Galilean boosts preserve the Galilean Structure

up:: Galilean Coordinate Space

Let $(\mathbb{R}\times {\mathbb{R}}^3,t,⟨\cdot ,\cdot ⟩)$ be a galilean coordinate space. Let ${\beta }_{\vec{v}}(t,\vec{x})=(t,\vec{x}+\vec{v}t)$ be the Galilean Boost of the galilean coordinate space by a uniform velocity $\vec{v}$.

Then for any two points $P,Q\in {\mathbb{R}}^4$, note that t(Q - P) = t(\beta_\vec{v}(Q) - \beta_\vec{v}(P)), thus it preserves the time distance.

For Simultaneous Events, note that their distance will remain the same: let $P=(t,\vec{p}),Q=(t,\vec{q})$. Then

||Q - P|| = ||\vec{q} - \vec{p}|| = ||\vec{q} + \vec{v}t - (p + \vec{v}t)|| = ||\beta_\vec{v}(Q) - \beta_\vec{v}(P)||

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