Spacelike or timelike

Problem

In the laboratory SR, four events have coordinates (in seconds and light-seconds) $E_1=(0,0)\quad E_2=(5,3)\quad E_3=(2,7)\quad E_4=(4,4).$ For each of the pairs $\{E_1,E_2\}$, $\{E_1,E_3\}$, $\{E_1,E_4\}$, $\{E_2,E_3\}$: (a) classify the interval; (b) compute $\Delta\tau$ or $\Delta\sigma$; (c) find the velocity of the SR of co-spatiality (for timelike intervals) or of simultaneity (for spacelike ones).

Solution

We work in natural units with $c=1$ (times in s, distances in light-seconds). For two events with separation $\Delta t$ and $\Delta s$ the invariant is $\Delta t^2 - \Delta s^2 \;\begin{cases}>0 & \text{timelike}\\ =0 & \text{lightlike}\\ <0 & \text{spacelike}\end{cases}$ For a timelike interval the proper time is $\Delta\tau=\sqrt{\Delta t^2-\Delta s^2}$ and the SR in which the two events are co-spatial (same location) travels at $v_\text{cosp}=\Delta s/\Delta t$. For a spacelike interval the proper distance is $\Delta\sigma=\sqrt{\Delta s^2-\Delta t^2}$ and the SR in which the two events are simultaneous travels at $v_\text{sim}=\Delta t/\Delta s$.

Pair $\{E_1,E_2\}$: $\Delta t=5$, $\Delta s=3$. $\Delta t^2-\Delta s^2 = 25-9 = 16 > 0 \;\Rightarrow\; \text{timelike}$ $\Delta\tau_{12}=\sqrt{16}=4\ \text{s}, \qquad v_\text{cosp}=\frac{\Delta s}{\Delta t}=\frac{3}{5}=0.6\,c$

Pair $\{E_1,E_3\}$: $\Delta t=2$, $\Delta s=7$. $\Delta t^2-\Delta s^2 = 4-49 = -45 < 0 \;\Rightarrow\; \text{spacelike}$ $\Delta\sigma_{13}=\sqrt{45}\ \text{ls}\approx 6.7\ \text{ls}, \qquad v_\text{sim}=\frac{\Delta t}{\Delta s}=\frac{2}{7}\approx 0.29\,c$

Pair $\{E_1,E_4\}$: $\Delta t=4$, $\Delta s=4$. $\Delta t^2-\Delta s^2 = 16-16 = 0 \;\Rightarrow\; \text{lightlike}$ $\Delta\tau_{14}=0$ The two events can be connected by a light ray: they lie on the bisector of the light cone (the dashed line in the figure).

Pair $\{E_2,E_3\}$: $\Delta t=2-5=-3$, $\Delta s=7-3=4$. $\Delta s^2-\Delta t^2 = 16-9 = 7 > 0 \;\Rightarrow\; \text{spacelike}$ $\Delta\sigma_{23}=\sqrt{7}\ \text{ls}\approx 2.65\ \text{ls}, \qquad v_\text{sim}=\frac{|\Delta t|}{|\Delta s|}=\frac{3}{4}=0.75\,c$

Summary. $\ev{\{12\}\ \text{timelike},\ \Delta\tau=4\,\text{s};\quad \{13\}\ \text{spacelike},\ \Delta\sigma=\sqrt{45}\,\text{ls};\quad \{14\}\ \text{lightlike},\ \Delta\tau=0;\quad \{23\}\ \text{spacelike},\ \Delta\sigma=\sqrt{7}\,\text{ls}}$

Links

Topics: Relatività ristretta Concepts: Invariante spazio-temporale · Relatività della simultaneità Skills: Costruzione di diagrammi di Minkowski · Cambio di sistema di riferimento Methods: Diagramma di Minkowski · Unità naturali