Structure of the Galilean group
Theorem 5.4 (Structure of the Galilean group). Every Galilean transformation \(g\) of \(\R \times \R ^3\) can be written uniquely as
where \(g_1\) is a uniform motion, \(g_2\) a translation and \(g_3\) a rotation (Exercise 5.3).
Source: [1, ch. 1]
Proof. A Galilean transformation is affine, so \(g(t, \mathbf x) = L(t, \mathbf x) + (s, \mathbf s)\) for a linear map \(L\) and a fixed \((s, \mathbf s) \in \R \times \R ^3\).
Time. \(g\) preserves time intervals, so the time component of \(L(\tau , \boldsymbol \xi )\) is \(\tau \).
Space. Hence \(L\) sends a simultaneous displacement \((0, \boldsymbol \xi )\) to a simultaneous displacement \((0, G\boldsymbol \xi )\) for some linear \(G : \R ^3 \to \R ^3\). Distances between simultaneous events are preserved, so \(\abs {G\boldsymbol \xi } = \abs {\boldsymbol \xi }\), and \(G\) is orthogonal.
Velocity. Write \(L(1, \mathbf 0) = (1, \mathbf v)\). By linearity \(L(\tau , \boldsymbol \xi ) = (\tau , \mathbf v\tau + G\boldsymbol \xi )\), so
Factorisation. For \(g_1(t, \mathbf x) = (t, \mathbf x + \mathbf v t)\), \(g_2(t, \mathbf x) = (t + s, \mathbf x + \mathbf u)\) and \(g_3(t, \mathbf x) = (t, G\mathbf x)\),
This equals \(g\) exactly when \(\mathbf u = \mathbf s - \mathbf v s\).
Uniqueness. The composite determines \(s\) (its time shift), \(G\) (its action on \(\mathbf x\)), \(\mathbf v\) (its coefficient of \(t\)) and then \(\mathbf u\). So two factorisations of the same \(g\) coincide.