= Solution
The coordinate <derivatives> of the smooth two-parameter map commute. With a torsion-free <Levi-Civita connection>,
$$
\frac{DY^a}{Ds}=\frac{\partial^2x^a}{\partial s\partial\tau}+\Gamma^a{}_{bc}V^bY^c
=\frac{DV^a}{D\tau},
$$
because $\Gamma^a{}_{bc}$ is symmetric in its lower indices. Each member of the family is a <geodesic>, so $DV/Ds=0$. Consequently
$$
\frac{D^2Y^d}{Ds^2}=\frac D{Ds}\frac D{D\tau}V^d
=\left(\frac D{Ds}\frac D{D\tau}-\frac D{D\tau}\frac D{Ds}\right)V^d.
$$
The <commutator> of <covariant derivatives> is the action of the <Riemann curvature tensor>. With the source's index and sign convention this gives
$$
\boxed{\frac{D^2Y^d}{Ds^2}=R_{abc}{}^dV^aY^bV^c.}
$$
This is <geodesic deviation>: <free fall> relative acceleration measures <curvature>, even where an individual observer can eliminate the connection at one point.
For Earth's exterior Newtonian field, $\phi=-GM/r$ and acceleration is $a_i=-\partial_i\phi$. Nearby freely falling particles have relative acceleration $\delta a_i=-\phi_{,ij}Y^j$. At the observation point align $z$ radially upward and $x,y$ horizontally. Since $GM/R_E^2=g$,
$$
-\phi_{,ij}=\frac g{R_E}\operatorname{diag}(-1,-1,2).
$$
The weak-field <geodesic>-deviation equation therefore identifies
$$
\boxed{R_{010}{}^1\simeq-\frac g{R_E},\quad R_{020}{}^2\simeq-\frac g{R_E},\quad
R_{030}{}^3\simeq\frac{2g}{R_E}.}
$$
In these units time is measured in seconds as in the stated Newtonian comparison; if the zeroth coordinate is $ct$ instead, the <curvature> components acquire the corresponding $c^{-2}$ factor. A small freely falling cloud is squeezed in the two horizontal directions and stretched radially. These are tidal effects, not the uniform acceleration removed by a freely falling frame.
Their characteristic inverse-square time is $g/R_E$, so $\boxed{t_{\rm tide}=\sqrt{R_E/g}\simeq8.1\times10^2\ {\rm s}\simeq13.4\ {\rm min}}$. One concrete thought experiment is an ideal satellite moving in a <circular orbit> just above a spherical Earth, ignoring atmosphere and terrain. Its orbital <angular frequency> is $\sqrt{GM/R_E^3}=\sqrt{g/R_E}$ and its period is $2\pi t_{\rm tide}\simeq84$ minutes. Equivalently, initially nearby <free fall> trajectories develop order-one tidal changes on this characteristic scale. This is the <surface tidal time scale of a spherical gravitating body>.
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