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The Null Raychaudhuri equation is the light-ray version of the Raychaudhuri equation associated with Amal Kumar Raychaudhuri. His original work concerned relativistic cosmology.^[ref] For the normal null geodesic congruence before its first focal point, null twist vanishes. With an affine parameter $\lambda$, the four-dimensional equation reduces to the displayed ordinary differential equation. The null shear contraction is nonnegative on the transverse screen space, and the null convergence condition makes the other part of $Q$ nonnegative. Consequently ${\theta }^{\prime }\le -{\theta }^2/2$.

For $\theta (0)={\theta }_0<0$, put $w=-1/\theta >0$. ThenA regular negative null expansion therefore cannot persist beyond $\lambda =2/|{\theta }_0|$ if the null geodesic extends that far. The null focusing theorem turns this estimate into a focal-point bound. The number refers to an affine parameter, not the proper time of a light ray, which is zero.

For an original soluble model, prescribe constant $Q$ along one twist-free beam; this does not specify an entire solution of the Einstein field equations. Introduce a positive affine scale $L$ and the dimensionless variablesFor an infinitesimal area $A$ define $a=\sqrt{A/A_0}$ before the first zero. Since $y=2a^{\prime }/a$, where primes now mean $d/du$, the same model becomesTwo independently checkable solutions areThe nonnegative extra focusing term moves the first zero earlier. Direct substitution into both ordinary differential equations verifies the solutions; numerical integration of the nonlinear equation provides a separate comparison with the linear beam equation.

A focal point makes this beam description degenerate, but individual null geodesics may continue smoothly through it. Even converging light in Minkowski spacetime can form a caustic. Converting local focusing into null geodesic incompleteness requires the global step below.

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.

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.

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 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.

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.

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.

A future trapped surface is a smooth compact spacelike two-surface without boundary whose two future-directed orthogonal null expansions are strictly negative. In four-dimensional general relativity, the Penrose singularity theorem states: a time-oriented globally hyperbolic spacetime with a noncompact Cauchy hypersurface, a trapped surface, and the null convergence condition $R_{ab}{k}^a{k}^b\ge 0$ for every null vector is future null-geodesically incomplete. With the Einstein field equations, the null energy condition implies this null convergence condition; a cosmological constant drops out of the null contraction.

The Kruskal spacetime is an example. In its black hole interior, use future null normals in ingoing Eddington-Finkelstein coordinates:For a round sphere of areal radius $r$, its area is $4\pi r^2$ and its null expansions areBoth are negative for $0<r<2M$. The vacuum Einstein field equations give $R_{ab}=0$, and a two-ended Kruskal Cauchy hypersurface is noncompact. The future radial null geodesics reaching $r=0$ in finite affine parameter provide precisely the incompleteness predicted by the Penrose singularity theorem. A trapped surface at $r<2M$ is strictly trapped; the horizon sphere has one zero null expansion instead.

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 Penrose singularity theorem states that a globally hyperbolic spacetime with a noncompact Cauchy hypersurface, a closed trapped surface, and the null convergence condition is future null-geodesically incomplete. Time reversal replaces the trapped surface by an anti-trapped surface and gives past incompleteness.

In his 1965 paper Gravitational Collapse and Space-Time Singularities, Roger Penrose used trapped surfaces and global causal structure to address whether nonspherical perturbations could avert a breakdown during gravitational collapse. Quasars supplied the opening astrophysical motivation. The paper used metric signature $(+,-,-,-)$ and a stronger timelike local-energy assumption; the statement above uses the modern null convergence condition.^[ref]

For this version, take a smooth connected four-dimensional spacetime with a time orientation. A Cauchy hypersurface meets every inextendible causal curve exactly once; its existence expresses global hyperbolicity. The hypersurface must be noncompact, meaning not compact; this is a global spatial condition, not an assumption about the size of the collapsing body. The null convergence condition is $R_{ab}{k}^a{k}^b\ge 0$ for every null vector $k$. Through the Einstein field equations it follows from the null energy condition; a cosmological constant drops out of the null contraction.^[ref]

The conclusion is that at least one future-inextendible null geodesic has a finite upper endpoint of its affine parameter. It neither asserts infinite mass density or divergent curvature at every endpoint, nor says that such an endpoint is visible. Locating an event horizon requires additional global information; the theorem is not a horizon-finding rule for arbitrary spacetimes. Weak cosmic censorship conjecture is an additional issue.^[ref]

The supermassive black hole account describes the motivating observational evidence from stellar orbits at the Galactic centre. Such measurements infer a compact gravitating mass. They do not observe a trapped surface, measure a geodesic's complete future, or experimentally prove the Penrose singularity theorem.