BMS Group and Asymptotic Symmetries at $\mathscr{I}^+$
Bondi Coordinates & the Bondi–Sachs Metric
Near future null infinity $\mathscr{I}^+$, we adopt retarded Bondi coordinates$(u,r,x^A)$ where $u$ is retarded time, $r$ the luminosity distance, and $x^A = (\theta,\phi)$ angular coordinates on the celestial sphere $S^2$. The Bondi gauge conditions $g_{rr}=0$, $g_{rA}=0$, and$\det(g_{AB}/r^2)=\det(\gamma_{AB})$ fix the coordinate freedom up to the residual asymptotic symmetries.
The Bondi–Sachs metric expansion in inverse powers of $r$ takes the form:
$$ds^2 = -du^2 - 2\,du\,dr + r^2 \gamma_{AB}dx^A dx^B + \frac{2m_B}{r}\,du^2 + r\,C_{AB}dx^A du + O(r^{-2})$$
Here $\gamma_{AB}$ is the round metric on $S^2$,$m_B(u,x^A)$ is the Bondi mass aspect, and $C_{AB}(u,x^A)$ is the symmetric trace-free shear tensor encoding gravitational wave data at $\mathscr{I}^+$.
News Tensor & Bondi Mass Loss
The Bondi news tensor is the retarded-time derivative of the shear:
$$N_{AB} \;:=\; \partial_u C_{AB}$$
Gravitational radiation is present if and only if $N_{AB}\neq 0$. The Bondi mass loss formula encodes the irreversible energy radiated to $\mathscr{I}^+$:
$$\partial_u m_B = -\frac{1}{8}\,N_{AB}\,N^{AB} + \frac{1}{4}\,D_A D_B N^{AB}$$
The first term is manifestly non-positive (upon integration over $S^2$), establishing that the total Bondi mass $M(u) = \frac{1}{4\pi}\oint m_B\,d^2\Omega$ is monotonically non-increasing: gravitational waves always carry positive energy.
BMS Group Structure
The residual diffeomorphisms preserving Bondi gauge at $\mathscr{I}^+$ form the Bondi–van der Burg–Metzner–Sachs (BMS) group. A general BMS vector field is:
$$\xi = \left[f + \frac{u}{2}D_A Y^A\right]\partial_u + Y^A\partial_A - \frac{1}{2}\left(D_A Y^A - \frac{r}{2}D_A \xi^A\right)\partial_r$$
where $f(x^A)$ parametrises supertranslations (an infinite-dimensional abelian ideal) and $Y^A(x^B)$ generates Lorentz transformations (rotations and boosts) on $S^2$.
BMS Algebra
$$\mathfrak{bms}_4 \;=\; \mathfrak{supertrans} \;\rtimes\; \mathfrak{lorentz}$$
Extended BMS (Barnich–Troessaert):
$$\mathfrak{bms}_4^{\mathrm{ext}} \;=\; \mathfrak{supertrans} \;\rtimes\; (\mathfrak{Vir} \oplus \overline{\mathfrak{Vir}})$$
Supertranslation Action on Shear
Under a supertranslation $f(x^A)$, the asymptotic shear transforms as:
$$C_{AB} \;\to\; C_{AB} + 2\,D_{\langle A}D_{B\rangle}f$$
where $D_{\langle A}D_{B\rangle} = D_A D_B - \frac{1}{2}\gamma_{AB}D^2$ denotes the symmetric trace-free covariant derivative on $S^2$. This infinite-dimensional degeneracy of gravitational vacua (labelled by $C_{AB}$ modulo supertranslations) is the origin of soft graviton theorems.
Superrotation Charges
The charge conjugate to a superrotation generated by the vector field $Y^A$ on the celestial sphere is:
$$Q_Y = \frac{1}{8\pi G}\oint_{S^2}\left[Y^A N_A + \frac{u}{2}D_A Y^B\,N_{AB} + \frac{1}{4}C_{AB}\left(2D^A Y^B + D_C Y^C\,\gamma^{AB}\right)\right]d^2\Omega$$
where $N_A$ is the angular momentum aspect. For global Lorentz transformations ($Y^A$ restricted to $\ell=1$ harmonics), $Q_Y$ reduces to the standard ADM angular momentum. For the full Virasoro extension, these charges generate the superrotation Ward identity central to the spin memory effect.