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In four spacetime dimensions, the deformation tensor decomposes aswhere $\theta =B^{ab}{h}_{ab}$ is the expansion scalar, ${\sigma }_{ab}=B_{(ab)}-\theta h_{ab}/3$ is the symmetric trace-free shear tensor of a timelike congruence, and ${\omega }_{ab}=B_{[ab]}$ is the vorticity tensor of a timelike congruence.

A timelike geodesic vector field is a smooth unit timelike field $u^a$ whose integral curves obey $u^b{\nabla }_b{u}^a=0$. Its expansion scalar, shear tensor of a timelike congruence, and vorticity tensor of a timelike congruence evolve according to the timelike Raychaudhuri equation.