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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 a null vector, the perfect fluid in general relativity tensor givesbecause $g_{\alpha \beta }{V}^{\alpha }{V}^{\beta }=0$. For phantom energy, $\rho +P=(1+w)\rho <0$. A nonzero $u_{\alpha }{V}^{\alpha }$ therefore makes the contraction negative, so phantom energy violates the null energy condition.

The four laws of black-hole mechanics closely parallel thermodynamics. The zeroth law says that the surface gravity $\kappa$ is constant over a stationary Killing horizon. The first law isThe second law is Hawking's area theorem, $\delta A\ge 0$ under the null energy condition, and the third law of black-hole mechanics says that a regular extremal horizon with $\kappa =0$ cannot be reached by a finite physical process. These match constancy of temperature in equilibrium, $dE=TdS+$ work terms, entropy increase, and unattainability of zero temperature.

Quantum field theory makes the analogy literal. A stationary horizon emits at the Hawking temperatureComparing $T_{H}dS$ with the area term in the first law gives the Bekenstein-Hawking entropyFor a Schwarzschild black hole, $\kappa =c^4/(4GM)$ and $T_H=\hslash c^3/(8\pi GM{k}_B)$.

To see why particles are produced, quantize a real scalar field using the conserved Klein-Gordon inner product. In the asymptotically Minkowski past choose positive-frequency modes $u_i^{\mathrm{in}}$ and writeThe asymptotically Minkowski future supplies another positive-frequency basis $u_j^{\mathrm{out}}$ and operators $a_j^{\mathrm{out}}$. In a nonstationary middle region there is no preferred timelike Killing vector and therefore no invariant positive-frequency split: the notion of particle is observer- and basis-dependent. The two mode bases are related by a Bogoliubov transformation,so the corresponding operators mix annihilation and creation operators. The in-vacuum then containsout-particles whenever $\beta \ne 0$.

For a black hole formed by collapse, late outgoing modes traced backwards toward the event horizon undergo an exponentially large blueshift. This produces a universal Bogoliubov mixing withthe Planck distribution at $T_H$. Greybody scattering outside the horizon modifies the flux reaching infinity but not its characteristic temperature.

Hawking radiation gives a black hole negative heat capacity: it heats up as it loses mass and can evaporate in finite time. The generalized second law assigns entropy $S_{\mathrm{BH}}$ to the hole and states that this plus exterior entropy does not decrease. The enormous area entropy suggests microscopic horizon degrees of freedom. If semiclassical evaporation ends with only thermal radiation, an initially pure state appears to become mixed, producing the black hole information paradox. Resolving the endpoint, information recovery, and the microscopic origin of the area law are central constraints on quantum gravity.

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

The result is weaker than the pointwise null energy condition used in the elementary proof of the second law of black-hole mechanics: $T_{ab}{U}^a{U}^b$ may be negative over a finite interval, so the expansion and area can have local behavior that the pointwise argument does not control. Nevertheless, the averaged null energy condition for a nonminimally coupled scalar supplies the integrated positivity required by strengthened global focusing and area theorems when their completeness, endpoint, and genericity hypotheses hold. Thus the calculation supports a suitable global second law for $\xi <0$, but the displayed averaged inequality alone does not prove pointwise area monotonicity without those additional assumptions.

The second law of black-hole mechanics states that the area of spatial cross-sections of a future event horizon cannot decrease toward the future, provided the null energy condition and the standard global assumptions hold.

Let $k^a$ be an affinely parametrized horizon generator. It is hypersurface-orthogonal, so its null twist vanishes. The Null Raychaudhuri equation in $D$ dimensions and the Einstein equation giveIf $\theta ({\lambda }_0)<0$, integration implies that $\theta$ diverges to $-\infty$ within affine distance at most $(D-2)/|\theta ({\lambda }_0)|$. The resulting focal point would make the generator leave the achronal boundary that defines the event horizon, contradicting its assumed future completeness. Hence $\theta \ge 0$ everywhere. Since null expansion obeysevery horizon area element, and therefore every complete cross-section area, is nondecreasing.

In the orthonormal frame choose the null vector $k=e_0+e_2$, which has a nonzero angular component. The curvature forms giveFor a null vector, the trace term in the Einstein field equations drops out, so the null energy condition implies $R_{ab}{k}^a{k}^b\ge 0$. Therefore

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.

The two future null directions are proportional to $k_{+}=(1,1)$ and $k_{-}=(1,-1)$. The null energy condition requiresEquivalently,

For $T_{ab}=(\rho +p)u_a{u}_b+p{g}_{ab}$ and nonzero null $k$, contraction gives $T_{ab}{k}^a{k}^b=(\rho +p)(u\cdot k{)}^2$. A timelike fluid velocity has nonzero scalar product with every nonzero null vector, so the null energy condition is exactly the displayed inequality. Individual nonnegativity of pressure or density is unnecessary for null focusing. A cosmological constant term has zero null contraction.

In a regular double-null region satisfying the null energy condition, an initially negative $r_{,U}$ on a Cauchy hypersurface remains negative in its future domain of dependence. If both spherical null expansions are negative at one sphere, the exchanged-coordinate focusing inequality preserves the other sign along its future outgoing null direction. Every later sphere that still exists there remains a trapped surface.

Under the null energy condition and the usual global assumptions excluding naked pathologies, the area of every cross-section of a future event horizon is nondecreasing toward the future. The Null Raychaudhuri equation and future completeness rule out negative expansion of its generators.

A trapping horizon is a hypersurface foliated by marginal two-surfaces with one null expansion zero and the other nonzero. A future outer trapping horizon has ${\theta }_{+}=0$, ${\theta }_{-}<0$ and ${\mathcal{L}}_n{\theta }_{+}<0$ for a future ingoing null normal $n$. Unlike an event horizon, this is a quasi-local definition. Under the null energy condition, a regular future outer horizon is spacelike or null. The ingoing derivative condition distinguishes its outer type from an inner trapping horizon.