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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 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.^[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 $R_{ab}{k}^a{k}^b\ge 0$ 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.^[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.^[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.

In an antisymmetric vibration of a symmetric linear triatomic molecule, the outer atoms move oppositely and the central atom does not couple to that mode.

For $g_{\mu \nu }={\phi }^2{\eta }_{\mu \nu }$,For a null curve, the last term drops from the geodesic equation; the remaining acceleration is parallel to the tangent and can be removed by reparametrization.

A metric is conformally flat when it is locally a positive scalar multiple of a flat metric. Multiplication by the conformal factor preserves null cones and unparametrized null geodesics.

A conformally flat metric is locally a positive scalar multiple of a flat metric. It makes its domain a conformally flat manifold.

Nonlinear Schrödinger blowup analysis studies whether a solution of the Focusing nonlinear Schrodinger equation remains bounded in its natural Sobolev space or develops an unbounded gradient norm in finite time.

The Emden-Fowler transformation is a change of variables for scaling-invariant radial equations, including the Lane-Emden equation. The Emden-Fowler transformation replaces a radial variable $r$ by logarithmic time $s=\log r$ and rescales the dependent variable by a power of $r$. It converts many scale-invariant radial equations into autonomous ordinary differential equations.

A proper two-sided ideal $P$ of a noncommutative ring is prime when $AB\subseteq P$ for two-sided ideals $A,B$ implies $A\subseteq P$ or $B\subseteq P$.

Ore localization extends localization to suitable multiplicative subsets of noncommutative rings.

An injective module has the extension property for homomorphisms defined on submodules.

The Jacobson radical is the intersection of all maximal left ideals, equivalently the intersection of the annihilators of all simple left modules.

Morita-equivalent rings have equivalent module categories. In particular, $R$ and ${\mathrm{Mat}}_n(R)$ are Morita equivalent.

A left primitive ring has a faithful simple left module.

The first Weyl algebra is $A_1(k)=k⟨x,y⟩/(yx-xy-1)$, the algebra of polynomial-coefficient differential operators in one variable.

A ring is right Noetherian when every ascending chain of right ideals stabilizes.

A ring is left Noetherian when every ascending chain of left ideals stabilizes, equivalently when every left ideal is finitely generated.

Suspension rheology relates particle concentration, particle pressure, shear stress, deformation rate, and fluid migration in a mixture of particles and a liquid.

Viscoelasticity combines viscous dissipation with elastic storage and relaxation of deformation.

A power-law fluid has shear stress $\tau =K|\dot{\gamma }{|}^{n-1}\dot{\gamma }$. It is shear thinning for $n<1$, Newtonian for $n=1$, and shear thickening for $n>1$.

A yield-stress fluid behaves as a rigid material below a critical shear stress and flows once that stress is exceeded.

A local field is a nondiscrete locally compact topological field. A non-Archimedean local field is a finite extension of ${\mathbb{Q}}_p$ or of ${\mathbb{F}}_p((t))$.