Spin Memory Effect โ Full Derivation
1. E/B Decomposition of the Shear Tensor
The symmetric trace-free (STF) tensor $C_{AB}$ on $S^2$ admits a Helmholtz decomposition into electric (E) and magnetic (B) parity parts. Any STF tensor on the 2-sphere can be written as:
$$C_{AB} = \underbrace{D_{\langle A}D_{B\rangle}\Phi}_{\text{E-mode}} + \underbrace{\epsilon_{C(A}\,D^C D_{B)}\Psi - \frac{1}{2}\gamma_{AB}\epsilon^{CD}D_C D_D\Psi}_{\text{B-mode}}$$
where $D_{\langle A}D_{B\rangle} = D_{(A}D_{B)} - \frac{1}{2}\gamma_{AB}D^2$ is the STF derivative. The last term vanishes since $\epsilon^{CD}D_CD_D$ is antisymmetric on a scalar, so the B-mode simplifies to:
$$C_{AB}^{(B)} = \epsilon_{C(A}\,D^C D_{B)}\Psi$$
Proof of completeness: In the complex null dyad formalism with $m^A$, $\bar{m}^A$, define the complex shear $\sigma = C_{AB}m^A m^B$. This has spin weight $s=-2$. Expanding in spin-weighted spherical harmonics $ {}_{-2}Y_{\ell m}$:
$$\sigma(u,x^A) = \sum_{\ell \geq 2}\sum_{m=-\ell}^{\ell}\left[E_{\ell m}(u) - i\,B_{\ell m}(u)\right]\,{}_{-2}Y_{\ell m}(x^A)$$
The real and imaginary parts correspond precisely to the electric and magnetic scalars:
$$\Phi = -\sum_{\ell,m}\frac{2E_{\ell m}}{(\ell-1)\ell(\ell+1)(\ell+2)}\,Y_{\ell m}, \qquad \Psi = -\sum_{\ell,m}\frac{2B_{\ell m}}{(\ell-1)\ell(\ell+1)(\ell+2)}\,Y_{\ell m}$$
The factor $(\ell-1)\ell(\ell+1)(\ell+2)$ arises from applying $D_{\langle A}D_{B\rangle}$ twice to $Y_{\ell m}$, or equivalently from the spin-raising/lowering operators: $\eth^2 Y_{\ell m} = \sqrt{(\ell-1)\ell(\ell+1)(\ell+2)}\,{}_{-2}Y_{\ell m}$.
2. Angular Momentum Aspect Evolution
The angular momentum aspect $N_A(u,x^B)$ appears at $O(r^{-2})$ in the Bondi-Sachs metric. The Einstein equation $G_{uA} = 0$ at this order yields, after careful computation of all Christoffel symbols:
$$\partial_u N_A = D_A m_B - \frac{1}{4}\partial_u\!\left(C_{AB}D_CC^{BC}\right) + \frac{1}{4}D_B\!\left(C^{BC}N_{CA} - N^{BC}C_{CA}\right) - \frac{1}{16}D_A\!\left(N_{BC}C^{BC}\right)$$
Let us track each term. The first, $D_Am_B$, is the gradient of the mass aspect; using the Bondi mass loss formula, $\partial_u m_B = -\frac{1}{8}N_{AB}N^{AB} + \frac{1}{4}D_AD_BN^{AB}$. The second term involves the product of shear and its divergence. The third is a non-linear shear-news coupling. At linearised order (weak field), only $D_Am_B$ survives:
$$\partial_u N_A\Big|_{\rm linear} = D_A m_B + O(C^2)$$
The non-linear terms are essential for the spin memory, as they source the B-mode component of $\Delta N_A$ even when the source has no intrinsic magnetic moment.
3. Integration Across the Gravitational Wave Burst
We integrate from $u = u_i \to -\infty$ to $u = u_f \to +\infty$. The net change in angular momentum aspect is:
$$\Delta N_A = \int_{-\infty}^{+\infty}\partial_u N_A\,du = \Delta(D_Am_B) - \frac{1}{4}\left[C_{AB}D_CC^{BC}\right]_{-\infty}^{+\infty} + \int_{-\infty}^{+\infty}\frac{1}{4}D_B(C^{BC}N_{CA} - N^{BC}C_{CA})\,du$$
The boundary term $[C_{AB}D_CC^{BC}]_{-\infty}^{+\infty}$ generically does not vanish, because the shear takes different vacuum values before and after the burst (this is exactly the displacement memory). Decomposing $\Delta N_A$ into gradient (E) and curl (B) parts:
$$\Delta N_A = D_A\alpha + \epsilon_A^{\;\;B}D_B\beta$$
The gradient part $\alpha$ encodes displacement memory (E-mode), and the curl part $\beta$ encodes spin memory (B-mode). Taking $\epsilon^{AB}D_A$ of the evolution equation projects onto the B-mode:
$$D^2\beta = \epsilon^{AB}D_A\Delta N_B = \int_{-\infty}^{+\infty}\epsilon^{AB}D_A\partial_u N_B\,du$$
4. The Spin Memory Formula and the $\Delta_2(\Delta_2+2)$ Operator
The magnetic potential $\Psi$ of the shear is related to $\beta$ by the angular momentum flux. Using $C_{AB}^{(B)} = \epsilon_{C(A}D^CD_{B)}\Psi$ and the identity $D^B\epsilon_{C(A}D^CD_{B)}\Psi = \frac{1}{2}\epsilon_A^{\;\;B}D_B(D^2+2)\Psi$, the divergence of the B-mode shear gives:
$$D^BC_{AB}^{(B)} = \frac{1}{2}\epsilon_A^{\;\;B}D_B(D^2+2)\Psi$$
The spin memory $\Delta\Psi = \Psi(u_f) - \Psi(u_i)$ is obtained by inverting this relation. Taking the curl of $\Delta N_A$ and using $\Delta N_A^{(B)} = \frac{1}{2}\epsilon_A^{\;\;B}D_B(D^2+2)\Delta\Psi$, we get:
$$\frac{1}{2}D^2(D^2+2)\Delta\Psi = \epsilon^{AB}D_A\Delta N_B$$
Using $\Delta_2 = D^2$ (the scalar Laplacian on $S^2$), and noting that $Y_{\ell m}$ are eigenfunctions with $D^2 Y_{\ell m} = -\ell(\ell+1)Y_{\ell m}$, the operator $\Delta_2(\Delta_2+2)$ has eigenvalues $\ell(\ell+1)[\ell(\ell+1)-2] = (\ell-1)\ell(\ell+1)(\ell+2)$, which is invertible for $\ell \geq 2$:
$$\boxed{\Delta\Psi = -\frac{2}{\Delta_2(\Delta_2+2)}\int_{-\infty}^{+\infty}\epsilon^{AB}D_A\partial_u N_B\,du}$$
The $\ell=0$ mode is absent (scalars have no curl) and $\ell=1$ gives zero eigenvalue of $\Delta_2+2$, corresponding to global rotations (not physical memory). Physical spin memory starts at $\ell=2$.
5. Physical Holonomy: Spin Connection and Curvature
The spin memory is measured by the holonomy of the gravitational spin connection on the celestial sphere. Define the spin connection 1-form:
$$A_A^{\rm spin}(u) = \frac{1}{2}\epsilon^{BC}D_B C_{CA}$$
This is the connection through which a gyroscope spin vector is parallel-transported on $S^2$ in the presence of gravitational wave shear. The curvature (Berry-like phase) is:
$$F_{AB}^{\rm spin} = \partial_A A_B^{\rm spin} - \partial_B A_A^{\rm spin} = \frac{1}{2}\epsilon^{CD}D_C D_{[A}C_{B]D} + \frac{1}{2}\epsilon_{AB}(D^2+2)\Psi$$
By Stokes theorem, the holonomy angle around a closed loop $\mathcal{C}$ enclosing a region $\Sigma$ is:
$$\Delta\alpha_{\mathcal{C}} = \oint_{\mathcal{C}} A_A^{\rm spin}\,dx^A = \int_\Sigma F_{AB}^{\rm spin}\,d\Sigma^{AB}$$
The change in holonomy after the burst passes is:
$$\boxed{\Delta\alpha_{\mathcal{C}} = \frac{1}{2}\int_\Sigma (D^2+2)\Delta\Psi\;\epsilon_{AB}\,d\Sigma^{AB}}$$
This is a gauge-invariant, physically measurable quantity: a ring of gyroscopes around $\mathcal{C}$ undergoes a net relative rotation $\Delta\alpha_{\mathcal{C}}$ after the gravitational wave burst.
6. Mode Expansion in Vector Spherical Harmonics
Expand $\Delta N_A$ in vector spherical harmonics $Y_A^{(\ell m)} = D_A Y_{\ell m}$ (E-type) and $S_A^{(\ell m)} = \epsilon_A^{\;\;B}D_B Y_{\ell m}$ (B-type):
$$\Delta N_A = \sum_{\ell \geq 2}\sum_m \left[\Delta N_{\ell m}^E\,D_A Y_{\ell m} + \Delta N_{\ell m}^B\,\epsilon_A^{\;\;B}D_B Y_{\ell m}\right]$$
The B-type coefficients determine the spin memory through:
$$\Delta\Psi_{\ell m} = \frac{2\,\Delta N_{\ell m}^B}{(\ell-1)\ell(\ell+1)(\ell+2)}$$
The rapid fall-off with $\ell$ (as $\ell^{-4}$) means the spin memory is dominated by the $\ell = 2$ quadrupole mode for compact binary sources.
7. Post-Newtonian Estimate from the Quadrupole Formula
For a compact binary with total mass $M$, reduced mass $\mu$, and orbital separation $a$, the angular momentum flux is related to the mass quadrupole moment $I_{ij}$ by:
$$\frac{dL}{du} = \frac{G}{5c^5}\epsilon_{ijk}\,\ddot{I}_{ja}\,\dddot{I}_{ka}$$
The total angular momentum radiated during inspiral and merger is $\Delta L \sim \eta\,\frac{GM^2}{c}$ where $\eta = \mu/M$ is the symmetric mass ratio ($\eta = 1/4$ for equal masses). The spin memory at distance $D$ is dominated by $\ell=2$, with angular dependence $\sin^2\theta$ from $Y_{2,0}$:
$$\boxed{\Delta\Psi_{\ell=2} \sim \frac{G}{c^3}\,\frac{\Delta L}{D}\,\sin^2\theta \sim \frac{G^2\eta M^2}{c^4 D}\,\sin^2\theta}$$
For a $60\,M_\odot$ BBH merger at $D = 400\,\mathrm{Mpc}$ with $\Delta L \sim 0.05\,GM^2/c$, the peak spin memory strain is $h_{\rm spin} \sim 10^{-25}$. This is roughly $10^{-3}$ times the displacement memory ($h_E \sim 10^{-22}$), making spin memory a target for next-generation detectors (Einstein Telescope, Cosmic Explorer, LISA).
Simulation: Spin Memory for a Binary Black Hole Merger
We compute the E-mode (displacement) and B-mode (spin) memory patterns for a $60\,M_\odot$ BBH merger at 400 Mpc, comparing angular profiles and multipole spectra.
Spin Memory: E-mode vs B-mode for BBH Merger
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