#### Ibix

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Originally, Einstein didn't come up with it that way - he used his train thought experiment. But then people showed that the transforms are simply an expression of symmetries, and this is a more abstract but much more powerful way of thinking, that underlies all fundamental physics post-Einstein.Yes, that is exactly what I am asking.

Dale: It's a difficult concept for me. I understand what the general form of an equation is. I just don't see how people originally came up with the idea that the general equation took that form.

The transforms simply relate coordinates on spacetime, so they can only depend on the coordinates and the relative velocity of the frames. There are no other physical concepts that are relevant. The transforms have to be linear because if they aren't there's either a special place or a special time or both. We can arbitrarily choose an origin that the coordinates share, and we can arbitrarily choose our x axis to lie parallel to the relative velocity of the frames.

With those assertions, the most general transformation we can possibly write is$$\begin{eqnarray*}t'&=&At+Bx+Cy+Dz\\

x'&=&Et+Fx+Gy+Hz\\

y'&=&It+Jx+Ky+Lz\\

z'&=&Mt+Nx+Oy+Pz\end{eqnarray*}$$where the capital letters are constant. We are

*not*making any assumptions about these constants. Many of them may be zero - but we'll prove that, we won't assume it.

All the off-diagonal elements including ##y## or ##z## must be zero from symmetry. Simply rotate your coordinate axes 180° about the x axis and ##y## becomes ##-y## - but this shouldn't have any effect on ##x'## or ##t'##, and the only way for it to have no effect if those cross terms are zero. We

*cannot*make this argument about off diagonal terms with ##x## because there is a difference if we flip the x axis (the velocity changes sign) and we can't make this argument about ##t## because flipping the direction of time also flips the direction of velocity.

That tells us that our four equations are just

$$\begin{eqnarray*}t'&=&At+Bx\\

x'&=&Et+Fx\\

y'&=&Ky\\

z'&=&Pz\end{eqnarray*}$$Adding that the inverse transforms must look the same as the forward transforms except for the sign of the velocity (i.e., the principle of relativity) means that we want ##y'=Ky## and ##y=y'/K## to look the same which means ##K=1##, and similarly ##P=1##.

This is where you started. Sweetsprings already posted the derivation from here.