Screen-space projector ($P^a{}_b$)

Given null vectors $k$ and $n$ with $k\cdot n=-1$, the screen-space projector isIt annihilates $k$ and $n$ and, in $D$ spacetime dimensions, projects onto the $(D-2)$-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: $\theta =A^{-1}dA/d\lambda$.

For a four-dimensional spherical metric $d{s}^2=-fd{v}^2+2dvdr+r^{2}d{\Omega }^2$, a radial null vector normal $K$ to a round sphere has null expansion ${\theta }_K=\frac{1}{2}{q}^{AB}{\mathcal{L}}_K{q}_{AB}=2K(r)/r$, where $q_{AB}=r^2{\gamma }_{AB}$. The angular metric changes only by a common scale, so radial congruences have zero null shear; hypersurface orthogonality also gives zero null twist. The convention here is the trace, not the screen-averaged expansion.

For a four-dimensional spherically symmetric metric $d{s}^2=-2fdUdV+r^{2}d{\Omega }^2$, the affine null gradients are $f^{-1}{\partial }_V$ and $f^{-1}{\partial }_U$. Their null expansions are $2r_{,V}/(fr)$ and $2r_{,U}/(fr)$. Their null shear and null twist vanish because the transverse metric changes only by a common scale and the covectors are exact gradients.

The Null Raychaudhuri equation and Einstein field equations give ${\partial }_U(f^{-1}{r}_{,U})=-r{R}_{UU}/(2f)=-4\pi r{T}_{UU}/f$, with an exchanged-coordinate identity in $V$. Under the null energy condition, each rescaled radial derivative is nonincreasing along its corresponding future null direction.

In a regular double-null region satisfying the null energy condition, an initially negative $r_{,U}$ on a Cauchy hypersurface remains negative in its future domain of dependence. If both spherical null expansions are negative at one sphere, the exchanged-coordinate focusing inequality preserves the other sign along its future outgoing null direction. Every later sphere that still exists there remains a trapped surface.

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

For an affine null congruence in four spacetime dimensions with a parallel-transported screen, the displayed equation follows by taking the symmetric trace-free part of the Sachs optical equations. The Ricci contribution to the optical tidal tensor is $P_{ab}{R}_{cd}{U}^c{U}^d/2$, so it changes the expansion but contributes no screen-trace-free source. The projected Weyl tensor supplies the tidal source of shear.

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 affine null geodesic congruence, $B_{ab}={\nabla }_b{U}_a$ annihilates $U$ in both slots. Relative to a null partner, its antisymmetric part is its screen projection plus a term of the form $U_{[a}{q}_{b]}$. Hence zero null twist implies $U\wedge dU=0$, and Frobenius theorem makes $U_a$ locally proportional to a hypersurface gradient. Conversely, if $U_a=h{\nabla }_{a}S$, its antisymmetric derivative has a factor of $U$, so the screen projection vanishes. Geodesicity is essential to the converse from vanishing screen twist to the full Frobenius condition.

For an affinely parametrized four-dimensional null geodesic congruence, the screen-projected evolution of the null twist isConsequently zero twist remains zero along every generator on which the optical description remains regular.