Galilean Isomorphism

up:: Galilean Space

A map between galilean spaces $\phi :\left(E,V,t,⟨\cdot ,\cdot ⟩\right)\to \left(E^{\prime },V^{\prime },t^{\prime },{⟨\cdot ,\cdot ⟩}^{\prime }\right)$ is a galilean isomorphism if

• it is an Affine Isomorphism with underlying linear isomorphism $L:V\to V^{\prime }$, such that
• $L$ preserves time: $t^{\prime }\circ L=t$
  • Note that any two points $P,Q\in E$ which are Simultaneous Events (i.e. $\overrightarrow{PQ}\in \ker (t)$) are also simultaneous in $E^{\prime }$, since $$
    \overrightarrow{PQ} \in \ker(t) \implies t(\overrightarrow{PQ}) = t’(L(\overrightarrow{PQ})) \implies L(\overrightarrow{PQ}) \in \ker(t’)
• $L$ preserves the inner product for Simultaneous Events in $E$ (i.e. for all $v\in \ker (t)$): ${⟨L(v_1),L(v_2)⟩}^{\prime }=⟨v_1,v_2⟩$ for any $v_1,v_2\in V$

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References