# Mathematically rigorous derivation of Einstein-Lorentz transformation

Many elementary explanations of the Einstein-Lorentz transformation derive it using Einstein's special relativity postulates, combined with some thought experiments considering light sources and observers. However, I have read that these thought experiments often do not hold when subject to rigorous mathematical scrutiny, and that Einstein himself did not use them when making his arguments. Further, it is obvious that Lorentz did not use Einstein's postulates when deriving his transformation because at that time he did not trust in Einstein's postulates. Rather, from what I have read, the actually rigorous way that the transformations were derived is using more complicated math such as hyperbolic geometry. I cannot seem to find a good resource that walks through the derivation in a rigorous way; especially one that does not assume prior knowledge and explains the math being used. Can someone please either provide a rigorous derivation here, or point me to a good resource for me to find it? I am especially interested in learning the approaches of both Einstein and Lorentz.

• Landau and Lifshitz, Classical Theory of Fields, Chapter 1. Commented Jun 25, 2023 at 22:43
• You should note that very often the goal of a physicist proposing a new theory, is not to be rigorous but rather to be right ;-) Where by right I just mean, construct a model that agrees with observations in a more complete and beautiful way than what existed before. I'm saying this because you shouldn't conflate things: the historical derivation that Einstein used may not be the most rigorous one!
– Amit
Commented Jun 25, 2023 at 22:59

However, I have read that these thought experiments often do not hold when subject to rigorous mathematical scrutiny

Those that are cited in textbooks today are the ones that are good. The ones that are definitely bad are those that actually send the observer or massive object exactly to the speed of light.

Einstein himself did not use them when making his arguments.

Citation needed. It would be weird for Einstein to not use his own arguments.

Further, it is obvious that Lorentz did not use Einstein's postulates when deriving his transformation because at that time he did not trust in Einstein's postulates.

Here you misunderstand. Lorentz got to the transformations before Einstein came along. Lorentz never understood what Einstein was saying. His successful derivation of the transformations seemed to hinder his understanding of what Einstein was saying. In the end, he died before learning.

the actually rigorous way that the transformations were derived is using more complicated math such as hyperbolic geometry.

This is technically correct, but it is not necessary. Hyperbolic functions simply make the results look beautiful. You can, however, understand the theory rigorously using old algebraic methods.

What I actually suggest for students who want a better understanding, is to try a resource that covers the topic from Minkowski diagrams. You can derive the whole transformation rigorously from it, and answer all the important spacetime questions in terms of it. However, the drawback is that the energy-momentum relations would seem like magic, and for that I suggest doing maths, not diagrams.

A Very good book is Sander Bais's Very Special Relativity.