Relativistic western duel

Problem

In the desert, the Ugly (at $-2000$~ls) and the Bad (at $+2000$~ls) shoot at the Good (in the middle). For the spectator (SRS), the two shots leave at the same instant $t=0$. What type of interval is there between the two shots? A vulture flies over the scene at $v=0{,}8\,c$ to the left: who fires first in its frame? Argue without explicit calculations, using the invariant and the relativity of simultaneity.

Solution

Natural units $c=1$. Call $E_B$ the Ugly’s shot and $E_C$ the Bad’s shot. In SRS both occur at $t=0$: $E_B=(0,-2000)$, $E_C=(0,+2000)$.

Type of interval. Between the two shots $\Delta t = 0$ and $\Delta s = 4000$~ls: $\Delta t^2-\Delta s^2 = 0 - 4000^2 < 0 \;\Rightarrow\; \ev{\text{spacelike interval}}$ Neither shot can cause the other (to do so a signal would have to travel faster than light). For a spacelike interval, the temporal order is not absolute: it depends on the frame.

Who fires first for the vulture. Since the interval is spacelike, the relativity of simultaneity tells us that two observers in relative motion can see the shots in a different order; the spectator’s order ($t=0$ for both, simultaneous) is only a special case. To establish the direction, we need the qualitative rule of Lorentz transformations: when moving to a frame that travels in a given direction, the event that is further ahead in the direction of motion happens earlier than the one behind it.

The vulture moves to the left (toward decreasing $s$), i.e. in the direction where the Ugly is ($-2000$~ls). So, in the vulture’s frame, the shot that is “ahead” in the direction of motion — the Ugly’s — happens first, and the Bad’s after. $\ev{\text{for the vulture, the Ugly fires first (the event in the direction of motion)}}$ This does not contradict causality: since the interval is spacelike, the two shots have no cause-effect relationship, and their order is legitimately relative to the frame.

Links

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