 Solution

The fields $X$ and $S$ arise by pushing forward the two coordinate vectors on the two-parameter family of geodesics, so $[X,S]=0$. Part (i), or equivalently the torsion-free property, givesIn components,ThereforeThis says that the deformation tensor of a timelike geodesic congruence gives the relative velocity of neighboring geodesics represented by the geodesic deviation vector.