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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.

With the curvature convention $[{\nabla }_{\mu },{\nabla }_{\beta }]X_{\alpha }=-R^{\rho }{}_{\alpha \mu \beta }{X}_{\rho }$, the deformation tensor along a unit timelike geodesic field satisfiesTaking its spatial trace and inserting the kinematic decomposition of a timelike geodesic congruence leads to the timelike Raychaudhuri equation.