= Penrose singularity theorem
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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 https://doi.org/10.1103/PhysRevLett.14.57[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>.<Gravitational Collapse and Space-Time Singularities by Roger Penrose>[pp. 57–58, motivation and trapping]{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 $R_{ab}k^ak^b\geq0$ 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.<The 1965 Penrose singularity theorem by José M. Martín Senovilla and David Garfinkle>[Theorem 2, printed p. 10; convergence condition (6)]{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.<The 1965 Penrose singularity theorem by José M. Martín Senovilla and David Garfinkle>[section 4.1, printed pp. 11–12; section 6.1, cosmic censorship]{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>.
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