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First differentiate the unit normalization $X^{\alpha }{X}_{\alpha }=-1$. Metric compatibility givesThe timelike geodesic vector field equation gives the other contraction,For the spatial projector of a timelike congruence,and symmetry also gives $h_{\alpha \beta }{X}^{\beta }=0$. Thus

Splitting the deformation tensor of a timelike geodesic congruence into symmetric and antisymmetric parts and substituting the definitions givesMoreover, the orthogonality of $B_{\alpha \beta }$ implieswhile in four dimensionsIt follows thatHencewhich is the kinematic decomposition of a timelike geodesic congruence.

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.