Of course you can always change the dummy variable in any equation and get the same answer. But then you're stuck having to interpret the new dummy variable as if it were the exact same thing as the old dummy variable.
I don't think you can change x0
without specifying a function y(x). So I don't think you can go from the integral on the left to the middle integral. But if you do, then it is no longer a dummy variable change; you're actually specifying a function from x to y.
What I think I've shown is when there is a relationship between x and y, namely, y=y(x), you get the same equation back again. Then we can interpret x and y as different coordinates. And x and y can not have any arbitrary relationship but must be a diffeomorphism. I think this means that the Dirac delta function is diffeomorphism invariant, right? So if the dirac delta is required for some reason (TBD), then that requirement also specifies the existence of some properties of the space on which it resides, namely, that it admitts a diffeomorphic coordinate patches, right?
All this does is render a mathematical description of the observations. It does not explain it. What I'm trying to understand is WHY space would have the properties it does that makes necessary these kinds of observations.
Even the author of the paper is trying to derive the Lorentz transformations from properties of spacetime. He starts with the a priori existence of diffeomorphic coordinate transformations on patches of an underlying manifold (paraphrased). I'd like to go even further back and explain the need of for this property of spacetime to begin with.