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.
- space and time shifts, because their space origin and time origin (time they consider 0, i.e. when they started their timers) are not synchronized. This can be modelled with a 4-vector addition.
- their space axes are rotated relative to one another. This can be modelled with a 4x4 matrix multiplication.
- and they are moving relative to each other, which leads to the usual spacetime interactions of special relativity. Also modelled with a 4x4 matrix multiplication.
The Poincare group is the set of all matrices such that such a relationship like this exists between two frames of reference.