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I was looking at the Galilean transformation matrix and came across a matrix for Galilean of the form.

$$\begin{pmatrix}x'\\ y'\\ z'\\ t'\end{pmatrix}=\begin{pmatrix}1&0&0&-\beta c\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}\begin{pmatrix}x\\ y\\ z\\ t\end{pmatrix},$$

where $\beta=v/c$. What I don't understand is what is the point in using the $\beta$ term why not just $-v$.

Is there something I am missing, I just can't think of a reason for using this. Does it have to do with it giving some indication when $v$ is equal to some fraction of the speed of light it, demonstrates how the Galilean transformation breaks down, and there need to the a correction factor involved?

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  • $\begingroup$ See here en.wikipedia.org/wiki/Special_relativity#Physics_in_spacetime. $\endgroup$
    – eranreches
    Commented Dec 12, 2018 at 14:53
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    $\begingroup$ Just for convenience. The author probably felt it was didactically or otherwise helpful to express speeds in units of $c$. The speed of light is not special in Galilean spacetime. In fact Galilei transformations are obtained from Lorentz transformations in the limit $c\to\infty$ $\endgroup$
    – Nephente
    Commented Dec 12, 2018 at 14:55

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It could be that this form allows one to easily verify that the Lorentz transform reduces to the Galilean transform for small relative velocities, i.e. when beta -->0.

Once realized that the speed of light was fixed in vacuum in all inertial frames of reference, this does not immediately mean that particles obey the same symmetries as light. This was a hypothesis imposed by Einstein. He reasoned that since everything we know, we come to know by electromagnetic processes (via the nervous system, light, electric current, etc), that the laws of mechanics and gravity would also have to be bound by the same symmetry principle. This led to the derivation of the Lorentz transform and a covariant version of Newton's laws of mechanics.

The speed of light is certainly special. Even if you tried to develop a theory of moving particles that was not Lorentz invariant while Maxwell's equations for EM fields is valid, c is special. One arrives at an approximate match between the Galilean and Lorentz transforms for small relative motion between inertial observers. The relevant parameter is beta = v/c. Taking v/c <<< 1 in the Lorentz transform will recover the form of the Galilean transform you are looking at.

Taking c-->infinity would be illogical as there is nothing to prevent v from going to infinity as well without some other constraints in place.

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