# How did Lorentz transformations get their modern definition?

Historically, Special Relativity was motivated by apparent inconsistencies between Maxwell's Electrodynamics and Newtonian Mechanics. In Einstein's well known paper "On the electrodynamics of moving bodies" he explains quite well his motivations.

Central objects of the theory are the Lorentz transformations. If one forgets motivations, history and intuition the Lorentz transformations are formally defined as the linear transformations $\Lambda:\mathbb{R}^4\to \mathbb{R}^4$ such that

$$\eta(\Lambda v,\Lambda w)=\eta(v,w),$$

where $\eta = \operatorname{diag}(-1,1,1,1)$. Furthermost, it seems that before this definition they were defined as the transformations which keep the speed of light the same in all frames.

My question is: how did Lorentz transformations get this modern definition?

How were they first defined, how did they relate to Einstein's paper, and how did they get the modern definition as "transformations which preserve the spacetime inner product"? Specifically I'm interested in knowing how from the motivations for relativity physicists got to the definition of Lorentz transformations as the transformations $\Lambda$ such that $\eta(\Lambda x,\Lambda y) = \eta(x,y)$

• This question doesn't really make much sense. You discuss two mathematically equivalent definitions, and ask when one gave way to the other. Since they're mathematically equivalent, there is no reason that one has to give way to the other. This is just a matter of a particular author's preferences regarding how to present the subject. – Ben Crowell May 3 '16 at 2:19
• I believe that the wording came out in a confuse manner. I'm not asking why would one pick the latter instead of the former. I agree that it is a matter of preference. But as far as I know, the first definition used was that based on Einstein's postulates which appear on his paper. The other definition, equivalent to the first, I believe appeared latter. What I'm asking here, is how physicists got to the second definition. How, from the first approach, which is what Einstein presented, it was discovered that this other definition could do the same? It is not a question regarding which to pick. – user1620696 May 3 '16 at 2:24