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Linearity of Lorentz transformations from Homogeneity

  1. Sep 17, 2014 #1
    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?
     
  2. jcsd
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