Kinematic decomposition of a timelike geodesic congruence
= Kinematic decomposition of a timelike geodesic congruence
In four spacetime dimensions, the deformation tensor decomposes as
$$
B_{ab}=\frac13\theta h_{ab}+\sigma_{ab}+\omega_{ab},
$$
where $\theta=B^{ab}h_{ab}$ 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>.