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Poincaré group by Ciro Santilli 36 Updated +Created
In simple and concrete terms. Suppose you observe N particles following different trajectories in Spacetime.
There are two observers traveling at constant speed relative to each other, and so they see different trajectories for those particles:
Note that the first two types of transformation are exactly the non-relativistic Galilean transformations.
The Poincare group is the set of all matrices such that such a relationship like this exists between two frames of reference.