Galilean principle of relativity

[Wox]

A Galilean transformation consists of a rotation (in space), a boost (in space) and a translation (in space and time). This can be represented for homogeneous coordinates as

<br /> \left[\begin{matrix}t&#039;\\x&#039;\\y&#039;\\z&#039;\\1\end{matrix}\right]=<br /> <br /> \left[\begin{matrix}<br /> 1&amp;0&amp;0&amp;0&amp;t_{t}\\<br /> u_{x}&amp;R_{11}&amp;R_{12}&amp;R_{13}&amp;t_{x}\\<br /> u_{y}&amp;R_{21}&amp;R_{22}&amp;R_{23}&amp;t_{y}\\<br /> u_{z}&amp;R_{31}&amp;R_{32}&amp;R_{33}&amp;t_{z}\\<br /> 0&amp;0&amp;0&amp;0&amp;1<br /> \end{matrix}\right]<br /> <br /> \cdot\left[\begin{matrix}t\\x\\y\\z\\1\end{matrix}\right]<br />

To me there seem to be two principles of relativity in frames that are related by a Galilean transformation. The first says that all physical laws described in Galilean space-time have the same form in frames related by a Galilean transformation. Newton's second law of motion for example given by F=m.a in one frame becomes F&#039;=m.a&#039; in the second frame, while F and F&#039; transform under a Galilean transformation.

The second says that all physical laws are the same in frames that are related by a Galilean transformation with R=id (i.e. inertial frames of reference). Again Newton's second law of motion: F=F&#039; and a=a&#039;.

Is this a correct understanding of Galilean relativity?

 

[Wox]

Actually I'm really having problems with the concept of Galilean relativity and I think it is because I don't understand Galilean spacetime properly. Consider a world line and its underlying spatial trajectory
<br /> \bar{w}\colon \mathbb{R}\to \mathbb{R}^{4}\colon t \mapsto (t,\bar{x}(t))<br />
<br /> \bar{x}\colon \mathbb{R}\to \mathbb{R}^{3}\colon t \mapsto (x,y,z)<br />
where \mathbb{R}^{3} with the Euclidean structure and \mathbb{R}^{4} with the Galilean structure. The acceleration of the world line is given by
<br /> \bar{a}\colon \mathbb{R}\to \mathbb{R}^{4}\colon t\mapsto \frac{d^{2}\bar{w}}{dt^{2}}=(0,\frac{d^{2}\bar{x}}{dt^{2}})\equiv(0,\tilde{a}(t))<br />
A force field is given by \bar{F}\colon \mathbb{R}^{4}\to \mathbb{R}^{3} and it can be evaluated along a world line by using \bar{F}(\bar{w}(t))=m \tilde{a}(t) When change frame using a Galilean transformation
<br /> t=t&#039;+t_{t}\quad\quad \bar{x}=t&#039;\bar{u}+R\cdot\bar{x}&#039;+\bar{t}_{\bar{x}}<br />
we find that
<br /> \tilde{a}(t)=R\cdot \tilde{a}&#039;(t&#039;)\Leftrightarrow \bar{F}(\bar{w}(t))=m\tilde{a}(t)=m R\cdot \tilde{a}&#039;(t&#039;)=R\cdot \bar{F}(\bar{w}&#039;(t&#039;))<br />

So for inertial frames (R=id) we find that \bar{F}(\bar{w}(t))=\bar{F}(\bar{w}&#039;(t&#039;)). Is this then the second aspect of Galilean relativity? And what about the other aspect that states that laws have the same form after a Galilean transformation?

Both aspects of invariance are for example discussed here.