Illustrated Physics
physics, drawn
Concept

Light cone coordinates — where it appears

The pair u = t − x and v = t + x, in units where light has speed one, whose lines of constant value are light rays. An event lies in another's future exactly when it is later in both coordinates, so the causal order becomes two separate orderings.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A map that keeps every cone and bends every worldline. Left: a grid of inertial worldlines, drawn solid, lines of simultaneity, faint, and light lines at 45°, dashed, in one space dimension. Right: the same grid after a map that stretches the light-cone coordinates u = t − x and v = t + x by two different increasing functions, u + 0.45·tanh(1.5u) and sinh(0.45v)/0.45. Light lines go to light lines, still at 45° to within a part in a billion, and the causal order of every one of 4000 sampled pairs of events is unchanged: whatever could influence what still can, and nothing new can. But the straight worldlines are bent — the one through x = 1 by 0.19 across the window — and the lines of simultaneity are no longer straight. In one space dimension the light cones cannot tell this picture from the inertial one.

What the light cones alone can decide

Keep nothing of spacetime but its light cones — which events could influence which — and ask how much geometry survives. With one dimension of space, almost none: any pair of increasing stretches of the two families of light lines preserves every cone and bends every straight worldline. With two or more, almost all of it: the only maps that keep every cone are Lorentz transformations, shifts and a uniform stretch, and nothing about straightness has to be assumed.

relativity · Spacetime diagram All of flat spacetime in a diamond. The whole of flat spacetime with one space dimension, squeezed into a finite diamond by applying arctan separately to the two light-cone coordinates u = t − x and v = t + x. Solid curves are the worldlines of observers at rest at x = −3, −2, −1, 0, 1, 2, 3; faint curves are the instants t equal to the same values; the dashed lines are the two light rays through the origin, still at 45°, as every light line is — checked to a part in a billion — and the causal order of 4000 sampled pairs is unchanged. Infinity is not one place. Every worldline at rest runs from the bottom corner i⁻ to the top corner i⁺; every instant runs between the side corners i⁰; and light rays begin on the lower edges ℐ⁻ and end on the upper edges ℐ⁺. Each of these limits is checked at ten million units out.

The five places infinity turns out to be

Flat spacetime goes on for ever in every direction, and it can still be drawn whole on a page. Squeeze each family of light rays with a function that keeps their order and the infinite plane becomes a diamond with every light cone still at 45°. The price is distance, which the picture no longer shows. What it shows instead is that infinity is not one place: observers slower than light all end at a single point, instants end at another, and light ends along a whole edge of its own.

relativity · Spacetime diagram

Named alongside it

The objects these essays reach for when they reach for this one.

Causal orderCausalityLight coneCompactificationConformal symmetryDilationInvariantThe Lorentz transformationMinkowski spaceNull infinityPenrose diagramPostulate

All concepts