Angular Momentum Flux and Memory
1. GW Angular Momentum Flux from the Isaacson Formula
Begin with Isaacson's effective stress-energy tensor for gravitational waves, obtained by averaging the second-order Einstein equations over several wavelengths:
$$T_{\mu\nu}^{\rm GW} = \frac{c^4}{32\pi G}\left\langle \partial_\mu h_{\alpha\beta}^{\rm TT}\,\partial_\nu h^{\alpha\beta}_{\rm TT}\right\rangle$$
The angular momentum flux through a 2-sphere at large $r$ is obtained by contracting with the rotational Killing vectors $\phi_i^a$ of the background:
$$\frac{dJ^i}{du} = -\lim_{r \to \infty} r^2 \int_{S^2} T_{ur}^{\rm GW}\,\phi_i^A\,d^2\Omega = -\frac{1}{32\pi G}\int_{S^2} \dot{h}_{\alpha\beta}^{\rm TT}\,\mathcal{L}_{\phi_i}h^{\alpha\beta}_{\rm TT}\,d^2\Omega$$
In terms of Bondi shear $C_{AB}$ and news $N_{AB} = \partial_u C_{AB}$, this becomes:
$$\frac{dJ^i}{du} = -\frac{1}{16\pi G}\int_{S^2} N_{AB}\,\mathcal{L}_{\phi_i}C^{AB}\,d^2\Omega$$
2. Wald-Zoupas Charge for Super-rotations
For a generic vector field $Y^A$ on $S^2$ (generating a super-rotation), the Wald-Zoupas prescription gives the associated charge:
$$J(Y) = \frac{1}{16\pi G}\int_{S^2}\left[Y^A N_A + \frac{1}{2}C^{AB}\mathcal{L}_Y \gamma_{AB}\right]d^2\Omega$$
The Lie derivative of the shear under $Y^A$ is:
$$\mathcal{L}_Y C_{AB} = Y^C D_C C_{AB} + C_{CB}D_A Y^C + C_{AC}D_B Y^C - \gamma_{AB}C^{CD}D_C Y_D$$
The flux of this charge (the rate of change due to radiation) is:
$$\frac{dJ(Y)}{du} = -\frac{1}{16\pi G}\int_{S^2} N_{AB}\,\mathcal{L}_Y C^{AB}\,d^2\Omega + \frac{1}{4\pi G}\int_{S^2} Y^A T_{uA}^{\rm matter}\,d^2\Omega$$
When $Y^A$ is one of the three rotational Killing vectors $\phi_i^A$ of $S^2$, this reduces to the standard angular momentum flux. For general $Y^A$ (including superboosts and super-rotations), it gives the generalized angular momentum/center-of-mass charges.
3. Ward Identity: Linking Memory to Angular Momentum
Integrate the charge conservation equation from $u = -\infty$ to $u = +\infty$:
$$\Delta J(Y) = J(Y)\big|_{+\infty} - J(Y)\big|_{-\infty} = \frac{1}{16\pi G}\int_{S^2} Y^A \Delta N_A\,d^2\Omega + \frac{1}{32\pi G}\int_{S^2} \Delta C^{AB}\,\mathcal{L}_Y\gamma_{AB}\,d^2\Omega$$
The radiated angular momentum (the flux integral) satisfies:
$$\int_{-\infty}^{+\infty}\frac{dJ(Y)}{du}\,du = -\frac{1}{16\pi G}\int_{S^2}\int_{-\infty}^{+\infty} N_{AB}\,\mathcal{L}_Y C^{AB}\,du\,d^2\Omega + \text{matter}$$
Combining charge conservation $\Delta Q = Q_{\rm final} - Q_{\rm initial} = -\text{(radiated)}$ and rearranging:
$$\boxed{\int_{S^2} Y^A \Delta N_A \, d^2\Omega = -8\pi G \, \Delta J(Y) + \text{(shear boundary terms)}}$$
This is the Ward identity for super-rotations. It states that the change in the angular momentum aspect $\Delta N_A$, projected onto any vector field $Y^A$, equals the change in the corresponding BMS charge. When $Y^A$ has odd parity (B-mode), this gives the spin memory directly.
4. Three Memory Effects from First Principles
Each memory effect is associated with a BMS symmetry and its Ward identity:
Displacement memory (supertranslation): For a supertranslation$f(\theta,\phi)$, the Ward identity gives:
$$\int_{S^2} f \left(D^A D^B \Delta C_{AB}\right) d^2\Omega = -16\pi G \int_{S^2} f \int_{-\infty}^{+\infty}\left(\frac{1}{4}N_{AB}N^{AB} + 4\pi T_{uu}^{\rm matter}\right)du\,d^2\Omega$$
The source is the energy flux (positive definite), giving a purely E-mode signal.
Spin memory (super-rotation): For an odd-parity vector field$Y^A = \epsilon^{AB}D_B g$:
$$\int_{S^2} g\,\epsilon^{AB}D_A \Delta N_B \, d^2\Omega = -8\pi G \, \Delta J_{\rm odd}(g)$$
The source is the angular momentum flux, giving a B-mode signal.
Center-of-mass memory (superboost): For a boost generator$W = n^i x_i$ ($l=1$ harmonics):
$$\Delta \dot{\xi}^i \sim \frac{G}{c^4 D_L}\,\Delta p^i_{\rm GW}$$
The source is the linear momentum flux (GW recoil), giving an E-mode kick memory.
5. Order-of-Magnitude Estimates
Start from the quadrupole formula. The angular momentum radiated by a binary with symmetric mass ratio $\eta = m_1 m_2/(m_1+m_2)^2$ and orbital velocity $v$:
$$\frac{dJ}{du} = -\frac{32}{5}\,\frac{G}{c^5}\,\eta^2 M^2 v^5\,\hat{L}$$
The spin memory strain at distance $D_L$ scales as:
$$h_{\rm spin} \sim \frac{G\,\Delta J}{c^3 D_L} \sim \frac{G^2 \eta M^2 v}{c^6 D_L} \cdot \frac{v^4}{c} \sim h_{\rm disp} \times \frac{v}{c}$$
Numerical values for specific sources:
GW150914 ($M = 65\,M_\odot$, $\eta = 0.247$, $D_L = 410\,\text{Mpc}$, $v/c \sim 0.5$):
$h_{\rm disp} \sim 2.5 \times 10^{-22}$, $\quad h_{\rm spin} \sim 3 \times 10^{-23}$
GW190521 ($M = 150\,M_\odot$, $\eta = 0.22$, $D_L = 5.3\,\text{Gpc}$, $v/c \sim 0.6$):
$h_{\rm disp} \sim 8 \times 10^{-22}$, $\quad h_{\rm spin} \sim 1.5 \times 10^{-22}$
LISA SMBH ($M = 2 \times 10^7\,M_\odot$, $\eta = 0.25$, $D_L = 3\,\text{Gpc}$, $v/c \sim 0.8$):
$h_{\rm disp} \sim 10^{-16}$, $\quad h_{\rm spin} \sim 5 \times 10^{-17}$
6. Leading Post-Newtonian Correction to Spin Memory
At 1PN order, the spin memory acquires corrections from spin-orbit coupling and tail effects. The angular momentum flux gains a correction:
$$\frac{dJ}{du}\bigg|_{\rm 1PN} = \frac{dJ}{du}\bigg|_{\rm leading}\left[1 + \left(\frac{743}{336} + \frac{11}{4}\eta\right)\frac{v^2}{c^2} + 4\pi\frac{v^3}{c^3} + \ldots\right]$$
For precessing binaries with spins $\chi_1$, $\chi_2$, there is an additional spin-orbit contribution to the B-mode angular momentum flux at 1.5PN:
$$\frac{dJ_{\rm B-mode}}{du}\bigg|_{\rm 1.5PN} \supset \frac{32G\eta^2 M^2 v^8}{5c^8}\left[\frac{4}{3}\delta\chi_s + \frac{4}{3}\chi_a\right]\sin\iota$$
where $\chi_s = (\chi_1 + \chi_2)/2$, $\chi_a = (\chi_1 - \chi_2)/2$,$\delta = (m_1 - m_2)/M$, and $\iota$ is the inclination. This spin-orbit term provides the dominant contribution to spin memory for precessing systems, since the leading Newtonian-order spin memory requires nonlinear GW self-interaction.
Three Memory Effects: Parameter Dependence and Detector Comparison
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