Penrose singularity theorem

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.

The globally hyperbolic maximal Cauchy development of appropriate Reissner-Nordstrom spacetime data obeys the Penrose singularity theorem when it contains a closed trapped surface. Its resulting null geodesic incompleteness can include generators reaching a smoothly extendible inner Cauchy horizon in finite affine parameter. The full analytic extension is not globally hyperbolic and cannot be substituted into that version of the theorem. Incompleteness alone does not establish curvature blowup at every endpoint.

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.

A concrete example separates a trapped surface from a failure of coordinates. Use geometrized units $G=c=1$, metric signature $(-,+,+,+)$, and the regular ingoing Eddington-Finkelstein coordinates of a Schwarzschild black hole with mass $M>0$:For a round sphere, choose future null vector normalsThe spherical null expansion in ingoing coordinates follows directly from $A=4\pi r^2$: ${\theta }_k=2k(r)/r$. At $r=3M$, $({\theta }_{+},{\theta }_{-})=(1/(9M),-2/(3M))$; at $r=3M/2$, it is $(-2/(9M),-4/(3M))$. Both future light fronts shrink in the second case. At $r=2M$, ${\theta }_{+}=0$, so this sphere is marginally outer trapped surface, not a strict trapped surface. In the spherical model this boundary is an event horizon; that identification does not extend to arbitrary geometries.

Although $k_{-}$ is affine as written, $k_{+}$ generally obeys ${\nabla }_{k_{+}}{k}_{+}=(M/r^2)k_{+}$. These are valid initial normals for the sign test; use affine rescaling of a null normal before applying the affine Null Raychaudhuri equation. Regularity of the displayed metric tensor at $r=2M$ shows why a divergence in another chart would not establish null geodesic incompleteness.

On a compact trapped surface, both null expansions have a uniform negative margin. Sufficiently small smooth perturbations controlling the relevant metric tensor and surface derivatives preserve trapping. This conditional robustness does not prove that every nonspherical gravitational collapse creates a trapped surface.^[ref]

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.

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]