the singularity theorems
For a long time it was hoped that the singularities in solutions of Einstein's equations — the infinitely dense center of a black hole, the Big Bang at the start of the universe — were artifacts of assuming perfect symmetry, and would smear out in any realistic, lumpy spacetime. The singularity theorems of Penrose and Hawking (1965 onward) shattered that hope. They prove, with no symmetry assumptions at all, that under broad and physically reasonable conditions a spacetime must be geodesically incomplete: some freely falling observer or light ray runs out of spacetime in finite proper time. Singularities are generic, not artifacts.
The theorems are theorems of Lorentzian geometry, and their shape is always the same three-ingredient recipe. (1) An energy condition: matter gravitates attractively, formalized as a curvature inequality like Ric(v, v) >= 0 for all timelike v (the strong/null energy condition via the Einstein equations). This forces nearby geodesics to focus — the Raychaudhuri equation says the expansion of a bundle of geodesics must decrease and reach -infinity (a focal/conjugate point) within bounded proper time, the Lorentzian echo of Bonnet-Myers. (2) A causality condition (e.g. no closed timelike curves) so that focusing actually produces a contradiction rather than a recurrence. (3) A boundary or initial condition capturing 'gravity has become strong somewhere': a closed trapped surface (a 2-surface whose outgoing AND ingoing light both converge — the geometric signature of being inside a black hole's grip), or a moment of universal expansion for the cosmological version. The conclusion: a causal geodesic of finite affine length that cannot be extended. Penrose's 1965 theorem used a trapped surface to predict black-hole singularities; Hawking's reversed time to predict the Big Bang.
Why they matter: they show classical general relativity predicts its own breakdown — singularities are unavoidable consequences of the theory plus mild physics, which is the strongest argument that a quantum theory of gravity is needed. The honest cautions are important and often glossed over. First, 'singularity' here means geodesic incompleteness, NOT that any curvature scalar blows up; the theorems prove a geodesic stops, not that 'curvature equals infinity'. There exist incomplete spacetimes with bounded curvature, and the theorems are silent about what the singularity 'looks like'. Second, they are existence/incompleteness theorems — they do not describe the nature, location, or strength of the singularity, and they say nothing about cosmic censorship (whether singularities are hidden behind horizons), which remains a major open conjecture. Third, the energy conditions, while physically motivated, are genuine hypotheses that some matter models (and quantum effects, dark energy) can violate, so the theorems constrain but do not forbid singularity-free spacetimes.
Penrose's theorem: if a spacetime satisfies the null energy condition, is globally hyperbolic with a noncompact Cauchy surface, and contains a closed trapped surface, then it has an incomplete null geodesic. A collapsing star that has shrunk inside its Schwarzschild radius forms a trapped surface, so the theorem forces a singularity — no perfect symmetry needed, the collapse cannot avoid it.
A trapped surface in collapsing matter forces an incomplete geodesic: a singularity with no symmetry assumed.
'Singularity' in these theorems means geodesic incompleteness, not that a curvature scalar diverges; the theorems prove a geodesic cannot be extended, say nothing about the singularity's strength or whether it is hidden (cosmic censorship is a separate open problem), and depend on energy conditions that exotic matter can violate.