# Long live Galileo?

Classical Mechanics Level 4

Generically, both $$x_B$$ and $$t_B$$ are functions of both $$x_A$$ and $$t_A$$. In Galilean relativity, $$x_B=X(t_A,x_A,v)$$ while $$t_B=t_A$$. However there is no mathematical reason why $$t_B$$ cannot also be a function of $$x_A$$ and $$v$$. There is also no reason why $$x_A=x_B$$ or $$x_A \neq x_B$$ as $$v\rightarrow 0$$, as I can shift my origin or not between two coordinate systems/inertial frames.

We now mention an observational fact: the speed of light is experimentally equal to one in every inertial reference frame. Can the following transformation law for $$t_B,x_B$$ as a function of $$t_A,x_A$$ be true?

$$x_B=x_A-vt_A, t_B=t_A$$

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