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- TL;DR Summary
- The two known textbook axioms of Special Relativity are generally thought to be independent. Or are they not, really?

Sorry if this is discussed here previously, but I just stumbled upon an article from 1911 which I would like to bring forth to you.

Is it correct? Well, two Austrian physicists, Philipp Frank and Hermann Rothe, write an article of 30 pages published in Vol. 34 of the 4th series of the Annalen der Physik (Publication year = 1911) whose conclusion is (original in German, then my

"Unter allen Transformationsgleichungen, die eingliedrigen linearen homogenen Gruppen entsprechen, gibt es drei Typen, bei denen der Betrag der Kontraktion nicht von der Richtung der Bewegung I am absoluten Raume abhaengt. Darunter hat nur ein Typus eine tatsaechliche Kontraktion der Laengen zur Folge, naemlich die Lorentztransformation [Gleichung (1)], die beiden anderen Typen, die Galilei- und die Dopplertransformation (Gleichung (2) bzw. (129)] lassen die Laengen unverandert. Bei der Lorentztransformation hat die Lichtgeschwindigkeit in allen bewegten Systemen bei beliebiger Fortpflanzungsrichtung denselben endlichen Wert

"From all the transformations, which correspond to monomial linear homogenous groups, there are three types for which the amount of the contraction does not depend on the direction of motion in absolute space. From them only type shows a contraction of lengths, namely the Lorentz equation [eqn (1)], while the other two types, the Galilei and Doppler transformation, leave lengths unchanged. For the Lorentz transformation, the speed of light in all systems in motion has the the same finite value c for an arbitrary direction of propagation. For the Doppler transformation, on the other hand, (the finite value

It appears that the first axiom is supplemented by the request that space-time transformations between systems are linear monomials. Then, because SR appeared after Lorentz contraction was known, it was required by F & R that this contraction does not depend on the direction of movement, which offered three possible transformations > Lorentz was the only one which showed length contraction (the other two didn't) and, moreover, showed that the speed of light is invariant for propagation under any direction, while for the Doppler one, only under one direction (namely the direction of light > source - receiver).

Was this article known to you? What do you think of this?

**Preamble**: it is generally thought that Einstein's (refined) two axioms of SR (1. The laws of physics are invariant upon shifting from one IRF to another. 2. The one-way speed of light in vacuum is constant in any IRF) are independent, in the sense that, no matter how much physical/mathematical theory you can derive from 1., axiom 2. can never be reached (proven).Is it correct? Well, two Austrian physicists, Philipp Frank and Hermann Rothe, write an article of 30 pages published in Vol. 34 of the 4th series of the Annalen der Physik (Publication year = 1911) whose conclusion is (original in German, then my

*perhaps imperfect*translation to English below)"Unter allen Transformationsgleichungen, die eingliedrigen linearen homogenen Gruppen entsprechen, gibt es drei Typen, bei denen der Betrag der Kontraktion nicht von der Richtung der Bewegung I am absoluten Raume abhaengt. Darunter hat nur ein Typus eine tatsaechliche Kontraktion der Laengen zur Folge, naemlich die Lorentztransformation [Gleichung (1)], die beiden anderen Typen, die Galilei- und die Dopplertransformation (Gleichung (2) bzw. (129)] lassen die Laengen unverandert. Bei der Lorentztransformation hat die Lichtgeschwindigkeit in allen bewegten Systemen bei beliebiger Fortpflanzungsrichtung denselben endlichen Wert

*c*. Bei der Dopplertransformation hingegen nur bei Fortpflanzung nach einer Richtung, bei der Galileitransformation ueberhaupt nur, wenn die Lichtgeschwindigkeit unendlich waere"."From all the transformations, which correspond to monomial linear homogenous groups, there are three types for which the amount of the contraction does not depend on the direction of motion in absolute space. From them only type shows a contraction of lengths, namely the Lorentz equation [eqn (1)], while the other two types, the Galilei and Doppler transformation, leave lengths unchanged. For the Lorentz transformation, the speed of light in all systems in motion has the the same finite value c for an arbitrary direction of propagation. For the Doppler transformation, on the other hand, (the finite value

*c*is true) only for one direction of propagatioin, while for the Galilei transformation, actually only when the speed of light is infinite".It appears that the first axiom is supplemented by the request that space-time transformations between systems are linear monomials. Then, because SR appeared after Lorentz contraction was known, it was required by F & R that this contraction does not depend on the direction of movement, which offered three possible transformations > Lorentz was the only one which showed length contraction (the other two didn't) and, moreover, showed that the speed of light is invariant for propagation under any direction, while for the Doppler one, only under one direction (namely the direction of light > source - receiver).

Was this article known to you? What do you think of this?