TT coordinate separation and measured separation (${\xi }^{\hat{A}}={\xi }_{\mathrm{coord}}^A+\frac{1}{2}{h}_{AB}{L}^B$)

In transverse-traceless gauge, particles initially at rest have fixed spatial coordinates because ${\Gamma }^{\mu }{}_{00}=0$. Their separation measured in a parallel-propagated orthonormal frame changes: to first order $S^{\hat{A}}=S^A+h_{AB}{S}^B/2$. Hence the physical geodesic deviation is ${\ddot{\xi }}^{\hat{A}}={\ddot{h}}_{AB}{L}^B/2$, whereas the coordinate displacement obeys ${\ddot{\xi }}_{\mathrm{coord}}^A=0$. The extra ${\partial }_0{\Gamma }^A{}_{0B}{L}^B$ term in the coordinate expansion of the second covariant derivative reconciles the two results. This applies to infinitesimal separations and linear metric perturbations.