Hawking–Penrose Singularity Theorems

The classical singularity theorems turn focusing plus global causal information into a precise failure of geodesic completeness. They do not produce a singular point. The Penrose, Hawking, and Hawking–Penrose results below use different initial geometry and have different causal types and time directions in their conclusions.

Helpful background. Causality and Global Structure supplies chronology, Cauchy hypersurfaces, trapped sets, and geodesic completeness; 3+1 Formulations of General Relativity supplies the hypersurface-normal and expansion language used in the Hawking variant.

Detailed treatments. See The Penrose Singularity Theorem, Hawking Cosmological Singularity Theorems, The Hawking–Penrose Theorem, and What Singularity Theorems Conclude for proofs and the logical limits of incompleteness.

Three classical statements

Use the following smooth formulations on a connected, time-oriented spacetime $(M,g)$ of dimension $n\geq3$, without boundary. The arguments require at least the classical $C^2$-level curvature and causal-geodesic structure; lower-regularity versions are separate theorems.

Result	Hypotheses	Typed conclusion
Penrose	$M$ has a noncompact Cauchy hypersurface, hence is globally hyperbolic; $R_{ab}k^ak^b\geq0$ for every null $k$; and there is a closed, achronal, future-trapped spacelike codimension-two surface $P$. Here closed means compact without boundary, and both future null expansions are strictly negative.	At least one future-inextendible null geodesic has finite affine length: $M$ is future null geodesically incomplete.
Hawking	$M$ has a compact smooth spacelike Cauchy hypersurface $S$; $R_{ab}v^av^b\geq0$ for every timelike $v$; and the future unit normal has expansion $\Theta\geq\Theta_0\gt0$ everywhere on $S$.	$M$ is past timelike geodesically incomplete. Past proper-time separation from $S$ is bounded above by $(n-1)/\Theta_0$. Time reversal, with $\Theta\leq-\Theta_0\lt0$, gives future timelike incompleteness.
Hawking–Penrose	$M$ is chronological; $R_{ab}v^av^b\geq0$ for every causal $v$; every inextendible causal geodesic encounters a point where its transverse tidal operator $[X]\mapsto[R(X,\dot\gamma)\dot\gamma]$ is nonzero on the appropriate transverse space; and one initial alternative below holds.	Some inextendible timelike or null geodesic has a finite proper- or affine-parameter endpoint. The basic statement fixes neither causal type nor time direction.

The Penrose formulation and its compactness argument are given by Penrose 1965, pp. 57–59 and in modern form by Steinbauer 2023, Theorem 3.9, arXiv v2 PDF pp. 16–18. Hawking’s compact-slice theorem and its time-oriented translation are supported by Hawking 1967, pp. 187–201 and Steinbauer 2023, Theorem 3.8, arXiv v2 PDF pp. 15–16.

For the genericity clause, the transverse space is $\dot\gamma^\perp$ along a timelike geodesic and $\dot\gamma^\perp/\operatorname{span}\{\dot\gamma\}$ along a null geodesic. The displayed Ricci inequalities are geometric convergence conditions. Matter null or strong energy conditions imply them only after applying the Einstein equation with the cosmological term treated explicitly; the labels are not interchangeable assumptions Steinbauer 2023, §§ 2.1 and 3.2, arXiv v2 PDF pp. 4–5 and 10–13.

Hawking–Penrose initial alternatives

The combined theorem requires at least one of these three alternatives:

1. a compact achronal set without edge;
2. a closed future- or past-trapped spacelike codimension-two surface; or
3. a trapped point, meaning that every null geodesic from the point in one chosen time direction eventually acquires negative expansion.

Together with convergence, the generic tidal condition forces conjugate points on every complete causal geodesic. The selected initial alternative supplies a trapped causal set, while chronology enables the global contradiction. The original causal lemmas and theorem occupy Hawking and Penrose 1970, pp. 531–544; a modern formulation and proof map appear in Steinbauer 2023, Theorem 3.11, arXiv v2 PDF pp. 18–20.

What the conclusions do not say

Each conclusion is existential: at least one causal geodesic is incomplete in the stated sense. None of these theorems establishes curvature blowup, inserts a singular point into the manifold, proves event-horizon formation, or proves inextendibility in any regularity class. Penrose’s result does not imply timelike incompleteness; Hawking’s displayed result does not imply null incompleteness. Correspondingly future- or past-oriented Hawking–Penrose initial alternatives admit a conclusion in that direction, but still do not select timelike rather than null incompleteness.

References

• Hawking, Stephen W. “The Occurrence of Singularities in Cosmology. III. Causality and Singularities.” Proceedings of the Royal Society A 300 (1967): 187–201. DOI.
• Hawking, Stephen W., and Roger Penrose. “The Singularities of Gravitational Collapse and Cosmology.” Proceedings of the Royal Society A 314 (1970): 529–548. DOI.
• Penrose, Roger. “Gravitational Collapse and Space-Time Singularities.” Physical Review Letters 14 (1965): 57–59. DOI.
• Steinbauer, Roland. “The Singularity Theorems of General Relativity and Their Low Regularity Extensions.” Jahresbericht der Deutschen Mathematiker-Vereinigung 125, no. 2 (2023): 73–119. DOI. Open preprint: arXiv:2206.05939v2.