Spin Memory & BMS Symmetries
The infinite-dimensional symmetry group of asymptotically flat spacetimes and its physical consequences
1. The BMS Group
The Bondi-van der Burg-Metzner-Sachs (BMS) group is the asymptotic symmetry group of asymptotically flat spacetimes at null infinity $\mathscr{I}^+$. Instead of the expected Poincare group, one finds an infinite-dimensional enhancement:
$$\mathrm{BMS} = \mathrm{Lorentz} \ltimes \mathrm{Supertranslations}$$
Supertranslations are angle-dependent translations of retarded time $u$: for any smooth function $f(\theta, \phi)$ on $S^2$,
$$u \to u + f(\theta, \phi), \quad C_{AB} \to C_{AB} + 2D_A D_B f - \gamma_{AB} D^2 f$$
The ordinary translations form the $\ell = 0, 1$ subspace of supertranslations (4-dimensional). The extended BMS group further promotes the Lorentz factor to superrotations, generated by all conformal Killing vectors on $S^2$ (Virasoro algebra).
2. Displacement vs. Spin Memory
The displacement memory is the permanent shift in $C_{AB}$ sourced by the energy flux $T_{uu}$ at $\mathscr{I}^+$:
$$\Delta C_{AB}^{\rm disp} \sim \int_{-\infty}^{+\infty} N_{CD} N^{CD}\,du$$
The spin memory is sourced by the angular momentum flux. Define the magnetic part of the shear via the dual:
$$\Sigma_{AB} = \epsilon_{(A}{}^C\,C_{B)C}$$
The spin memory is the permanent shift in the magnetic shear, related to the angular momentum aspect $N_A$:
$$\boxed{\Delta\Sigma_{AB} = \int_{-\infty}^{+\infty} \epsilon_{(A}{}^C\,N_{B)C}\,du \;\sim\; \Delta J_{\rm rad}}$$
While displacement memory shifts the distance between test masses, spin memory induces a relative rotation of a gyroscope frame, detectable through the time integral of a differential frame-dragging effect.
3. The Infrared Triangle
Strominger discovered a remarkable equivalence between three seemingly unrelated phenomena, forming a triangle:
$$\text{Asymptotic symmetries} \;\longleftrightarrow\; \text{Soft theorems} \;\longleftrightarrow\; \text{Memory effects}$$
For supertranslations, the three corners are:
$$\text{Supertranslation Ward identity} \;\longleftrightarrow\; \text{Weinberg soft graviton theorem}$$
$$\text{Weinberg soft graviton} \;\longleftrightarrow\; \text{Displacement memory}$$
$$\text{Displacement memory} \;\longleftrightarrow\; \text{Supertranslation Ward identity}$$
For superrotations, the analogous triangle connects superrotation symmetry, the Cachazo-Strominger subleading soft theorem, and the spin memory effect.
4. Weinberg Soft Graviton Theorem
When a graviton with momentum $q$ is emitted with $q \to 0$, the scattering amplitude factorizes:
$$\lim_{\omega \to 0}\mathcal{A}_{n+1}(q, p_1, \ldots, p_n) = \frac{\kappa}{2}\sum_{k=1}^{n}\frac{p_k^\mu p_k^\nu \varepsilon_{\mu\nu}(q)}{p_k \cdot q}\;\mathcal{A}_n(p_1, \ldots, p_n)$$
This leading soft theorem is equivalent to the supertranslation Ward identity. The subleading soft theorem at order $\omega^0$ involves the angular momentum operator:
$$\lim_{\omega \to 0}(1 + \omega\,\partial_\omega)\mathcal{A}_{n+1} = \frac{\kappa}{2}\sum_{k=1}^{n}\frac{p_k^\mu \varepsilon_{\mu\nu} q_\lambda J_k^{\nu\lambda}}{p_k \cdot q}\;\mathcal{A}_n$$
This is the superrotation Ward identity and connects to spin memory.
5. Celestial Holography
The celestial sphere at null infinity carries the structure of a 2D conformal field theory. Scattering amplitudes, when Mellin-transformed in the boost eigenvalue $\Delta$, become correlation functions of conformal primary operators on $S^2$:
$$\widetilde{\mathcal{A}}_n(\Delta_k, z_k, \bar{z}_k) = \prod_{k=1}^{n}\int_0^\infty d\omega_k\,\omega_k^{\Delta_k - 1}\;\mathcal{A}_n(\omega_k, z_k, \bar{z}_k)$$
The BMS symmetries become the symmetries of this celestial CFT: supertranslations generate a $\hat{u}(1)$ Kac-Moody symmetry, and superrotations generate two copies of the Virasoro algebra. The OPE of celestial operators encodes collinear splitting functions:
$$\mathcal{O}_{\Delta_1}(z_1)\,\mathcal{O}_{\Delta_2}(z_2) \sim \frac{B(\Delta_1, \Delta_2)}{(z_1 - z_2)^{h_1 + h_2 - h_{12}}}\,\mathcal{O}_{\Delta_1 + \Delta_2 - 1}(z_2) + \ldots$$
For the full treatment of BMS symmetries and gravitational wave memory, see GR Part VI: Gravitational Wave Memory & BMS Symmetries.
Simulation: Spin Memory from BBH Merger
We model the spin memory accumulation from a binary black hole merger, comparing displacement and spin memory for different mass ratios and spin parameters. Spin memory is sourced by the angular momentum flux $\sim N_{AB}\,\epsilon^{AC}\,C_{BC}$:
Spin memory accumulation from BBH merger
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