Long live Galileo?
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?