Null geodesic congruence

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

For affine null tangent $U$, choose an auxiliary null vector $N$ with $U\cdot N=-1$ and parallel transport it along the rays. The screen-space projector is $P_{ab}=g_{ab}+U_a{N}_b+N_a{U}_b$. The optical matrix of a null congruence describes evolution of screen-projected deviation vectors. Its trace is the null expansion, its symmetric trace-free part is the null shear, and its antisymmetric part is the null twist. In four dimensions $\hat{B}=\theta P/2+\hat{\sigma }+\hat{\omega }$.

For a parallel-transported screen and affine null geodesic tangent $U$, the optical matrix satisfiesIts trace gives the null Raychaudhuri equation. In a two-dimensional screen, symmetric trace-free matrices have scalar squares and anticommute with antisymmetric matrices. The trace-free symmetric and antisymmetric parts consequently yield the null shear and twist transport equations without additional quadratic trace-free terms.

In the Ingoing Eddington-Finkelstein coordinates metric $d{s}^2=-f(r)d{v}^2+2dvdr+r^{2}d{\Omega }^2$, where $f=1-2M/r$, the tangent $U=-E{\partial }_r$ is affine when $E>0$ is constant along each ray. The displayed radial partner obeys $N^2=0$ and $U\cdot N=-1$. Since ${\Gamma }^v{}_{rv}={\Gamma }^a{}_{rr}=0$ and ${\Gamma }^r{}_{rv}=-f^{\prime }/2$, its derivative ${\partial }_r{N}^r=f^{\prime }/(2E)$ cancels the connection term, giving ${\nabla }_{U}N=0$. Its screen is the angular two-plane, with optical tensor ${\hat{B}}_{AB}=-Er{\gamma }_{AB}$, null expansion $-2E/r$, and zero null shear and null twist.

If a null vector tangent satisfies ${\nabla }_{K}K=\kappa K$, then $U=hK$ is affine when $K(\log h)=-\kappa$. Choose $h=1$ on the initial transverse surface. The screen-space projector kills derivative terms proportional to $K$, so the initial optical tensor and its null expansion agree with those of $K$. In ingoing spherical coordinates, $K={\partial }_v+(f/2){\partial }_r$ has $\kappa =f^{\prime }/2$.

For an affine null geodesic congruence with future tangent $U$, choose a future unit timelike vector $T$ on a transverse section, put $E=-U\cdot T$, and set $N=T/E-U/(2E^2)$ there. It has $N^2=0$ and $U\cdot N=-1$. Parallel transport along each generator preserves these contractions by metric compatibility and ${\nabla }_{U}U=0$. The construction is local up to caustics and supplies the auxiliary normal for a screen-space projector.

If $k_a={\nabla }_{a}S$ is a null gradient, symmetry of the Hessian gives $k^a{\nabla }_a{k}_b=k^a{\nabla }_b{k}_a=\frac{1}{2}{\nabla }_b(k^2)=0$. Its integral curves are therefore affinely parametrized null geodesics, rather than merely null curves with a nonaffinity term.

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.

For an affinely parametrized null congruence in $D$ spacetime dimensions,

For an affinely parametrized twist-free null geodesic congruence, let $A$ be local transported screen area and ${\sigma }^2={\hat{\sigma }}_{AB}{\hat{\sigma }}^{AB}/2$. The Null Raychaudhuri equation gives ${\theta }^{\prime }=-{\theta }^2/2-2{\sigma }^2-R_{ab}{k}^a{k}^b$ and $A^{\prime }/A=\theta$. Combining these gives the displayed equation. Positive shear focuses the beam and has a negative sign. For nonuniform expansion this is a local area identity, not the same differential equation for the entire integrated horizon area.

In flat spacetime take a twist-free beam's two screen Jacobi factors to be $1+\lambda$ and $1-\lambda$. They solve the zero-curvature geodesic deviation equation. Their product is the displayed area, and subtracting the trace from their optical matrix gives the displayed nonnegative shear. Direct differentiation yields $(A^{1/2}{)}^{\prime \prime }=-(1-{\lambda }^2{)}^{-3/2}=-{\sigma }^2{A}^{1/2}$, an explicit check of the focusing sign.

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

A conjugate point to a spacelike surface occurs where normal geodesics develop a focal degeneracy: the transverse Jacobi field map from initial points on the surface ceases to be invertible. Collapse of the transverse area makes the null expansion diverge. Beyond a first such point, the normal null geodesic cannot continue as a generator of the achronal boundary of the surface's future.