Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 1 / d

Parallel propagation means ${\nabla }_T{e}_{\alpha }=0$, where $T=\dot{\lambda }$. By metric compatibility,The inner products therefore retain their initial values ${\eta }_{\alpha \beta }$, so the parallel-propagated orthonormal frame remains orthonormal.

In the propagated frame $T=e_0$, and the constant sectional curvature formula reduces geodesic deviation toThe stated temporal initial data give $y^0(t)=0$. Writing $\omega =\sqrt{-K}$ for $K<0$ and $\kappa =\sqrt{K}$ for $K>0$, the spatial displacement isThese formulas are valid to first order in the initial separation and relative velocity.