The relativity of simultaneity

One of the most counter-intuitive results of special relativity is that “at the same instant” is not an absolute notion. Two events that occur simultaneously for one observer may occur at different times — and even in reversed order — for another observer moving relative to the first.

Principle — The relativity of simultaneity

Two events that occur at the same instant in one reference frame are not in general simultaneous in another frame. The temporal order between two spatially separated events depends on the reference frame, unless the two events can be connected by a light signal or a slower one (time-type interval).

An effective way to feel its force is the “relativistic western duel”: in the desert the Ugly (on the left) and the Bad (on the right) shoot at the Good, who stands in the middle. For the spectator on the ground the two shots leave at the same instant. The two shots are spatially separated but simultaneous for the spectator, so $\Delta t=0$ and $\Delta s\neq 0$: their interval is space-type. It follows that the order of the two shots is not absolute. A vulture flying over the scene in motion (say, leftward at $0.8\,c$) will attribute different times to the two events: in its frame one of the two gunmen fires before the other. The conclusion is drawn without any calculation, relying only on the space-type nature of the interval and on the relativity of simultaneity.

That this generates no causal paradoxes is guaranteed by the structure of the light cone. The ambiguity of order concerns only space-type pairs, which by definition cannot exchange any physical signal: whether the Ugly fires “before” or “after” the Bad is irrelevant, because neither shot can influence the other. By contrast, a time-type interval preserves causal order: if $E_1$ can be the cause of $E_2$, it is so in every frame, and no observer will ever see the effect precede the cause. The relativity of simultaneity bends “before” and “after” exactly where it is harmless to do so, and leaves them rigid wherever causality is at stake.

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

Related exercises: Space-type or time-type · Relativistic western duel · No signals faster than light