Suppose that we write a function that changes coordinates from one inertial frame to another in the form(adsbygoogle = window.adsbygoogle || []).push({});

[tex]x\mapsto\Lambda x+a[/tex]

where [itex]\Lambda[/itex] is linear, with components

[tex]\Lambda=\gamma\begin{pmatrix}1 & \alpha\\ -v & \beta\end{pmatrix}[/tex]

in the standard basis. (This is the most general form of a 2×2 matrix with the upper left component non-zero. I'm writing the matrix this way because this v can be interpreted as the velocity difference between two inertial frames). Suppose that we now impose the requirement that the substitution [itex]v\rightarrow -v[/itex] must give us [itex]\Lambda^{-1}[/itex]. (This can be thought of as a mathematical statement that expresses one aspect of Galileo's principle of relativity). We can show that

[tex]\Lambda^{-1}=\frac{1}{\gamma(\beta+\alpha v)}\begin{pmatrix}\beta & -\alpha\\ v & 1\end{pmatrix}[/tex]

so the requirement I just mentioned gives us the following information:

[tex]\beta=1[/itex]

[tex]\gamma=\pm\frac{1}{\sqrt{1+\alpha v}}[/tex]

[itex]\alpha[/itex] is an odd function of v, andnotequal to -1/v.

I would like to impose one more mathematical requirement that also expresses an aspect of the principle of relativity and is sufficient to imply that [itex]\alpha(v)=-Kv[/itex], where K is a constant (and the sign is just a convention). I would appreciate if someone could help me find an appropriate axiom, or to justify one that I already know would work. For example, it's sufficient to require that "velocity addition" is commutative, but is there a way to think of that as a consequence of the principle of relativity?

Once we have the condition [itex]\alpha(v)=-Kv[/itex], it's not too hard to show that K=0 gives us the Galilei group, that K=1 gives us the Poincaré group, and that all other choices of K gives us a group that's isomorphic to the Poincaré group.

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# Only Galilei and Poincaré

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