# One more talk about the independence of Einstein's SR axioms

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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.
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).

## Answers and Replies

I didn't know that one, although I knew from comments on here that people had started poking at so-called "single postulate" derivations of relativity fairly soon after it was discovered.

Here's a more recent article saying much the same thing in English.

Gold Member
Summary:: The two known textbook axioms of Special Relativity are generally thought to be independent. Or are they not, really?

2. The one-way speed of light in vacuum is constant in any IRF
This is not the 2nd premise of SR.
It is more along the lines of "The measurement of the speed of light in a vacuum will be independent of the inertial frame in which it is measured"

In particular, there is no assumption of the one-way speed of light being constant since this would have been a true assumption, not something that came from empirical findings.

Mind you, I am speaking of the wording of the SR theory, and not of this 1911 article.

phinds and Ibix
Was [the Frank & Rothe] article known to you?
Yes.
What do you think of this?
Heh, congratulations! You are now the 10,000'th person to become interested in the (so-called) 1-postulate derivation(s) of SR. Collect your prize..., oh wait, there isn't any prize, just a whole lot of fun.

I've talked about this quite a few times in PF threads -- see links below. But more recently, I tend to say something like:
strangerep said:
Mmmpff! Must... resist... temptation to talk about 1-postulate derivations...

Essentially, my position is that the usual 2nd postulate of SR is unnecessary, but you need another principle, which I call "Physical Regularity" (described in one of the threads linked below) to get all the way to SR and its physically-plausible generalizations.

https://www.physicsforums.com/threa...t-from-only-one-postulate.754310/post-4750899

https://www.physicsforums.com/threa...r-than-the-speed-of-light.988473/post-6336781

If you're interested to delve deeper, PM me and I'll send you some (still-unpublished) work which I can't discuss on the PF public forums

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PeroK, robphy and dextercioby
Mentor
The two known textbook axioms of Special Relativity are generally thought to be independent. Or are they not, really?
We’ve had this discussion before, but it remains interesting to me as a question about the history of science if not the science itself.

The discussion is complicated by two factors. First, although Einstein called them “postulates” (did he? Or is that an artifact of translation into English from German?) no mathematician would characterize them as such because they’re not phrased with sufficient rigor (that‘s how more than a century later we can still have disagreements over exactly what they mean). Thus, the question of whether they are independent may not have an unambiguous answer, it may depend on our interpretation of them.

Second, I doubt that Einstein attached much weight to the question of the independence of the postulates, at least in 1905. His argument to his contemporaries relied on two claims and putting them forth as "postulates" allowed him to use these claims without further argument. No mathematician would be completely happy with a structure in which one axiom is on closer examination implied by another - but if you're trying to demonstrate an internally consistent and experimentally supported alternative to the Galilean transforms that's not really a problem.

The second postulate in particular was phrased to address the most likely contemporary concerns and it is not at all clear that a modern presentation of special relativity would cater to the same objections in teh same way.

Delta2, David Lewis, vanhees71 and 3 others
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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.
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).

I don't know that paper, and I've to read it, but from what you say it's pretty much consistent with the derivation of the Lorentz transformations from the assumptions that the special principle of relativity is valid (existence and indistinguishability of inertial reference frames), Euclidicity of space for any inertial observer (including the corresponding isotropy and homogeneity of Euclidean affine space). Making these assumptions you get (up to isomorphy) either the Galilei or the Lorentz (or rather Poincare) group as the symmetry group. AFAIK the first investigation in this direction is due to Ignatowsky (also around 1910).

Here it seems the authors also consider other possibilities when the isotropy assumption about space for any inertial observer is not made.

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We’ve had this discussion before, but it remains interesting to me as a question about the history of science if not the science itself.

The discussion is complicated by two factors. First, although Einstein called them “postulates” (did he? Or is that an artifact of translation into English from German?) no mathematician would characterize them as such because they’re not phrased with sufficient rigor (that‘s how more than a century later we can still have disagreements over exactly what they mean). Thus, the question of whether they are independent may not have an unambiguous answer, it may depend on our interpretation of them.
Einstein in the German original called the "two postulates" simply "das Relativitätsprinzip" (the principle of relativity).

Nugatory
Personally, I like to have both postulates separate:
1 is about laws that apply within a frame
2 is about how a quantity transforms between frames

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As the alternative derivation of the Poincare group as the symmetry group of special-relativistic spacetime (aka Minkowski space) shows, the "invariance of the speed of light" is not about transformation properties of a physical quantity but about a universal constant of Nature, i.e., relevant not only for electromagnetic fields (and thus light) but to all phenomena. It's the universal "limiting speed" of the space and can just be arbitrarily chosen, as is the case since 1983 in the SI. You can even use "natural units" and set it equal to 1, then measuring temporal and spatial lengths in the same unit.

Mentor
Personally, I like to have both postulates separate:
1 is about laws that apply within a frame
2 is about how a quantity transforms between frames

But 1 isn't just about laws within a frame; it's saying that the laws take the same form in all frames, which means the quantities appearing in the laws have to transform between frames in the right way.

The view that there is only one postulate is basically saying that 2 is just a special case of the above--the speed of light is a quantity that appears in the laws so it has to transform properly, and since it's a scalar, "transform properly" means "stays unchanged".

Guillermo Navas, cianfa72 and vanhees71
Mentor
The view that there is only one postulate is basically saying that 2 is just a special case of the above--the speed of light is a quantity that appears in the laws so it has to transform properly, and since it's a scalar, "transform properly" means "stays unchanged".
That one-postulate view is my favorite - you may have seen me paraphrase the second postulate as "And we really mean that first postulate, even when it comes to E&M".

To get from there to SR we still must take Einstein's next step: a proof by construction that there is an internally consistent and experimentally supported theory in which the scalar ##c## is the speed of light in vacuum, as distinct from the speed of light relative to an unobserved hypothetical ether. The wording of the second postulate suggests that Einstein won't be introducing an ether ("and we don't need no steenkin' ether either!") but that doesn't make the postulate necessary - if a proof by construction doesn't need an assumption we just don't introduce it.

But as a matter of history, this one-postulate view is a lot more easily sold to a late 20th-century audience than a late 19th-century one.

Mentor
To get from there to SR we still must take Einstein's next step: a proof by construction that there is an internally consistent and experimentally supported theory in which the scalar ##c## is the speed of light in vacuum, as distinct from the speed of light relative to an unobserved hypothetical ether.

I would put this a little differently; I would say that the proof by construction is that there is an internally consistent and experimentally supported theory in which a finite scalar ##c## appears in the transformation law between inertial frames. Since it is a scalar, its numerical value must be the same in all frames; and we observe experimentally that that numerical value is the speed of light in vacuum. The "its numerical value must be the same in all frames" part is how we know it's not a speed relative to an unobserved hypothetical ether; but its presence in the transformation law I think is the key observation that makes this theory distinct from Newtonian mechanics (where the transformation law contains no such scalar).

cianfa72
But 1 isn't just about laws within a frame; it's saying that the laws take the same form in all frames,
What I mean is:

P1 deals with "per-frame-laws". You can apply these laws in different frames, but they themselves to not deal with multiple frames.
- You pick a frame, then you apply these laws, and they work
- You pick a different frame, you apply these laws, and they still work
P2 is a different kind of rule, that itself deals with multiple frames.

Another way to put it:

P1 : Galiean Relativity is true...
P2 : ... if you replace the Galiean Transformation with the Lorentz Transformation

Dropping P2 here would mean that SR adds nothing to Gallieo.

I know that you can interpret and reformulate it all differently. But I like Einsteins formulation, and I don't think it is redundant, if you interpret it as I wrote above.

P1 : Galiean Relativity is true...
P2 : ... if you replace the Galiean Transformation with the Lorentz Transformation

Dropping P2 here would mean that SR adds nothing to Gallieo.
P1 contains more than Galiean Relativity, because it speaks not only about mechanical laws, but also about electromagnetic laws, which Galilei did not know.

Mentor
P1 deals with "per-frame-laws".

But it doesn't just deal with those; as you say:

P1 : Galilean Relativity is true...

"Galilean Relativity" (see further comment below on that term) puts requirements on the transformation laws between frames; they have to preserve the form of all the per-frame laws.

P2 : ... if you replace the Galiean Transformation with the Lorentz Transformation

Then you are switching meanings of "Galilean Relativity". In your P1 above it did not require the transformation law to be the Galilean transformation law; but in your P2 here it does. You can't have it both ways.

Then you are switching meanings of "Galilean Relativity". In your P1 above it did not require the transformation law to be the Galilean transformation law; but in your P2 here it does. You can't have it both ways.
Just stick to the explicit part, then we don't have to argue about what Galilean Relativity includes or doesn't:
P1 deals with "per-frame-laws". You can apply these laws in different frames, but they themselves do not deal with multiple frames.
- You pick a frame, then you apply these laws, and they work
- You pick a different frame, you apply these laws, and they still work
P2 is a different kind of rule, that itself deals with multiple frames.

Mentor
Just stick to the explicit part

The explicit part has the same issue I raised: P1 puts requirements on transformations between frames, so it's not just about "per-frame laws". And those requirements already imply P2.

cianfa72 and vanhees71
P1 puts requirements on transformations between frames,
How?

Mentor
How?

Because it says the laws take the same form in all frames. That means the transformations between frames have to keep the laws in the same form, which is a requirement on the transformations.

vanhees71
How?
P1 containing electromagnetic laws is incompatible to Galilei transformation. For example the Lorentz force ##\vec{F} = q (\vec{E} + \vec{v} \times \vec{B})## on the a "moving" point charge, created by the magnetic field of current in an electric wire: In the rest frame of the charge (##v = 0##), a magnetic field does not create a force on the charge. This can only be solved by the relativistic length contraction of the wire and the influence on it's charge density.

Mentor
I would put this a little differently; I would say that the proof by construction is that there is an internally consistent and experimentally supported theory in which a finite scalar c appears in the transformation law between inertial frames.
Yes, that’s a better way of putting it... and might be even harder to sell at the beginning of the last century.

Einstein in the German original called the "two postulates" simply "das Relativitätsprinzip" (the principle of relativity).

I think only the first postulate is "das Relativitätsprinzip" (the principle of relativity). The second one is "das Prizinp der Konstanz der Lichtgeschwindigkeit" (the principle of the constancy of the speed of light).
http://myweb.rz.uni-augsburg.de/~eckern/adp/history/einstein-papers/1905_17_891-921.pdf (p5 para 2)

vanhees71 and Sagittarius A-Star
(p5 para 2)
See also end of p1 (p 891), where he lists mechanical, electrodynamical and optical laws in the first postulate and begin of page 2 (p 892), were he describes the second postulate.

mitochan
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).

1. is fundamental. 2. is one example of physical laws. Say we include into the laws of physics which 1. says that "propagation of physical effects has maximum constant speed c", we may save number of axioms, i.e.

1. The laws of physics, which include that propagation of physical effects has maximum constant speed c, are invariant upon shifting from one IRF to another.

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P1 containing electromagnetic laws is incompatible to Galilei transformation.
But P1 doesn't list the specific laws that are true. It just says that whatever laws we use, they should take the same form in all inertial frames. That by itself, without assuming specific laws, doesn't imply a certain transformation. It's only P2 that rules out the Galilei transformation.

Sagittarius A-Star and dextercioby
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That one-postulate view is my favorite - you may have seen me paraphrase the second postulate as "And we really mean that first postulate, even when it comes to E&M".

To get from there to SR we still must take Einstein's next step: a proof by construction that there is an internally consistent and experimentally supported theory in which the scalar ##c## is the speed of light in vacuum, as distinct from the speed of light relative to an unobserved hypothetical ether. The wording of the second postulate suggests that Einstein won't be introducing an ether ("and we don't need no steenkin' ether either!") but that doesn't make the postulate necessary - if a proof by construction doesn't need an assumption we just don't introduce it.

But as a matter of history, this one-postulate view is a lot more easily sold to a late 20th-century audience than a late 19th-century one.
You can of course build SRT from different "postulates". My favorite is to have the symmetry properties of spacetime as basic assumptions. In case of SRT it's the special principle of relativity and Euclidicity of space and time-translation invariance for any inertial observer. From these assumptions you can derive that there are only the Galilei-Newton and the Einstein-Minkowski space times realizing the symmetries as the Galilei group for Galilei-Newton spacetime and as the (proper orthochronous) Poincare group for Einstein-Minkowski spacetime.

Which one is the better description is of course a question to be decided by empirical evidence, and it's clearly Einstein-Minkowski space time (which then has to be extended to a Lorentz or maybe an Einstein-Cartan manifold when the gravitational interaction is included.

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See also end of p1 (p 891), where he lists mechanical, electrodynamical and optical laws in the first postulate and begin of page 2 (p 892), were he describes the second postulate.
Sure, but the 2nd postulate follows from the assumption that the 1st one holds for electrodynamics (including optics), because for Einstein it's clear that Maxwell electrodynamics is the correct theory concerning electrodynamics.

Sagittarius A-Star
Einstein (1916) said:
The special relativity theory rests on the following postulate which also holds valid for the Galileo-Newtonian mechanics.

If a co-ordinate system K be so chosen that when referred to it the physical laws hold in their simplest forms, these laws would be also valid when referred to another system of co-ordinates K′ which is subjected to an uniform translational motion relative to K. We call this postulate "The Special Relativity Principle".
...
The Special Relativity Theory does not differ from the classical mechanics through the assumption of this postulate, but only through the postulate of the constancy of light-velocity in vacuum which, when combined with the special relativity postulate, gives in a well-known way, the relativity of synchronism as well as the Lorentz transformation, with all the relations between moving rigid bodies and clocks.
Source

vanhees71, A.T. and atyy
Sure, but the 2nd postulate follows from the assumption that the 1st one holds for electrodynamics (including optics),
Yes, you can say:
P1 + Maxwell -> P2

But the natural way is to start with the postulates:
P1 + P2 -> Maxwell

... because for Einstein it's clear that Maxwell electrodynamics is the correct theory concerning electrodynamics.
It's clear to him, but he expresses that in P2. P1 is completely agnostic on this. The idea that P2 is redundant comes from conflating P1 with P1 + what Einstein knew and expressed in P2.

vanhees71
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Of course from a modern point of view, you may say P1+P2->Maxwell, but fortunately special relativity works for everything, as far as we know, not only electrodynamics. It leads you to Minkowski space as the space-time model with the proper orthochronous Poincare transformations as the symmetry group. Gauging this symmetry then also very naturally leads to GR (or its extension to Einstein-Cartan theory when spin is taken into account).

Mentor
the natural way is to start with the postulates:
P1 + P2 -> Maxwell

But P1 + P2 does not give you Maxwell. The Lorentz transformations by themselves, which is basically what P1 + P2 amounts to, do not give you Maxwell's Equations; those are by no means the only possible equations which are Lorentz invariant.

vanhees71
But P1 + P2 does not give you Maxwell.
P1 + P2 tells you that Maxwell is correct and doesn't need modifications while Newton does.

Mentor
P1 + P2 tells you that Maxwell is correct

Yes, but if you already know Maxwell is correct, you can use that to deduce P2, as @vanhees71 said. But you cannot use P2 to deduce Maxwell.

More generally, either P1 by itself or P1 + P2 does not tell you what the laws of physics are. It only tells you that, whatever they are, they must be invariant under transformations between inertial frames (P1) and those transformations must be Lorentz transformations (P2). If all you're interested in is those general constraints on the laws of physics, then P1 + P2 is sufficient. But to know what the laws of physics actually are, you need to know more than just P1 + P2.

cianfa72 and vanhees71
But to know what the laws of physics actually are, you need to know more than just P1 + P2.

But regarding Maxwell not much, according to the following paper.
paper said:
Special Relativity and Maxwell’s Equations

19. Discussion
In the preceding sections we have derived Maxwell’s equations directly from special relativity and Coulomb’s law.
Source:
http://richardhaskell.com/files/Special Relativity and Maxwells Equations.pdf

From that you get P1 + P2 + Coulomb's law + conservation of 4-momentum -> Maxwell

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Mentor
From that you get P1 + P2 + Coulomb's law (+ conservation of 4-momentum) -> Maxwell

More precisely, P1 + P2 + Coulomb + 4-momentum conservation + the Lorentz force law -> Maxwell.

Whether the things that need to be added to P1 + P2 in this case are "not much" is a matter of opinion.