BMS Asymptotic Symmetry Group
The infinite-dimensional symmetry algebra governing null infinity
The BMS Group
The asymptotic symmetry group of asymptotically flat spacetimes at null infinity is the Bondi–van der Burg–Metzner–Sachs (BMS) group. In its extended form:
$$\text{BMS}_{\rm ext} = \underbrace{\mathcal{S}}_{\text{supertranslations}} \rtimes \underbrace{\text{Diff}(S^2)}_{\text{super-rotations}}$$
Supertranslations
A supertranslation is a retarded-time shift depending on direction:
$$u \mapsto u + f(x^A), \qquad f \in C^\infty(S^2)$$
Ordinary translations correspond to the $\ell=0,1$ modes; the $\ell\geq 2$ modes constitute the infinite-dimensional enhancement. The supermomentum charge is:
$$P(f) = \frac{1}{4\pi G}\int_{S^2} f(x^A)\,m_B\,d^2\Omega$$
Super-rotations
Generated by conformal Killing vector $Y^A$ on $S^2$:
$$\mathcal{L}_Y \gamma_{AB} = \mu(x)\,\gamma_{AB}$$
Acting on metric perturbations:
$$\delta_Y C_{AB} = \mathcal{L}_Y C_{AB} - \tfrac{1}{2}\mu\,C_{AB} + 2D_{\langle A}D_{B\rangle}f_Y$$
The super-rotation charge:
$$J(Y) = \frac{1}{8\pi G}\int_{S^2} Y^A N_A\,d^2\Omega + \text{(shear correction terms)}$$
Key Correspondence
Displacement Memory
↔ Supertranslation charge imbalance
Spin Memory
↔ Super-rotation charge imbalance
Charge Balance Law
$$J(Y)\big|_{u=+\infty} - J(Y)\big|_{u=-\infty} = -\frac{1}{16\pi G}\int_{-\infty}^{+\infty}\int_{S^2} N_{AB}\,\mathcal{L}_Y C^{AB}\,d^2\Omega\,du + (\text{matter flux})$$
A non-zero right-hand side forces $\Delta N_A \neq 0$ — the spin memory.