Singularity theorems

The singularity theorems arose from attempts to understand the inevitable collapse of massive stars and the initial state of the universe. In the 1930s, work by Subrahmanyan Chandrasekhar and J. Robert Oppenheimer suggested that sufficiently massive stars must collapse to a black hole, but the final state remained unclear. General relativity predicts that under extreme conditions, the curvature of spacetime becomes infinite, but whether this actually occurs in physical situations was debated. The theorems resolved this by showing that, under reasonable energy conditions, singularities are unavoidable.

Penrose's 1965 theorem applies to a spacetime containing a trapped surface — a closed surface from which light rays converge inward — and assumes the null energy condition and global hyperbolicity. It concludes that null geodesic incompleteness must occur, implying a singularity. Hawking's 1966 theorem applied similar reasoning to the entire universe, showing that if the universe is expanding and contains enough matter, a past singularity (the Big Bang) is inevitable. The 1970 Hawking–Penrose theorem generalized these results, relaxing assumptions and covering both black holes and cosmological singularities.

The theorems imply that classical general relativity breaks down at singularities, where the laws of physics cease to be predictive. This motivated the search for a quantum theory of gravity, such as string theory or loop quantum gravity, which might resolve singularities. However, the theorems rely on energy conditions that can be violated by quantum effects, such as the Casimir effect, so their conclusions may not hold in the quantum regime. The cosmic censorship conjecture, proposed by Penrose, suggests that singularities are always hidden behind event horizons, but this remains unproven.

While Penrose and Hawking are the most famous contributors, earlier work by Amal Kumar Raychaudhuri and Lev Landau laid crucial groundwork. The Raychaudhuri equation, which describes the focusing of geodesics, is central to the proofs. Also, the theorems apply to spacetimes with positive cosmological constant, but the 2003 work by Senovilla showed that a singularity can be avoided in certain cyclic cosmologies. The theorems also have implications for the chronology protection conjecture, which forbids time travel, as closed timelike curves often lead to singularities.

Geodesic incompleteness

A property of a spacetime where some geodesic (path of a free-falling particle or light ray) cannot be extended indefinitely in at least one direction, indicating a singularity.

Trapped surface

A closed, spacelike two-surface such that both outgoing and ingoing light rays converge, indicating a region of no escape, typical of black holes.

Energy conditions

Assumptions about the stress-energy tensor that ensure gravity is attractive and energy density is non-negative, used in singularity theorems.

Global hyperbolicity

A property of a spacetime that ensures causality is well-behaved and that the initial value problem is well-posed.