 Solution

Describe the geodesic variation by a smooth map $F(s,\tau )$, with each curve $\tau \mapsto F(s,\tau )$ a timelike geodesic parametrized by proper time. Its tangent vector is $T=F_{\ast }{\partial }_{\tau }$ and its deviation vector is $S=F_{\ast }{\partial }_s$. Thus $S$ measures the first-order separation of neighboring members at the same value of proper time.

Mixed coordinate derivatives on the parameter surface commute. Consequently, along the variation,For an embedded two-dimensional variation this is the usual Lie bracket of vector fields on its image; more generally the same identity is read along the map $F$ using its commuting parameter derivatives. Since the Levi-Civita connection is torsion-free, it follows that ${\nabla }_{T}S={\nabla }_{S}T$.