= Trapped spheres as a model for nonspherical collapse
{title2=$\theta_+=\frac{1-2M/r}{r},\quad\theta_-=-\frac2r$}
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> $M>0$:
$$
ds^2=-f(r)\,dv^2+2\,dv\,dr+r^2d\Omega^2,\qquad f(r)=1-\frac{2M}{r}.
$$
For a round sphere, choose future <null vector> normals
$$
k_+=\partial_v+\frac f2\partial_r,\qquad k_-=-\partial_r,
\qquad g(k_+,k_-)=-1.
$$
The <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.
\Image[/penrose-trapped-null-congruences.png]
{title=Both future null families contract at a trapped surface}
{description=The schematic compares signs of <null expansion>. The horizontal separation represents transverse beam width, not a radial <spacetime> coordinate; the ellipses represent two-dimensional sections. No horizon location or endpoint of a <null geodesic> is encoded by the drawing.}
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>.<Gravitational Collapse and Space-Time Singularities by Roger Penrose>[p. 58, small departures from symmetry]{ref}
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