin with a future-directed orthonormal frame $(e_0,e_r,e_\theta,e_\phi)$, with $e_r$ outward, and set

$$\ell^a=\frac{e_0^a+e_r^a}{\sqrt2}, \qquad n^a=\frac{e_0^a-e_r^a}{\sqrt2}, \qquad m^a=\frac{e_\theta^a+i e_\phi^a}{\sqrt2}.$$

Then $\ell\mathbin{\cdot}n=-1$, $m\mathbin{\cdot}\bar m=1$, and all other inner products vanish. With the site’s curvature convention, define

$$\Psi_4:=-C_{abcd}n^a\bar m^b n^c\bar m^d, \qquad \Psi_4^0:=\lim_{r\to\infty}r\Psi_4.$$

For an outgoing wave in this tetrad, the complete derivative chain is

$$\boxed{ \Psi_4^0=\partial_u\mathcal N=\partial_u^2H }.$$

The sign is not portable by the symbol $\Psi_4$ alone: changing the Riemann sign, exchanging null legs, reversing the Newman–Penrose definition, or using $h_++ih_\times$ changes the displayed dictionary. Bishop and Rezzolla 2016, §§3.3 and 6.2, pp. 21–27 and 54–58 give the extraction conventions and translations. The canonical curvature definition is Newman–Penrose Radiation Scalars.

Fourier, modes, and positive outward flux

Use the transform pair

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

Therefore $\partial_u\leftrightarrow-i\omega$ and

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

In a time-independent Bondi angular frame, use unit-normalized spin-weight $-2$ harmonics with the declared Condon–Shortley and $e^{im\phi}$ phases:

$$H(u,x^A) =\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell} H_{\ell m}(u)\,{}_{-2}Y_{\ell m}(x^A),$$ $$\int_{S^2}{}_{-2}Y_{\ell m}\, {}_{-2}\bar Y_{\ell' m'}\,d\Omega =\delta_{\ell\ell'}\delta_{mm'}.$$

Projection commutes with $\partial_u$ only when the basis is not itself rotating, giving

$$N_{\ell m}=\dot H_{\ell m}, \qquad \psi_{4,\ell m}=\dot N_{\ell m}=\ddot H_{\ell m}.$$

Boyle 2016, §§II–VI, pp. 2–17 gives the spin-weighted and BMS transformations, including mode mixing under rotations, boosts, translations, and supertranslations. The exact modal dictionary is collected in Strain and Curvature Mode Decompositions.

For the outward orientation, the energy flux at null infinity is

$$\frac{dE_{\rm out}}{du\,d\Omega} =\frac{1}{32\pi G}N_{AB}N^{AB} =\frac{1}{16\pi G}\left\lvert\mathcal N\right\rvert^2,$$

and hence

$$\frac{dE_{\rm out}}{du} =\frac{1}{16\pi G}\int_{S^2} \left\lvert\mathcal N\right\rvert^2d\Omega =\frac{1}{16\pi G}\sum_{\ell,m} \left\lvert\dot H_{\ell m}\right\rvert^2.$$

The corresponding outward linear-momentum flux uses the same density weighted by the unit radial direction $\hat r^i$ on the sphere,

$$\frac{dP^i_{\rm out}}{du} =\frac{1}{16\pi G}\int_{S^2} \hat r^i\left\lvert\mathcal N\right\rvert^2d\Omega.$$

This is an asymptotic balance quantity, not a local gravitational energy density. The short-wavelength wave-zone stress tensor agrees with the leading flux after averaging under its additional scale-separation assumptions Flanagan and Hughes 2005, § 5.3, printed pp. 27–29. The canonical Bondi formula and source-loss sign are given in Energy Flux at Null Infinity.

What curvature does not determine

Knowing $\Psi_4^0$ on an interval beginning at $u_0$ determines strain only up to two integration constants:

$$H(u)=H(u_0)+(u-u_0)\mathcal N(u_0) +\int_{u_0}^{u}(u-u')\Psi_4^0(u')\,du'.$$

The constant term chooses a shear vacuum or strain offset. The coefficient of $u-u_0$ chooses the initial news and can also represent numerical drift. A physical endpoint condition, a matched strain calculation, or another justified low-frequency prescription must fix them. A memory step is physical low-frequency information and must not be removed merely because division by $\omega^2$ is ill-conditioned near zero frequency.

A supertranslation can add a retarded-time-independent, angle-dependent term to $H$. News removes that offset, but a full BMS transformation also relabels cuts and angles, so fields must be compared by the correct pullback rather than at equal coordinate labels. A time-dependent screen rotation contributes a basis-rotation term to $\partial_uH$; it is not new radiation. These qualifications are part of the BMS transformation law, not optional waveform post-processing Boyle 2016, §§III–VI, pp. 5–17.

From curvature to a detector channel

Layer	Object and boundary
Local tidal field	$R_{0i0j}$ controls relative acceleration in a declared observer frame. It is not a detector calibration model.
Radiation-zone strain	$h_+$ and $h_\times$ require a weak-field radiation frame; under the long-wavelength idealization an instrument contracts them with its detector tensor.
Null-infinity data	$H$, $\mathcal N$, and $\Psi_4^0$ require an asymptotically flat Bondi frame, areal radius, retarded time, dyad, and normalized tetrad.
Finite-radius extraction	A coordinate sphere leaves gauge, tetrad, radius, near-zone, and backscatter corrections. Extrapolation or characteristic extraction must expose their convergence and frame choices.
Calibrated detector channel	A measured channel contains antenna response, propagation, instrument transfer, calibration, sampling, and noise; it is not identical to one source-side polarization or mode.

For an ideal long-wavelength channel,

$$s=F_+h_++F_\times h_\times,$$

but real inference additionally uses the instrument response and calibration model. Flanagan and Hughes 2005, § 3, printed pp. 13–17 derives the tidal response, while Bishop and Rezzolla 2016, § 3.3.2 and §§ 5–7 reviews extrapolation, perturbative extraction, and characteristic extraction. For the detailed derivations, read Curvature and Detector Observables and Gauge-Invariant Radiative Quantities.

Compact checks

• A monochromatic mode $H=Ae^{-i\omega u}$ gives $\Psi_4^0=-\omega^2H$ and a nonnegative flux proportional to $\omega^2\lvert A\rvert^2$.
• $N_{AB}=0$ gives zero outward luminosity even when a time-independent shear remains.
• The sphere integral and harmonic mode sum must agree; disagreement diagnoses a normalization, phase, omitted-mode, or frame error.
• Differentiating strain and independently extracting curvature should agree only after retarded time, areal radius, tetrad, dyad, and BMS frame are aligned.

The chapter Polarizations and Gauge-Invariant Wave Observables provides the common route into the local and detector-side treatments.

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.
• Flanagan, Éanna É., and Scott A. Hughes. “The Basics of Gravitational Wave Theory.” New Journal of Physics 7 (2005): 204. DOI. Open PDF.
• Mädler, Thomas, and Jeffrey Winicour. “Bondi–Sachs Formalism.” arXiv:1609.01731v3, 2018. Open PDF.