Spacetime diagram — the series · Illustrated Physics

A spacetime diagram at β = 0.5. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.5 of the speed of light.

Two axes, one speed, and a diagram that does the arguing

Put position across and time up, insist that light travels at forty-five degrees for everyone, and nearly every result in special relativity becomes something to read off rather than derive.

part 1 · relativity

What puts a scale on a tilted axis. A spacetime diagram with a second observer's axes at β = 0.6. The curves are the sets of events at a fixed interval from the origin — c²t² − x² = s², one branch each for s = 0.5, s = 1, s = 1.5 — and the whole point of them is where they cross. A unit of the moving observer's time is wherever the s = 1 curve meets the tilted time axis, and on the page that point is 1.250 times as far from the origin as the stationary observer's own unit. Without the hyperbolae the tilted axes carry no scale at all, and every argument about which of two clocks is behind is unreadable off the diagram. Each drawn crossing reads back as its own interval to 0.0e+0.

The quantity nobody argues about

Relativity takes away the length of a rod and the duration of an event and hands back exactly one thing in their place. Its hyperbolae are what put a scale on the tilted axes of a spacetime diagram — without which the diagram is a picture with no units on it.

part 2 · relativity

Where the second tick actually goes. A spacetime diagram with a second observer's axes at β = 0.6. The hyperbolae are the sets of events one second and one metre from the origin — invariantly, by the interval — and every observer's unit tick is where their own axis crosses them. That is checked here rather than drawn by eye. The moving observer's one-second mark sits 1.458 times further from the origin on the page than the stationary one's, so a ruler laid on this picture reads the two frames on different scales. The picture is not distorted; the page is Euclidean and spacetime is not. The Lorentz factor here is 1.2500.

The diagram a ruler cannot read

A spacetime diagram is drawn on flat paper, and the geometry it depicts is not flat. The tick marking one second on a moving observer's axis sits further from the origin than the stationary observer's, by an amount that is not the Lorentz factor and means nothing at all.

part 3 · relativity

The transformation that never mentions light

Assume space and time are homogeneous, that space is isotropic, that two changes of frame compose into a third, and that the relativity principle holds. Those four leave exactly one free constant — and three possible worlds, one of them Galileo's and one of them Einstein's. Light appears nowhere in the derivation; it enters only when the constant has to be measured.

part 4 · relativity

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.

part 5 · relativity

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.

part 6 · relativity