= Solution
<tidal forces> measure differences in <gravitational acceleration> across an extended region. Neighboring freely falling particles each have zero <four-acceleration>, but their separation need not accelerate uniformly. Let $u^\mu$ be the tangent to a reference <timelike geodesic> and $\xi^\mu$ its infinitesimal separation from a neighboring one, chosen from an affinely parametrized <geodesic congruence>. Using the <Riemann curvature tensor> convention specified in Solution 3,
$$
\frac{D^2\xi^\alpha}{D\tau^2}
=-R^\alpha{}_{\beta\gamma\delta}
u^\beta u^\gamma\xi^\delta .
$$
Changing the overall <curvature> convention changes this displayed sign. The <covariant derivatives> compare <vectors> by <parallel transport>, avoiding a coordinate-dependent subtraction of <accelerations>.
In the <Newtonian limit>, expand the force field at two nearby points:
$$
\ddot\xi_i=-\partial_i\partial_j\Phi\,\xi_j.
$$
For $\Phi=-GM/r$, a local <orthonormal frame> aligned with the radius gives
$$
\boxed{\ddot{\boldsymbol\xi}
=\frac{GM}{r^3}\operatorname{diag}(2,-1,-1)\boldsymbol\xi.}
$$
There is radial stretching and transverse compression; the trace vanishes outside the <mass>. Two freely falling test <masses> and a measurement of their changing separation implement <gravity gradiometry>. Their common <acceleration> drops out, while a relative <acceleration> proportional to separation remains. A <local inertial frame> removes <Christoffel symbols> at a point, but cannot remove nonzero <curvature> over a finite region.
\Image[/past-exam-of-the-relativity-course-of-the-university-of-cambridge/2012/paper-1-tidal-deviation.png]
{title=Point-mass tidal deviation}
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