# Lorentz Transformation Derivation

## Main Question or Discussion Point

Hi everyone, I am looking into relativity as preparation for university and I was wondering if anyone could help me out with this.

I am reading Six Not-So-Easy-Pieces and in it Feynman uses the Lorentz transformation and 'derives' it, by showing that Lorentz contraction is necessary to account for the null result of the Michelson-Morley experiment. He then shows that in a primed system that has contracted due to its motion, the distance to a fixed point will also contract and so an observer can calculate the 'real' distance as:
\begin{align}
x &= x'{\sqrt{1-\frac{u^2}{c^2}}}
\end{align}
As the primed system is moving towards the fixed point:
\begin{align}
x &= x'{\sqrt{1-\frac{u^2}{c^2}}} + ut
\end{align}
Thus:
\begin{align}
x' &= \frac{x - ut}{\sqrt{1-\frac{u^2}{c^2}}}
\end{align}
Now I don't like this way for two reasons:
1) It presupposes the Lorentz contraction rather than showing it as a result of the Lorentz transformation
2) It is seemingly invented to 'fix' the experiment

Although I have yet to completely follow through with their reasoning, it too uses Lorentz contraction and time dilation.

So my question is firstly, is it OK to derive the Lorentz transformations using length contraction and time dilation. Surely that is a 'circular derivation'! If not, then how can it be derived by other means.

Thanks

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Bill_K
Yes, Einstein originally obtained it as the transformation that preserved Maxwell's Equations, or more specifically the wave equation.

jtbell
Mentor
In Einstein's book Relativity: The Special and General Theory, he presents a derivation of the Lorentz transformation starting from the assumption that the speed of a light-signal is the same in any inertial reference frame:

http://www.bartleby.com/173/a1.html

If you know PDEs, then check out the original article by Einstein:

http://www.fourmilab.ch/etexts/einstein/specrel/www/

For Einstein, the concept of synchronization of clocks plays essential role.

Also, time dilation, length contraction and velocity composition may be derived without using Lorentz transformation. This is shown by the use of the 'light clock'.

Thank you all for your replies, it all makes sense to me now (or at least the order)!

@dickfore - Unfortunately I don't know PDEs, but that might be another port of call for me this summer!