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Contract the Einstein field equations with a null vector. Both the trace term and the cosmological constant term multiply $g_{ab}{\ell }^a{\ell }^b=0$ and disappear. The null energy condition therefore still implies the null convergence condition, independently of the sign of the cosmological constant. A black-hole area theorem still needs the appropriate global horizon assumptions as well as this local focusing condition.

The Null Raychaudhuri equation and Einstein field equations give ${\partial }_U(f^{-1}{r}_{,U})=-r{R}_{UU}/(2f)=-4\pi r{T}_{UU}/f$, with an exchanged-coordinate identity in $V$. Under the null energy condition, each rescaled radial derivative is nonincreasing along its corresponding future null direction.

Hawking's area theorem states that the total area of classical future event horizons cannot decrease when the null energy condition and the required global assumptions hold. It is the geometric content of the second law of black-hole mechanics.

The null convergence condition requires $R_{ab}{k}^a{k}^b\ge 0$ for every null vector. Through the Einstein field equations, it follows from the null energy condition, since all terms proportional to the metric tensor vanish on a null vector.

The classical stress-energy tensor of a canonical scalar field has a gradient-product term and a metric-proportional term. Contracting with a real null vector removes the latter, leaving the displayed square for a real field. This proves the null energy condition for a minimally coupled canonical scalar, including the massless case. It is not a statement about arbitrary curvature couplings, wrong-sign kinetic terms or a renormalized quantum stress-energy tensor.

In a planar warped spacetime, contraction of the Ricci tensor with a null vector in an orthonormal frame gives $R_{ab}{k}^a{k}^b=-2(\log A{)}^{\prime \prime }(k^3{)}^2$. The Einstein field equations identify this with $8\pi T_{ab}{k}^a{k}^b$; metric terms, including a cosmological constant, vanish in a null contraction. Thus the null energy condition is equivalent to concavity of $\log A$ in the proper transverse coordinate $z$. Null directions confined to a planar slice saturate the condition.

For a hypersurface-orthogonal null congruence under the null energy condition in $D$ spacetime dimensions, an initially negative expansion ${\theta }_0$ diverges to $-\infty$ within affine distance at most $(D-2)/|{\theta }_0|$.

Use $G=\hslash =c_{\mathrm{light}}=k_B=1$ and normalize the stationary time at infinity. For a regular stationary, asymptotically flat electrovacuum black hole, the Kerr-Newman metric hasInitially assume $M>0$ and $d>0$, so the outer Killing horizon is nondegenerate. Its surface gravity, angular velocity, electric potential and area areThese are horizon quantities for the generator $\chi ={\partial }_t+{\Omega }_H{\partial }_{\varphi }$.

The laws of black-hole mechanics provide the first reason to regard $\kappa$ and area as thermodynamic variables. The Zeroth law of black-hole mechanics makes $\kappa$ constant on an equilibrium horizon. The first law for the Kerr-Newman family readsFor example, write the horizon relation as $M=(r_{+}^2+a^2+Q^2)/(2r_{+})$ and use $J=Ma$; differentiating eliminates $d{r}_{+}$ and $da$ to give this law. The last two terms are rotational and electric work. The classical second law of black-hole mechanics says that area cannot decrease under the null energy condition and appropriate global horizon assumptions. The third law of black-hole mechanics concerns unattainability of a zero-surface-gravity regular horizon by a finite physical process.

This analogy alone does not determine the temperature scale: if entropy were $\eta A_H$, the first law would only give $T=\kappa /(8\pi \eta )$. The decisive input is quantum field theory in the collapsing or stationary background. In a collapse vacuum that is regular for freely falling observers, the late outgoing retarded time $u$ and a regular affine null coordinate $U$ satisfy the Hawking exponential ray mapAn outgoing mode $e^{-i\omega u}$ therefore behaves as $(-U/C{)}^{i\omega /\kappa }$. Continuing this power through the horizon changes its amplitude by the factor $e^{-\pi \omega /\kappa }$. Its positive- and negative-frequency decomposition consequently has the thermal ratio of Hawking Bogoliubov coefficientsCombining that ratio with bosonic normalization $|\alpha {|}^2-|\beta {|}^2=1$ gives a Planck occupation $1/(e^{2\pi \omega /\kappa }-1)$. The fermionic normalization instead gives the corresponding Fermi factor. Thus Hawking radiation fixesFor rotating charged modes, the horizon energy is $\tilde{\omega }=\omega -m_{\varphi }{\Omega }_H-q{\Phi }_H$, so the emission spectrum contains the same temperature with angular-momentum and charge chemical potentials. Exterior scattering supplies greybody factors; it changes the received flux, not the horizon temperature. Superradiant bosonic modes require the usual signed absorption factor, and a globally regular thermal bath need not exist throughout an asymptotically flat rotating exterior. The collapse-state emission argument is the relevant one.

A complementary check is the Euclidean black-hole regularity condition. The local corotating nondegenerate horizon geometry is Rindler-like:Smoothness at $\rho =0$ requires $\tau$ to have period $2\pi /\kappa$. Imaginary-time periodicity is precisely inverse temperature, agreeing with the radiation calculation. Matching the resulting temperature to the first law fixes the Bekenstein-Hawking entropy to $S=A_H/4$. The generalized second law then uses $S_{\mathrm{outside}}+A_H/4$: the classical area theorem alone does not apply to the negative-energy quantum flux responsible for evaporation.

Substitution gives the Kerr-Newman horizon temperatureIt reduces to $1/(8\pi M)$ for a Schwarzschild black hole. The nonextremal limit towards $d=0$ gives zero temperature. At exact extremality the Euclidean horizon is degenerate, so the elementary conical-period argument does not itself fix a period; zero temperature here is the limiting semiclassical result. The reasoning assumes ordinary Einstein–Maxwell dynamics, a regular horizon, the stated normalization at infinity and a regime where quantum fields on a slowly evolving classical geometry are a useful approximation. The mechanical laws, radiation spectrum and Euclidean regularity agree under these assumptions, which is substantially stronger evidence than the classical analogy alone.

A horizon separates regions according to a specified causal, dynamical or geometrical criterion. The criterion must be stated: an event horizon, an apparent horizon, a Killing horizon and a Cauchy horizon answer different questions. Some horizons describe a black hole, others an observer's acceleration or the universe's expansion. A vanishing coordinate metric component alone does not establish any of these definitions.

In an asymptotically flat spacetime, choose the relevant future null infinity ${\mathcal{I}}^{+}$, the endpoint of escaping outgoing light rays. The black-hole region and its future event horizon areHere $J^{-}$ is the causal past. Events in $\mathcal{B}$ cannot send a causal signal to the chosen infinity. This is a global definition: identifying the horizon requires the full future development, so a local measurement need not reveal whether an event lies on it. Smooth portions of an event horizon are null hypersurfaces generated by null geodesics. Generators may join at past caustics or crease sets, so the horizon need not be smooth everywhere. A past event horizon is defined by the time-reversed escape criterion; in the maximally extended Schwarzschild solution it bounds a white-hole region.

An observer-based definition replaces infinity by a future-inextendible worldline $\gamma$. Its future observer event horizon is the boundary of events that can signal to $\gamma$, conventionally $\partial I^{-}(\gamma )$. A past observer horizon similarly bounds events reachable from its past worldline. Such horizons depend on the observer and need not imply a black hole.

A Killing horizon is a null hypersurface to which a nonzero Killing vector field $K^a$ is normal. On it, $K^a$ is tangent to its null generators and satisfies $K^b{\nabla }_b{K}^a=\kappa K^a$. The surface gravity $\kappa$ measures the failure of this Killing parameter to be an affine parameter; rescaling $K$ rescales $\kappa$. A degenerate Killing horizon has $\kappa =0$. Stationary black-hole horizons often are Killing horizons, whereas a general evolving event horizon need not have any Killing symmetry. The condition $K^a{K}_a=0$ alone is insufficient: the null vector must also be normal to a null hypersurface.

The Schwarzschild spacetime illustrates several of these notions. In units $G=c=1$, let $f=1-2M/r$. Its metric is $d{s}^2=fd{t}^2-f^{-1}d{r}^2-r^{2}d{\Omega }^2$, and $r=2M$ is the future event horizon of the black-hole region in the appropriate extension. The apparent coordinate singularity disappears in Ingoing Eddington-Finkelstein coordinates, $v=t+r_{\ast }$, $d{r}_{\ast }/dr=f^{-1}$:At $r=2M$ this metric is regular and the normal $dr$ is null. Future outgoing radial rays satisfy $dr/dv=f/2$; inside the horizon their areal radius decreases even on the outgoing family. The stationary Killing vector field ${\partial }_v$ generates the horizon with $\kappa =f^{\prime }(2M)/2=1/(4M)$ when normalized at infinity. The Kretschmann scalar is $48M^2/r^6$, finite at the horizon and divergent at $r=0$. Thus the event horizon is not the curvature singularity. The photon sphere at $r=3M$, which supports circular null orbits, is also not the horizon.

Quasi-local horizon notions use null expansions. For a closed spacelike two-surface, choose future outgoing and ingoing null normals $\ell ,n$, with $\ell \cdot n=1$ in signature $(+---)$. Each expansion is the logarithmic rate of its positive area element along the corresponding null normal. A future trapped surface has ${\theta }_{\ell }<0$ and ${\theta }_n<0$: both future light fronts initially shrink. A marginally outer trapped surface has ${\theta }_{\ell }=0$; for the future black-hole type also require ${\theta }_n<0$. On a chosen spacelike slice, the outer boundary of the future trapped region is an apparent horizon, ordinarily an outermost marginally outer trapped surface when regularity permits this characterization. It depends on the slice, and a slicing need not display an apparent horizon even when the global spacetime contains an event horizon.

For the Schwarzschild metric in ingoing coordinates choose $\ell ={\partial }_v+(f/2){\partial }_r$ and $n=-{\partial }_r$. The spherical null expansion in ingoing coordinates givesThe round spheres inside $2M$ are future trapped, while $r=2M$ is marginal. On standard horizon-penetrating slices its sections give the apparent horizon, coinciding with the stationary event and Killing horizon. This coincidence in a symmetric stationary example does not identify the notions in a general spacetime.

A trapping horizon is a hypersurface foliated by marginal surfaces. Its future outer type obeys ${\theta }_{\ell }=0$, ${\theta }_n<0$ and ${\mathcal{L}}_n{\theta }_{\ell }<0$. Unlike an event horizon, this can be diagnosed from local expansion data and their variation. The null energy condition explains its causal character. At a marginal surface, the Null Raychaudhuri equation, with the present curvature sign and vanishing null twist, givesin units $c=1$. A possible nonaffinity term vanishes here because it is proportional to the zero expansion. Write a horizon evolution vector as $V=\ell -Cn$. Preserving ${\theta }_{\ell }=0$ gives $C=({\mathcal{L}}_{\ell }{\theta }_{\ell })/({\mathcal{L}}_n{\theta }_{\ell })\ge 0$. Hence $V^2=-2C\le 0$: the future outer horizon is spacelike or null. Its area expansion is ${\theta }_V=-C{\theta }_n\ge 0$. These conclusions use the outer condition and the energy condition, not merely marginality.

For the global event horizon, the future-complete horizon focusing proof of the area theorem uses the same Null Raychaudhuri equation. Under the null energy condition, a negative generator expansion forces a future caustic in finite affine parameter. Future completeness and the usual regularity or predictability assumptions exclude such a caustic on the future event-horizon boundary. Its generator expansion is therefore nonnegative and its classical area cannot decrease. These assumptions matter: this is not an unconditional result for every null surface or every matter model.

Rotating and charged solutions provide further distinctions. A nonextremal Kerr black hole has $r_{\pm }=M\pm \sqrt{M^2-a^2}$, with $a=J/M$. Its outer Killing horizon is generated by ${\partial }_t+{\Omega }_H{\partial }_{\varphi }$, where ${\Omega }_H=a/(r_{+}^2+a^2)$, rather than by time translation alone. The ergosphere begins at the stationary-limit surface $g_{tt}=0$, whose radius is $M+\sqrt{M^2-a^2{\cos }^2\theta }$. Away from the axis this lies outside the event horizon: it forbids static observers but does not forbid escape to infinity. In Reissner-Nordstrom spacetime, the nonextremal radii are $r_{\pm }=M\pm \sqrt{M^2-Q^2}$. The outer surface is an event horizon, whereas the inner surface is a Cauchy horizon in the exact analytic extension.

A Cauchy horizon concerns predictability from initial data. The future domain of dependence $D^{+}(\Sigma )$ of a partial Cauchy surface consists of events for which every past-inextendible causal curve intersects $\Sigma$. Its future Cauchy horizon isBeyond it, causal influences can arrive without meeting the specified initial surface; data on that surface no longer determine the solution there. This is distinct from the inability to send signals to infinity. The inner horizon of the exact charged black hole supplies an example, and the definition necessarily specifies which initial surface is being used.

Flat spacetime itself has acceleration horizons. In the right Rindler wedge, set $t=\xi \sinh (a\tau )$ and $x=\xi \cosh (a\tau )$, with $\xi >0$. The Minkowski spacetime metric becomes $d{s}^2=a^2{\xi }^{2}d{\tau }^2-d{\xi }^2-d{y}^2-d{z}^2$. The null boundaries $t=x$ and $t=-x$ are Rindler horizons of the boost Killing flow. A uniformly accelerated observer cannot receive signals from one side of its future horizon, even though the curvature vanishes and inertial observers encounter no such boundary. These are observer and Killing horizons, not a black-hole event horizon defined using the full Minkowski null infinity.

In a spatially flat Friedmann-Lemaître-Robertson-Walker metric with noncompact spatial sections, radial light rays have $d\chi /dt=\pm 1/a(t)$. The particle horizon measures how far signals could have travelled since the initial cosmic time; the cosmological event horizon measures how far signals emitted now could ever travel toward a comoving observer. Their comoving and proper radii areThe respective horizon is present only if its integral is finite. The event-horizon integral uses the actual future endpoint, often $t_f=\infty$, so it is global and observer-dependent. For $a(t)=t^q$ on $0<t<\infty$, a finite particle horizon exists for $0<q<1$, while a finite event horizon exists for $q>1$. Neither integral is finite for $q=1$. In the expanding flat patch of de Sitter spacetime, $a=e^{Ht}$ and the proper event-horizon radius is $H^{-1}$. The static form $d{s}^2=(1-H^2{r}^2)d{t}^2-(1-H^2{r}^2{)}^{-1}d{r}^2-r^{2}d{\Omega }^2$ identifies it as a cosmological Killing horizon; its surface-gravity magnitude is $H$ with the central static time normalization.

Finally, the Hubble radius must not automatically be called an event horizon. For a flat expanding cosmology, the two future radial null expansions of a sphere of proper radius $R$ are ${\theta }_{\pm }=2(H\pm 1/R)$. At $R=H^{-1}$ the ingoing expansion vanishes, while outside it both expansions are positive: this is the cosmological marginal, past-trapped type rather than a future black-hole trapped surface. A dust universe with $a\propto t^{2/3}$ has such a Hubble sphere but no future event horizon. Global causal boundaries, slice-dependent trapped-surface boundaries, symmetry horizons and initial-data boundaries must therefore be distinguished, even when several coincide in a special example.

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.

A stationary black hole behaves mechanically like an equilibrium thermodynamic system. Its horizon area A, asymptotic energy E, angular momentum J and electric charge Q are linked by the laws of black-hole mechanics. Let ${\chi }^a=t^a+{\Omega }_H{\varphi }^a$ generate its Killing horizon, with the time translation normalized at infinity; its surface gravity is defined by ${\chi }^b{\nabla }_b{\chi }^a=\kappa {\chi }^a$ on that horizon. We use $c=1$ and keep G explicit. The horizon electrostatic potential ${\Phi }_H$ is measured relative to infinity.

The Zeroth law of black-hole mechanics states that kappa is constant on each connected stationary horizon, under the Einstein equations and suitable energy/regularity assumptions. It parallels the zeroth law of thermodynamics, which makes temperature uniform in equilibrium. Kappa need not be constant in a dynamical geometry. For an asymptotically flat static hole, local observers see gravitationally redshifted temperatures; the equilibrium quantity comparable to kappa is the temperature referenced to infinity.

The First law of black-hole mechanics relates nearby stationary solutions:The same leading relation describes a small physical process which perturbs a horizon and subsequently settles to equilibrium, with the appropriate hypotheses on fluxes and final state. The work terms correspond to rotational and electromagnetic work. The structural analogy with the first law of thermodynamics, $dE=TdS+\text{work}$, identifies area as an entropy-like variable and surface gravity as a temperature-like variable. Classically this fixes their product in the first law, rather than their separate normalizations.

The second law of black-hole mechanics, or black-hole area theorem, states that future event-horizon area does not decrease classically. Its essential geometric ingredient is the Raychaudhuri equation for affinely parametrized null generators. In four dimensions a hypersurface-orthogonal congruence obeysThe null energy condition and Einstein equations make the last term nonpositive. If the expansion became negative, the inequality would focus the generators to a conjugate point in finite affine parameter. Under future completeness and predictability assumptions, that would conflict with their remaining on an achronal event horizon. Thus $\vartheta \ge 0$, and the area element, satisfying $d\log dA/d\lambda =\vartheta$, cannot decrease. For a merger the final area is at least the sum of the initial component areas. These assumptions matter: the theorem is a classical statement, not a prohibition of quantum evaporation.

The third law of black-hole mechanics is an unattainability statement: a nonextremal hole cannot be driven to zero surface gravity in a finite physical process satisfying the usual bounded-stress and regularity conditions. This parallels the unattainability form of the Third law of thermodynamics. Extremal stationary solutions exist, so the law concerns how an initially nonextremal hole approaches them. The stronger entropy form of Nernst's theorem does not transfer unchanged: extremal holes can have finite area and entropy, depending on their conserved charges.

Quantum field theory turns the mechanical analogy into a physical thermodynamic identification. Hawking radiation is thermal at infinity to leading semiclassical order, withThese give $T_H\delta S_{\mathrm{BH}}=\kappa \delta A/(8\pi G)$ exactly, so the mechanical first law becomes the thermodynamic first law with Bekenstein-Hawking entropy. Scattering through the exterior adds greybody factors to the emitted spectrum but leaves this temperature identification intact. Equilibrium with an external bath and isolated evaporation are different physical states of the field.

During black-hole evaporation, the area can decrease because the semiclassical stress tensor need not satisfy the classical null energy condition. The proposed generalized second law applies instead to $S_{\mathrm{gen}}=S_{\mathrm{BH}}+S_{\mathrm{outside}}$, asserting nondecrease of the combined entropy. A Schwarzschild hole illustrates an unusual thermodynamic property: $T_H\propto E^{-1}$, so its heat capacity is negative. It becomes hotter as it loses energy and is unstable in a simple canonical heat bath, while entropy and energy laws still describe an isolated system. The quantum temperature, area entropy and generalized second law supply the correspondence; the mechanical laws alone do not establish a thermal emission spectrum or a microscopic entropy count.

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.

Take the screen trace of the optical matrix of a null congruence equation. The cross-contractions of symmetric null shear with antisymmetric null twist vanish, andThe minus sign in the null twist contribution follows from tracing the square of an antisymmetric matrix. The tidal trace is $P^{ef}{R}_{cedf}{U}^c{U}^d=R_{cd}{U}^c{U}^d$: the additional null terms in $P^{ef}$ vanish by curvature antisymmetry. Thus the Null Raychaudhuri equation isIn particular, null shear focuses the null geodesic congruence while null twist enters with the opposite sign. With vanishing null twist and the null energy condition, the Einstein field equations imply $U\cdot \nabla \theta \le -{\theta }^2/2$. These signs and the factor $1/2$ depend on using the full-trace null expansion in a two-dimensional screen.

Use geometrized units $G=c=1$. The Physical-process first law of black-hole mechanics concerns a small perturbation of a rotating black hole in an initially stationary spacetime by infalling matter, followed by relaxation to another stationary spacetime. For an uncharged black hole, or neutral accretion with charge held fixed, its first-order statement isHere $\kappa >0$ is the background surface gravity, ${\Omega }_H$ is the background horizon angular velocity, and $\delta A$ is the change in event horizon area. The energy $\delta E$ and angular momentum $\delta J$ are the fluxes carried into the event horizon; when these account for the changes of the black hole, they equal $\delta M$ and its change of angular momentum. The result is a linearized law, not an exact identity for a finite violent accretion event. The perturbation must be small enough that focusing does not produce new caustics or invalidate the expansion about the existing generators.

Let $k^a$ be the stationary Killing vector field normalized at infinity, $m^a$ the axial Killing vector field with $2\pi$-periodic orbits, andOn the background Killing horizon, ${\chi }^a$ is its future-directed null generator and satisfies ${\chi }^b{\nabla }_b{\chi }^a=\kappa {\chi }^a$. Choose a parameter $v$ with $\chi ={\partial }_v$. The null expansion in this parameter is $\theta ={\partial }_v\log dA$. The generator is not affinely parametrized; if $\ell ={\partial }_{\lambda }$ is affine, then $\chi =\kappa \lambda \ell$ after an appropriate choice of affine origin, with $\lambda \propto e^{\kappa v}$. Rescaling the Null Raychaudhuri equation givesThe null twist vanishes because the generators are normal to the event horizon. By the Einstein field equations, $R_{ab}{\chi }^a{\chi }^b=8\pi T_{ab}{\chi }^a{\chi }^b$. Background null expansion and null shear vanish on a Killing horizon of the background stationary spacetime. Thus $\theta$ and $\hat{\sigma }$ are first order, their squares are second order, and the linearized equation isThe final stationary spacetime supplies the boundary condition $\theta \to 0$ in the future. Explicitly,This future boundary condition reflects the global definition of the event horizon. For localized infall, $\theta$ also tends to zero in the remote past. Integrating the linearized equation along the generators, and then over a background cross-section with area element $d{A}_0$, givesUsing the background area element introduces only second-order corrections.

With orientations chosen so that future infalling positive energy has positive flux, the conserved Killing energy and axial angular momentum giveConsequently $\delta E-{\Omega }_H\delta J={\int }_{\mathcal{H}}\delta T_{ab}{\chi }^a{\chi }^{b}dvd{A}_0$, proving the displayed Physical-process first law of black-hole mechanics. Restoring $G$ replaces $8\pi$ by $8\pi G$. The null energy condition makes the integral nonnegative and therefore implies area increase in this regime; that condition is needed for the sign conclusion, not for the linearized flux identity itself. Charged infall would require the additional electrostatic work term, and an extremal background needs separate treatment because the proof uses $\kappa >0$.