Deformation tensor of a timelike geodesic congruence ($B_{ab}={\nabla }_b{u}_a$)

For a unit geodesic vector field $u^a$, differentiation of $u^a{u}_a=-1$ and the geodesic equation show that $B_{ab}{u}^a=B_{ab}{u}^b=0$. The tensor therefore acts entirely in the local rest space.

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.

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.