Radiation at Null Infinity and Bondi–Sachs Structure

Future null infinity turns the radiation zone into geometry. In outgoing Bondi coordinates, the $1/r$ shear coefficient becomes finite boundary data, its retarded-time derivative is the news, and a leading Newman–Penrose Weyl scalar supplies the corresponding curvature. The asymptotic Einstein equations then propagate mass and angular-momentum aspects and convert news into flux balance. The BMS group explains which parts of the shear depend on the asymptotic frame and which statements about radiation survive every allowed frame change. This chapter develops that chain without attempting rigorous peeling proofs or detector-specific waveform reconstruction.

Helpful background. Einstein Equations supplies the vacuum field equations; Waves, Green Functions, and Geometric Optics supplies outgoing characteristics; Causality and Global Structure supplies conformal null infinity and the geometric BMS group; Gravitational Radiation and Waveforms supplies the volume-wide strain and flux conventions; and Formulations of General Relativity supplies characteristic constraints and asymptotic charge machinery.

One asymptotic dictionary

Use outgoing Bondi coordinates $(u,r,x^A)$, with $u=t-r$ in the Minkowski limit, $r$ an areal radius, and $q_{AB}$ the unit-sphere metric. The angular metric begins

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

Choose a complex sphere dyad with $m^A\bar m_A=1$. Every page then uses

$$\begin{aligned} H&=C_{AB}\bar m^A\bar m^B =\lim_{r\to\infty}r(h_+-ih_\times),\\ N_{AB}&=\partial_uC_{AB}, \qquad \mathcal N=\partial_uH,\\ \Psi_4^0&=\lim_{r\to\infty}r\Psi_4 =\partial_u^2H. \end{aligned}$$

The null tetrad is ordered $(\ell,n,m,\bar m)$ with $\ell\!\cdot n=-1$, and $\Psi_4=-C_{abcd}n^a\bar m^bn^c\bar m^d$. With the inherited $e^{-i\omega u}$ convention, $\widetilde\Psi_4^0=-\omega^2\widetilde H$. Mädler and Winicour define their tensor news with an extra factor $1/2$; in this chapter $N^{\rm MW}_{AB}=N_{AB}/2$. The explicit dictionary prevents a correct source formula from acquiring an incorrect factor or complex conjugation during translation.

The six-page route

The pages proceed from the boundary itself to its constrained radiative data:

1. Radiative Limits at Future Null Infinity shows how $\Omega=1/r$ makes the outgoing limit finite, checks nullness in compactified Minkowski space, and separates a regular radiative limit from a strong peeling claim.
2. Bondi–Sachs Coordinates and Expansion imposes null, angular, and determinant gauge conditions, expands the metric, and distinguishes free shear from radial and supplementary coefficients.
3. Bondi Shear and News defines the spin-weight $-2$ strain, differentiates it to news, and connects news to positive outward flux.
4. BMS Symmetry Acting on Radiative Data applies translations, rotations, boosts, and pure supertranslations to shear and news, explaining what a waveform alignment must fix.
5. Newman–Penrose Radiation Scalars derives the sign of $\Psi_4^0=\partial_u^2H$ from a TT plane wave and states the tetrad and regularity qualifications.
6. Radiative Data and Asymptotic Constraints propagates the mass and angular-momentum aspects and recovers energy–momentum loss by integrating the constraint over sphere harmonics.

A reader interested mainly in waveforms can read pages 2, 3, 5, and 6, but should not skip the BMS page when comparing data from different frames. A reader focused on asymptotic geometry should begin with pages 1 and 2, then follow the constraint and BMS applications.

Free data, frame data, and constrained data

The main objects have different logical status:

Object	Role	Essential qualification
$C_{AB}(u,x^A)$	Free radiative shear in the formal expansion	Shifts inhomogeneously under a pure supertranslation
$N_{AB}=\partial_uC_{AB}$	Radiation and flux data	Transforms by geometric pullback; zero news is frame independent
$m(u,x^A)$	Bondi mass aspect	Initial value is supplied and evolution is constrained
$L_A(u,x^A)$	Angular-momentum aspect	Evolution is constrained and charge interpretation needs a BMS frame
$\Psi_4^0$	Leading outgoing Weyl curvature	Requires the declared asymptotic tetrad and curvature sign

The formal Bondi expansion prescribes $C_{AB}$ through a retarded-time interval and evolves $m$ and $L_A$ from an initial cut. It is not by itself a well-posed construction of the interior spacetime. Characteristic evolution and charge definitions are developed by GR formulations, while conformal-boundary existence and the geometric derivation of BMS symmetry belong to global structure.

Checks that join the descriptions

Several short calculations catch most convention errors:

• Compactified Minkowski space has a regular unphysical metric and a null boundary at $\Omega=0$.
• The determinant condition makes $C_{AB}$ trace-free and fixes the trace of the next angular coefficient.
• A TT plane wave with $R_{0i0j}=-\tfrac12\ddot h^{\rm TT}_{ij}$ gives $\Psi_4=\ddot h_+-i\ddot h_\times$ in the declared tetrad.
• A time-independent dyad rotation gives $H$, $\mathcal N$, and $\Psi_4^0$ the same spin weight $-2$ and leaves $\lvert\mathcal N\rvert^2$ unchanged.
• Integrating the mass constraint removes the sphere divergence and leaves a nonpositive change of Bondi energy.
• For $\ell=0,1$ supertranslations, the trace-free Hessian vanishes, recovering ordinary translations without an added vacuum shear.

The conformal construction and Bondi hierarchy are developed in Mädler and Winicour 2018, §§3–4, pp. 10–20, PDF. The tetrad-to-strain translation is checked independently in Bishop and Rezzolla 2016, eqs. (95)–(103), pp. 22–23, and (271)–(276), p. 55, PDF, while full waveform transformations under the BMS group are given in Boyle 2016, §§II–IV, pp. 2–12, PDF.

Chapter diagnostic

Suppose

$$C_{AB}(u,x^A)=C^-_{AB}(x^A)+F(u)Q_{AB}(x^A),$$

where $Q_{AB}$ is real, symmetric, trace-free, $F(-\infty)=0$, and $F(+\infty)=1$. Identify the news, memory, radiative curvature, and sign of the total Bondi-energy change. Then apply one common pure supertranslation to both stationary endpoints.

Solution

The news is $N_{AB}=F'(u)Q_{AB}$, the displacement memory is $\Delta C_{AB}=Q_{AB}$, and the leading curvature is $\Psi_4^0=F''(u)Q_{AB}\bar m^A\bar m^B$. The energy change is nonpositive because it is minus a positive constant times $\int du\,d\Omega\,[F'(u)]^2Q_{AB}Q^{AB}$. A common supertranslation adds the same time-independent trace-free Hessian to both endpoint shears, so it leaves their difference, the news, the curvature, and the flux unchanged after pullback.

References

• Michael Boyle, “Transformations of Asymptotic Gravitational-Wave Data,” Physical Review D 93 (2016) 084031, arXiv:1509.00862v3, DOI, Open PDF.
• Nigel T. Bishop and Luciano Rezzolla, “Extraction of Gravitational Waves in Numerical Relativity,” Living Reviews in Relativity 19 (2016) 2, arXiv:1606.02532v2, DOI, Open PDF.
• Thomas Mädler and Jeffrey Winicour, “Bondi–Sachs Formalism,” Scholarpedia 11 (2016) 33528, updated arXiv version (2018), arXiv:1609.01731v3, Open PDF.