Timelike, spacelike, lightlike

Given the invariant $(\Delta\tau)^2 = (\Delta t)^2 - (\Delta s)^2$ (in natural units, with $c=1$), it is the sign of this quantity that divides all pairs of events into three qualitatively distinct classes. The classification is intrinsic: it does not depend on the reference frame, because $\Delta\tau$ is by construction the same in every inertial frame. And it is crucial, because it establishes whether two events can “talk to each other” — that is, exchange a physical signal — and in what order.

The underlying idea is simple. The ratio $|\Delta s/\Delta t|$ is the “velocity” that would be needed to join the two events, starting from the first and arriving at the second. If this velocity is less than $c$, a material object or a signal can do it; if it is greater than $c$, no physical signal can; if it is exactly $c$, only light can.

Principle — Three types of interval

Given two events $E_1=(t_1,s_1)$ and $E_2=(t_2,s_2)$, with $\Delta t = t_2-t_1$ and $\Delta s = s_2-s_1$, the pair $\{E_1,E_2\}$ is:

• timelike if $(\Delta t)^2 > (\Delta s)^2$, i.e. $|\Delta s/\Delta t| < 1$ (a signal slower than $c$ can connect them). The invariant $\Delta\tau$ is real and positive.
• spacelike if $(\Delta t)^2 < (\Delta s)^2$, i.e. $|\Delta s/\Delta t| > 1$ (a signal faster than $c$ would be needed). The invariant is imaginary; instead we work with the proper distance $\Delta\sigma\equiv\sqrt{(\Delta s)^2-(\Delta t)^2}$ (real and positive).
• lightlike if $(\Delta t)^2 = (\Delta s)^2$, i.e. $|\Delta s/\Delta t|=1$. Only a light beam can connect them, and $\Delta\tau = 0$.

The lightlike case deserves attention: two events connected by a light beam have a zero proper interval. Along a light beam “no proper time flows” — this is the geometric translation of the fact that light travels exactly on the boundary between the possible and the impossible.

Graphical convention in the Minkowski plane

To represent this classification visually we adopt a graphical convention in the $(t,s)$ plane, with $t$ on the horizontal axis and $s$ on the vertical axis:

• two events in a timelike relation are joined by a solid arrow, oriented from the event that precedes to the one that follows (the order is absolute, the same in all RFs);
• two events in a spacelike relation are joined by a dashed line, with no arrow (the temporal order depends on the RF, so there is no absolute “before”);
• two events in a lightlike relation are joined by a line at $45^\circ$, parallel to the light cone.

Light, with $c=1$, always travels at $45^\circ$: the light cone is the set of such lines emanating from the reference event.

The three interval classes relative to event $O$: $A$ is timelike (solid arrow, inside the cone), $B$ is spacelike (dashed, outside the cone), $L$ is lightlike (on the cone, at $45^\circ$).

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