# Derivation of the Lorentz Transformation

Let us consider a particle moving with the speed of light in the coordinate system and the coordinate system moving with the velocity with respect to it. The trajectory equation in this system is and in the coordinate system it is moving with the velocity with respect to the coordinate system . Assuming that is travels the same distance with respect to in both systems and in both it travels with the speed of light it must be . So there is a time dilation of its time at a given position in the system :

- .

Analogically we can write

- .

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

- ,
- .

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

Reverting the transformation for we obtain however

- ,
- .

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

- ,
- .

The reverse transformation in an obvious way becomes immediately:

- ,
- .

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

or

- ,

that is

- .

The obtained transformation is therefore the Lorentz transformation.