Displacement Memory Effect (Review)
Review displacement memory to set up the structural parallel with spin memory
Geodesic Deviation
The displacement memory effect arises from geodesic deviation in the presence of gravitational radiation:
$$\ddot{\xi}^i = -R^i{}_{0j0}\,\xi^j = \tfrac{1}{2}\ddot{h}_{ij}^{\rm TT}\,\xi^j$$
Integrating twice yields a permanent displacement:
$$\Delta\xi^i = \tfrac{1}{2}\Delta C_{ij}^{\rm TT}\,\xi^j_0, \qquad \Delta C_{AB} = C_{AB}(+\infty) - C_{AB}(-\infty)$$
Bondi Mass Evolution
The evolution of the Bondi mass aspect governs how energy is radiated to null infinity:
$$\partial_u m_B = -\tfrac{1}{8}N_{AB}N^{AB} + \tfrac{1}{4}D^A D^B N_{AB} + 4\pi G\,T_{uu}^{\rm matter}$$
Christodoulou Nonlinear Memory
The $N_{AB}N^{AB}$ term sources a DC offset — the Christodoulou nonlinear memory:
$$(D^2 + 2)D^2\,\Delta C_{AB}^{\rm mem} = -16\int_{-\infty}^{+\infty}\left(N_{CA}N^C{}_B - \tfrac{1}{2}\gamma_{AB}N_{CD}N^{CD}\right)du$$
Summary
Observable
$\Delta\xi^i$ — permanent separation
Source
Energy flux $\int N_{AB}N^{AB}\,du$
Symmetry
Supertranslation $\delta_f$
Detector Signal
Step in $h_+(t)$ or $h_\times(t)$