The Minkowski plane and the light cone

The diagram with which we classified intervals is a special case of the Minkowski plane (Hermann Minkowski, 1908): the canonical geometric tool for thinking about special relativity. The rule for constructing it is extremely simple, and it turns subtle physical questions into matters of elementary geometry.

Convention of the Minkowski plane

Each event is represented as a point on a Cartesian plane in which:

• the horizontal axis is time $t$ (or $ct$ if natural units are not used);
• the vertical axis is a spatial coordinate $s$;
• light beams, with unit slope ($|\Delta s/\Delta t| = 1$), form two lines at $45^\circ$ emanating from the event.

With this convention every inertial frame has its own pair of $45^\circ$ light lines (Einstein’s postulate), and the light cone is an invariant property of space-time.

Many textbooks show the scheme with $t$ on the vertical axis and $s$ on the horizontal axis, with the light arrows still at $45^\circ$: it is entirely equivalent, you just rotate the page. Here we keep $t$ on the horizontal axis for consistency with the motion-time graphs from earlier years.

World line

The trajectory of a particle in the Minkowski plane is called a world line. For an inertial observer at rest in the laboratory RF, the coordinate $s$ remains constant: their world line is a vertical straight line, parallel to the $s=0$ axis. For an observer moving at velocity $v$, instead, $s = v\cdot t$ holds: their world line is tilted relative to the vertical. The faster the observer, the more the world line “lies down” towards the light cone; but never beyond $45^\circ$, because $|v|<c=1$. The light cone is therefore an impassable geometric limit for every material world line.

Past, future, elsewhere

The light cone through an event $O$ divides the Minkowski plane into three qualitatively different regions.

The light cone divides the Minkowski plane into future, past and elsewhere. Material world lines (observer at rest and in motion) always remain inside the cone.

• Future of $O$: the region to the right, enclosed within the light cone towards increasing values of $t$. All the events inside this region are timelike relative to $O$, and $O$ can influence them with a physical signal, a material particle, or a light beam. The temporal order is unambiguous: $O$ precedes all of them.
• Past of $O$: the region to the left. These are events timelike relative to $O$, but which may have influenced it: $O$ comes after them, in any RF.
• Elsewhere: the two regions above and below, outside the light cone. All the events inside these regions are spacelike relative to $O$. No physical signal connects them; the temporal order depends on the RF, and there is no causality.

Summary

Light cone: future / past / elsewhere. Timelike $\Leftrightarrow$ inside the cone. Spacelike $\Leftrightarrow$ outside the cone. Lightlike $\Leftrightarrow$ on the cone.

Historical note — Minkowski

Minkowski had been Einstein’s mathematics teacher in Zurich, and in 1908 he proposed a geometric reformulation of special relativity in a four-dimensional space-time. His famous phrase, in the Cologne lecture of 21 September 1908, is: “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality” (Kumar 2010, p. 104; see Riferimenti bibliografici). Einstein, who already had all the physical results, initially received the reformulation with scepticism (“superfluous mathematical erudition”, he said), but changed his mind when, ten years later, he needed it to build general relativity: there, the Minkowski plane becomes the “flat” special case of a curved space-time, and it becomes indispensable.

Links

Topics: Special relativity Concepts: Space-time invariant Skills: Constructing Minkowski diagrams Methods: Minkowski diagram Objects: Light clock

Related exercises: Spacelike or timelike · Problem — Particle or photon · Mr Rossi goes to the theatre