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.

For a unit timelike vector field $u^a{u}_a=-1$, the tensor $h_{ab}=g_{ab}+u_a{u}_b$ projects onto the three-dimensional subspace orthogonal to $u^a$. In particular, $h_{ab}{u}^a=h_{ab}{u}^b=0$ and $g^{ab}{h}_{ab}=3$ in four spacetime dimensions.

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.

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.