Nash Frank

2022-07-16

Three events $A,B,C$ are seen by observer $O$ to occur in the order $ABC$. Another observer $O\prime$ sees the events to occur in the order $CBA$. Is it possible that a third observer sees the events in the order $ACB$?
I could draw spacetime diagram for three arbitrary-ordered events for $O$ and $O\prime$. But I couldn't draw spacetime diagram for third observer.

Ragazzonibw

Expert

One can design a situation that obeys your conditions – both ABC and CBA ordering may be realized in some inertial systems – but the ordering ACB is impossible.
Simply imagine that A, B, C are three events in spacetime that lie on a spacelike line in the spacetime. Then B is inevitably in the middle between A, C, chronologically and spatially, regardless of the frame, so only ABC and CBA orderings are possible.
However, when A,B,C do not belong to the same line, and we add this proposition as an extra assumption, then one may show that your additional assumptions are enough for any ordering to be realized in some inertial systems.
When both ABC and CBA ordering is possible, it proves that A-B have to be spatially separated (because the ordering of A,B appears in both ways, depending on the frame), B-C are spatially separated (the same reason), A-C are spatially separated (the same).
With the extra assumption that A, B, C don't lie at the same line in the spacetime, we see that ABC is a spacelike triangle in the spacetime. You may therefore go to the inertial system in which all events A, B, C occur simultaneously. Now, when you boost the system in directions along the vertices of the triangle, or the opposite direction, you may get all 3!=6 orderings.
You probably failed to draw a spacetime diagram because one needs at least 2 spatial dimensions plus 1 time to realize all the situations: it's not a problem that may be fully solved and visualized in the 1+1-dimensional spacetime.

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