Penrose Diagrams Explained: All of Spacetime on One Page

A Penrose diagram is a map of spacetime that fits infinite distances and infinite times inside a finite shape, usually a diamond or a triangle, while keeping the paths of light at 45°. Our Three Radii of a Black Hole poster maps the geometry around a non-spinning black hole, and the Relativity & Cosmos collection gathers the rest of our spacetime designs. A Penrose diagram is the tool physicists reach for when they talk about black holes; for what happens to a black hole over time, see our explainer Do Black Holes Evaporate?.

Start with an ordinary spacetime diagram

In a normal spacetime diagram, time runs up the page and one direction of space runs across. If time is measured as ct (in metres), a flash of light travels along lines at 45°. Everything slower than light, like you or a planet, moves on a steeper line. The trouble is size: the diagram goes on forever in every direction, so you can never draw the far future or the far edge of space.

Penrose solved this with a conformal transformation. The Nobel Committee for Physics explains that such transformations can change the scale but always keep angles, so events infinitely far away can be brought in to fit inside a diagram of finite size. A light ray drawn at 45° stays at 45° (Nobel Committee, 2020 scientific background^[1]). Penrose introduced the method in 1963 (Penrose, 1963^[2]).

The two steps that shrink infinity

Sean Carroll’s widely used general relativity notes walk through the construction for flat, empty spacetime (Carroll, Lecture Notes on General Relativity^[3]). He uses units where c = 1, so time t and radius r are both measured in metres. First, switch to “null” coordinates that follow incoming and outgoing light rays:

u = (t + r)/2, v = (t − r)/2

Each of u and v still runs from minus infinity to plus infinity. Second, squeeze each one with the arctangent function, which turns the whole number line into a finite interval:

U = arctan u, V = arctan v

Now U and V only run from −π/2 to +π/2. Infinity has become an edge you can draw (Carroll, §7^[4]). Because the squeeze acts separately along the two light-ray directions, lines of constant u or v stay as 45° lines.

Worked example

Take an event at t = 0 and r = 1 metre. Then u = 0.5 and v = −0.5, so U = arctan(0.5) ≈ 0.464 and V ≈ −0.464. Move it to r = 1,000 m and U ≈ arctan(500) ≈ 1.569, already very close to the edge value π/2 ≈ 1.571. Distances far away get packed tightly near the boundary, which is exactly how all of space fits on the page.

How to read a Penrose diagram

For empty, flat spacetime the result is a diamond (shown as a triangle when only positive r is drawn). Carroll names the parts of its boundary, which together are called conformal infinity (Carroll, §7^[4]):

• i⁺, future timelike infinity: the top corner, where every object moving slower than light ends up.
• i⁻, past timelike infinity: the bottom corner, where those paths begin.
• i⁰, spatial infinity: the side corner, infinitely far away in space.
• ℐ⁺ and ℐ⁻, future and past null infinity: the 45° edges, pronounced “scri-plus” and “scri-minus”, where light rays end and begin.

Carroll sums up the pattern: all timelike geodesics begin at i⁻ and end at i⁺, all null geodesics begin at ℐ⁻ and end at ℐ⁺, and all spacelike geodesics begin and end at i⁰. He also notes that drawing flat spacetime this way teaches us little new. The diagrams earn their keep with black holes. Each point in the diagram stands for a whole sphere of space, because the angles around the centre are not drawn.

Penrose diagrams and black holes

In 1965 Penrose published a short paper, “Gravitational Collapse and Space-Time Singularities” (Penrose, 1965^[5]). Its key idea was the trapped surface, a closed surface where all light rays heading outward still converge. The Nobel background explains that, within general relativity and for matter with positive energy density, once a trapped surface forms the collapse towards a singularity cannot be stopped, and shows the collapse of a star as a Penrose diagram (Nobel Committee, 2020 scientific background^[1]). Penrose received half of the 2020 Nobel Prize in Physics “for the discovery that black hole formation is a robust prediction of the general theory of relativity” (NobelPrize.org, Physics 2020^[6]). The committee’s popular summary notes that not even Einstein thought black holes could actually exist, and that ten years after Einstein’s death Penrose showed they can form (NobelPrize.org, 2020 popular background^[7]).

In a Penrose diagram of a black hole, the horizon is a 45° line. Anything inside can only move up the page toward the singularity at the top of the diagram. What a singularity really is remains an open question: the Stanford Encyclopedia reports a consensus that no entirely satisfactory, strict definition has yet been formulated (Curiel, SEP “Singularities and Black Holes”^[8]).

Why the light cone matters here

Every point in a Penrose diagram has its own light cone, and because the angles are kept, the cones are still drawn at 45°. Our Null Beacon tee plays with the same causal structure on cosmic scales: in an accelerating universe, light that some distant galaxies emit today will never reach us. To see how Stephen Hawking used these diagrams, read What Did Stephen Hawking Discover?, and for the puzzle they helped sharpen, the black hole information paradox.

Quick FAQ

Is a Penrose diagram to scale?

No. It keeps angles, not distances. Far-away regions are squeezed more and more toward the edges.

Why are light rays at 45°?

Because time is drawn as ct, light covers one unit of space per unit of time, and the conformal squeeze keeps that angle.

Can I draw a Penrose diagram of any spacetime?

Diagrams have been drawn for many important spacetimes, including black holes, but each one needs its own construction, and the full four-dimensional picture is reduced to two dimensions.

References

1. Nobel Committee for Physics (2020), Scientific background: “Theoretical foundation for black holes and the supermassive compact object at the galactic centre”. https://www.nobelprize.org/uploads/2020/10/advanced-physicsprize2020.pdf
2. R. Penrose (1963), “Asymptotic Properties of Fields and Space-Times,” Phys. Rev. Lett. 10, 66–68 (DOI 10.1103/PhysRevLett.10.66; metadata verified via Crossref). https://doi.org/10.1103/PhysRevLett.10.66
3. S. M. Carroll (1997), “Lecture Notes on General Relativity,” arXiv:gr-qc/9712019. https://arxiv.org/abs/gr-qc/9712019
4. S. M. Carroll (1997), “Lecture Notes on General Relativity,” arXiv:gr-qc/9712019 (full text; conformal diagram of Minkowski space and light-cone terminology). https://arxiv.org/pdf/gr-qc/9712019
5. R. Penrose (1965), “Gravitational Collapse and Space-Time Singularities,” Phys. Rev. Lett. 14, 57–59 (DOI 10.1103/PhysRevLett.14.57; metadata verified via Crossref). https://doi.org/10.1103/PhysRevLett.14.57
6. NobelPrize.org, “The Nobel Prize in Physics 2020” (R. Penrose; R. Genzel & A. Ghez). https://www.nobelprize.org/prizes/physics/2020/summary/
7. NobelPrize.org, Popular science background: “Black holes and the Milky Way’s darkest secret” (2020). https://www.nobelprize.org/prizes/physics/2020/popular-information/
8. E. Curiel, “Singularities and Black Holes,” Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/spacetime-singularities/

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.