Riccati equation for 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.