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Locally write the null hypersurface as $u=0$, $du\ne 0$, with normal ${\ell }_{\mu }=f{\partial }_{\mu }u$, $f\ne 0$. A vector $V$ is tangent exactly when $V(u)=0$. Nullness gives $\ell (u)={\ell }^2/f=0$ on the hypersurface. Hence $\boxed{\ell \in T\mathcal{N}}$: its normal direction lies inside its tangent space.

Put $n=du$ and $q=n_{\mu }{n}^{\mu }$. Torsion freedom gives $n^{\mu }{\nabla }_{\mu }{n}_{\nu }=\frac{1}{2}{\partial }_{\nu }q$. Since $q$ vanishes on $\mathcal{N}$, all tangential derivatives vanish there and $dq=Cdu$ there for a scalar $C$. ConsequentlyThis is the unparametrized null geodesic equation. The integral curves are therefore generators of $\mathcal{N}$. An affine rescaling of a null normal removes the proportionality coefficient locally: if ${\nabla }_{\ell }\ell =\kappa \ell$, set $K=h\ell$ with $\ell (\log h)=-\kappa$.

For each of the two normal directions, launch a null geodesic congruence orthogonally from the sphere with initial tangent $U=X$ or $U=Y$. For $U=X$ take initial auxiliary $N=Y$; for $U=Y$ take initial $N=X$, and then parallel transport $N$ as in part (a). On the sphere the resulting screen-space projector projects onto its angular tangent space, whose metric is $q_{AB}=r^2{\gamma }_{AB}$.

For any radial normal $K$, the spherical null expansion in ingoing coordinates follows directly from area variation:Equivalently $\mathcal{A}\propto r^2$ gives $K(\log \mathcal{A})=2K(r)/r$. ConsequentlySpherical symmetry also makes the null shear and null twist zero for these radial congruences.

There is a small parametrization distinction. The smooth field $X=-{\partial }_r$ is affine, since ${\Gamma }^a{}_{rr}=0$. The natural smooth field $Y={\partial }_v+(f/2){\partial }_r$ obeys ${\nabla }_{Y}Y=(f^{\prime }/2)Y$. To meet part (a)'s affine convention, use an affine rescaling of a null normal along the outgoing generators, with scaling equal to one on $S$. Its angular derivatives on $S$ then leave the projected derivative unchanged. The displayed null expansions are therefore precisely those for the affinely launched congruences, even though that convenient global expression for $Y$ is nonaffine away from its initial sphere.

A concrete example separates a trapped surface from a failure of coordinates. Use geometrized units $G=c=1$, metric signature $(-,+,+,+)$, and the regular ingoing Eddington-Finkelstein coordinates of a Schwarzschild black hole with mass $M>0$:For a round sphere, choose future null vector normalsThe spherical null expansion in ingoing coordinates follows directly from $A=4\pi r^2$: ${\theta }_k=2k(r)/r$. At $r=3M$, $({\theta }_{+},{\theta }_{-})=(1/(9M),-2/(3M))$; at $r=3M/2$, it is $(-2/(9M),-4/(3M))$. Both future light fronts shrink in the second case. At $r=2M$, ${\theta }_{+}=0$, so this sphere is marginally outer trapped surface, not a strict trapped surface. In the spherical model this boundary is an event horizon; that identification does not extend to arbitrary geometries.

Although $k_{-}$ is affine as written, $k_{+}$ generally obeys ${\nabla }_{k_{+}}{k}_{+}=(M/r^2)k_{+}$. These are valid initial normals for the sign test; use affine rescaling of a null normal before applying the affine Null Raychaudhuri equation. Regularity of the displayed metric tensor at $r=2M$ shows why a divergence in another chart would not establish null geodesic incompleteness.

On a compact trapped surface, both null expansions have a uniform negative margin. Sufficiently small smooth perturbations controlling the relevant metric tensor and surface derivatives preserve trapping. This conditional robustness does not prove that every nonspherical gravitational collapse creates a trapped surface.^[ref]