I'm reading Einstein's paper on special relativity (On the electrodynamics of moving bodies 1905), it gives a derivation of the lorentz transformation. In the derivation, he compares the stationary and the moving system. There is one important thing that I don't understand. It says
"In the first place it is clear that the equations must be linear on account of the properties of homogeneity which we attribute to space and time."
Why the equations must be linear?
A linear transformation maps straight lines to straight lines, that is a straight line in the $(t, x)$ coordinates maps to a straight line in the $(t', x')$ coordinates.
This is required because a body moving freely travels in a straight line, and any curvature would imply some external force was being applied. If the Lorentz transformations were non-linear then the trajectory of the body would be a straight line in some frames but not in others. That is, it would appear that an external force was being applied in some frames but not in others.
Lunearity means any time one can take portion of motion as measurement and predicts past or future. So if space and time aren't condensed ir dilutued then intervals are uniform.
If equation of motion is linear then, $x=vt$ and if non-linear then $x=at^2$. So any time interval in linear motion gives any prediction of motion. For non-linear motion, at higher value of time space becomes less dense. Linearity ensures similar spacing of space and time.
