Einstein’s derivation of the Lorentz Transformation is purely theoretical. This study shows how it is related to the physical phenomenon of time dilation and length contraction. The Lorentz Transformation was first derived using the conditions of time dilation and length contraction. Later, Albert Einstein has given a different derivation of the Lorentz Transformation by using constancy of the speed of light only, making time dilation and length contraction subsequent to the Lorentz Transformation which then acquired the status of fundamental law.

Einstein exposed his derivation in the Appendix 1 of his 1920 book « Relativity: The Special and General Theory ». But the physical significance of this derivation is blurry. I will analyze Einstein’s theoretical derivation to find out how it is related to the physical conditions of time dilation and length contraction.

In his derivation Einstein only used the fact that the speed of light is c in K and K’. However, another claim was used without being properly declared, which is: (x,t) the co-ordinates of an event in K and (x’,t’) the co-ordinates of the same event in K’ satisfy equations (3) and (6) in general. As these equations are in direct proportional form, we give this claim the name “Proportionality assumption”. This claim is an assumption because it is not proven.

I analyze this assumption in the article below.
https://www.academia.edu/41712834/Analysis_of_Einsteins_derivation_of_the_Lorentz_Transformation
https://pengkuanonphysics.blogspot.com/2020/01/analysis-of-einsteins-derivation-of.html