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In Taylor's Classical Mechanics text, he derives the Lorentz transformation from length contraction which, in turn, uses time dilation. But doesn't the use of length contraction necessitate that the time transformation is just the time dilation?

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  • $\begingroup$ You mean $t' = t/\gamma$ and $x'=\gamma(x-v t)$? $\endgroup$
    – user154997
    Oct 29, 2017 at 20:23
  • $\begingroup$ Yes. Using both of those facts seems to indicate to me that the time transformation is just t'=gamma*t $\endgroup$ Oct 29, 2017 at 20:58
  • $\begingroup$ I got hold of Taylor's book and I am not sure I understand your question anymore actually. He uses length contraction as observed from $\mathcal{S}'$ (I hope we have the same edition so that the notations make sense to you) to get $x'=\gamma(x-Vt)$. Then he uses the same argument from the point of view of $\mathcal{S}$ to get $x=\gamma(x'+Vt')$, and then he can conclude. Why do you ask about time dilation here? $\endgroup$
    – user154997
    Oct 30, 2017 at 11:51
  • $\begingroup$ Exactly! The moment he applied length contraction to get gamma(x-Vt) didn't he not force the relationship of t=gamma*t' ? (Because he proved length contraction from time dilation) $\endgroup$ Oct 31, 2017 at 0:11
  • $\begingroup$ I see: I needed to read further back! Sorry, that book was unknown to me. Yes, I agree, his reasoning is a bit circular. Well, to be generous, let's say that the two arguments are disconnected. He showed that just with length contraction, one can get Lorentz transforms, from which one can then get time dilation. Conversely, he showed earlier that just with time dilation, one can get length contraction. And therefore that one goes with the other, within the hypotheses he made. $\endgroup$
    – user154997
    Oct 31, 2017 at 0:22

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If the time coordinate transformation was given simply by

$$ t' = \gamma t, $$

all events simultaneous in the $x,t$-frame would be simultaneous also in the $x',t'$-frame. That would be in contradiction to relativity of simultaneity, which is an important part of special relativity.

For example, imagine a spherical expanding light wave that reaches two distant observers at the same time as observed in the $x,t$-frame. In a different frame $x',t'$ that moves along the line joining the observers, the wave cannot reach both observers at the same time $t'$, because one is moving towards the wave and the other is moving away from it. So $t'$ cannot depend on $t$ coordinate of the event and $\gamma$ only. In fact it depends on the spatial coordinates of the event too.

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    $\begingroup$ Yes, I understand that it is incorrect. I am merely asking if Taylor's use of length contraction in the derivation of the transform is a contradiction and if it isn't why isn't it? $\endgroup$ Oct 29, 2017 at 22:25
  • $\begingroup$ Relativity of simultaneity is an artefact of the chosen synchronisation. There is nothing fundamental in it. $\endgroup$
    – user154997
    Oct 29, 2017 at 22:29

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