= 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 $T=F_*\partial_\tau$ and its <deviation vector> is $S=F_*\partial_s$. 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,
$$
\boxed{[S,T]=0.}
$$
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 $\nabla_TS=\nabla_ST$.
Back to article page
