Derivation of the Lorentz Transformation

Let us consider a particle moving with the speed of light $$c$$ in the coordinate system $$K$$ and the coordinate system $$K'$$ moving with the velocity $$v$$ with respect to it. The trajectory equation in this system is $$x=ct$$ and in the coordinate system $$K$$ it is moving with the velocity $$c-v$$ with respect to the coordinate system $$K'$$. Assuming that is travels the same distance with respect to $$K'$$ in both systems and in both it travels with the speed of light it must be $$x'=(c-v) t = c t'$$. So there is a time dilation of its time at a given position in the system $$K$$:


 * $$t'= \left(1 - \frac{v}{c}\right) t = t - \frac{v}{c^2} c t = t - \frac{v}{c^2} x$$.

Analogically we can write


 * $$x'=(c-v) t = x - v t$$.

The transformation sought in this way turns out to be a transformation of Galileo with time dilation:


 * $$t'= t -  \frac{v}{c^2} x$$,
 * $$x'= x - v t$$.

We will now assume that it is valid for any event coordinates, not only for coordinates of the photon trajectory.

Reverting the transformation for $$x, t$$ we obtain however


 * $$t= \frac{t' +  \frac{v}{c^2} x'}{1-\frac{v^2}{c^2}}$$,
 * $$x= \frac{x' + v t'}{1-\frac{v^2}{c^2}}$$.

However the physical situation seen from the coordinate system $$K'$$ is identical as seen from $$K$$ but only the system $$K$$ is moving with respect to the system $$K'$$ with the velocity $$-v$$. So we will try to improve the transformation with the scaling factor $$\kappa$$ which naturally preserves the speed of light:


 * $$t'= \left(t -  \frac{v}{c^2} x \right)\kappa$$,
 * $$x'= (x - v t)\kappa$$.

The reverse transformation in an obvious way becomes immediately:


 * $$t= \frac{t' +  \frac{v}{c^2} x'}{\left(1- \frac{v^2}{c^2}\right) \kappa }$$,
 * $$x= \frac{x' + v t'}{\left(1- \frac{v^2}{c^2}\right) \kappa}$$.

In order for the two transformations to be identical except for the physical change of the relative velocity sign it therefore must be:


 * $$\frac{1}{\left(1-\frac{v^2}{c^2} \right)\kappa}=\kappa$$

or
 * $$\kappa^2=\frac{1}{1 - \frac{v^2}{c^2}}$$,

that is


 * $$\kappa=\gamma=\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$.

The obtained transformation is therefore the Lorentz transformation.