= Solution
The <black-hole area theorem> is a statement about classical future <event horizons>, with both a local focusing condition and global horizon regularity assumptions. Assume the <null energy condition>, an appropriately predictable exterior and no future termination of horizon generators before the focusing argument can be applied. Future-complete generators provide a particularly direct sufficient version of the last assumption. <Strong asymptotic predictability> is the usual global condition used to exclude the pathological alternatives; the energy condition by itself is not the whole theorem.
On a smooth part of the horizon, choose a future-directed affinely parametrized generator $\ell^a$ and <affine parameter> $\lambda$. Its expansion $\theta$ is the fractional rate of change of an infinitesimal transverse area element:
$$
\frac{d}{d\lambda}\log dA=\theta.
$$
The horizon is a null hypersurface, so its generator congruence is hypersurface-orthogonal and has zero twist. The transverse screen metric is positive definite, hence $\sigma_{ab}\sigma^{ab}\geq0$. In four dimensions the <Null Raychaudhuri equation> is
$$
\frac{d\theta}{d\lambda}=-\frac12\theta^2-\sigma_{ab}\sigma^{ab}
-R_{ab}\ell^a\ell^b.
$$
The <Einstein field equations> and the <null energy condition> imply the <null convergence condition> $R_{ab}\ell^a\ell^b\geq0$, so $\theta'\leq-\theta^2/2$.
Suppose $\theta_0<0$ at $\lambda_0$. As long as the congruence remains smooth, integrating $d(1/\theta)/d\lambda\geq1/2$ shows focusing within affine distance at most $2/|\theta_0|$. Equivalently, comparison with the equality solution gives
$$
\theta(\lambda)\leq\frac{\theta_0}{1+\tfrac12\theta_0(\lambda-\lambda_0)}.
$$
Its denominator tends to zero at finite future affine parameter. The corresponding transverse area collapses, giving a <conjugate point to a spacelike surface>. Beyond a focal point a generator cannot remain on an <achronal boundary>, since nearby points can then be joined by a timelike curve. But a regular future <event horizon> is such a boundary, and its generators cannot leave it to the future. With the stated future-extension assumption this is a contradiction. Therefore $\theta\geq0$ on the smooth horizon.
Following generators from one horizon cross-section to a later one now gives a noncontracting area map. Generators entering at past endpoints or merger crease sets can add area, whereas the global hypotheses exclude generators disappearing from the future horizon. Consequently
$$
\boxed{A_{\rm later}\geq A_{\rm earlier}.}
$$
This is the <future-complete horizon focusing proof of the area theorem>. In $D$ spacetime dimensions, the coefficient $1/2$ is replaced by $1/(D-2)$ and the same proof gives a focusing distance at most $(D-2)/|\theta_0|$. It therefore also applies to the three-dimensional horizon cross-sections of the preceding five-dimensional example.
\b[A <cosmological constant> does not invalidate the local focusing step.] Contract the modified <Einstein field equations> with the null generator. Since $g_{ab}\ell^a\ell^b=0$, both the scalar-curvature term and the $\Lambda$ term disappear:
$$
\boxed{R_{ab}\ell^a\ell^b=8\pi T_{ab}\ell^a\ell^b\geq0.}
$$
Thus a <cosmological constant preserves null convergence>, whatever its sign. Under suitable global assumptions for the relevant asymptotic region, the black-hole area argument still works. A nonzero $\Lambda$ changes the asymptotic geometry and can introduce cosmological horizons, so the statement concerns the appropriate future black-hole horizon; it is not an assertion that every horizon-like surface automatically obeys the same area ordering.
\b[A classical real canonical <massless scalar field> also preserves the area theorem's energy condition.] Its stress tensor has
$$
T_{ab}\ell^a\ell^b
=(\ell^a\partial_a\phi)^2
-\frac12(g_{ab}\ell^a\ell^b)g^{cd}\partial_c\phi\partial_d\phi
=\boxed{(\ell^a\partial_a\phi)^2\geq0.}
$$
This proof is independent of whether the field gradient is spacelike, timelike or null: the metric term vanishes and the remaining real square is nonnegative. It is the <null energy condition for a canonical scalar field>. Adding this field to other matter satisfying the <null energy condition> retains null convergence. With the same global regularity assumptions, area still cannot decrease; when the derivative along a generator is nonzero, it supplies an additional nonnegative focusing source.
The conclusion uses the given minimally coupled classical stress tensor. A wrong-sign kinetic term or a nonminimal curvature coupling would require another calculation. Likewise, a renormalized quantum <stress-energy tensor> need not obey the classical pointwise condition; the quantum radiation considered next is not covered by this classical proof.
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