Spin Memory Effect — Full Derivation
Gyroscope Setup at $\mathscr{I}^+$
Consider a network of freely falling gyroscopes stationed at large radius $r$ near future null infinity $\mathscr{I}^+$. Each gyroscope carries a spin vector$S^a$ that is parallel-transported along its worldline. As a burst of gravitational radiation passes, the connection on the celestial sphere is modified by the time-varying shear$C_{AB}(u,x^C)$. The net effect after the burst is a permanent rotation of the gyroscope's polarisation direction: the spin memory effect.
Spin-Weighted Spherical Harmonics Decomposition
The symmetric trace-free shear tensor $C_{AB}$ on $S^2$ is decomposed into electric and magnetic parts using spin-weighted spherical harmonics$ {}_{s}Y_{\ell m}$. Defining the complex shear scalar:
$$\sigma(u,x^A) = C_{AB}\,m^A m^B = \sum_{\ell\geq 2}\sum_{m}\left[E_{\ell m}(u) - i\,B_{\ell m}(u)\right]\,{}_{-2}Y_{\ell m}(x^A)$$
where $m^A$ is a complex null dyad on $S^2$,$E_{\ell m}$ are the electric (parity-even) modes and $B_{\ell m}$are the magnetic (parity-odd) modes. The spin memory is sourced by the magnetic sector.
Angular Momentum Aspect Evolution
The angular momentum aspect $N_A(u,x^B)$ satisfies an evolution equation dictated by the Einstein equations at subleading order in the $1/r$ expansion:
$$\partial_u N_A = D_A m_B - \frac{1}{4}\partial_u\!\left(C_{AB}\,D_C C^{BC}\right) + \frac{1}{4}D_B\!\left(C^{BC}N_{CA} - N^{BC}C_{CA}\right)$$
Integrating across a complete gravitational wave burst (from $u\to -\infty$ to$u\to +\infty$), we obtain the net change:
$$\Delta N_A = \int_{-\infty}^{+\infty}\partial_u N_A\,du$$
The curl of $\Delta N_A$ on $S^2$ isolates the magnetic component responsible for spin memory.
The Spin Memory Formula
Extracting the magnetic parity component and inverting the angular Laplacian on $S^2$, the spin memory is governed by:
$$\boxed{\Delta\Psi = -\frac{1}{\Delta_2(\Delta_2+2)}\int_{-\infty}^{+\infty}\epsilon^{AB}D_A\partial_u N_B\,du}$$
Here $\Delta_2 = D^A D_A$ is the scalar Laplacian on $S^2$,$\epsilon^{AB}$ is the Levi-Civita tensor on $S^2$, and$\Delta\Psi$ is the net angular displacement experienced by a gyroscope due to the magnetic component of gravitational radiation.
Physical Measurement: Holonomy Angle via Stokes' Theorem
The observable spin memory is the holonomy of the connection around a closed loop$\mathcal{C}$ on the celestial sphere. By Stokes' theorem, the rotation angle is given by the integrated curvature over the enclosed region $\Sigma_{\mathcal{C}}$:
$$\Delta\alpha_{\mathcal{C}} = \int_{\Sigma_{\mathcal{C}}}\epsilon^{AB}D_A N_B\,d^2\Omega$$
This relates the holonomy of gyroscope transport directly to the curl of the angular momentum flux, providing a gauge-invariant, physically measurable observable. A ring of gyroscopes enclosing a solid angle $\Sigma_{\mathcal{C}}$ experiences a net relative rotation$\Delta\alpha_{\mathcal{C}}$ after the passage of the wave burst.
Mode Expansion & Detectability Estimate
Expanding $\Delta\Psi$ in vector spherical harmonics, the leading$\ell = 2$ mode dominates for compact binary mergers. The amplitude estimate is:
$$\Delta\Psi_{\ell=2} \;\sim\; \frac{G}{c^3}\,\frac{\Delta L}{D}\,\sin^2\theta$$
where $\Delta L$ is the angular momentum radiated during the burst,$D$ is the luminosity distance, and $\theta$ is the inclination angle. For a binary black hole merger at $D \sim 400\,\mathrm{Mpc}$ with$\Delta L \sim 0.05\,M^2 c$, the spin memory strain is of order$h_{\mathrm{spin}} \sim 10^{-25}$, marginally within reach of next-generation detectors such as the Einstein Telescope and Cosmic Explorer.