# Linearity of Lorentz transformations from Homogeneity

1. Sep 17, 2014

### center o bass

In this paper: http://www.jstor.org/stable/25170907?seq=1 the authors claim to show that linearity of the lorentz transformaiton follows from homogeneity of space and time. They consider two reference systems K and K' with coordinates (x,t) and (x',t') and write down the transformation laws as
$$f(x',x,t) = 0; h(t',x,t) = 0.$$
They then say that homogeneity of space implies
$$f(x' + \epsilon, x + \epsilon,t) = 0; h(t', x+ \epsilon, t) = 0$$
which they say represent a translation of the origin, or equivalently the translation of the place at which an event occurs, and where $\epsilon$ is an arbitrary parameter of dimension length.

Now, is not this transformation assuming that if the event, according to K, originally placed at $x$ away from the origin, is moved to $x + \epsilon$, then that same event, according to K', will be moved to $x' + \epsilon$?

If so would not that contradict the fact that K and K' does not agree on lengths? It seems that if an event is moved by $\epsilon$ in K, then it should be moved by $\epsilon'$ in K' with (in general) $\epsilon \neq \epsilon'$.

So are the authors making an erroneous assumption? Or where does my reasoning go wrong?