Asymptotic Expansion of the Gravitational Field
Bondi–Sachs Form of the Metric
Work in Bondi coordinates $(u, r, x^A)$ where $u$ is retarded time, $r$ is luminosity distance, and $x^A = (\theta, \phi)$ parametrize the 2-sphere. The metric near future null infinity $\mathscr{I}^+$ takes the Bondi–Sachs form:
$$ds^2 = -du^2 - 2\,du\,dr + r^2\,\gamma_{AB}\,dx^A dx^B + \frac{2m_B}{r}\,du^2 + r\,C_{AB}\,dx^A du + \mathcal{O}(r^{-1})$$
Here $\gamma_{AB}$ is the round metric on $S^2$, $m_B(u, x^A)$ is the Bondi mass aspect, and $C_{AB}(u, x^A)$ is the shear tensor (symmetric, trace-free).
The News Tensor
The news tensor encodes the radiative degrees of freedom at null infinity:
$$N_{AB}(u, x^A) = \partial_u C_{AB}$$
Subleading Component: the $g_{uA}$ Expansion
$$g_{uA} = -\tfrac{1}{2}D^B C_{AB} + \frac{1}{r}\left(\tfrac{2}{3}N_A + \tfrac{1}{16}\partial_A(C_{BC}C^{BC})\right) + \mathcal{O}(r^{-2})$$
where $D_A$ is the covariant derivative on $S^2$ and $N_A(u, x^A)$ is the angular momentum aspect. Its evolution equation is:
$$\boxed{\partial_u N_A = \frac{1}{4}D_A D^B D^C C_{BC} - \frac{1}{4}D^B(D_A D^C C_{BC} - D_B D^C C_{AC}) + \frac{1}{4}N_{BC}D^B C^C{}_A + 4\pi G\,T_{uA}^{\rm matter}}$$
Note: This equation is the cornerstone of the spin memory derivation.
Spin-Weight Decomposition
$$\Psi_4^0 = -\ddot{C}_{AB}\bar{m}^A\bar{m}^B, \qquad \Psi_4^0 = \ddot{h}_+ - i\,\ddot{h}_\times$$
Spin memory is encoded in the magnetic parity (odd, B-mode) sector of $C_{AB}$.