= Geodesic deviation
{title2=$\nabla_U\nabla_UJ=R(U,J)U$}
{wiki}
For a family of affinely parametrized geodesics with tangent $U$ and connecting field $J$, the Jacobi equation in the convention used here is
$$
\nabla_U\nabla_UJ=R(U,J)U.
$$
It describes the relative acceleration produced by spacetime curvature.
Here $J$ describes an infinitesimal separation and the <covariant derivatives> compare it in a frame transported along the reference <geodesic>. The <Riemann curvature tensor> thus becomes something measured with neighboring trajectories. The Newtonian version follows by differentiating <gravitational acceleration> with respect to position. In the exterior of a spherical mass, let $\xi_r$ be radial separation and $\xi_\perp$ separation in either transverse direction. <Newton's law of universal gravitation> gives
$$
\ddot\xi_r=\frac{2GM}{r^3}\xi_r,\qquad
\ddot\xi_\perp=-\frac{GM}{r^3}\xi_\perp.
$$
The upper particle falls less strongly than the lower one, stretching their radial separation; side-by-side particles accelerate toward converging radial directions, compressing their transverse separation. See https://sites.science.oregonstate.edu/physics/coursewikis/GGR/book/ggr/tidal.html[Oregon State's derivation of tidal acceleration].
For a spherical <Earth> model with $r=R_\oplus=6.371\times10^6\,\mathrm m$, $g=9.81\,\mathrm{m\,s^{-2}}$ and $GM=gR_\oplus^2$, a one-metre radial separation gives $\ddot\xi_r=2g/R_\oplus=3.07958\times10^{-6}\,\mathrm{m\,s^{-2}}$; a one-metre transverse separation gives $-1.53979\times10^{-6}\,\mathrm{m\,s^{-2}}$. Initially comoving particles therefore change separation after one second by approximately $\tfrac12\ddot\xi\,t^2$: $+1.54$ micrometres radially or $-0.770$ micrometres transversely. This short-time calculation holds the reference radius fixed to leading order. Its beauty is that common falling motion drops out, leaving a measurable stretching pattern that reveals curvature.
\Image[/gravitational-tidal-field.png]
{title=Radial stretching and transverse compression in a small freely falling laboratory near Earth's surface}
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