I'm trying to find the first paper in which the concept of four-vectors was introduced. I read "Principle of Relativity" by H. Minkowski but he only presents the notion of metric and invariant space-time interval, but not specifically four-vectors. I'm searching this because I want to find the physical motivation for four-vectors other than "the components must transform by Lorentz transformations" because I think this is a random statement. If this articles exists, please tell me the title, thanks

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    $\begingroup$ It is hard to understand what you are looking for. Use of higher dimensional vectors was not specific to relativity, or motivated by physics. Hamilton's quaternions were "$4$-vectors", he invented them by trying to generalize complex numbers and understand rotations in $3$-space (he also invented the word "vector"). Lorentz invariance of Maxwell's equations was discussed by Poincare before Minkowski. Applying $4$-vectors there was a natural move requiring no special "motivation from physics". $\endgroup$
    – Conifold
    Feb 13, 2020 at 1:14
  • $\begingroup$ Recently I figured out that some topics in theoretical physics was motivated only by computational problem solving, obtain covariant form of some equations and not by physical intuition. Since the most physics I've studied are Classical Mechanics, wich most of contents are intuitively motivated, I'm having some problems to make the transition for the "theoretical way of thinking" of 20th century and recent physics topicos $\endgroup$ Feb 13, 2020 at 1:39
  • $\begingroup$ @Conifold I don't wish to be pedantic, and you do place 4-vectors in quotes, but is it correct to say that Hamilton's quaternions were 4-vectors, as in elements of a vector space. They were certainly 4-tuples, but I thought that Hamilton's definition was purely formal, as was his definition of complex numbers as ordered pairs of real numbers. He was creating a number system rather than a vector space. Grassmann was developing his notion of a vector space at about the same time, though not published until much later. $\endgroup$
    – nwr
    Feb 13, 2020 at 4:44
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    $\begingroup$ @Nick The formal notion of vector space was not in much use until 1910s, see When did people start viewing a matrix as a linear transformation between two vector spaces?, hence my scare quotes. But Hamilton did define what is now called vector space operations on quaternions (although he only called imaginary quaternions "vectors"). Vector calculus of Gibbs and Heaviside was developed without treating vectors as elements of a "vector space", and under direct influence of quaternions. $\endgroup$
    – Conifold
    Feb 13, 2020 at 5:12
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    $\begingroup$ @Lil'Gravity What you call "physical intuition" is an old form of it tied to mechanistic models, like ether. Covariance, gauge invariance, etc., are part of the new, equally physical, intuition that emerged after ether's abolition by Einstein. Einstein himself and Weyl did much to clarify and promote the physical import of invariance requirements, such as frame and calibration independence. Perhaps you should look at Einstein's 1905 SR paper, §3 for how "$4$-vectors" naturally come up in SR. $\endgroup$
    – Conifold
    Feb 13, 2020 at 5:32


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