Timelike geodesic vector field

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.

The expansion scalar of a unit timelike congruence with tangent $u^a$ is $\theta ={\nabla }_a{u}^a$. It gives the fractional rate of change of an infinitesimal comoving spatial volume.

The shear tensor is the symmetric trace-free spatial part of ${\nabla }_b{u}_a$. It changes the shape of an infinitesimal comoving volume without changing that volume to first order.

The vorticity tensor is the antisymmetric spatial part of ${\nabla }_b{u}_a$. It vanishes exactly when the timelike congruence is locally hypersurface orthogonal.

A focal point of a hypersurface along an orthogonal geodesic is a point where a nonzero normal Jacobi field arising from variations of the initial point vanishes. Beyond the first focal point, that geodesic no longer locally maximizes proper time from a spacelike initial hypersurface.