Past exam of the general relativity and cosmology course of the University of Oxford / 2012 / 1 / c

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 becomesReplacing 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 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 $R\delta \Omega t\simeq -(3/2)\Omega \delta Rt$. 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.