= Solution
<Geodesic deviation> compares neighboring members of a family of freely falling trajectories. Subtracting their <geodesic equations> to first order in the connecting vector gives a covariant relative-acceleration equation governed by the <Riemann curvature tensor>. For a slowly moving satellite, $u^0=c\dot t\simeq c$ dominates the spatial components. Choose simultaneous spatial separation in a locally transported frame, take the <weak-field approximation>, and neglect terms quadratic in the small separation and higher velocity corrections. The leading spatial equation becomes
$$
\boxed{\frac{d^2\Delta^i}{d\tau^2}+c^2R^i{}_{0j0}\Delta^j\simeq0.}
$$
Replacing the transported covariant derivatives by ordinary derivatives is a leading local or Newtonian approximation; a rotating orbital frame would introduce additional inertial terms. In the convention established above, the instantaneous <tidal acceleration> stretches radial separations and compresses transverse ones.
The growth expected for nearby \i[different-radius] circular orbits is especially transparent from <Kepler's third law>. If the radii are $R$ and $R+\delta R$, their coordinate angular frequencies differ by $\delta\Omega\simeq-(3/2)\Omega\delta R/R$. Initially aligned orbits therefore acquire an along-track separation $R\delta\Omega\,t\simeq-(3/2)\Omega\delta R\,t$. Over the interval in which the nearby-orbit approximation remains valid, this produces a growing separation without needing to solve the deviation equation.
This growth is not universal or indefinitely unbounded. Equal-radius co-orbiting geodesics with a fixed phase offset have constant separation, and slightly inclined equal-radius circular orbits have bounded, periodically varying separation. Even different-radius circular orbits have bounded exact positions; their relative angle eventually wraps around. The secular drift statement assumes distinct radii and describes the local differential-orbit regime.
\Image[/past-exam-of-the-general-relativity-and-cosmology-course-of-the-university-of-oxford/2012-nearby-circular-orbit-drift.png]
{title=Physical tidal stretching and the early differential drift of neighboring circular orbits}
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