Could SR not be built from only one postulate?

[guitarphysics]

Hello, I have a doubt regarding the postulates of SR.

The two postulates, according to Schutz, are:

1) No experiment can measure the absolute velocity of an observer; the results of any experiment performed by an observer do not depend on his speed relative to other observers who are not involved in the experiment.
2) The speed of light relative to any unaccelerated observer is c, regardless of the motion of the light's source relative to the observer.


Couldn't we combine these two postulates into one? I'm thinking of something along the lines of:

All the laws of physics are the same in every inertial frame of reference.


With this, I'm thinking you reach both of the original postulates- this new postulate implies that Maxwell's equations must hold in every inertial frame, so c must be the same in every inertial frame (which takes care of original postulate 2). Also, from this you can conclude that for an observer who is in an inertial frame of reference, the same laws of physics will hold as for another inertial observer moving at a different speed. Therefore, the first observer's experimental results will not be affected by their speed relative to the other observer (this takes care of original postulate 1).

What do you all think?

 

[Dale]

All the laws of physics are the same in every inertial frame of reference.


With this, I'm thinking you reach both of the original postulates- this new postulate implies that Maxwell's equations must hold in every inertial frame

Only if you also postulate that Maxwell's equations is a law of physics.

 

[guitarphysics]

Seriously? But that seems sort of superfluous to me; would it really be necessary?

 

[Matterwave]

Seriously? But that seems sort of superfluous to me; would it really be necessary?

Can you derive Maxwell's equations from your one postulate?

 

[homeomorphic]

That's just the usual theoretical argument that establishes postulate 2. It's just that we'd rather not mention Maxwell's equations in some contexts because then we can just start with your 2) and you don't have to know E and M to understand. So, nothing really new here.

 

[xox]

Hello, I have a doubt regarding the postulates of SR.

The two postulates, according to Schutz, are:

1) No experiment can measure the absolute velocity of an observer; the results of any experiment performed by an observer do not depend on his speed relative to other observers who are not involved in the experiment.
2) The speed of light relative to any unaccelerated observer is c, regardless of the motion of the light's source relative to the observer.Couldn't we combine these two postulates into one? I'm thinking of something along the lines of:

No, you cannot combine the two postulates into one but there are a lot of formulations of SR that drop the second postulate. You can do a google search for "single postulate formulation of SR". The most famous one dates from 1910(!), by Ignatowski.
A word of caution, SR is based on a lot more that the two postulates you listed, so , what you are really looking is for formulations that drop the principle of constancy of light speed.

 

[bcrowell]

You can certainly combine the two postulates into one. We all know that Maxwell's equations are a law of physics, so there's no need to state explicitly that they're included. If Schutz's #2 were violated, then his #1 would also automatically be violated.

There is nothing special or sacred about Einstein's 1905 axiomatization of SR. From the modern point of view, it's awkward and archaic.

There's a more detailed description of this sort of thing in ch. 2 of my SR book: http://www.lightandmatter.com/sr/

 

[guitarphysics]

Alright, thanks everyone for your responses! Homeomorphic, your argument made the most sense to me, thanks :) (Matterwave, I don't think you can derive Maxwell's equations from the two original postulates either, nor do I find it a necessary requirement of an SR postulate). And thank you xox, I looked a bit into it and it's all pretty interesting; and yeah, I did know about other assumptions made in SR (I'd seen in wiki before about things like the need to assume spatial homogeneity and isotropy).

 

[guitarphysics]

Oh sorry Ben, just saw your post. Thanks :). I'll think of the postulates of SR in terms of only one principle, then- it seems simpler.

Ps. Your book looks great! I might read it alongside Schutz and Hartle.

 

[Matterwave]

Alright, thanks everyone for your responses! Homeomorphic, your argument made the most sense to me, thanks :) (Matterwave, I don't think you can derive Maxwell's equations from the two original postulates either, nor do I find it a necessary requirement of an SR postulate). And thank you xox, I looked a bit into it and it's all pretty interesting; and yeah, I did know about other assumptions made in SR (I'd seen in wiki before about things like the need to assume spatial homogeneity and isotropy).

You can't derive Maxwell's equations from only the original 2 postulates. But my point was not that it could be. My point was that if you are getting rid of one postulate, and you state that Maxwell's equations can be used to get rid of that postulate, then you have to postulate Maxwell's equations as a substitute unless you can derive Maxwell's equations from postulate 1.

In more formal language. Let's start with postulates A and B (the 1 and 2 originally we had), and you claim that A+C (where C is Maxwell's equations) implies B, then certainly you can use A and C as your fundamental postulates. However, unless you can also show A implies C, you cannot reduce to just A.

In other words, my argument was that the following argument is invalid: "A+C implies B, therefore to derive all the derivable facts given A and B, I only need A".

 

[guitarphysics]

You can't derive Maxwell's equations from only the original 2 postulates. But my point was not that it could be. My point was that if you are getting rid of one postulate, and you state that Maxwell's equations can be used to get rid of that postulate, then you have to postulate Maxwell's equations as a substitute unless you can derive Maxwell's equations from postulate 1.

In more formal language. Let's start with postulates A and B (the 1 and 2 originally we had), and you claim that A+C (where C is Maxwell's equations) implies B, then certainly you can use A and C as your fundamental postulates. However, unless you can also show A implies C, you cannot reduce to just A.

In other words, my argument was that the following argument is invalid: "A+C implies B, therefore to derive all the derivable facts given A and B, I only need A".

But like Ben said, we already know Maxwell's equations and that they're laws of physics, so they're not required as an additional postulate. What I'm saying is we have postulates A and B, and an extra fact (Maxwell's equations) that isn't used in either postulate, but we know is true. So let's make a postulate C, which when considered along with that extra fact, will imply both A and B.

 

[Dale]

Oh sorry Ben, just saw your post. Thanks :). I'll think of the postulates of SR in terms of only one principle, then- it seems simpler.

You should probably actually read the chapter. It involves much more than 1 principle. I think it is 5 or so.

 

[Dale]

You can certainly combine the two postulates into one. We all know that Maxwell's equations are a law of physics, so there's no need to state explicitly that they're included.

The reason that I don't like this approach is because it is circular in motivation, if not in formulation.

If you just want to reduce the number of postulates you can always simply postulate the Lorentz transforms. But the point was to justify the Lorentz transforms on the basis of principles that physicists could be persuaded to accept.

The motivation for justifying the Lorentz transforms was that they were the symmetry group of Maxwell's equations. So including Maxwell's equations in the derivation (either directly or indirectly) makes the whole derivation silly. You may as well just state the fact that Maxwell's equations are invariant under the Lorentz transform and be done with it. That much was already recognized.

That said, your list is much cleaner and more thorough.

 

[guitarphysics]

You should probably actually read the chapter. It involves much more than 1 principle. I think it is 5 or so.

I did. And by thinking of the SR postulates in terms of one principle, I was referring to the original post (which combined the original two into one).

Anyway, from Ben's book, I think the postulates P2, P4, and P5 are all implied by the definition of an inertial frame (which isn't a postulate itself). P3 regarding the isotropy and homogeneity of space I had mentioned previously and I'm not thinking of it as a postulate of SR, because (I could be completely wrong, but) I think it's a postulate for all physical theories. So for now I'll just think of SR as a physical theory built from one postulate.

 

[Dale]

Sure, you can always build any theory from one postulate simply by postulating the theory with all of the underlying constructs. There is nothing wrong with that. Just think about your purpose in establishing a set of postulates (I understand Einstein's motivation, but I am not sure what yours is), and whether your choice of postulate accomplishes that purpose.

 

[Matterwave]

But like Ben said, we already know Maxwell's equations and that they're laws of physics, so they're not required as an additional postulate. What I'm saying is we have postulates A and B, and an extra fact (Maxwell's equations) that isn't used in either postulate, but we know is true. So let's make a postulate C, which when considered along with that extra fact, will imply both A and B.

But this is the same as postulating A and C (Maxwell's equations). I never said you could not do this. Just because you called it an "extra fact" and not a "postulate" does not mean you have removed it as a postulate...-.-

 

[guitarphysics]

Ok, so the purpose of the postulates in a theory is to basically be the basis from which you can form arguments and make conclusions. My point is that having the rest of physics as a base (i.e. Maxwell's equations, the assumption that space is homogenous and isotropic, etc.), the only *additional* postulate required by SR is what I said in the original question, as opposed to the two that are usually cited.

 

[xox]

And thank you xox, I looked a bit into it and it's all pretty interesting; and yeah, I did know about other assumptions made in SR (I'd seen in wiki before about things like the need to assume spatial homogeneity and isotropy).

Correct. the interesting postulate to omit is the principle of constancy of light. As you can see, it has been done.

 

[Matterwave]

But how do you know which physics to base your theory on? After all, if instead of Maxwell's equations, we took Newton's laws to be the "rest of physics as a base" you mention, we get the wrong answer.

 

[strangerep]

Guitarphysics,

Let's go back to your original question...

The two postulates, according to Schutz, are:

1) No experiment can measure the absolute velocity of an observer; the results of any experiment performed by an observer do not depend on his speed relative to other observers who are not involved in the experiment.
2) The speed of light relative to any unaccelerated observer is c, regardless of the motion of the light's source relative to the observer.

Couldn't we combine these two postulates into one? I'm thinking of something along the lines of:

All the laws of physics are the same in every inertial frame of reference.

Yes, but some other things are required (which I'll explain below).

With this, I'm thinking you reach both of the original postulates- this new postulate implies that Maxwell's equations must hold in every inertial frame,

No, it doesn't. The most general transformation that preserves inertial motion for a given observer (located at the origin of his coordinate system) is fractional linear, i.e., of the form:
$$t' = \frac{At + Bx}{Ct + Dx} ~,~~~~~ x' = \frac{Et + Fx}{Ct + Dx} ~.$$The most general transformations that preserves the Maxwell wave eqns are conformal transformations -- which have a quadratic denominator in general. See special conformal transformation.

If one asks for a common subset of transformations that do both, one is reduced to ordinary linear transformations. If one assumes spatial isotropy, and a principle of "physical regularity", (i.e., that physical transformation must map finite values of observables to finite values), then the usual Lorentz transformations can be derived without further assumptions, and a universal constant limiting speed (called "c") is an additional output of the derivation. By examining the properties of material bodies whose relative speed is very close to "c", and taking a limit, one can deduce properties that coincide with those usually observed in light. I.e., one can use experiment to identify that "c" corresponds to lightspeed.

So let us drop the assumption that inertial-motion-preserving transformations ("IMTs") should also preserve Maxwell's eqns. Long ago, Bacry and Levy-Leblond[1] figured out that the most general such algebras (larger than the Poincare algebra) are the deSitter algebras, and an additional universal constant with dimensions of length^2 is a further output of the derivation. This has lead to a modern exploration of ways to use this method to "derive" the cosmological constant $\Lambda$ -- since that's essentially what GR without matter boils down to: a deSitter universe.

Others have approached it in different ways. Kerner[2a,2b], and more recently Manida[3a,3b], explored different, more physically-motivated, generalizations -- by seeking the most general form of IMT that could reasonably be interpreted physically as a velocity boost. They arrived at deSitter geometries (surprise, surprise).

In these approaches, the local speed of light is still the usual "c", and Poincare-invariance is retained up to distance scales where cosmological effects become significant. Indeed, the apparent speed of light can vary over (large) times and distances -- but this is already familiar in cosmology, arising from expansion of space over time.

Buried within these approaches are different assumptions about time-reversal invariance. Bacry and Levy-Leblond assumed it explicitly. Manida initially didn't assume it, but later returned to it by embracing deSitter algebras. A slightly more general approach (relaxing the tacit demand for a co-moving transformed frame) might also be possible -- but that's not yet published (afaik) so I can't talk about it on PF.

References:

[1] H. Bacry, J.-M. L\'evy-Leblond, "Possible Kinematics",
J. Math. Phys., vol 9, no 10, 1605, (1968)

[2a] E. H. Kerner,
An extension of the concept of inertial frame and of Lorentz transformation,
Proc. Nat. Acad. Sci. USA, Vol. 73, No. 5, pp. 1418-1421, May 1976

[2b] E. H. Kerner,
Extended inertial frames and Lorentz transformations. II.
J. Math. Phys., Vol. 17, No. 10, (1976), p1797.

[3a] S. N. Manida,
Fock-Lorentz transformations and time-varying speed of light,
Available as: arXiv:gr-qc/9905046

[3b] S. N. Manida,
Generalized Relativistic Kinematics,
Theor. Math. Phys., vol 169, no 2, (2011), pp1643-1655.
Available as: arXiv:1111.3676 [gr-qc]

 

[bcrowell]

But how do you know which physics to base your theory on? After all, if instead of Maxwell's equations, we took Newton's laws to be the "rest of physics as a base" you mention, we get the wrong answer.

This is a nice way of stating what's unsatisfactory about Einstein's 1905 axiomatization. It assumes the state of the art in 1905, which was that there were two main theories of physics: Newton's laws and Maxwell's equations. If you want an axiomatization that reads more like the modern view of how relativity works -- as a theory of the geometry of spacetime -- then you probably want something more like Ignatowsky's axiomatization.

 

[guitarphysics]

strangerep, thanks very much for your detailed explanation, and the references! Unfortunately, I don't know much algebra so there's some of what you said that was beyond me, not to mention the papers you referenced (I could follow them pretty much through the introduction but nothing more :\ ). I had heard a bit about de Sitter and anti-de Sitter space, but didn't know what it was about. You made that a bit clearer for me, so thanks for that as well!

Matterwave, that's a good point- there's no guarantee that the current physics is correct either, so it would probably be better for *every* postulate of SR to be stated (like in Ben's book- again, Ben thanks for that, it looks like a very refreshing take on SR :D).

Thanks for the interesting responses everyone, you've given me a lot to think about.

 

[Fredrik]

All the laws of physics are the same in every inertial frame of reference.


With this, I'm thinking you reach both of the original postulates- this new postulate implies that Maxwell's equations must hold in every inertial frame, so c must be the same in every inertial frame

I like Matterwave's reply (post #19) the best. You could also say that it implies that the velocity of a massive particle influenced by a constant force must satisfy the formula $v=(F/m)t+v_0$ in every inertial coordinate system. This implies that c is not the same in every inertial coordinate system.

I have a lot more to say about this subject, but unfortunately I don't have time right now.

 

[atyy]

Ok, so the purpose of the postulates in a theory is to basically be the basis from which you can form arguments and make conclusions. My point is that having the rest of physics as a base (i.e. Maxwell's equations, the assumption that space is homogenous and isotropic, etc.), the only *additional* postulate required by SR is what I said in the original question, as opposed to the two that are usually cited.

As Matterwave, DaleSpam and others have pointed out - you need to specify what the "laws of physics" are. If the laws of physics include Maxwell's equations then 2 is contained in 1. If Maxwell's equations are not in the "laws of physics" then 2 is not contained in 1.

The reason that Maxwell's equations are stated explicitly in 2 in most books is that for many years between Newton and Maxwell, Maxwell's equations were not in the "laws of physics". For example, if Newton's universal gravitation but not Maxwell's equations are in the "laws of physics", then 1 alone would produce Galilean relativity.

Whether you want to count the axioms as 1 or 2 is a matter of taste, depending on what you include in the "laws of physics".

 

[xox]

As Matterwave, DaleSpam and others have pointed out - you need to specify what the "laws of physics" are. If the laws of physics include Maxwell's equations then 2 is contained in 1. If Maxwell's equations are not in the "laws of physics" then 2 is not contained in 1.

The reason that Maxwell's equations are stated explicitly in 2 in most books is that for many years between Newton and Maxwell, Maxwell's equations were not in the "laws of physics". For example, if Newton's universal gravitation but not Maxwell's equations are in the "laws of physics", then 1 alone would produce Galilean relativity.

Whether you want to count the axioms as 1 or 2 is a matter of taste, depending on what you include in the "laws of physics".

The above is an excellent way of stating the answer.

 

[guitarphysics]

Yeah, I liked that. But then there's the problem of subtleties that Ben mentioned in his book, isn't there?
I think I prefer strangerep's way of looking at it best; as far as I can tell, it doesn't rely on the rest of physics, and is very precise. However, it seems like a much more mathematical way of looking at the problem and I feel there should be more physics (like in Ben's approach).

 

[atyy]

Yeah, I liked that. But then there's the problem of subtleties that Ben mentioned in his book, isn't there?
I think I prefer strangerep's way of looking at it best; as far as I can tell, it doesn't rely on the rest of physics, and is very precise. However, it seems like a much more mathematical way of looking at the problem and I feel there should be more physics (like in Ben's approach).

Yes. The most modern way is to simply postulate that Poincare symmetry is a symmetry of the laws of physics (whatever those may be).

However, it is worth remembering the old ways, especially because the Principle of Relativity ("axiom 1" in the old way) goes all the way back to Galileo, and is still very useful. It says one can drink coffee in an aeroplane that is moving very fast, just as well as on the ground.

Also, in Einstein's formulation of general relativity, the Principle of Relativity can be said to fail as a global principle, but hold as a local principle. This "hold as a local principle" is the Principle of Equivalence, which again goes back to Galileo: bodies of different masses (as long as their mass is small relative to the earth's) will fall and reach the ground at the same time.

So GR can be seen as reconcilation of 2 important "principles" of Galileo, and the fact that both Newtonian gravitation and Maxwell's equations are "laws of physics" in some regime.

Edit: I'm not sure I agree with Ben's criticism of Einstein postulates in 2.4.1. of http://www.lightandmatter.com/sr/. It is true that special relativity can be handle accelerated frames. However, one can think of the first postulate as stating the existence of global inertial frames. Stating the postulate in this way does not depend on not being able to handle accelerated frames, but merely states the existence of a special class of frames which we call "inertial". I feel that Ben's criticism based on accelerated frames is not valid criticism of Einstein's SR postulates, but is valid criticism of Einstein's (initial) postulates for GR.

I do agree with Ben's criticism that "the speed of light is the same in all inertial frames" is a slightly less general postulate than is possible, in the sense that if the photon were found to have a mass, then the speed of light would not be the same in all inertial frames. However, special relativity (Poincare symmetry of the laws of physics) could still hold, even if the photon were found to have a mass.

Roughly speaking, the Principle of Relativity says global inertial frames exist. However, we know that Newton's law of gravitation also obeys the Principle of Relativity - Galilean relativity. So to specify special relativity, we must add either (a) Maxwell's equations are a law of physics, or (b) speed of light if the same in all inertial frames, or (3) Poincare symmetry is the symmetry of the laws of physics.

 

[Meir Achuz]

I just noticed this thread. The best single postulate for relativity was almost that of Galileo in 1632:
"any two observers moving at constant speed and direction with respect to one another will obtain the same results for all mechanical experiments".
If he hadn't been so careful to include the word "mechanical", his quote would include special relativity.
Removing that word leaves a better single postulate than the two of Einstein.
Since measuring the speed of light is an experimental measurement, no additional postulate is necessary.

 

[PAllen]

I just noticed this thread. The best single postulate for relativity was almost that of Galileo in 1632:
"any two observers moving at constant speed and direction with respect to one another will obtain the same results for all mechanical experiments".
If he hadn't been so careful to include the word "mechanical", his quote would include special relativity.
Removing that word leaves a better single postulate than the two of Einstein.
Since measuring the speed of light is an experimental measurement, no additional postulate is necessary.

No, you are making the assumption that c is frame invariant. Newton would have proposed that, like bullets, light speed would be frame variant in precisely the right way for the Galilean transform to be valid. You cannot arrive at the Lorentz transform without some additional fact. You could add the second postulate as an experimental fact (circa 1900, not much earlier) rather than a postulate, but either way it has to be added (or something equivalent, e.g. Maxwell's equations). Let me add, since the speed of bullets is arbitrary, you would have to go from measurements seeming to come out the same for light to proposing that it is (or follows from) a law that it is constant.

 

[robphy]

Probably not along the OP's line of thinking...
but quite interesting (and is motivation for some approaches to quantum gravity)

http://en.wikipedia.org/wiki/Alfred_Robb and his "after" (causality) relation

http://books.google.com/books?id=io...a=X&ei=4IN6U5_9KMq2yATjwYHYCA&ved=0CEoQ6AEwBg

http://www.mcps.umn.edu/assets/pdf/8.7_Winnie.pdf
http://www.renyi.hu/conferences/nemeti70/LR12Talks/lefever.pdf

 

[guitarphysics]

No, not along my line of thinking but interesting nevertheless- at least the parts that I could understand.

Nice 'signature', by the way :D.

 

[strangerep]

I think I prefer strangerep's way of looking at it best; as far as I can tell, it doesn't rely on the rest of physics, and is very precise. However, it seems like a much more mathematical way of looking at the problem and I feel there should be more physics (like in Ben's approach).

Well, the physical content (using only a very small set of intuitive assumptions), and its mathematical development proceed along the following lines:

Following Rindler [1],

Postulate 1 (Principle of Relativity):
The laws of physics are identical in all inertial frames,
or, equivalently,
the outcome of any physical experiment is the same when performed with identical initial conditions relative to any inertial frame.

This requires a definition of inertial frames. Still following Rindler, an Einsteinian inertial frame is a reference frame in which spatial relations, as determined by rigid scales at rest in the frame, are Euclidean and in which there exists a universal time in terms of which free particles remain at rest or continue to move with constant speed along straight lines (i.e., in terms of which free particles obey Newton's first law).

The boundary between the physics and the maths lies in this: an observer can reasonably possesses local length scales (a very short rigid rod), and a local clock (measuring short time intervals). That much is physical. One then imagines that the rod could be successively laid end-over-end indefinitely to create a spatial coordinate grid. Similarly, one imagines that the clock could be duplicated endlessly, with the duplicates moved to spatially remote locations.

The (abstract) space of dynamical parameters needed to describe such an arrangement is assumed to correspond to (possibly a subspace of) $R^4$, i.e., 3 space and 1 time. Similarly, velocities are assumed to correspond to (possibly a subspace of) $R^3$. Thus we imagine a 7-dimensional velocity extended phase space of parameters. (We need not extend any further to higher dimensional phase spaces involving acceleration, jerk, etc, since the requirement of inertial motion restricts acceleration to be zero.)

These imaginings are made more precise using group theory. E.g., the basic physical spatial displacement defined by the rod is expressed in terms of a transformation of these dynamical parameters. Demanding that such transformations preserve inertial motion (i.e., that zero acceleration is mapped to zero acceleration), and that they form a Lie group, one can derive quite strong restrictions for the possible form that such transformations may take.

It's similar for temporal displacements, and velocity boosts (preserving the origin). One takes the basic physical operation of a small temporal displacement expressed via a local clock, or a velocity difference (holding the spatiotemporal origin invariant), expresses these as transformations, and imposes the same group theoretic requirements when composing multiple such transformations.

Summarizing, the physical content consists of the concept of inertial motion of an observer, and the availability of means for measuring very local spatial displacements, very local temporal delay, and relative velocities of other such observers who are momentarily at the 1st observer's origin. One also assumes spatial isotropy: that there is no preferred spatial direction.

Then one asks for the most general transformation of the parameters in the (abstract) velocity phase space which preserve the condition of zero acceleration. That encapsulates all the ways in which 2 inertial observers could be labelling the 7-dimensional abstract phase space in "different" (though physically equivalent) ways. The parameters of the transformation between 2 such observers may then be identified as the physically meaningful relative observables (spatial displacement, temporal delay, relative velocity, spatial orientation, etc). One naturally adopts an additional principle of "physical regularity": that finite values of these relative observables must be mapped by the transformations to finite values. That restricts the allowable transformations a bit further.

Then it's just a matter of grinding through the math of Lie group theory applied to such transformations to find the most general possibility, as I described earlier. The method is firmly grounded in realistic physics, which can be expressed quite concisely. The detailed math is extensive, of course, taking many pages if one performs all calculations explicitly. But overall it's a good thing: from a small set of physical concepts based on intuitive local operations, one derives an extensive theory. The possibility of an invariant limiting speed constant, and an invariant length constant, emerge as derived consequences. Only their values need be determined experimentally.References:

[1] W. Rindler, Introduction to Special Relativity,
Oxford University Press, 1991 (2nd Ed.), ISBN 0-19-85395-2-5.

 

[Fredrik]

I will briefly describe my view on the postulates and "derivations" of the Lorentz transformation here.

The postulates are nowhere near as significant as most texts will make you think. It should be emphasized that they're not even part of the theory. SR is defined by (purely mathematical) definitions of terms like "Minkowski spacetime" and "proper time", and a few correspondence rules that tell us how to interpret the mathematics as predictions about results of experiments.

The "derivations" of the Lorentz transformation that start with the postulates are certainly interesting and fun, but they shouldn't be viewed as proofs. They should be viewed as ways to guess how to define a new theory, or rather, a new framework in which to define theories. Once we have defined the mathematics of the theory properly, we can prove theorems that resemble the postulates.

The proper way to turn the "derivations" into actual derivations (i.e. proofs) is to first interpret the postulates as mathematical statements. Then you can take those statements as the starting point of a proof. The question is, what are we really proving? There's no obvious answer to the question of what mathematical statement best corresponds to the principle of relativity. So we still won't be able to say that we have (rigorously) derived the Lorentz transformation from the postulates. We have derived it from one mathematical interpretation of the postulates.

Because of this, I prefer to do those "derivations" in a way that's not completely rigorous, and to use language that indicates what parts of the argument are really just clever guesses. See e.g. this post. (Start reading at the line that starts with "The explicit". The "numbered statements" that I'm referring to in that post are the postulates).

I also think that some of the theorems (with rigorous proofs) that take a mathematical interpretation of the principle of relativity as a starting point are very interesting. The ones I've looked at can be interpreted as saying that SR and Newtonian mechanics are the only possible theories of physics in which $\mathbb R^4$ is the underlying set of spacetime (the mathematical structure that represents real-world space and time), and inertial coordinate systems can be defined on that set.

 

[Sugdub]

... Then one asks for the most general transformation of the parameters in the (abstract) velocity phase space which preserve the condition of zero acceleration. That encapsulates all the ways in which 2 inertial observers could be labelling the 7-dimensional abstract phase space in "different" (though physically equivalent) ways. The parameters of the transformation between 2 such observers may then be identified as the physically meaningful relative observables (spatial displacement, temporal delay, relative velocity, spatial orientation, etc). One naturally adopts an additional principle of "physical regularity": that finite values of these relative observables must be mapped by the transformations to finite values. That restricts the allowable transformations a bit further...

It would seem that the physical world can be split into two classes of objects: those which have an inertial state of motion and those which have a non-inertial state of motion. Neglecting gravitational effects, the Lorentz transformation of SR could be essentially derived from two postulates:

1) the difference between the inertial and non-inertial states of motion is absolute, so that any "7-dimensional abstract phase space" which matches the boundary between both classes of objects (i.e. which correctly assigns its inertial or non-inertial state of motion to any object) can be said providing an “inertial frame of reference”;

2) all inertial frames of reference are physically equivalent, so that the laws of physics (i.e. the internal rules belonging to a physics theory) are invariant through a change of the inertial frame of reference.

Is this correct?

 

[PAllen]

It would seem that the physical world can be split into two classes of objects: those which have an inertial state of motion and those which have a non-inertial state of motion. Neglecting gravitational effects, the Lorentz transformation of SR could be essentially derived from two postulates:

1) the difference between the inertial and non-inertial states of motion is absolute, so that any "7-dimensional abstract phase space" which matches the boundary between both classes of objects (i.e. which correctly assigns its inertial or non-inertial state of motion to any object) can be said providing an “inertial frame of reference”;

2) all inertial frames of reference are physically equivalent, so that the laws of physics (i.e. the internal rules belonging to a physics theory) are invariant through a change of the inertial frame of reference.

Is this correct?

For (2), the result you get depends on what laws of physics you include. If you use Newtonian machanics, you get the Galilean transform from these assumptions. If you use Maxwell's equations, you get the Lorentz transform. Note, part of the genesis of SR was working out what dynamical laws of mechanics replace Newtonian mechanics. If you include as the laws of physics Newtonian mechanics + Maxwell, you have a self contradictory system.