Geodesic congruence

A geodesic congruence is a family of nonintersecting geodesics filling an open spacetime region. Its derivative tensor describes the relative motion of neighboring geodesics.

A null geodesic congruence has an affinely parametrized null tangent $k^a$ satisfying $k^a{k}_a=0$ and $k^b{\nabla }_b{k}^a=0$.

Given null vectors $k$ and $n$ with $k\cdot n=-1$, the screen-space projector isIt annihilates $k$ and $n$ and projects onto the two-dimensional spacelike space transverse to them.

For $B_{ab}={\nabla }_b{k}_a$, the optical tensor is the screen projection ${\hat{B}}_{ab}=P_a{}^c{P}_b{}^d{B}_{cd}$. Its trace, symmetric trace-free part, and antisymmetric part are the expansion, shear, and twist of the null congruence.

The null expansion is $\theta =P^{ab}{\hat{B}}_{ab}$. It is the logarithmic derivative of an infinitesimal transverse area: $\theta =A^{-1}dA/d\lambda$.

The null shear is the symmetric trace-free part ${\hat{\sigma }}_{ab}={\hat{B}}_{(ab)}-\frac{1}{2}\theta P_{ab}$. It changes the shape of a transverse beam while preserving its area to first order.

The null twist is ${\hat{\omega }}_{ab}={\hat{B}}_{[ab]}$. It vanishes for null generators orthogonal to a hypersurface by the Frobenius theorem.

For an affinely parametrized null congruence in four spacetime dimensions,

For a hypersurface-orthogonal null congruence under the null energy condition, an initially negative expansion ${\theta }_0$ diverges to $-\infty$ within affine distance at most $2/|{\theta }_0|$.