BMS Symmetry Acting on Radiative Data

The Bondi–Metzner–Sachs group is the residual symmetry of asymptotically flat null infinity after Bondi gauge has been imposed. Its action is not merely a change of labels: a transformation moves retarded-time cuts, rotates or boosts the celestial sphere, changes the polarization dyad, and can shift the Bondi shear by an angle-dependent vacuum term. News and curvature transform more simply, but their samples and harmonic modes are still compared at transformed points. Consequently, two asymptotic waveforms require a common BMS frame before pointwise differences, mode mismatches, or memory offsets have invariant meaning. The geometric derivation of the group and the construction of its charges remain with global structure and GR formulations; rigorous peeling proofs and detector-specific reconstruction are not treated here.

Helpful background. Einstein Equations supplies the asymptotic vacuum equations; Waves, Green Functions, and Geometric Optics supplies retarded propagation; Causality and Global Structure supplies the geometric BMS group; Gravitational Radiation and Waveforms supplies strain, news, and waveform comparisons; and Formulations of General Relativity supplies asymptotic constraints and charge constructions.

The BMS action on null infinity

In a Bondi frame with unit-sphere metric $q_{AB}$, an infinitesimal BMS generator restricts to

$$\xi =\left[\alpha(x^A)+\frac u2D_AY^A\right]\partial_u +Y^A(x^B)\partial_A,$$

where $Y^A$ is a conformal Killing field of the sphere and $\alpha$ is an arbitrary smooth real function. The $Y^A$ sector generates the connected Lorentz group. The functions $\alpha$ form an abelian normal subgroup of supertranslations. Their spherical-harmonic components with $\ell=0,1$ are ordinary time and space translations; components with $\ell\ge2$ are pure supertranslations. This structure and the absence of a canonical Lorentz subgroup in a generic radiative spacetime are derived in Mädler and Winicour 2018, §4, pp. 17–20, PDF.

A finite Lorentz transformation acts conformally on the sphere and rescales the affine parameter along its generators. A finite supertranslation changes the cut by an angle-dependent amount. The group is a semidirect product rather than a direct product: applying a boost and a supertranslation in the opposite order generally changes the effective supertranslation.

Pure supertranslations of shear and news

Fix the passive convention

$$u'=u+\alpha(x^A), \qquad x'^A=x^A.$$

After the subleading coordinate adjustments required to restore Bondi gauge, define

$$S_{AB}[\alpha] =2D_AD_B\alpha-q_{AB}D^2\alpha.$$

In this sign convention, the transformed radiative data are

$$\begin{aligned} C'_{AB}(u',x^A) &=C_{AB}(u'-\alpha,x^A)+S_{AB}[\alpha],\\ N'_{AB}(u',x^A) &=N_{AB}(u'-\alpha,x^A). \end{aligned}$$

Writing the coordinate change with the opposite sign reverses the displayed inhomogeneous term; the invariant content is the pullback plus a time-independent electric-parity shear shift. Differentiating the first line proves the second because $\alpha$ is independent of $u$. Thus a supertranslation can change which radiative vacuum is called $C_{AB}=0$, but it cannot turn nonzero news into zero news.

For the complex strain $H=C_{AB}\bar m^A\bar m^B$,

$$H'(u',x^A) =H(u'-\alpha,x^A) +2\bar m^A\bar m^B D_AD_B\alpha.$$

The trace term drops out of the dyad contraction. The same transformation mixes spherical-harmonic modes because the retarded-time shift depends on angle. Boyle gives the complete spin-weighted BMS transformations, including boosts, dyad phases, and mode coupling Boyle 2016, §§II–IV, pp. 2–12, PDF.

Translations, rotations, and boosts

The familiar Poincaré operations sit inside the larger action but affect waveform data in distinct ways:

• A constant $\alpha$ shifts the retarded-time origin and has $S_{AB}=0$.
• An $\ell=1$ supertranslation is a spatial translation. It also has $S_{AB}=0$, but its direction-dependent time shift mixes modes when a translated source is expanded about the old origin.
• A rotation changes angles and the complex dyad. The strain acquires its spin-weight $-2$ phase while modes at fixed $\ell$ mix through a rotation matrix.
• A boost changes the sphere by a conformal map, rescales retarded time, and changes frequency and amplitude weights. It is not equivalent to a constant time or phase shift.
• A pure $\ell\ge2$ supertranslation both shifts cuts and adds the inhomogeneous shear term. It changes the vacuum offset relevant to memory.

These are frame effects, not extra radiation. The curvature, news, and strain descriptions agree only after all are transformed with their correct spin and conformal weights.

What a waveform comparison must fix

A reproducible comparison states the retarded-time origin, spatial origin or center-of-mass prescription, orientation, asymptotic rest frame, polarization dyad, and supertranslation frame. Aligning only time and phase is sufficient only when the remaining BMS freedom has already been controlled. In numerical work, a residual translation can masquerade as higher-mode content, while a pure supertranslation can create a strain offset without creating news.

Memory illustrates the boundary sharply. If early and late stationary regions have shears $C^-_{AB}$ and $C^+_{AB}$, then their difference is obtained by integrating news. A common supertranslation shifts both endpoints by the same $S_{AB}[\alpha]$, so the difference is unchanged after the correct pullback. Applying different frame choices to the two endpoints would instead manufacture a false memory.

Exercise

Prove that ordinary translations do not produce the inhomogeneous shear $S_{AB}[\alpha]$.

Solution

For an $\ell=0$ harmonic, $\alpha$ is constant, so both derivatives vanish. For an $\ell=1$ harmonic on the unit sphere, $D_AD_B\alpha=-q_{AB}\alpha$ and $D^2\alpha=-2\alpha$. Substitution gives $S_{AB}=2(-q_{AB}\alpha)-q_{AB}(-2\alpha)=0$. Hence the four-dimensional translation subgroup changes cuts by pullback but adds no trace-free vacuum shear; only $\ell\ge2$ supertranslations can do so.

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.