Could SR not be built from only one postulate?

I'll answer that only after you've studied the whole thread, and references given therein. Otherwise, I'm just rehashing things that have gone before.

The rest of your post is either speculative, or too far off-topic for this thread, so I won't pursue it.

 

I would not mind too much if SR could be derived from many more than one postulate: the real issue is the status of these postulates: are they symmetry principles (homogenity , isotropy, a relativiy principle) or are they something else that nobody would have expected and that looks just ugly such as the principle taht there exists an universal invariant speed , which then will be the speed of any staff with zero mass hence probably the speed of light if photons have indeed zero mass.

The reason why we like symmetry principles is not an arbitrary one : for instance the principle of invariance under spacetime translations just means that the laws are the same averywhere and any time and this has to do with the intuition that science itself is possible.

So i'm indeed interested in trying to avoid the second principle of SR: the existence of a universal invariant speed but not necessarily by trying to deduce everything from a single principle but rather by replacing this second principle (invariance of c) by something that more looks like a symmetry principle and would exclude the galilean group.

One usually agrees that eventually SR really unifies space and time : it mixes x and ct in just the same way i.e
in the transformation you can make a permutation of x and ct and at the same time x' and ct' and this will leave the transformation laws invariant : in the galilean case where c is infinite the previous sentence would make no sense at all and this is a reason why i believe galilean transformation cannot just be considered as a special case of Lorentz transformation with c infinite.

Now if i wanted to derive the lorentz transformation i would start by just a transformation transforming a given (x,ct) into another (x',c't') where c and c' are not necessarily equal but just needed from the begining becaus one cannot mix in a vector compnents which are not expressed in the same units , so the speeds c and c' are needed. Then i would impose a perturmation symmetry i.e invariance of the transformation laws under the simultaneous exchange of x and ct and at the sametime x' and c't' ... and from this i would demonstrate that necessarily c=c' if the transformation we are speaking about represents the physical transformation between two frames at constant relative speed. Actually it seems that it works and is quite straightforward and would allow to replace the speed of light constancy by a permutation principle...

 

Not in the same sense as isotropy , homogeneity or relativity principle : those symmetry principles are based on the idea that there is no privileged staff : all positions, speeds, directions must be treated in the same way. The permutation symmetry as i think about would have to be a kind of generalisation of isotropy to 4d : the transformation must treat on the same footing the fourth coordinate (ct) and the three others for instance x and ct in a boost along x.

You could notice that for instance even in simple rotations for instance about x, there is no such permutation symmetry under the exchange y <-> z and y' <-> z' but of course this is due to the fact that in this case the permutation symmetry must be accompanied by the angle reversal because rotations are physically oriented... so the invariance under permutation is actually also satisfied in the case of rotations which i believe is a consequence of isotropy.

Eventually the minkowskian vs the euclidian signature might be a consequence of a difference between oriented (boost) and non oriented transformation (rotation).

I would summarize in that way: suppose you already have 3 coordinates x, y, z then you add the fourth coordinate x4 ; there are two ways for x4 to satisfy a permutation symmetry when performing a transformation which mixes x4 with any of the 3 others : either a rotation (oriented) and then x4 is a spacelike coordinates as x,y,z , or a boost (non oriented) transformation and then x4 is a timelike coordinate with flipped signature.

Does all this makes sense ;-) i hope so, i'd like! waiting for other helpful critical comments

 

I still think the best approach for a postulate besides POR, isotropy, and homogeneity (which leaves exactly 2 choices: SR or Galilean relativity) is that you must pick some additional invariance:

t invariance -> Galilean relativity
c invariance -> SR

This is leaner than "laws of physics". The other major approach is you don't postulate at all. You do any experiment which will distinguish. For example, SR predicts you can keep muons in a storage ring for for some time if they are very energetic. Galilean relativity says you would never succeed in this given the low energy decay time of 2 microseconds. Of course, you wouldn't get very far building your accelerator or storage ring without using laws that already imply c invariance.

 

The second principle deals with the invariance of the speed of light. Since in the absence of gravitation SR must encompass all phenomena, this specific reference to the propagation of light in empty space is problematic insofar it artificially injects an asymmetry in SR foundations. What is missing is a different, more general, justification for the existence of an invariant speed c. I suggested a way forward in #70 : a physics theory which excludes instantaneous actions at a distance necessarily imposes the existence of a finite maximum speed. True or False?

What is your proposed justification for injecting a constant c in your equations? What role does it play?

 

Here an old post of mine from a now-closed thread on Galilean relativity:

https://www.physicsforums.com/showthread.php?p=4112167#post4112167

Essentially copied from my earlier self...

A maximum signal speed corresponds to a null-directed (in that metric) eigenvector.
If that signal speed is infinite, then spacelike[=orthogonal to timelike] coincides with null,
which leads to a "t-invariance".

This is in the spirit of the Cayley-Klein formulation of Euclidean and the constant-curvature non-euclidean geometries.


As I mentioned earlier in this thread, there is another approach which could be regarded as more primitive: using a causality axiom (the causal structure) to obtain the Minkowski spacetime
https://www.physicsforums.com/showpost.php?p=4751681&postcount=30

 

False. There is no contradiction between every interaction having some speed of propagation (different for different types of interaction, or perhaps different for different energy domains), and t invariance. To conclude a finite maximum speed, you must assume an upper bound on such speeds. Further, you must conclude no relative speed can exceed this upper bound (rather than assuming that objects with very large relative speed cannot interact).

 

Not specifically into the SR foundations, but into the common framework in which both SR and pre-relativistic classical mechanics are defined. That framework admits two different groups of functions that describe a coordinate change from one inertial coordinate system to another. To choose the group is to choose the theory.

Edit: I wouldn't use the term "asymmetry" either. It's simply a choice of which symmetries to include in the theory.

I don't think it can be some idea that can be arrived at purely by intellectual means. In the future, we may be able to think of this as a prediction made by some future theory. At the moment, I think we have to rely on experiments.

You will certainly have to limit your attention to some class of theories to make such an argument. If we focus on theories that use ##\mathbb R^4## as a model of space and time, and is consistent with (mathematical statements corresponding to) the principle of relativity, I suppose that your statement is true. But I don't think it's an improvement over the simple idea that we can use experiments to distinguish between the two possibilities.

The thing is, when we set out to find the group of functions that "translate" between inertial coordinate systems on ##\mathbb R^4##, that constant shows up without being "injected". You start out with several undetermined parameters, and then you find that you can get rid of all but one by using the principle of relativity and symmetry principles.

 

Isn't it the other way around? When the Lagrangian doesn't change under translations, translations are a symmetry of the Lagrangian. I can't think of a reason to call the invariant lines (or the speed they represent) a symmetry.

Edit: I think it's OK to call the group of functions that change coordinates from one inertial coordinate system to another a symmetry group, since the presence of that group in the theory reflects invariance properties of spacetime. So each of those coordinate change functions (Galilean or Poincaré transformations) can be considered a "symmetry".

 

As fas as i know, it never works that way: from a methematical point of view a genuine symmetry is a transformation that lets the laws of physics unchanged even though it might either transform or not various objects that enter the equations. (BTW c is not a scalar field, it is just a fundamental constant, so the invariance of c is also not the same kind of staff as the invariance of a scalar field). For instance once you have admitted the existence of a constant c and thanks to that you have your lorentz transformation, then you can ask the question : are my equations of physics going to be invariant under this transformation, if yes, this is a symmetry.

But of course behind the maths there is also the physical meaning of this invariance : it means that there is no privileged speed in the universe, in the sense that if you make an experiment on a table at rest in frame A and do the same experiment on a table at rest in frame B you will get the same results whatever the relative speed of A and B .

And the physical requirement is even more fundamental than the mathematical one which is just one of its translations in the language of maths: for instance the physical requirement of isotropy is not only translated in the invariance mathematical requirement according to which the laws of physics should be invariant under rotations but is also used to establish the Lorentz tranformations themselves for instance when one demands that the lorentz contraction should be the same for a boost at speed -u and for a boost at speed u (may be this is not used in all derivations of the Lorentz transformations but in the books i have studied it was). In both mathematical requirements there is behind the same and unic physical idea that should remain clear : we dont want any privileged direction so we must treat all of them in the same way.

The permutation symmetry is a kind of generalisation of this : i want to treat x and ct in the same way in the transformation: the physical idea is thus the same as the idea behind any other well known symmetry though the mathematical translation (invariance under permutation) is here a bit unusual.

 

c is not a priori a constant and it might be transformed into another c' in a boost: thus (x,ct) transforms (x',c't'). At the begining we dont know what is the meaning of c and c' but we know these should be there because one cannot add time and space coordinates, thus c and c' at the begining should merely be considered as conversion factors. But if i demand an invariance under permutation then the most general form of the boost that was:

x' =b_11 x + b_12 ct
c't' = b_21 x + b_22 ct

now must be

x' =b_1 x + b_2 ct
c't' = b_2 x + b_1 ct

and then i can derive c=c' and at the same time the usual values of b_1 and b_2 it's quite straightforward if i didnt make any mistake.

 

But , i was trying to explain exactly the same thing, please read again my previous message where i explain that a symmetry is actually more a physical idea which can have many possible translations in maths.

The real issue is that the constancy of c as a principle has nothing to do with a symmetry principle, physically speaking, though mathematically we might be illusioned by the fact that indeed there is a transformation and some thing invariant (c) under this transformation.

Again, since we dont see behind the constancy of c, the physical idea that is the genuine characteristic of a symmetry (no privileged staff) , it's not , physically speaking, a symmetry!

I's not a symmetry but it can be derived from a symmetry : permutation invariance because the requirement is then to treat x and ct in the same way (no privileged staff)

 

ok, if you are not impressed by my rationnal arguments , i might just remind you that you will hardly find any textbook in relativity saying that the constancy of c is a symmetry principle. And most of the time this is the reason why many people have tried to axiomatized SR just in the hope of avoiding the arbitrariness of demanding from the begining a constant c: some of them will make use of causality principles (to avoid action at a distance) for instance and exclude the galilean option ... it remains that i dont know many physicist which feel really confortable with the second principle of SR: most of them will just say: ok it's disturbing but the principles of physics are totally arbitrary after all: it's just experimental results that imposes us the principles, and we have to accept them wether we like them or not... (this is not my way of thinking as you may have understood)