I have this silly doubt in my head and it's bugging me for a real long time now. Let us consider the Galilean transformation $x=x'+vt$ for two frames measuring coordinates $x$ and $x'$. For simplicity, I'll call these frames YOU(measuring $x'$) and ME (measuring $x$).Now, there are two ways to look at this(I'll mention both so that the reader gets the drift-changing one concept of Galilean relativity gives Lorentz transforms)-
(1) I (i.e. frame ME) sees the origin of YOU at a distance of $vt$ after a time $t$ ( the regular stuff about origins coinciding at $t=0$ holds of course). Now, there is an event, $E$, which occurs at a coordinate $x'$ in you frame. So,
(Distance of E from ME)=(Distance of E from YOU) + (Distance of YOU from ME)
So $x=x'+vt$, the Galilean transformation law.
Now, in SR, we just let go of the idea that all frames measure the same length. Since it is possible to show the existence of time dilation and length contraction in SR from purely physical reasoning (i.e. without invoking Lorentz transforms), we may write
(Distance of E from ME)=(Distance of E from YOU 'as seen by ME') + (Distance of YOU from ME)
And we get $x=x'/\gamma+vt$, using the result of lorentz contraction for the 'length' $x'$ YOU measured. And we have derived the lorentz transformation rule.
Now, the problem is here-
(2) The way I am now trying to look at this is to imagine the following. Suppose YOU measures a coordinate $x'$, and communicates his measurement to me. At this instant, YOU is at a distance $vt$ from me. If YOU could communicate this INSTANTLY, ME could geometrically add the information about distances he knows to get his coordinate- $x=x'+vt$. Thus, Galilean transformations can sort of be ascribed to this 'instantaneous' communication of measurements.
Now, if we account for the finite speed at which a signal can be transmitted, so that NO instantaneous communication is possible, can a similar line of reasoning as above lead to Lorentz Transforms? Or are Lorentz transformations much more fundamental than that? I have a feeling that there is a very trivial point I have overlooked and gotten myself in this mess.
I tried to work it out (not too diligently, I admit), but did not get anywhere close. So is my guess that the Lorentz transformations are simply a correction induced in SR due to finiteness of signal speed incorrect? This is annoying. Any help would be appreciated.