Galilean Space-time Structure

up:: Euclidean Affine Space

A Galilean Space-time structure consists of:

1. Event space: An Affine Space $A$ over a four-dimensional vector space $V$ over $\mathbb{R}$
2. Time map: A linear mapping $t:V\to \mathbb{R}$ which measures time intervals between parallel displacements
   1. Given events $P,Q\in A$ for which $t(Q-P)=0$, then $P$ and $Q$ are said to be Simultaneous Events (Galilean)
3. Distance between simultaneous events: A distance between simultaneous events $P,Q\in A$ can be defined as $\rho (P,Q)=||Q-P||=\sqrt{⟨Q-P,Q-P⟩}$. I.e. it uses the Vector Space Norm over $V$ which is induced by the Euclidean Structure’s Inner Product

An affine space with this structure is said to be a Galilean Space. Transformations which preserve this structure are called Galilean Transformations. Transformations which preserve the galilean structure are Galilean Isomorphisms.

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References

• ARNOL’D, Vladimir Igorevich. Mathematical methods of classical mechanics. Springer Science & Business Media, 2013.