The Galilean transformations of the coordinate space

Coordinate space. Let $C:=\mathbb{R}\times {\mathbb{R}}^3$, whose elements are written $(t,x)$ with $t\in \mathbb{R}$ and $x=(x_1,x_2,x_3)\in {\mathbb{R}}^3$. The Euclidean inner product and norm on the second factor are those of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$; addition and scalar multiplication on $C$ are componentwise, and $I_3$ is the identity matrix of Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes.

The transformations. For a quadruple $\theta =(b,a,v,R)\in \mathbb{R}\times {\mathbb{R}}^3\times {\mathbb{R}}^3\times SO(3)$ (The orthogonal group $O(3)$ and the rotation group $SO(3)$) define the map $g_{\theta }:C\to C$ by

$g_{(b,a,v,R)}(t,x):=(t+b,Rx+vt+a),$

where $Rx$ and $vt$ are the usual matrix–vector products with $R=(R_{ij})$ and with the column vectors $x,v,a$. The four components of $\theta$ are called the time shift $b$, the spatial translation $a$, the boost velocity $v$ and the rotation $R$, and $\theta$ is called the parameter quadruple of $g_{\theta }$. Write

$G:=\{g_{(b,a,v,R)}:(b,a,v,R)\in \mathbb{R}\times {\mathbb{R}}^3\times {\mathbb{R}}^3\times SO(3)\}$

for the set of all these maps and $\mathrm{\Theta }:=\mathbb{R}\times {\mathbb{R}}^3\times {\mathbb{R}}^3\times SO(3)$ for the parameter set. The elements of $G$ are the Galilean transformations of the coordinate space, and the assignment $\theta \mapsto g_{\theta }$ is the parameter map.

Elementary readings. For $R=I_3$ and $v=0$ the map $g_{(b,a,0,I_3)}$ is the pure time-and-space translation $(t,x)\mapsto (t+b,x+a)$; for $b=0$, $a=0$ and $R=I_3$ it is the pure boost $(t,x)\mapsto (t,x+vt)$; and for $b=0$, $a=0$, $v=0$ it is the rotation $(t,x)\mapsto (t,Rx)$ acting only on the spatial factor. The time coordinate of $g_{\theta }(t,x)$ depends only on $t$ and $b$, in accordance with the absolute-time convention of the model that consumes this definition.

What is and is not asserted. This definition introduces the maps and names their parameters. It asserts no property of the parameter map (such as injectivity), no closure of $G$ under composition or inversion, no continuity, smoothness or differentiability of the maps, and no physical interpretation: whether these maps describe changes between inertial frames of a physical model is a separate, explicitly adopted assumption of the physics items that consume this definition. The group structure of $G$ is proved in the group lemma of this page, and the units in which the parameters are read are fixed by the model's unit conventions, not here.

• The definition is purely mathematical. Its content is the set of maps displayed above, the naming of the parameters and the reading of the elementary cases; nothing is assumed about which coordinate systems a physical model uses.
• The composition and inverse computations of The Galilean transformations form a group of bijections of the coordinate space show that $G$ is a group of bijections of $C$ with respect to composition, with the parameters of a composite given by the displayed polynomial expressions. In the model, the composition of frame changes is used only through that lemma.
• Notation $Rx$ requires the identification of matrices over $\mathbb{R}$ with linear maps of ${\mathbb{R}}^3$; the products displayed are the usual ones of Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes read componentwise, and the case $R\in SO(3)$ is the one of The orthogonal group $O(3)$ and the rotation group $SO(3)$.