Bondi Shear and News

The Bondi shear records the leading shape distortion of outgoing null wavefronts; the Bondi news records how that distortion changes from one retarded-time cut to the next. The distinction is decisive. A time-independent shear can reflect an asymptotic frame or a radiative vacuum, but nonzero news produces positive outward gravitational-wave energy flux. With the asymptotic frame and normalization declared, the retarded-time derivative of shear is therefore the invariant marker of outgoing radiation at future null infinity. This is an asymptotic statement, not a detector reconstruction prescription.

Helpful background. Einstein Equations supplies the asymptotic vacuum constraints; Waves, Green Functions, and Geometric Optics supplies retarded propagation; Causality and Global Structure supplies the geometry of future null infinity; Gravitational Radiation and Waveforms supplies strain and flux interpretation; and Formulations of General Relativity supplies characteristic gauge and data.

Shear as asymptotic shape data

In an inertial Bondi frame, the angular metric has the expansion

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

The real symmetric trace-free tensor $C_{AB}(u,x^A)$ is the Bondi shear coefficient. Geometrically, it is the finite leading shear of the congruence generating the outgoing null hypersurfaces after the universal radial expansion has been removed. It has two components, exactly matching the two tensor polarizations.

Choose a complex dyad $m^A$ on the unit sphere with

$$m^Am_A=0, \qquad m^A\bar m_A=1, \qquad q_{AB}=m_A\bar m_B+\bar m_A m_B.$$

The complex shear amplitude is

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

A screen-basis rotation $m^A\mapsto e^{i\chi}m^A$ sends $H\mapsto e^{-2i\chi}H$. Thus $H$ has spin weight $-2$. The tensor $C_{AB}$ is basis independent; only its complex component changes.

News is change, not offset

This volume defines

$$N_{AB}:=\partial_uC_{AB}, \qquad \mathcal N:=N_{AB}\bar m^A\bar m^B=\partial_uH.$$

Mädler and Winicour use $N^{\rm MW}_{AB}=N_{AB}/2$ and the opposite complex polarization, so their scalar is $N_{\rm MW}=\overline{\mathcal N}/2$. Their construction shows both that news is a retarded-time derivative of radiation strain and that it has an intrinsic conformal definition on $\mathscr I^+$ Mädler and Winicour 2018, eqs. (40)–(43), p. 12, and eq. (73), pp. 18–19, PDF.

The derivative removes the time-independent inhomogeneous shift of the shear produced by a supertranslation. A full BMS transformation also relabels the cut and angle, so components are compared by pullback rather than by equating values at identical coordinate labels. After that geometric comparison, the statement $N_{AB}=0$ is frame independent. Boyle develops these spin-weighted waveform transformations and the required pullbacks explicitly Boyle 2016, §§II–IV, pp. 2–12, PDF.

News and positive outward flux

For the declared normalization, the gravitational energy flux per unit solid angle is

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

The second equality follows from $N_{AB}N^{AB}=2\lvert\mathcal N\rvert^2$. It provides two independent checks: the flux is nonnegative for the outward orientation, and its dimensions are energy per time per solid angle because $C_{AB}$ has dimensions of length. Integrating the asymptotic mass constraint over the sphere gives the corresponding decrease of Bondi energy. The detailed balance law is developed with the flux pages; here it establishes why nonzero news cannot be dismissed as a coordinate oscillation.

No news does not require zero shear. If $C_{AB}$ is independent of $u$, then $N_{AB}=0$ and there is no radiative energy flux, even when $C_{AB}\ne0$ in the chosen Bondi frame. Likewise, a burst may have $N_{AB}=0$ before and after it but different early and late shears. That difference is displacement memory. The news identifies the radiative interval; the integrated news records the permanent change.

Relation to radiative curvature

With the null tetrad and Weyl sign fixed on Newman–Penrose Radiation Scalars, the leading outgoing curvature is

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

The derivative chain makes the roles transparent: $H$ is asymptotic strain, $\mathcal N$ controls flux, and $\Psi_4^0$ is radiative curvature. Integration constants prevent one from silently identifying these quantities. In particular, knowing curvature over a finite interval does not by itself fix the constant and linear pieces of an integrated strain.

Exercises

1. Basis rotation. Let $H'=e^{-2i\chi}H$ under a time-independent rotation of the polarization dyad. Show that the flux is unchanged, and explain what fails if $\chi$ is allowed to depend on retarded time.

Solution

For $\partial_u\chi=0$, the news transforms as $\mathcal N'=\partial_uH'=e^{-2i\chi}\mathcal N$, so $\lvert\mathcal N'\rvert^2=\lvert\mathcal N\rvert^2$ and the flux is unchanged. If $\chi=\chi(u,x^A)$, differentiation adds $-2i(\partial_u\chi)H$. That term is rotation of the basis itself, not new curvature radiation; one must use a properly transported frame rather than interpreting $\partial_uH'$ as physical news.

2. One smooth memory pulse. Let $H=A F(u)\,{}_{-2}Y_{20}$, with $\int\lvert{}_{-2}Y_{20}\rvert^2d\Omega=1$, $F(u)=[1+\tanh(u/\tau)]/2$, and $\tau\gt0$. Here $A$ has dimensions of length. Find the change in strain coefficient and the total radiated energy. How does the energy change if the same memory is accumulated more slowly?

Solution

The endpoint difference is $\Delta H=A\,{}_{-2}Y_{20}$ and the news is $\mathcal N=A\operatorname{sech}^2(u/\tau)\,{}_{-2}Y_{20}/(2\tau)$. Thus

$$E_{\rm out} =\frac{\lvert A\rvert^2}{64\pi G\tau} \int_{-\infty}^{\infty}\operatorname{sech}^4x\,dx =\frac{\lvert A\rvert^2}{48\pi G\tau}.$$

To evaluate the integral, put $y=\tanh x$; it becomes $\int_{-1}^1(1-y^2)dy=4/3$. Increasing $\tau$ leaves memory unchanged but reduces the energy. This is a prescribed radiative-data example: compatible mass and angular-momentum aspects must still satisfy the asymptotic constraints, and these data alone do not construct a complete interior source.

References

• Michael Boyle, “Transformations of Asymptotic Gravitational-Wave Data,” Physical Review D 93 (2016) 084031, arXiv:1509.00862v3, DOI, Open PDF.
• Thomas Mädler and Jeffrey Winicour, “Bondi–Sachs Formalism,” Scholarpedia 11 (2016) 33528, updated arXiv version (2018), arXiv:1609.01731v3, Open PDF.