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


Timelike geodesic vector field

{title2=$\omega_{ab}$}

The vorticity tensor is the antisymmetric spatial part of $\nabla_bu_a$. It vanishes exactly when the timelike congruence is locally hypersurface orthogonal.


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 Incoming links: Vorticity tensor of a timelike congruence