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A conjugate point to a spacelike surface occurs where normal geodesics develop a focal degeneracy: the transverse Jacobi field map from initial points on the surface ceases to be invertible. Collapse of the transverse area makes the null expansion diverge. Beyond a first such point, the normal null geodesic cannot continue as a generator of the achronal boundary of the surface's future.

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

Assume future null geodesic completeness. The compactness of the future horismos of a trapped surface makes $E^{+}(S)$ a nonempty compact achronal boundary without edge. Projection along a complete timelike vector field into the Cauchy hypersurface is injective, open by invariance of domain, and closed by compactness. Connectedness forces surjectivity, contradicting the noncompact Cauchy hypersurface. This global step completes the contradiction.^[ref]

Invariance of domain says that a continuous map that is an injective function between topological manifolds without boundary of the same dimension is an open map. It is useful in the Penrose singularity theorem: projecting a compact achronal boundary along timelike curves onto a Cauchy hypersurface gives a nonempty subset that is both open and closed, forcing that connected Cauchy hypersurface to be compact.

Interpret $A$ first as an infinitesimal screen area transported along generators of a future event horizon, not as the total area of an arbitrary trapped surface. Horizon generators form an affinely parametrized, hypersurface-orthogonal null geodesic congruence. The two-dimensional screen metric is positive definite. Its optical tensor has zero twist and decomposes asThe scalar here agrees with the supplied derivative definition: nullness and the affine geodesic equation remove the longitudinal terms in the contracted norm, leaving $B_{AB}{B}^{AB}-{\theta }^2/2$.

To derive the focusing equation, vary neighbouring geodesics and let $J$ be their screen Jacobi map. The geodesic deviation equation is $J^{\prime \prime }=-\mathcal{R}J$, where ${\mathcal{R}}_{AB}=R_{AcBd}{p}^c{p}^d$. Hence $B=J^{\prime }{J}^{-1}$ satisfies $B^{\prime }=-B^2-\mathcal{R}$. Taking its screen trace yields the Null Raychaudhuri equationSet $a=A^{1/2}$. Since $a^{\prime }/a=\theta /2$, the area-square-root optical focusing equation isThe PDF's plus sign before ${\sigma }^2$ is inconsistent with its own positive shear definition. This is an actual source error, not an antisymmetrization or curvature-sign convention.

For an explicit check, take a flat-space twist-free beam with screen Jacobi factors $1+\lambda$ and $1-\lambda$, for $|\lambda |<1$. ThenIts Ricci term is zero, so the result equals $-{\sigma }^2{A}^{1/2}$ and has the opposite sign to the printed equation. This is flat-space anisotropic beam shear focusing.

Under the null convergence condition $R_{ab}{p}^a{p}^b\ge 0$, the correct equation gives concavity of each local area square root. Equivalently ${\theta }^{\prime }\le -{\theta }^2/2$. If a horizon generator had $\theta ({\lambda }_0)={\theta }_0<0$, integration gives a focal point within affine distance at most $2/|{\theta }_0|$. Such a generator could not remain on an achronal boundary beyond the focal point. Under the usual global hypotheses that horizon generators remain regular and future complete, this contradicts their being generators of the event horizon. Hence $\theta \ge 0$ and $A^{\prime }=\theta A\ge 0$. Integrating over the horizon patches, and including newly joining generators which can add area, proves Hawking's area theorem. Global predictability/completeness assumptions are essential; the local differential inequality alone is not a theorem about arbitrary trapped-surface areas.

For the perfect fluid stress tensor, null contraction eliminates the pressure-metric term:A nonzero null vector cannot be orthogonal to a timelike fluid velocity. Thus the perfect-fluid null energy condition is preciselySeparate requirements $\rho \ge 0$ or $p_{\mathrm{fluid}}\ge 0$ are not needed for this null focusing argument. With the Einstein equation, $R_{ab}{p}^a{p}^b=8\pi T_{ab}{p}^a{p}^b$.

A cosmological constant contributes only a term proportional to $g_{ab}$, whose null contraction is zero:Therefore a nonzero cosmological constant does not change the fluid condition or the local area-focusing argument, though it can change the global asymptotics and which horizons and completeness hypotheses are appropriate.

Hawking's area theorem states that the area of later cross-sections of a classical future event horizon cannot be smaller than that of earlier cross-sections, including the sum over initially disconnected components. Its hypotheses include the null energy condition, the Einstein equations, and the global regularity/predictability conditions that exclude future naked pathologies of the horizon generators. A convenient sufficient version assumes those generators are future complete. Without the energy and global hypotheses the assertion is not unconditional.

Let $k^a$ be an affinely parametrized null generator, with expansion $\theta$. Because the horizon is a null hypersurface, the generator congruence has zero null twist. The Null Raychaudhuri equation in four dimensions givesThe null shear square is nonnegative and the null energy condition makes $R_{ab}{k}^a{k}^b=8\pi G{T}_{ab}{k}^a{k}^b\ge 0$; a cosmological term has zero null contraction. Suppose $\theta ({\lambda }_0)<0$. Integration of this inequality forces $\theta \to -\infty$ within affine distance at most $2/|\theta ({\lambda }_0)|$. This produces a focal point to the initial horizon cross-section. Beyond it the null geodesic cannot remain on an achronal boundary, whereas an event horizon is precisely such a boundary. Future completeness, or the corresponding standard predictability condition ensuring continuation over this interval, gives the contradiction. HenceThis proves the local area increase along every smooth generator bundle. Generators can enter the horizon at past crease points, but cannot leave through future endpoints on a regular horizon under these hypotheses. New generators add area, rather than subtracting it, so the result extends across the nonsmooth merger set and to whole horizon cross-sections. This is the future-complete horizon focusing proof of the area theorem.

Choose an early cross-section with the two separate components of areas $A_1,A_2$, and a later cross-section of the settled merged hole. Following the old generators forward maps their area elements injectively into the later horizon; a future crossing/focal point would violate the same achronality argument. Each element has nondecreased area. Any additional generators contribute nonnegative area, givingThis is a comparison of event-horizon cross-sections, not an assertion that set containment alone bounds the areas of arbitrary apparent horizons.

For a Schwarzschild black hole, $r_h=2Gm/c^2$ and $A=4\pi r_h^2=16\pi G^2{m}^2/c^4$. Applying the merger inequality to the two equal initial masses and the final Schwarzschild mass givesFor initially well-separated holes at rest, with no extra incoming energy or external work, energy conservation gives $E_{\mathrm{rad}}=(2m-M)c^2$. The area bound on equal-mass Schwarzschild merger radiation is consequentlyThis is the maximum permitted by the area/energy inequalities, not a demonstration that a dynamical merger attains it. A reversible limiting area change would be needed to saturate the bound. If initial separation, binding energy or external agents change the initial total energy, replace $2m{c}^2$ by that energy when computing escaped radiation. Quantum evaporation does not contradict the theorem: the classical energy-condition hypotheses are not retained unchanged in that setting.

Use Planck units for the following formulas. The black-hole area theorem is a classical statement with hypotheses. Assume the Einstein field equations, the null energy condition and appropriate global future regularity/predictability. For a clean version of the focusing sketch, take the future horizon generators to remain on a regular horizon and to be future complete in affine parameter. The usual strong asymptotic predictability formulation supplies the global causal control needed to exclude the same focusing pathology; an arbitrary spacetime with naked future breakdown is not covered by the theorem.

Let ${\ell }^a$ be an affinely parametrized generator of a smooth portion of the event horizon, and let $\theta =d\log dA/d\lambda$ be its null expansion. The horizon is a null hypersurface, so its generators have zero null twist. In four dimensions the Null Raychaudhuri equation isThe Einstein field equations give $R_{ab}{\ell }^a{\ell }^b=8\pi T_{ab}{\ell }^a{\ell }^b\ge 0$, since the metric terms vanish when contracted with a null vector. The screen-space null shear term is also nonnegative, so ${\theta }^{\prime }\le -{\theta }^2/2$. If ${\theta }_0<0$ at ${\lambda }_0$, integrating this inequality while the congruence remains regular givesIt focuses within affine distance at most $2/|{\theta }_0|$. The vanishing area element is a focal point to an earlier spacelike horizon cut. Past such a point the null geodesic can no longer generate an achronal boundary, but the event horizon is precisely such a boundary. Future completeness and the global hypotheses exclude its escaping this contradiction by ending prematurely at a pathology. Therefore $\theta \ge 0$ everywhere the smooth horizon description applies.

Integrate $d(dA)/d\lambda =\theta dA$ between ordered horizon cuts. Every continuing generator contributes a nondecreasing area element. New generators can enter at past endpoints or merger crease sets, adding area; generators cannot leave a regular future horizon through future endpoints under these hypotheses. Thus, including the total area of disconnected components,This is the future-complete horizon focusing proof of the area theorem. A smooth elementary congruence proof requires the stated regularity; the general theorem also treats the horizon's nonsmooth joining set rather than assuming mergers have a globally smooth horizon.

Hawking radiation changes the situation because the quantum field state in a collapsing geometry is not a classical positive-energy fluid. For a large isolated Schwarzschild black hole,in natural units. Outgoing quantum modes at future null infinity have a nearly thermal occupation, modified by greybody factors; an accompanying negative Killing energy flux near the horizon reduces the hole's mass. On timescales short compared with the evaporation time but long compared with $M$, its geometry is approximately Schwarzschild spacetime with slowly decreasing $M(t)$. It is not an exactly stationary vacuum solution with a parameter changed by hand: the radiation's stress-energy tensor sources the time dependence through the semiclassical Einstein equation. The distant luminosity is carried by the outgoing radiation, while the horizon-area decrease accompanies the inward negative-energy contribution.

As the black hole loses mass, its horizon scale and Bekenstein-Hawking entropy shrink, its temperature rises and its luminosity increases. This is the negative heat capacity of a Schwarzschild black hole. Large astrophysical holes evaporate extremely slowly in isolation; small holes evaporate more rapidly, and increasingly massive particle species become accessible as the temperature rises. An incoming radiation bath or accretion can instead offset the loss, so the pure evaporation law assumes negligible incoming energy. A formal complete evaporation also raises the black hole information paradox; the low-curvature calculation alone does not determine how information or the final state is resolved.

For an elementary luminosity estimate, treat the horizon as emitting photon blackbody radiation. In natural units the Stefan–Boltzmann law has $\sigma ={\pi }^2/60$, givingThe horizon area used here is an estimate of emitting area, not the exact frequency-dependent absorption cross-section. More generally write $P=\alpha /M^2$, where $\alpha >0$ includes the greybody factors and the active particle species. For constant effective $\alpha$, energy balance and direct integration giveHere $t$ is time measured at infinity. The formal evaporation timescale is $M_0^3/(3\alpha )$; with the photon blackbody estimate it is $5120\pi M_0^3$. Restoring constants givesfor that specified estimate. If particle thresholds make $\alpha$ mass dependent, the precise leading law is instead $t={\int }_{M(t)}^{M_0}{m}^{2}dm/\alpha (m)$. The robust result is the inverse-square mass-loss rate for a fixed species regime and the cubic mass scaling of its lifetime. This is the semiclassical cubic mass law for Schwarzschild evaporation. Extrapolating its zero to a definite end state is unjustified once $M$ approaches the Planck mass and curvature requires quantum gravity.

There is no contradiction with the classical area theorem: the renormalized quantum stress-energy tensor can violate the null energy condition used in the focusing inequality. In the evaporation regime,which is permitted when that hypothesis fails. The thermodynamic replacement is the generalized second law, concerning $S_{\mathrm{outside}}+A/(4G\hslash )$ rather than horizon area alone. Outside radiation can carry entropy while the hole's Bekenstein-Hawking entropy decreases.

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: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 ${\sigma }_{ab}{\sigma }^{ab}\ge 0$. In four dimensions the Null Raychaudhuri equation isThe Einstein field equations and the null energy condition imply the null convergence condition $R_{ab}{\ell }^a{\ell }^b\ge 0$, so ${\theta }^{\prime }\le -{\theta }^2/2$.

Suppose ${\theta }_0<0$ at ${\lambda }_0$. As long as the congruence remains smooth, integrating $d(1/\theta )/d\lambda \ge 1/2$ shows focusing within affine distance at most $2/|{\theta }_0|$. Equivalently, comparison with the equality solution givesIts 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 $\theta \ge 0$ 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. ConsequentlyThis 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.

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: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.

A classical real canonical massless scalar field also preserves the area theorem's energy condition. Its stress tensor hasThis 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.

The second law of black-hole mechanics states that the total area of a classical future event horizon cannot decrease toward the future, under the appropriate causal and energy assumptions. Use the null convergence condition $R_{ab}{U}^a{U}^b\ge 0$; in general relativity the Einstein field equations imply it from the null energy condition. Assume, as allowed here, that the horizon's null generators remain on the horizon and are complete to the future. The horizon is an achronal boundary, and its smooth generators are hypersurface-orthogonal, so their null twist is zero. The Null Raychaudhuri equation therefore givesIf some smooth horizon point had ${\theta }_0<0$, then while the optical description remains regular, $\theta$ stays negative andThusforcing focusing no later than ${\lambda }_0+2/|{\theta }_0|$.

The assumed geometrical result is that this focusing produces a conjugate point to a spacelike surface, namely a focal point relative to a small spacelike horizon cross-section through the original point. A null geodesic beyond such a focal point ceases to generate an achronal boundary: a timelike variation connects the cross-section to points just beyond it. Since the cross-section and the generator lie on the achronal horizon, this is impossible. Future completeness ensures that a singular endpoint cannot terminate the generator before the finite focusing parameter. This future-complete horizon focusing proof of the area theorem rules out ${\theta }_0<0$.

Therefore $\theta \ge 0$ on smooth horizon pieces, and the transverse-area identity givesIntegrating over generators proves area nondecrease between later cross-sections. At crease sets, new generators may enter the horizon from the past, adding area; horizon generators do not leave it toward the future under the stated assumptions. Hence the statement also applies to total horizon area through black-hole mergers. Hawking's area theorem is classical: quantum stress-energy can violate the null energy condition, so it does not forbid a shrinking evaporating black hole.

One standard form of the Penrose singularity theorem assumes a time-oriented globally hyperbolic spacetime with a noncompact Cauchy hypersurface, the null convergence condition $R_{ab}{k}^a{k}^b\ge 0$ for every null vector $k$, and a closed future trapped surface. It concludes future null geodesic incompleteness: some future-inextendible null geodesic has a finite upper endpoint of its affine parameter.

The focusing mechanism is the Null Raychaudhuri equation. The normal generators have zero null twist, soAn initial ${\theta }_0<0$ therefore gives a conjugate point to a spacelike surface within affine distance at most $2/|{\theta }_0|$, assuming the generator can be continued that far. Such a generator ceases to lie on the achronal boundary after its first focal point. Compactness of the trapped surface supplies a uniform bound for all normalized initial null normals. Future completeness would consequently make its future boundary compact. Projection along timelike curves to a connected Cauchy hypersurface is injective on the achronal boundary and has open image. Compactness makes the image closed as well; the nonempty image must therefore be the entire hypersurface, contradicting its noncompactness. This explains why the global assumptions supplement local focusing.

For Reissner-Nordstrom spacetime, the Maxwell stress-energy tensor satisfies the null energy condition; the Einstein field equations imply the required null convergence condition. The spheres in the band just found are closed and trapped. Apply the theorem to a maximal Cauchy development with a noncompact Cauchy hypersurface and containing one such sphere. That globally hyperbolic spacetime must be future null-geodesically incomplete.

The Penrose theorem at a Cauchy horizon needs care: the full maximal analytic Reissner-Nordstrom spacetime has inner Cauchy horizons and is not globally hyperbolic. It does not satisfy every hypothesis of the displayed theorem. In the exact solution some incomplete geodesics of the globally hyperbolic development reach a smoothly extendible Cauchy horizon in finite affine parameter. The theorem asserts incompleteness of the development, not that every such endpoint is a curvature singularity. The separate curvature singularity at $r=0$ does not justify silently dropping the theorem's global hypothesis.

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.

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.

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 $0\le \lambda \le L$, 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 lengthwhich is the incompleteness conclusion of the Penrose singularity theorem.