Black-Hole Area Theorem

The black-hole area theorem is a monotonicity statement for compact cuts of a classical future event horizon. Its conclusion is global: local null focusing supplies the sign of the expansion, while completeness or controlled access to future infinity prevents a horizon generator from escaping the focusing contradiction through a future endpoint.

Helpful background. Causality and Global Structure supplies event horizons, causal boundaries, and global hyperbolicity; 3+1 Formulations of General Relativity supplies spacelike cuts, normals, and induced area elements.

Detailed treatment. Hawking Area Theorem develops the proof and its relation to black-hole mechanics.

Area monotonicity with endpoint control

Let $(M,g)$ be a smooth, time-oriented spacetime of dimension $D\geq3$, and let $\mathcal H^+$ be a black-hole future event horizon. Let $\Sigma_1$ and $\Sigma_2$ be achronal, spacelike, embedded $C^2$ hypersurfaces. For a compact, finite-area specialization, set

$$S_a=\mathcal H^+\cap\Sigma_a, \qquad a=1,2,$$

and require the ordering

$$S_1\subset J^-(S_2).$$

Chruściel, Delay, Galloway, and Howard give three alternative sufficient global packages. The geometric hypothesis used below is null convergence, $R_{ab}k^ak^b\geq0$ for every null $k^a$. Under the Einstein equation this is equivalent to the matter null energy condition $T_{ab}k^ak^b\geq0$; a cosmological term drops out of the null contraction. Use one of the following complete packages:

1. $(M,g)$ is globally hyperbolic, it has a conformal completion $\overline M=M\cup\mathscr I^+$ with $\mathscr I^+$ H-regular, and null convergence holds throughout $I^-(\mathscr I^+;\overline M)\cap M$.
2. Every generator of $\mathcal H^+$ is future geodesically complete, and null convergence holds on $\mathcal H^+$.
3. The conformal completion $\overline M=M\cup\mathscr I^+$ is globally hyperbolic as a spacetime with boundary, and null convergence holds throughout $I^-(\mathscr I^+;\overline M)\cap M$.

The nonstandard H-regularity condition in package 1 means that some neighborhood $O$ of $\mathcal H^+$ has the following property. For every compact $C\subset O$ with

$$I^+(C;\overline M)\cap\mathscr I^+\ne\varnothing,$$

set $A_C=\partial I^+(C;\overline M)\cap\mathscr I^+$. There is a point $q\in A_C$ that is pastmost relative to $A_C$ within future infinity:

$$J^-(q;\mathscr I^+)\cap A_C=\{q\}.$$

This is Chruściel et al. 2001, Def. 4.3, p. 132.

Then the cross-sectional areas are well defined and

$$\operatorname{Area}_{D-2}(S_1) \leq \operatorname{Area}_{D-2}(S_2).$$

These hypotheses and the low-regularity conclusion are stated in Chruściel et al. 2001, Thm. 1.1, pp. 111–112. The alternatives are sufficient packages, not interchangeable fragments: in particular, horizon-only null convergence belongs to the future-complete generator package, whereas the two conformal-completion packages require it on the displayed causal past of future infinity.

The smooth expansion mechanism

On a $C^2$ portion of the horizon, choose a future-directed affinely parametrized generator $k^a$. Hypersurface orthogonality removes the twist, so Raychaudhuri’s equation is

$$\frac{d\theta}{d\lambda} =-\frac{\theta^2}{D-2} -\sigma_{ab}\sigma^{ab} -R_{ab}k^ak^b.$$

Null convergence makes the right-hand side no larger than $-\theta^2/(D-2)$. If $\theta$ were negative, it would diverge to $-\infty$ within finite affine parameter, producing a focal point. Beyond a focal point the generator cannot remain on an achronal boundary. The selected global package prevents it from ending first, hence $\theta\geq0$. Since the induced area element obeys

$$\frac{d}{d\lambda}\ln dA=\theta,$$

the generator flow cannot reduce area. This is the classical proof mechanism in Hawking 1972, § 3, pp. 156–160.

Equality need not mean that the hypotheses were unnecessary. If the two areas are equal, the swept open horizon portion

$$\Omega= \bigl(J^+(S_1)\setminus S_1\bigr) \cap \bigl(J^-(S_2)\setminus S_2\bigr)$$

is smooth (analytic when the metric is analytic), its null second fundamental form vanishes, and $R(X,k)k=0$ for every $X$ tangent to the horizon along the connecting generators. Thus on smooth portions $\theta=0$, the shear vanishes, and $R_{ab}k^ak^b=0$. See Chruściel et al. 2001, Thms. 1.2 and 6.1.

Creases, mergers, and the theorem’s boundary

A merger horizon need not be piecewise $C^2$. Generators can join the horizon at past endpoints, and several generators can meet at a crease where a classical pointwise $\theta$ is unavailable. The rigorous theorem instead uses the horizon’s almost-everywhere Alexandrov second derivatives and a measure-theoretic area comparison. The standard inequality above therefore does not assume smoothness at the crease. When $S_1\cap S_2=\varnothing$, Chruściel et al. 2001, Thm. 6.1 and Remark 6.2 also give the refined comparison in which the earlier cut is counted with generator multiplicity.

This theorem does not govern apparent, trapping, isolated, or dynamical horizons merely because they carry marginal surfaces. It does not establish cosmic censorship, future completeness, or predictability; one of the stated global packages must be supplied independently. It neither guarantees strict increase nor applies unchanged to quantum stress tensors that violate pointwise null convergence. Finally, it is not the generalized-entropy law: no statement about exterior von Neumann entropy or semiclassical evaporation follows from the classical area inequality. The thermodynamic analogy called the second law is formulated, with its classical energy-condition scope, by Bardeen, Carter, and Hawking 1973, pp. 168–169.

References

• Bardeen, J. M., B. Carter, and S. W. Hawking. “The Four Laws of Black Hole Mechanics.” Communications in Mathematical Physics 31 (1973): 161–170. DOI.
• Chruściel, Piotr T., Erwann Delay, Gregory J. Galloway, and Ralph Howard. “Regularity of Horizons and the Area Theorem.” Annales Henri Poincaré 2 (2001): 109–178. arXiv:gr-qc/0001003v2.
• Hawking, Stephen W. “Black Holes in General Relativity.” Communications in Mathematical Physics 25 (1972): 152–166. DOI.