= Poincaré group
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= Poincaré transformation
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Full set of all possible <special relativity> symmetries:
* translations in space and time
* rotations in space
* <Lorentz boosts>
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:
* 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.
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.