= Solution
Suppose, for contradiction, that every future null geodesic orthogonal to the compact <trapped surface> $T$ extends beyond $L=A/|\theta_0|$. Both families begin with expansion at most $\theta_0<0$, so part d gives a point conjugate to $T$ on every generator by affine length $L$. A generator of the <achronal boundary> $\dot J^+(T)$ cannot remain on that boundary beyond its first <conjugate point>, because afterward it can be deformed to a timelike curve from $T$.
The two bundles of initial null directions over compact $T$, restricted to $0\leq\lambda\leq L$, form a <compact set>. Their image under the geodesic <exponential map> contains all of $\dot J^+(T)$, so this <achronal boundary> is compact. Project it along a complete timelike flow onto the noncompact <Cauchy hypersurface> $\Sigma$. The projection is both open and closed in the connected Cauchy surface, hence would be all of $\Sigma$; compactness of the source would then make $\Sigma$ compact, a contradiction.
Therefore at least one orthogonal future null geodesic cannot extend to affine length $L$. Its maximal future development is future-inextendible with total affine length
$$
\boxed{\lambda_{\max}\leq\frac{D-2}{|\theta_0|}},
$$
which is the incompleteness conclusion of the <Penrose singularity theorem>.
Back to article page
