Sequentiality Restrictions in Special Relativity

TL;DR

This paper investigates the possible orderings of spacelike separated events in special relativity, revealing restrictions and possibilities for different numbers of events and observers in various spacetime dimensions.

Contribution

It characterizes which event orderings are realizable in different spacetime dimensions and observer configurations, extending understanding of relativistic event ordering constraints.

Findings

In 1+1 dimensions, certain event orderings are disallowed.

In 3+1 dimensions, all permutations of four events can be observed by four different observers.

For five events and observers, nearly all permutations are realizable, with only one impossible case identified.

Abstract

Observers in different inertial frames can see a set of spacelike separated events as occurring in different orders. Various restrictions are studied on the possible orderings of events that can be observed. In 1+1-dimensional spacetime {(1 2 3), (2 3 1), (3 1 2)} is a disallowed set of permutations. In 3+1-dimensional spacetime, any four different permutations on the ordering of n events can be seen by four different observers, and there is a set of four events such that any of the 4!=24 possible orderings can be observed in some inertial reference frame. A more complicated problem is that of five observers and five events, where of the 7,940,751 choices of five distinct elements from S_5 (containing the identity), all but at most one set of permutations can be realized, and it is shown that this remaining case is impossible. For six events and five observers, it is shown that there are at least 7970 cases that are unrealizable, of which at least 294 do not come from the forbidden configuration of five events.