The physical meaning of the three classes

The distinction between timelike, spacelike and lightlike is not a mathematical curiosity: each class corresponds to a precise physical situation, identified by the existence (or otherwise) of a particular privileged inertial observer.

Timelike. There exists an inertial observer for whom the two events happen at the same point in space: this is the RF in which $\Delta s'=0$. In that RF, the time elapsed between $E_1$ and $E_2$ is precisely the invariant $\Delta\tau$, called the proper time. This is the observer who “travels along with” the two events, for example the very particle that passes from one to the other. The event that is “first” in time is the same for all observers: the order is absolute.

Spacelike. There exists an inertial observer for whom the two events happen at the same instant: this is the RF in which $\Delta t'=0$. In that RF, the spatial distance between $E_1$ and $E_2$ is precisely $\Delta\sigma$, called the proper length. Here the temporal order is not absolute: for some observers $E_1$ precedes $E_2$, for others the reverse is true, and for one particular observer the two are simultaneous. This is precisely why there can be no “causality” between spacelike events: no physical signal can connect them, and allowing a reversal of the causal order would be a logical contradiction.

Lightlike. The two events are connected only by a beam of light. There is no RF in which they are co-spatial, nor any RF in which they are simultaneous: the limiting case between the two previous situations, on the boundary of the cone.

Light as a boundary: past, future and “elsewhere”

Read in causal terms, the classification says that the light cone of an event $E$ divides space-time into three regions:

• the past of $E$: all events that could have caused it (timelike, $\Delta t<0$);
• the future of $E$: all events that $E$ can cause (timelike, $\Delta t>0$);
• the elsewhere of $E$: events separated from $E$ in a spacelike way. No causal relationship, temporal order dependent on the RF.

The impassable wall between “can influence” and “cannot influence” is exactly the light cone. Relativistic causality can be summed up in a single sentence: no information travels faster than light. Everything else — from the relativity of simultaneity to the composition of velocities — is a consequence of it.

Links

Topics: Special relativity Concepts: Space-time invariant · Relativity of simultaneity Skills: Changing reference frame Methods: Minkowski diagram

Related exercises: Spacelike or timelike · Relativistic western duel · Problem — No signals faster than light