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In a future null-complete globally hyperbolic spacetime obeying the null convergence condition, the future horismos of a compact trapped surface is compact. Normalize its future null normals continuously using a timelike field. Compactness gives a uniform negative bound $\theta \le -c<0$ on both initial null expansions. The null focusing theorem bounds all boundary generators by affine length $2/c$. Their endpoints lie in the image under a continuous map of the compact bundle of normalized null normals times $[0,2/c]$. The future horismos is closed, so is a compact subset of that image.

For future-complete generators of an event horizon, the null convergence condition and hypersurface orthogonality reduce the Null Raychaudhuri equation to ${\theta }^{\prime }\le -{\theta }^2/2$. If ${\theta }_0<0$, integration gives focusing within affine distance $2/|{\theta }_0|$. The null focusing theorem identifies this with a conjugate point to a spacelike surface. A null generator cannot remain on an achronal boundary beyond such a focal point, contradicting future completeness on the horizon. Thus $\theta \ge 0$, and $dA/d\lambda =\theta A\ge 0$ gives the second law of black-hole mechanics. New horizon generators may join toward the future at crease sets, adding area, but generators have no future endpoints on a regular event horizon under these assumptions.

// Target: quantum-field-theory.bigb

The second law of black-hole mechanics says that the total area of future event horizon cross-sections cannot decrease toward the future, assuming the classical Einstein field equations, the null energy condition, suitable global predictability and regularity, and here the stipulated future completeness of its generators.

A future event horizon is an achronal boundary generated by null geodesics. If a smooth generator had $\theta <0$, the null focusing theorem would produce a conjugate point to a spacelike surface within finite affine parameter. Future completeness lets the generator reach it, but beyond such a point it could no longer remain on the achronal boundary, a contradiction. Hence $\theta \ge 0$ wherever the null expansion is defined. A transported area element obeysEvery generator present on an early cut continues to a later cut without decreasing its area contribution. New generators can join at past endpoints, adding area; future endpoints in a regular spacetime are excluded. ThereforeThis argument also covers the sum of initially disconnected event horizons of the black holes in a merger. Nonsmooth crease sets require the standard regularity treatment; they do not provide a classical area-decrease loophole under these hypotheses. Quantum negative-energy flux can violate the classical null energy condition, so this is a classical black-hole area theorem, not an unconditional law for evaporating black holes.

Use an affine parameter $\lambda$ along a generator of a null hypersurface. Its null twist vanishes by hypersurface orthogonality. The screen metric is positive definite, so ${\hat{\sigma }}^{ab}{\hat{\sigma }}_{ab}\ge 0$. The Einstein field equations and null energy condition imply $R_{ab}{U}^a{U}^b\ge 0$, giving from the Null Raychaudhuri equationStarting with ${\theta }_0<0$, the null expansion stays negative as long as the congruence remains regular. Set $z=-1/\theta >0$. Then $z^{\prime }={\theta }^{\prime }/{\theta }^2\le -1/2$, soA finite negative null expansion cannot persist to $\lambda -{\lambda }_0=2/|{\theta }_0|$. Thus the null focusing theorem givesThis is a focal breakdown of the transverse congruence, provided the generator and regular geometry extend that far; it is not by itself a curvature singularity or a statement that the individual null geodesic terminates there.

The Einstein field equations and the null energy condition imply the null convergence condition:The scalar-curvature and any cosmological constant terms vanish because $U^a{U}_a=0$. For generators of the null hypersurface, the supplied twist-free property gives $\hat{\omega }=0$. The null shear squared is nonnegative on the screen. Thus the Null Raychaudhuri equation impliesStarting from $\theta ({\lambda }_0)={\theta }_0<0$, the null expansion remains negative for as long as the regular congruence exists. HenceBefore the right-hand side reaches zero, inversion of the negative quantities givesSince the inverse of a finite negative null expansion cannot be nonnegative, the regular congruence cannot continue through the proposed upper limit. The null focusing theorem therefore givesProvided the geodesic itself extends this far, the transverse area collapses and $\theta \to -\infty$ at or before this bound. An earlier end of the affine geodesic would instead be geodesic incompleteness. A divergence of the null expansion marks a caustic or a conjugate point to a spacelike surface; it does not by itself establish a curvature singularity.

The relevant Penrose singularity theorem states: a connected time-oriented four-dimensional globally hyperbolic spacetime with a noncompact Cauchy hypersurface, a nonempty compact orientable boundaryless trapped surface, and the null convergence condition is future null-geodesically incomplete. Under the Einstein field equations, the null energy condition supplies the curvature hypothesis. A trapped surface here is spacelike and has both future normal null expansions strictly negative. The conclusion is an incomplete null geodesic, not necessarily a divergent curvature invariant at an identifiable point.

Suppose, for contradiction, that every future null geodesic is complete. Write $S$ for the trapped surface andThe equality follows because the causal future of a compact set is closed in a globally hyperbolic spacetime. This future horismos is a closed achronal boundary. It is nonempty: the restriction of a Cauchy time function to compact $S$ has a minimum, and a point at that minimum cannot be chronologically preceded by another point of $S$.

Normalize the two future null normal directions along $S$ against a smooth future timelike field, fixing the affine scale continuously. Their initial null expansions are continuous and strictly negative. Compactness of the normalized normal bundle gives a uniform $c>0$ with ${\theta }_{\pm }\le -c$ everywhere on $S$. The preceding null focusing theorem forces a conjugate point to a spacelike surface along each normal generator within affine length $L=2/c$.

By the supplied boundary-generator result, every point of $E^{+}(S)$ is reached by an orthogonal future null geodesic which has no earlier conjugate point. No such boundary generator can remain on the boundary beyond its first focal point. Every point of $E^{+}(S)$ therefore lies inThe parameter set is compact. Future null completeness makes its geodesic flow defined throughout this common finite interval, and smooth dependence on initial data makes its image $K$ compact. Since $E^{+}(S)$ is closed and contained in $K$, it is compact. This is the compactness of the future horismos of a trapped surface step; a merely pointwise finite bound would not suffice without compactness and uniform normalization.

Choose a smooth complete timelike vector field, obtained if necessary by positive rescaling against a complete auxiliary Riemannian metric. Its inextendible integral curves of a vector field meet a chosen smooth Cauchy hypersurface $C$ exactly once. Projection along those curves defines a continuous map ${\pi }_C:E^{+}(S)\to C$. The achronal set property makes this map an injective function: two boundary points on the same timelike integral curve of a vector field would be timelike related.

An achronal boundary is a topological hypersurface without boundary, even at nonsmooth generator junctions; the submanifold property allowed in the question supplies this fact. Thus domain and codomain are both three-dimensional topological manifolds without boundary. Invariance of domain makes ${\pi }_C(E^{+}(S))$ open in $C$. Compactness makes it closed in the Hausdorff space $C$, and it is nonempty. Connectedness of the spacetime gives connectedness of $C$, so this image is all of $C$. That would make $C$ compact, contradicting the noncompact Cauchy hypersurface hypothesis. The completeness assumption is false, proving the theorem.

The noncompactness hypothesis and strict trapping are essential to this version. Global hyperbolicity cannot simply be omitted in the compactness and projection steps, and nonpositive initial expansion with zeros does not supply the uniform focusing bound used here.

The hypersurface generators have zero null twist. Contracting the Einstein field equations twice with the null tangent removes the trace term, and the null energy condition gives $R_{ab}{U}^a{U}^b=8\pi T_{ab}{U}^a{U}^b\ge 0$. Since the squared null shear is nonnegative, the Null Raychaudhuri equation impliesWhile $\theta <0$ this is equivalent toStarting from $\theta (0)={\theta }_0<0$, the right-hand side reaches zero no later than $\lambda =2/|{\theta }_0|$. The reciprocal expansion must therefore vanish andThis is the null focusing theorem.

Suppose the closed trapped surface $T$ were not wholly in the black hole. The stated consequence of strong asymptotic predictability then supplies a point $p\in {\dot{J}}^{+}(T)\cap {\mathcal{I}}^{+}$ lying on a future null generator $\gamma$ orthogonal to $T$, with no point conjugate to $T$ before $p$. Both future null expansions at $T$ are negative. The null focusing theorem therefore forces the expansion along $\gamma$ to diverge after finite affine parameter, producing a conjugate point before the generator reaches future null infinity. A null geodesic with such a point cannot continue to generate the achronal boundary ${\dot{J}}^{+}(T)$, contradicting the property of $p$. Hence no point of $T$ can communicate with future null infinity, and

The second law of black-hole mechanics, or Hawking's area theorem, states that the total area of future-event-horizon cross-sections cannot decrease toward the future under the null energy condition and the usual predictability assumptions. If any horizon generator had $\theta <0$, the twist-free Null Raychaudhuri equation and the null focusing theorem would force a conjugate point at finite affine parameter. A generator complete to the future cannot pass through such a point while remaining on the achronal boundary of $J^{-}({\mathcal{I}}^{+})$. Hence $\theta \ge 0$ everywhere on the regular future horizon, and $dA/d\lambda =\theta A\ge 0$ proves the area law.

For generators of a null hypersurface, the Frobenius theorem gives ${\hat{\omega }}_{ab}=0$. The null energy condition and the Einstein field equations imply $R_{ab}{U}^a{U}^b\ge 0$, while the shear norm is nonnegative. With $A=D-2$, Raychaudhuri's equation givesAs long as $\theta <0$ remains finite,If $\theta (0)={\theta }_0<0$, integration givesThe right-hand side reaches zero at $\lambda =A/|{\theta }_0|$, which a finite negative $\theta$ cannot cross. Hence the null focusing theorem forcesprovided the generator extends that far.

The second law of black-hole mechanics states that, assuming the null energy condition and the usual predictability hypotheses, the area of cross-sections of a future event horizon never decreases toward the future.

Indeed, the horizon generators form a hypersurface-orthogonal null congruence, so their twist vanishes. If the expansion were negative anywhere, the null focusing theorem would make it diverge to $-\infty$ within finite affine parameter, provided the generator extends that far. This produces a conjugate point, after which the generator cannot remain on an achronal set such as an event horizon. Future completeness rules out the alternative that the generator simply ends first. Therefore $\theta \ge 0$ everywhere, and

Contracting the Einstein field equations with the null tangent eliminates the trace term and givesby the null energy condition. The screen metric is positive definite, so ${\hat{\sigma }}_{ab}{\hat{\sigma }}^{ab}\ge 0$. The Null Raychaudhuri equation consequently impliesWhile $\theta <0$ this is equivalent toIf $heta({\lambda }_0)={\theta }_0<0$, integration givesThe right-hand side reaches zero after affine distance $2/|{\theta }_0|$. A finite negative expansion cannot pass through this value, so $heta\to -\infty$ no later than that point. This is the null focusing theorem.

Apply the time-reversed Penrose singularity theorem. The two past-directed null congruences orthogonal to the compact anti-trapped surface have negative expansion. The null energy condition, through the Einstein field equations, supplies the null convergence condition, and the null focusing theorem forces each generator to acquire a conjugate point within finite affine length if it can be extended that far.

If every past-directed null generator were complete, the boundary of the causal past of the surface would therefore be generated only for a bounded affine interval. Compactness of the initial surface and continuous dependence of geodesics on initial data would make that achronal boundary compact. A globally hyperbolic spacetime provides a Cauchy hypersurface and a timelike flow projecting the boundary onto it. The standard Penrose argument then makes its image both open and closed, forcing the connected Cauchy hypersurface to be compact. This contradicts its stipulated topology ${\mathbb{R}}^3$.

Hence at least one past-directed null generator ends after finite affine parameter: the universe is null-geodesically incomplete to the past. Global hyperbolicity controls the causal boundary, the noncompact spatial topology supplies the contradiction, and the energy condition supplies focusing.