= Solution
A future <trapped surface> is a smooth compact spacelike two-surface without boundary whose two future-directed orthogonal <null expansions> are strictly negative. In four-dimensional <general relativity>, the <Penrose singularity theorem> states: a time-oriented <globally hyperbolic spacetime> with a noncompact <Cauchy hypersurface>, a <trapped surface>, and the <null convergence condition> $R_{ab}k^ak^b\ge0$ for every <null vector> is future <null-geodesically incomplete>. With the <Einstein field equations>, the <null energy condition> implies this <null convergence condition>; a <cosmological constant> drops out of the null contraction.
The <Kruskal spacetime> is an example. In its <black hole> interior, use future null normals in <Ingoing Eddington-Finkelstein coordinates>:
$$
l=-\partial_r,\qquad k=\partial_v+\frac f2\partial_r,\qquad g(l,k)=-1.
$$
For a round sphere of <areal radius> $r$, its area is $4\pi r^2$ and its <null expansions> are
$$
\theta_l=\frac{2l(r)}r=-\frac2r,\qquad \theta_k=\frac{2k(r)}r=\frac fr.
$$
Both are negative for $0<r<2M$. The vacuum <Einstein field equations> give $R_{ab}=0$, and a two-ended Kruskal <Cauchy hypersurface> is noncompact. The future <radial null geodesics> reaching $r=0$ in finite <affine parameter> provide precisely the incompleteness predicted by the <Penrose singularity theorem>. A <trapped surface> at $r<2M$ is strictly trapped; the horizon sphere has one zero <null expansion> instead.
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