= TT coordinate separation and measured separation
{c}
{title2=$\xi^{\hat A}=\xi_{\rm coord}^A+\tfrac12h_{AB}L^B$}
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 $\ddot\xi^A_{\rm coord}=0$. 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>.
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