= Solution
A <null geodesic congruence> is a smooth family of <null geodesics> filling a region without intersections there. For affine tangent $k$, the <null expansion> is the trace of the <optical tensor>: $\theta=d(\log A)/d\lambda$ for an infinitesimal pencil's area. Positive expansion means spreading; negative expansion means focusing.
For hypersurface-orthogonal generators in four dimensions, the <Null Raychaudhuri equation> reads
$$
\frac{d\theta}{d\lambda}=-\tfrac12\theta^2-\sigma_{ab}\sigma^{ab}-R_{ab}k^ak^b.
$$
The <null twist> vanishes. The <null energy condition> implies $R_{ab}k^ak^b\geq0$ through the <Einstein field equations>. An initial $\theta_0<0$ would force a focal point within affine distance at most $2/|\theta_0|$. A future-complete generator cannot develop such a point while remaining on an achronal <event horizon>. \b[Thus $\boxed{\theta\geq0}$] under the energy and global predictability/future-completeness hypotheses of <Hawking's area theorem>. Quantum energy-condition violations or failure of those global assumptions remove the conclusion.
The <event horizon> is the global boundary $\partial J^-(\mathscr I^+)$ and depends on the whole future. An <apparent horizon> on a chosen spacelike slice is the outermost marginally outer trapped surface, ordinarily $\theta_{\rm out}=0$, $\theta_{\rm in}<0$. It is quasi-local and slice-dependent; its tube need not be null.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-58-collapse-horizons.png]
{title=Ingoing Finkelstein diagram for a thin shell with an event horizon extending into flat spacetime}
{height=520}
For a shell arriving at $v=0$, the final horizon is $r=2M$. Before arrival, outgoing flat-space rays obey $dr/dv=1/2$. Tracing this horizon backward gives $r=2M+v/2$ until its beginning at $r=0$, $v=-4M$. No <apparent horizon> exists in that earlier flat region, explicitly illustrating the <event horizon>'s dependence on future collapse.
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