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 $x'=(c-v)t=ct'$. So there is a time dilation of its time at a given position in the system $K$:

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

Analogically we can write

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

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

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

$x'=x-vt$.

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

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

$x={\frac {x'+vt'}{1-{\frac {v^{2}}{c^{2}}}}}$.

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:

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

$x'=(x-vt)\kappa$.

The reverse transformation in an obvious way becomes immediately:

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

$x={\frac {x'+vt'}{\left(1-{\frac {v^{2}}{c^{2}}}\right)\kappa }}$.

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.