Penrose diagram

A Penrose diagram (also called a Penrose-Carter or conformal diagram) is a two-dimensional picture of the global causal structure of a spacetime, obtained by a conformal rescaling of the metric that drags coordinate infinities into a finite boundary. Distances on the diagram are heavily distorted, but the conformal map preserves angles, so light cones remain at a fixed 45-degree tilt everywhere. Because causality in relativity is entirely a matter of light cones, the diagram represents what can causally influence what with perfect fidelity even though it is useless for reading off lengths.

The standard construction for flat (Minkowski) space passes to null coordinates u = t - r and v = t + r, applies the arctangent to squeeze each from the infinite real line into the interval (-pi/2, pi/2), and plots the rescaled coordinates. The result is a finite triangle (a half-diamond) whose boundary consists of five distinct conformal infinities: future and past timelike infinity i+ and i- (where massive worldlines end and begin), spatial infinity i0 (where constant-time slices terminate), and future and past null infinity, written script-I-plus and script-I-minus and pronounced 'scri', where outgoing and ingoing light rays end and begin.

Applied to the maximally-extended Schwarzschild geometry, the Penrose diagram exposes structure invisible in the original coordinates: four regions comprising two asymptotically-flat exterior universes, a black-hole interior, and a white-hole interior, with the event horizons appearing as the two 45-degree lines crossing at a central bifurcation point. Most strikingly, the curvature singularity at r = 0 appears as a horizontal (spacelike) line capping the interior, demonstrating that the singularity is a moment in the infaller's future rather than a place in space. Penrose diagrams are the standard working language for global questions in general relativity, including the singularity theorems, asymptotic flatness, the definition of radiated mass and energy at null infinity, and the black-hole information paradox.

Roger Penrose introduced conformal compactification of spacetime in his 1963 paper 'Asymptotic Properties of Fields and Space-Times' and developed it through the mid-1960s; Brandon Carter applied related techniques to black-hole spacetimes, which is why the figures are sometimes called Penrose-Carter diagrams. The technique built on the 1960 Kruskal-Szekeres coordinates that first showed the Schwarzschild horizon to be a coordinate artifact rather than a physical edge, and it became the natural setting for Penrose's 1965 singularity theorem.