What diagram compactifies spacetime to show its causal structure?

A Penrose diagram compactifies spacetime to show its causal structure.

Named after mathematical physicist Roger Penrose, it uses a conformal transformation to represent an infinite spacetime inside a finite diagram. The transformation changes distances and angles in a controlled way but preserves the paths available to light. As a result, light rays are conventionally drawn at 45 degrees, making cause-and-effect relationships easy to inspect.

The vertical direction generally represents time and the horizontal direction represents one spatial dimension. Boundaries can represent different kinds of infinity, such as future or past timelike infinity, spatial infinity and null infinity. In spherically symmetric models, each point can stand for an entire two-dimensional sphere, so the drawing is a compressed representation rather than a literal map.

Penrose diagrams are especially useful for black holes. They can show an event horizon, the one-way nature of a black-hole interior and the spacelike character of a Schwarzschild singularity. The technique was introduced in the 1960s and is also called a conformal diagram or, more fully, a Penrose–Carter diagram. A Minkowski diagram is related but usually describes flat spacetime in special relativity; a Penrose diagram is designed to expose global causal structure, including curved spacetimes.