= Solution
\b[A <tidal force> is a difference of <gravitational acceleration> across an extended region.] In <Newtonian gravity>, $g_i=-\partial_i\Phi$. Two neighboring freely falling particles separated by $\xi^j$ have, to first order,
$$
\ddot\xi^i=\frac{\partial g_i}{\partial x^j}\xi^j
=-\frac{\partial^2\Phi}{\partial x^i\partial x^j}\xi^j.
$$
For a <point mass>, $\Phi=-GM/r$. In a local orthonormal radial/transverse frame,
$$
\boxed{\ddot\xi_r=\frac{2GM}{r^3}\xi_r,\qquad
\ddot\xi_{\perp1}=-\frac{GM}{r^3}\xi_{\perp1},\qquad
\ddot\xi_{\perp2}=-\frac{GM}{r^3}\xi_{\perp2}.}
$$
A small freely falling cloud stretches radially and compresses transversely. Outside the source the three eigenvalues sum to zero, consistently with $\nabla^2\Phi=0$; within matter the trace is $-4\pi G\rho$. These are accelerations of separations, not the much larger common acceleration of the cloud's centre.
\Image[/past-exam-of-the-relativity-course-of-the-university-of-cambridge/2015/paper-1-tidal-deviation.png]
{title=Original local tidal map: a freely falling cloud stretches along the radial direction and contracts in both transverse directions}
In <general relativity>, freely falling test particles follow <timelike geodesics>. A freely falling frame can eliminate the connection at one event, but not the <Riemann curvature tensor> throughout a finite laboratory. Let $u$ be the <geodesic> tangent and $\xi$ a connecting <vector> for neighboring <geodesics> with $[u,\xi]=0$. In the explicitly chosen <curvature> convention
$$
R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z,
$$
the <geodesic deviation> equation is
$$
\boxed{\frac{D^2\xi}{d\tau^2}=R(u,\xi)u.}
$$
It follows by commuting $\nabla_u\nabla_\xi u$, using $\nabla_u u=0$ and zero torsion. In a <parallel-propagated orthonormal frame>, its spatial components directly describe measured relative acceleration. The weak-field, slow-motion limit recovers the Newtonian Hessian above. Reversing the <curvature sign convention> reverses the displayed <curvature> side, without changing the physical stretching. The <equivalence principle> removes locally uniform gravity, while <geodesic deviation> measures the residual tidal field. The connection between <parallel transport>, <curvature> and neighboring <geodesics> is developed systematically in the standard geometric construction.<Lecture Notes on General Relativity by Sean M. Carroll>[Section 3, Curvature: parallel transport and geodesic deviation; translate the chosen curvature sign consistently]{ref}
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