BMS Asymptotic Symmetry Group
The infinite-dimensional symmetry algebra governing null infinity
1. Definition: Preserving the Bondi Gauge
A BMS transformation is defined as a diffeomorphism that preserves all Bondi gauge conditions at $\mathscr{I}^+$. These conditions are:
$$\text{(i)}\; g_{rr} = 0, \qquad \text{(ii)}\; g_{rA} = 0, \qquad \text{(iii)}\; \det(g_{AB}) = r^4 \det(\gamma_{AB})$$
$$\text{(iv)}\; g_{ur} = -1 + \mathcal{O}(r^{-1}), \qquad \text{(v)}\; g_{uA} = \mathcal{O}(1), \qquad \text{(vi)}\; g_{uu} = -1 + \mathcal{O}(r^{-1})$$
We seek the most general vector field $\xi^\mu = (\xi^u, \xi^r, \xi^A)$ such that $\mathcal{L}_\xi g_{\mu\nu}$ preserves conditions (i)-(vi). The Lie derivative of the metric is $\mathcal{L}_\xi g_{\mu\nu} = \nabla_\mu \xi_\nu + \nabla_\nu \xi_\mu$. Imposing each condition in turn generates the full BMS group:
$$\text{BMS}_{\rm ext} = \underbrace{\mathcal{S}}_{\text{supertranslations}} \rtimes \underbrace{\text{Diff}(S^2)}_{\text{super-rotations}}$$
2. Supertranslation Generator: Detailed Derivation
A supertranslation is parametrized by an arbitrary smooth function $f(x^A)$ on $S^2$. The generating vector field is:
$$\xi_f = f\,\partial_u - \frac{1}{r}D^A f\,\partial_A + \frac{1}{2}D^2 f\,\partial_r + \mathcal{O}(r^{-1})$$
Now compute the action on each metric component. For $g_{uu}$: since $g_{uu} = -1 + 2m_B/r + \ldots$, we need $\mathcal{L}_{\xi_f}g_{uu} = \xi_f^\mu\partial_\mu g_{uu} + 2g_{u\mu}\partial_u\xi_f^\mu$. At leading order:
$$\delta_f g_{uu}\big|_{r^{-1}} = \frac{2}{r}\left(f\,\partial_u m_B + \frac{1}{2}D^2 f\right) \quad \Longrightarrow \quad \delta_f m_B = f\,\partial_u m_B + \frac{1}{4}D^2(D^2 + 2)f$$
For $g_{ur}$: the condition $g_{ur} = -e^{2\beta} \approx -1$ is preserved automatically since $\mathcal{L}_{\xi_f}g_{ur} = \mathcal{O}(r^{-2})$. For $g_{AB}$ at order $r$:
$$\mathcal{L}_{\xi_f}g_{AB}\big|_r = f\,\partial_u(rC_{AB}) + r\left(-D_A\!\left(\frac{1}{r}D^B f\right)\cdot g_{BB'} + \ldots\right)$$
3. Transformation of the Shear Tensor
The key result is the supertranslation action on the shear. Starting from $g_{AB} = r^2\gamma_{AB} + rC_{AB} + \ldots$, the Lie derivative at order $r$ gives:
$$\mathcal{L}_{\xi_f}(rC_{AB}) = f\,\partial_u(rC_{AB}) + rC_{AB}\,\partial_r\!\left(\frac{1}{2}D^2 f\right) + r^2\gamma_{AB}\,\partial_r\!\left(\frac{1}{2}D^2 f\right) + 2r^2 D_{(A}\!\left(-\frac{1}{r}D_{B)} f\right)$$
Evaluating each term: the third term gives $-\frac{1}{2}r\gamma_{AB}D^2 f$ which is pure trace and drops out of the STF part. The fourth term gives $-2rD_{(A}D_{B)}f = -2rD_{\langle A}D_{B\rangle}f - r\gamma_{AB}D^2 f$. Collecting the STF pieces:
$$\boxed{\delta_f C_{AB} = f\,N_{AB} - 2D_{\langle A}D_{B\rangle}f}$$
For a vacuum region where $N_{AB} = 0$ (no radiation), this reduces to $\delta_f C_{AB} = -2D_{\langle A}D_{B\rangle}f$. This is precisely the pure gauge shift of the shear, analogous to a gauge transformation of the vector potential in electromagnetism.
4. Superrotation Generator: Conformal Killing Equation on $S^2$
A superrotation is generated by a vector field $Y^A(x^B)$ on $S^2$ satisfying the conformal Killing equation:
$$D_A Y_B + D_B Y_A = (D_C Y^C)\,\gamma_{AB}$$
In stereographic coordinates $z = e^{i\phi}\tan(\theta/2)$, $\bar{z} = e^{-i\phi}\tan(\theta/2)$, the round metric becomes $ds^2_{S^2} = \frac{4}{(1+z\bar{z})^2}dz\,d\bar{z}$. The conformal Killing equation splits into holomorphic and antiholomorphic parts:
$$\partial_{\bar{z}} Y^z = 0 \quad \Longrightarrow \quad Y^z = Y(z) \;\text{(holomorphic)}$$
The global conformal Killing vectors on $S^2$ are the Mobius transformations $Y(z) = a + bz + cz^2$ (6 real parameters = Lorentz group). The BMS group in its original form uses only these. The extended BMS group promotes $Y(z)$ to an arbitrary meromorphic function, enlarging to $\text{Diff}(S^2)$ or $\text{Vir} \times \overline{\text{Vir}}$ (two copies of the Virasoro algebra). The conformal weight $\mu = D_A Y^A$ measures local stretching.
5. BMS Algebra Commutators
The BMS algebra is a semidirect product. Let $\xi_f$ denote a supertranslation with parameter $f$ and $\xi_Y$ a superrotation with parameter $Y^A$. The commutators are:
Supertranslation-supertranslation: $[\xi_{f_1}, \xi_{f_2}] = 0$
This is because $\xi_f$ acts as $u \to u + f$ and two such shifts commute trivially.
Superrotation-supertranslation:
$$[\xi_Y, \xi_f] = \xi_{\hat{f}}, \qquad \hat{f} = Y^A\partial_A f - \frac{1}{2}(D_A Y^A)\,f$$
The $-\frac{1}{2}D_A Y^A f$ term arises because $f$ transforms as a conformal density of weight $-1/2$ under conformal transformations of $S^2$.
Superrotation-superrotation:
$$[\xi_{Y_1}, \xi_{Y_2}] = \xi_{[Y_1, Y_2]}, \qquad [Y_1, Y_2]^A = Y_1^B\partial_B Y_2^A - Y_2^B\partial_B Y_1^A$$
This is the Lie bracket of vector fields on $S^2$, giving a Virasoro algebra when expanded in Laurent modes $\ell_n = -z^{n+1}\partial_z$.
6. Supermomentum Charge from the Symplectic Form
The supermomentum charge associated with a supertranslation $f$ is constructed from the covariant phase space formalism. Starting from the presymplectic potential $\Theta[\delta g; g]$ on a Cauchy surface, the Noether charge 2-form on a cut of $\mathscr{I}^+$ is:
$$Q_f = \frac{1}{16\pi G}\oint_{S^2}\left(2f\,m_B + \frac{1}{16}f\,N_{AB}C^{AB} + \frac{1}{4}C^{AB}D_A D_B f\right)d^2\Omega$$
The first term is the Bondi mass weighted by $f$. For $f = 1$ (time translation, the $\ell=0$ mode), this reduces to the total Bondi mass. For the $\ell = 1$ harmonics, it gives the three components of Bondi linear momentum. The $\ell \geq 2$ modes are the genuine supertranslation charges with no Poincare analogue. The flux-balance law reads:
$$P(f)\big|_{u_2} - P(f)\big|_{u_1} = -\frac{1}{8}\int_{u_1}^{u_2}\!\oint_{S^2} f\,N_{AB}N^{AB}\,d^2\Omega\,du + \text{(soft terms)}$$
7. Super-Rotation Charge via Wald-Zoupas
The Wald-Zoupas prescription provides a systematic way to define finite, integrable charges. For a superrotation generated by $Y^A$, the charge on a cut $\mathcal{C}$ of $\mathscr{I}^+$ is:
$$J(Y) = \frac{1}{8\pi G}\oint_{\mathcal{C}} Y^A\!\left(N_A + u\,\partial_A m_B\right)d^2\Omega + \frac{1}{32\pi G}\oint_{\mathcal{C}} C^{AB}D_A D_B(D_C Y^C)\,d^2\Omega$$
The first integral contains the angular momentum aspect $N_A$ (the gravitational angular momentum) plus a correction from the Bondi mass gradient. The second integral is the Wald-Zoupas correction ensuring integrability. For the global Lorentz generators $Y^A = \epsilon^A{}_{B}x^B$, this reduces to the standard ADM angular momentum. The flux through a segment of $\mathscr{I}^+$ is:
$$\mathcal{F}_Y = -\frac{1}{16\pi G}\int_{u_1}^{u_2}\!\oint_{S^2} N_{AB}\!\left(\mathcal{L}_Y C^{AB} - \frac{1}{2}(D_C Y^C)C^{AB}\right)d^2\Omega\,du$$
8. Charge Algebra and Central Extensions
The Poisson bracket of the charges must reproduce the BMS algebra. For two supertranslations:
$$\{P(f_1), P(f_2)\} = 0$$
For a superrotation acting on a supermomentum:
$$\{J(Y), P(f)\} = P\!\left(Y^A\partial_A f - \tfrac{1}{2}D_A Y^A\,f\right)$$
For two superrotations, a possible central extension arises:
$$\{J(Y_1), J(Y_2)\} = J([Y_1, Y_2]) + K(Y_1, Y_2)$$
where $K(Y_1, Y_2)$ is a 2-cocycle. In the Virasoro basis $\ell_n = -z^{n+1}\partial_z$, the central extension takes the familiar form $K(\ell_m, \ell_n) = \frac{c}{12}m(m^2-1)\delta_{m+n,0}$, where the central charge $c$ depends on the gravitational theory. This connects to the Kerr/CFT correspondence and soft graviton theorems.
Simulation: Supertranslations & Superrotations on the Celestial Sphere
We visualize supertranslation profiles as deformations of $S^2$ (top row) and superrotation flow lines (bottom row):
BMS symmetry visualization on the celestial sphere
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