Relativity and gravitational theory Black holes Theorem

Penrose–Hawking singularity theorems

Builds on Lorentzian manifold Geodesic Ricci curvature

Statement

A spacetime here is a connected, time-oriented, four-dimensional Lorentzian manifold $(M, g)$ of signature $(-, +, +, +)$ . It is timelike (resp. null) geodesically complete if every inextendible timelike (resp. null) geodesic is defined for all values of its affine parameter. A closed trapped surface is a compact spacelike surface without boundary in $M$ . On it, both families of future-directed null geodesics orthogonal to it must have negative expansion. Let $\operatorname{Ric}$ be the Ricci curvature of $g$ .

(Hawking) Assume that $(M, g)$ is globally hyperbolic and that $\operatorname{Ric}(V, V) \ge 0$ for all timelike vectors $V$ . Let $\Sigma$ be a smooth spacelike Cauchy surface in $M$ . Let $\theta$ be the expansion on $\Sigma$ of the past-directed normal geodesics, that is, the mean curvature for the past-directed unit normal. Assume that $\theta \le C$ on $\Sigma$ for a constant $C < 0$ . Then every past-directed timelike curve starting on $\Sigma$ has length at most $3/\abs{C}$ . In particular, $(M, g)$ is not timelike geodesically complete.
(Penrose) Assume that $(M, g)$ is globally hyperbolic with a noncompact Cauchy surface. Assume that $\operatorname{Ric}(K, K) \ge 0$ for all null vectors $K$ , and that $M$ contains a closed trapped surface. Then $(M, g)$ is not null geodesically complete.
(Hawking and Penrose) Assume that $\operatorname{Ric}(V, V) \ge 0$ for all timelike and null vectors $V$ . Assume that $M$ has no closed timelike curves, and that every timelike or null geodesic satisfies the generic condition. Assume also that at least one of the conditions (a), (b), (c) below holds. (a) $M$ contains a compact achronal set without edge. (b) $M$ contains a closed trapped surface. (c) There is a point $p$ such that the null geodesics from $p$ all start to reconverge. This means that the expansion of the future-directed null geodesics from $p$ becomes negative along each of them. The same with past-directed geodesics also suffices. Then $(M, g)$ contains an incomplete timelike or null geodesic.

Remarks

The common core of the proofs is the Raychaudhuri equation. By it, the curvature conditions force the expansion of a congruence of geodesics to become $-\infty$ in finite affine parameter, which creates conjugate or focal points. Such points prevent geodesics from maximizing length beyond them. On the other hand, global causal assumptions, such as global hyperbolicity, guarantee the existence of maximizing geodesics. The two facts together show that some geodesic cannot be extended. The Schwarzschild black hole contains closed trapped surfaces inside its horizon, and the theorem of Penrose shows that the singularity there persists under perturbations that keep a trapped surface. The theorems prove incompleteness of geodesics; they do not say that the curvature becomes infinite.

Going deeper

For a student

The curvature conditions are energy conditions on matter, translated through the Einstein equations into inequalities on the Ricci curvature. The proofs use the comparison theory of geodesics, much like the proof of Myers's theorem in Riemannian geometry, where positive Ricci curvature bounds the length of minimizing geodesics. Penrose's theorem dates from 1965, and the joint theorem of Hawking and Penrose from 1970. The third theorem trades global hyperbolicity for the weaker chronology condition, at the cost of the generic condition.

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Geodesic concept
Lorentzian manifold concept
Ricci curvature concept
Penrose–Hawking singularity theorems statement

Tag	`WP4YAHJ6V4`
URI	`m2://WP4YAHJ6V4`
Kind	statement, theorem
Writing	W0 stub
MSC	83C75, 53C50
Status	This copy of the site has no build data. Run `m2 status WP4YAHJ6V4`.
About	Lorentzian manifold
Uses	Lorentzian manifold, Geodesic, Ricci curvature
Same as	Wikidata Q1452678 Wikipedia Penrose–Hawking singularity theorems
Changed	2 November 2026