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### S4. Galilean Transformation

How do we understand Galilean principle of relativity mathematically?
Ans: Make use of the Galilean transformation.

Well, that’s the simplest answer. The problem of the Galilean relativity is, connecting the two inertial frames of reference so that the mechanical law of physics is the same in both the frame concerned.

So, how do we do it? Here’s one of the following ways,

Fig. 1: Reference frame $S$ and ${S}'$

Suppose there are two reference frames (systems) designated by $S$ and ${S}'$ such that the co-ordinate axes are parallel (as in figure). In S, we have the co-ordinates $(x,y,z,t)$ and in ${S}'$ we have the co-ordinates $({x}', {y}',{z}',{t}')$. ${S}'$ is moving with respect to $S$ with velocity $v$ (as measured in $S$) in the $x$ direction. The clocks in both systems were synchronised at time $t=0$ and they run at the same rate.

One can clearly see that the position of the ${S}'$ frame as seen from the $S$ changes as the time passes and vice-versa. But how do we transform the coordinate of the ${S}'$ frame so that the mechanical laws of physics seems exactly as same a that observed in the $S$ frame of reference.

It is found that the following simple transformation does the job.

${x}'=x-vt$
${y}'=y$
${z}'=z$
${t}'=t$

And, these set of equations is called the Galilean Transformation.

It is important to note here that, we have assumed that the two clocks are set to agree with each other, and that the origin of the two frames are at the same point when ${t}'=t=0$. Using this relation, we can then translate the observations of the observer in inertial frame $S$ to that of the observer in inertial frame ${S}'$, and vice versa.

Does this really work? I mean do these transformation really preserve the invariance of laws of physics?

You can try out the following sequence of logic to decide for yourself.

“Consider two inertial frames $S$ and ${S}'$. A physical event in $S$ will have position coordinates $r=(x,y,z)$ and time $t$; similarly for ${S}'$. By using the universal time concept, one can synchronize the clock in the two frames and assume $t={t}'$. Suppose ${S}'$ is in relative uniform motion to $S$ with velocity $v$. Consider a point object whose position is given by $r=r(t)$ in $S$. We see that
$r^{‘}(t)=r(t)-vt$.
The velocity of the particle is given by the time derivative of the position:

Another differentiation gives the acceleration in the two frames:

It is this simple but crucial result that implies Galilean relativity. Assuming that mass is invariant in all inertial frames, the above equation shows Newton's laws of mechanics, if valid in one frame, must hold for all frames. But it is assumed to hold in absolute space, therefore Galilean relativity holds." wiki
To conclude, Galilean relativity as in words of the Galilean transformation can be fancifully restated as:

"The laws of physics must be invariant under the Galilei transformation."