Bondi supertranslations and their action on the shear

Assume Countable Choice for the smooth Lorentzian boundary conventions inherited from A smooth vacuum Bondi expansion. On its formal boundary ${\mathcal{I}}^{+}=I\times S^2$, let ${\mathrm{\ell }}_{\mathcal{I}}={\mathrm{\partial }}_u$ be the null-generator direction and let $q_{AB}$ be the unit round metric on the cuts; the degenerate boundary metric is $q_{AB}d{x}^{A}d{x}^B$ with kernel $\mathbb{R}{\mathrm{\ell }}_{\mathcal{I}}$. A fixed Bondi frame here means the chosen Bondi-Sachs chart and gauge $(u,r,x^A)$ of A smooth vacuum Bondi expansion, including $g_{rr}=g_{rA}=0$ and $\det \gamma =\det q$, with the round cut metric $q_{AB}$ held fixed. It is a chart/gauge choice, not an observer congruence or tetrad. An infinitesimal supertranslation is the asymptotic vector field parametrized by a smooth real function $f:S^2\to \mathbb{R}$ whose leading boundary action is ${\delta }_{f}u=f(x)$ and whose leading bulk generator is

${\xi }_f=f{\mathrm{\partial }}_u-\frac{1}{r}{D}^{A}f{\mathrm{\partial }}_A+\frac{1}{2}{D}^A{D}_{A}f{\mathrm{\partial }}_r+O(r^{-1}){\mathrm{\partial }}_u+O(r^{-2}){\mathrm{\partial }}_A+O(r^{-1}){\mathrm{\partial }}_r\mathrm{.}$

The displayed leading terms preserve Bondi gauge and the fixed round cut metric to the asymptotic orders used here; the omitted terms are not part of this definition. We use the active convention ${\delta }_f={\mathcal{L}}_{{\xi }_f}$. Its action on the shear and news of A smooth vacuum Bondi expansion is

${\delta }_f{C}_{AB}=f{N}_{AB}-2D_A{D}_{B}f+q_{AB}{D}^C{D}_{C}f,{\delta }_f{N}_{AB}=f{\mathrm{\partial }}_u{N}_{AB}\mathrm{.}$

The finite active boundary supertranslation with parameter $f$ is the map ${\mathrm{\Phi }}_f(u,x)=(u+f(x),x)$ on ${\mathcal{D}}_f:=\{(u,x)\in I\times S^2:u+f(x)\in I\}\mathrm{.}$ It is a diffeomorphism from ${\mathcal{D}}_f$ onto ${\mathcal{D}}_{-f}$, with inverse ${\mathrm{\Phi }}_{-f}$. If $I=\mathbb{R}$, it is a diffeomorphism of all of ${\mathcal{I}}^{+}$; for a shorter retarded-time interval the action is defined on this overlap only. In the same active convention, its induced action on the boundary shear and news is

$C_{AB}^{(f)}(u,x)=C_{AB}(u+f(x),x)-2D_A{D}_{B}f(x)+q_{AB}{D}^C{D}_{C}f(x),N_{AB}^{(f)}(u,x)=N_{AB}(u+f(x),x),(u,x)\in {\mathcal{D}}_f\mathrm{.}$

Differentiating these formulas for the parameter $tf$ at $t=0$ recovers the infinitesimal formulas above. This definition gives only the boundary action and induced shear/news; it does not specify the full finite bulk coordinate transformation.

The function $f$ carries length units when the retarded coordinate $u$ is read in SI geometric coordinates; the sphere derivatives and angular coordinates are dimensionless. The Hessian term is trace-free because $q^{AB}{q}_{AB}=2$. When $N_{AB}=0$, an infinitesimal supertranslation therefore changes a time-independent shear by the sphere trace-free Hessian while leaving the news zero.

A smooth vacuum Bondi expansion

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

This definition gives the infinitesimal action and the finite boundary action on its natural domain in one Bondi frame. Mädler–Winicour write the finite retarded-coordinate change passively; the active sign used here is fixed by the infinitesimal convention above. The full asymptotic symmetry group, its Lorentz extension and charge algebra remain outside scope. The ten-parameter subgroup referred to here is ${\mathbb{R}}^{1,3}⋊S{O}^{+}(1,3)$, where $S{O}^{+}(1,3)=\{A\in GL(4,\mathbb{R}):A^{\mathsf{T}}\eta A=\eta ,\det A=1,A\text{ preserves the future cone}\}$ and $\eta =\mathrm{diag}(-1,1,1,1)$. In a larger asymptotically flat framework, selecting a particular embedding of this subgroup requires additional frame or vacuum structure; different choices are related by supertranslations. The definition is classical and does not assert a soft-graviton quantization result.