Question on linearity of Lorentz transformations

[facenian]

Hello.The way the transformation of coordenates in Special Relativity are ussually derived presuposes linearity or try do demostrate such linearity using wrong arguments. For example some authors state that since linear and uniform motion remains linear and uniform after the transformation this fact imposes linearity, however this simply is not true as is demostrated(for instance) in J. Aharoni's "The Special Theory of Relattivity" where he shows a particular not linear transformation which transforms a uniform motion along a straight line into a similar kind of motion. Other authors simply state that the principle of relativity+homogeinity and isotropy of space-time imposes linearity but don't give details of how this come about.
Is there a correct,rigorous demostration for linearity from first principles, in which case where can one find it ?

 

[quZz]

Linearity is in fact a consequence of homogeneity.
Space and time intervals between two points should not depend on a particular point, so in
dx'^i = \frac{\partial x'^i}{\partial x^k} dx^k
matrix \partial x'^i/\partial x^k does not depend on x.

 

[George Jones]

Try

Zeeman, E. C. "Causality Implies the Lorentz Group", Journal of Mathematical Physics 5 (4): 490-493; (1964).

 

[Fredrik]

The way the transformation of coordenates in Special Relativity are ussually derived presuposes linearity or try do demostrate such linearity using wrong arguments.

I agree. The traditional "derivation" of the Lorentz transformation is mostly BS. It's not really a derivation at all. I've been complaining about that in lots of threads already, so I won't repeat everything here, but consider e.g. the claim that the principle of relativity implies that the group of functions that represent a change of coordinates from one inertial frame to another must be either the Galilei group or the Poincaré group. What we're really doing there is to take an ill-defined statement (the principle of relativity) and interpret it as representing a set of well-defined statements. Then we find out which of the well-defined statements that are consistent with all the other assumptions that we want to make. (One of those assumptions is linearity).

This is of course a perfectly valid way to find a set of statements that we can take as the axioms of a new theory, but to call it a "derivation" is preposterous.

Is there a correct,rigorous demostration for linearity from first principles, in which case where can one find it ?

No.

 

[Daverz]

Noel Doughty, in his book Lagrangian Interaction, refers to a paper by Mariwalla, Uniqueness of classical and relativistic systems, Phys. Lett. A79 143-146.

"Mariwalla shows that this result [linearity of a boost] may in fact be established without appealing to homogeneity. This follows because there are only three distinct 1-parameter groups acting in one spatial dimension [i.e. along the direction of the boost], and all imply linear transformation equations." (Bracketed comments are mine.)

He also refers to derivations of the LTs from Am. Journ. Phys. 43 434-437 and 44 271-277.

 

[atyy]

Details of what Fredrik said can be found in section 2.6 of Giulini's http://arxiv.org/abs/0802.4345

 

[atyy]

The traditional derivation is not BS if one does not add the conclusion that the Lorentz transformations are the only way that works.

Edit: Would anyone feel better if instead of "traditional derivation", we said "traditional construction"?

 

[meopemuk]

Usual linear Lorentz transformations are applicable to coordinates of non-interacting particles (with linear uniform motion). However, they are not valid for coordinates of particles interacting with each other (if this interaction is described by Poincare-invariant Hamiltonian dynamics). This has been proved a long time ago

Currie, D. G.; Jordan, T. F.; Sudarshan, E. C. G., Relativistic invariance and
Hamiltonian theories of interacting particles, Rev. Mod. Phys. 35, 350-375
(1963).

http://puhep1.princeton.edu/~mcdonald/examples/mechanics/currie_rmp_35_350_63.pdf

 

[atyy]

Usual linear Lorentz transformations are applicable to coordinates of non-interacting particles (with linear uniform motion). However, they are not valid for coordinates of particles interacting with each other (if this interaction is described by Poincare-invariant Hamiltonian dynamics). This has been proved a long time ago

Currie, D. G.; Jordan, T. F.; Sudarshan, E. C. G., Relativistic invariance and
Hamiltonian theories of interacting particles, Rev. Mod. Phys. 35, 350-375
(1963).

http://puhep1.princeton.edu/~mcdonald/examples/mechanics/currie_rmp_35_350_63.pdf

"two particles (not two particles and a field)"

 

[meopemuk]

"two particles (not two particles and a field)"

Yes, Currie-Jordan-Sudarshan theorem has been proven only for directly interacting particles. If you believe that particles interact via some "field" mediation, then the CJS result does not apply directly. However, the idea of their proof suggests that even in the case of field-mediated interactions Lorentz transformations of particle observables must be non-linear and interaction-dependent.

The idea is that in any Poincare-invariant interacting theory boosts are represented by interaction-dependent operators in the Hilbert space (or their appropriate analogs in the classical phase space). Therefore, it seems very likely that the action of boosts on particle observables must be interaction-dependent (rather than given by universal linear Lorentz transformations).

In any case, it seems more logical to derive boost transformations of particle observables from underlying dynamical theory rather than postulate them from the beginning, as has been done in special relativity.

 

[facenian]

Thank you for your answers.
First I want tell quZz that what he said is not correct otherwise the time inteval between two events would be the same for all observers which is what happens in Newtonian physics.
I see there is a little controversy, Fredrik says there is no such rigorous derivation and ohers cite papers dealing with such derivations. Regretfully I don't have access to papers, I will try the link atyy suggested though
Can someone tell me what BS stands for?
By the way I see the discussion went beyound SR

 

[robphy]

Try

Zeeman, E. C. "Causality Implies the Lorentz Group", Journal of Mathematical Physics 5 (4): 490-493; (1964).

Along those lines, there are also the theorems of A.D. Alexandrov
http://books.google.com/books?hl=en...ts=Gx4d8imeP5&sig=YZME-oNHVpD5hf-iwkWFov7Fsp8 [see theorem 1 and its corollary] (1967, with reference to one from 1953).

Of possible interest are the constructions of http://en.wikipedia.org/wiki/Alfred_Robb" ,
e.g. http://books.google.com/books?id=vp...YeY2EI&sa=X&oi=book_result&ct=result&resnum=6
http://www.archive.org/details/theoryoftimespac00robbrich (1914)

 

[facenian]

I'm sorry what I told quZz about why he is wrong is not correct. In fact I would like to know why his argument is incorrect in case it is.

 

[facenian]

I think I now know the answer why quZz's argument is wrong, what SR tells us is that only
ds^2=g_{ik}dx_idx_k have absolute meaning not the dx_i individually, what should be independent of space-time coordenates is ds^2 \,not\, dx_i

 

[Sam Park]

Some authors state that since linear and uniform motion remains linear and uniform after the transformation this fact imposes linearity, however this simply is not true as is demostrated(for instance) in J. Aharoni's "The Special Theory of Relattivity" where he shows a particular not linear transformation which transforms a uniform motion along a straight line into a similar kind of motion.

I assume when you wrote that Aharoni's non-linear transformation transforms "a" uniform motion along "a" straight line into a similar kind of motion, that you meant to say it transforms ALL uniform motions along straight lines into similar kinds of motion. This is what is required of a transformation between inertial coordinate systems, so your statement wouldn't make sense unless you replace A with ALL.

For those of us who don't have access to Aharoni's book, can you state the non-linear transformation that transforms ALL uniform motion in straight lines into similar kinds of motion?

 

[facenian]

you may be right perhaps I shouldn't have said "a uniform" pardon my english. The transformation equations are :
x_i^{'}=\frac{a_{ik}x_k+b_i}{c_kx_k+d},\quad i=0,1,2,3
where we've used Eintein's convention for repeated indices. Note that the denominator
does not depend on index "i".

 

[matheinste]

A derivation is given using the product of ""purely kinematic and non-kinematic transformations"" in:-

Torretti, Relativity and Geometry (1984). Chapter3 section 3.4--The Lorentz Transformation. Einstein's derivation of 1905.

Matheinste

 

[Fredrik]

Can someone tell me what BS stands for?

I recommend www.urbandictionary.com.

I see there is a little controversy, Fredrik says there is no such rigorous derivation and ohers cite papers dealing with such derivations.

There are plenty of rigorous mathematical arguments that end with the Lorentz transformation, but what do they prove, really? They always start with assumptions that are so strong that you might as well have started with the Lorentz transformation right away.

The "standard" derivation that I'm talking about is the one that starts with Einstein's postulates. You definitely can't prove anything from Einstein's postulates, since they are ill-defined. The biggest problem with them is that they talk about inertial frames as if that concept has been defined already.

Edit: Would anyone feel better if instead of "traditional derivation", we said "traditional construction"?

I don't think that works either. What I'd like to see is a clear statement at the start of the "derivation" that says that we're just trying to guess what might be appropriate axioms for a new theory.

 

[meopemuk]

There are plenty of rigorous mathematical arguments that end with the Lorentz transformation, but what do they prove, really? They always start with assumptions that are so strong that you might as well have started with the Lorentz transformation right away.

Hi Fredrik,

I agree with you that existing "derivations" of Lorentz transformations are inadequate. Here are some examples that I am talking about:

H. M. Schwartz, "Deduction of the general Lorentz transformations from a set of necessary assumptions", Am. J. Phys. 52 (1984), 346.

J. H. Field, "A new kinematical derivation of the Lorentz transformation and the particle description of light", Helv. Phys. Acta 70 (1997), 542.
http://www.arxiv.org/abs/physics/0410062

R. Polishchuk, "Derivation of the Lorentz transformations",
http://www.arxiv.org/abs/physics/0110076

D. A. Sardelis, "Unified derivation of the Galileo and the Lorentz transformations", Eur. J. Phys. 3 (1982), 96

J.-M. Levy-Leblond, "One more derivation of the Lorentz transformation", Am. J. Phys. 44
(1976), 271.

A. R. Lee and T. M. Kalotas, "Lorentz transformations from the first postulate", Am. J. Phys.
43 (1975), 434.

In my opinion, their major flaw is that they are not applicable to systems of interacting particles. For example, if one uses the 2nd Einstein's postulate (the constancy of the speed of light), then the obtained Lorentz transformations can be logically concluded to be valid for events associated with light pulses only. If one uses the uniformity and linearity of free moving particles, then there is no guarantee that obtained Lorentz transformations will remain valid for interacting particles whose movement in non-uniform and non-linear.

Fundamentally, we are interested in boost transformations of particle observables (positions, momenta, etc.). In quantum mechanics, such transformations can be unambiguously calculated by applying the operator of boost to particle observables. So, in order to rigorously derive Lorentz transformations one first needs to build a Poincare-invariant dynamical theory of interacting particles. In relativistic quantum mechanics this means construction of 10 Hermitian operators (that correspond to 10 generators of the Poincare group and satisfy commutation relations of the Poincare Lie algebra), which represent total observables of energy, momentum, angular momentum and boost. Then applying the unitary boost operator to observables of individual particles, we can calculate the boost transformation formulas.

If we follow this prescription, we can immediately realize that universal linear Lorentz transformations cannot be obtained in the interacting case, because it is well-known that the total boost operator must be interaction-dependent. So, Lorentz transformations of special relativity should be regarded as an approximation acceptable only for weakly interacting particles.

 

[quZz]

A lot of interesting links =)

I think I now know the answer why quZz's argument is wrong, what SR tells us is that only
ds^2=g_{ik}dx_idx_k have absolute meaning not the dx_i individually, what should be independent of space-time coordenates is ds^2 \,not\, dx_i

you misunderstood me. I'm not saying that dxⁱ are absolute (have the same values for all observers), that's incorrect. The thing is that the value of dx' depend only on the value of dx but not on x itself, that's what homogeneity is all about. Same dx, different x -> same dx'.

 

[quZz]

You definitely can't prove anything from Einstein's postulates, since they are ill-defined.

What's wrong with them?

 

[Sam Park]

If we follow this prescription, we can immediately realize that universal linear Lorentz transformations cannot be obtained in the interacting case, because it is well-known that the total boost operator must be interaction-dependent. So, Lorentz transformations of special relativity should be regarded as an approximation acceptable only for weakly interacting particles.

Well, we know that Lorentz invariance fails globally in general relativity, but it remains valid locally. Are you saying there is evidence of violation of Lorentz invariance locally? Many people have searched for such a thing, but as far as I know, there is no evidence that local Lorentz invariance ever fails. So I'm not sure what you mean when you say Lorentz transformations are only approximate.

 

[meopemuk]

Well, we know that Lorentz invariance fails globally in general relativity, but it remains valid locally. Are you saying there is evidence of violation of Lorentz invariance locally? Many people have searched for such a thing, but as far as I know, there is no evidence that local Lorentz invariance ever fails. So I'm not sure what you mean when you say Lorentz transformations are only approximate.

Unfortunately, Lorentz transformations themselves cannot be directly measured. However, we can measure some of their consequences, e.g., the slowing-down of the decays of moving particles. Rigorous calculations show that this slowdown deviates from the Einstein's time dilation formula. However the deviations are several orders of magnitude smaller than the accuracy of experiments.

E. V. Stefanovich, "Quantum effects in relativistic decays", Int. J. Theor. Phys. 35 (1996), 2539. http://www.geocities.com/meopemuk/IJTPpaper.html

M. I. Shirokov, "Decay law of moving unstable particle", Int. J. Theor. Phys. 43 (2004), 1541

M. I. Shirokov, "Evolution in time of moving unstable systems", Concepts of Physics 3
(2006), 193. http://www.arxiv.org/abs/quant-ph/0508087

E. V. Stefanovich, "Violations of Einstein's time dilation formula in particle decays", http://www.arxiv.org/abs/physics/0603043

 

[Saw]

Hmm… Meopemuk, I think we exchanged through email some years ago. Now my understanding of SR is a little better and I am arriving at this conclusion, which I think is in line with yours:

- SR postulates that all clocks, also mechanical clocks, suffer the same TD effect. But in principle the classical reasoning seemed logically flawless. If two observers moving wrt each other, when they meet, shoot down two bullets from their respective guns, it seems they should arrive at their respective targets at the same absolute time. The bullets were stationary with their respective holders before they were shot, since they are moving inertially with them; when they are shot, they are shot by guns at rest with their respective frames and, finally, their motion is measured in each case against the reference of the corresponding frame…
- Light is different, ok, because it does not take the motion of the source, but then it seems that the TD effect should apply to light clocks, not to mechanical clocks, whose ticker does take the motion of the source.
- If the classical reasoning fails, as it does, since experiments prove it, it must be because it is actually flawed and the only reason I can think of is that it disregards the importance of the interaction causing the bullet to accelerate. And if this interaction is to be relevant in the sense required by SR, it must be because the same bears a resemblance with light: what causes the acceleration of the bullet is an electromagnetic interaction. There is a light clock at the heart of the interaction that makes a mechanical clock tick.
- In any case, this reasoning assumes that the electromagnetic interaction is pure, develops in an ideal way that is immune to physical circumstances, which might vary from case to case. If we could account for that with very precise instruments, we might find out that the rules are more complicated, in accordance with certain patterns…

Is that more or less in line with what you hold? Of course, these are bold speculations, at least on my side. (And if I were thus breaking forum rules, please anyone in charge let me know. I enjoy the forum too much to be banned…)

 

[facenian]

A lot of interesting links =)


you misunderstood me. I'm not saying that dxⁱ are absolute (have the same values for all observers), that's incorrect. The thing is that the value of dx' depend only on the value of dx but not on x itself, that's what homogeneity is all about. Same dx, different x -> same dx'.

Yes quZz, I yust don't want to accept it could be that simple there mus be something wrong with it.
I want to tell Fredrik that I did not ask for somthing that rigorous. I yust wanted that some one tell me how first principles(relativity principle,homogeneity/isotropy of space-time) lead to Lorentz transformations,by the way according to L. D.Landau(Volumen I)
an inertial frame is one in which space is homogeneous and isotropic and time is homogeneus

 

[Fredrik]

What's wrong with them?

I mentioned their biggest problem immediately after the text you quoted.

 

[meopemuk]

Hmm… Meopemuk, I think we exchanged through email some years ago. Now my understanding of SR is a little better and I am arriving at this conclusion, which I think is in line with yours:

- SR postulates that all clocks, also mechanical clocks, suffer the same TD effect. But in principle the classical reasoning seemed logically flawless. If two observers moving wrt each other, when they meet, shoot down two bullets from their respective guns, it seems they should arrive at their respective targets at the same absolute time. The bullets were stationary with their respective holders before they were shot, since they are moving inertially with them; when they are shot, they are shot by guns at rest with their respective frames and, finally, their motion is measured in each case against the reference of the corresponding frame…
- Light is different, ok, because it does not take the motion of the source, but then it seems that the TD effect should apply to light clocks, not to mechanical clocks, whose ticker does take the motion of the source.
- If the classical reasoning fails, as it does, since experiments prove it, it must be because it is actually flawed and the only reason I can think of is that it disregards the importance of the interaction causing the bullet to accelerate. And if this interaction is to be relevant in the sense required by SR, it must be because the same bears a resemblance with light: what causes the acceleration of the bullet is an electromagnetic interaction. There is a light clock at the heart of the interaction that makes a mechanical clock tick.
- In any case, this reasoning assumes that the electromagnetic interaction is pure, develops in an ideal way that is immune to physical circumstances, which might vary from case to case. If we could account for that with very precise instruments, we might find out that the rules are more complicated, in accordance with certain patterns…

Is that more or less in line with what you hold? Of course, these are bold speculations, at least on my side. (And if I were thus breaking forum rules, please anyone in charge let me know. I enjoy the forum too much to be banned…)

Hi Saw,

In my reasoning I am trying to stick to well-established postulates and what can be deduced from them by rigorous logic.

The fundamental postulate of any relativistic theory is the requirement of invariance with respect to the Poincare group. Another obvious fact is that the Hamiltonian (the generator of time translations in the Poincare group) of any multiparticle system contains interaction terms. By group theory arguments it follows that the generator of boosts must also contain interaction terms (see P. A. M. Dirac, "Forms of relativistic dynamics", Rev. Mod. Phys. 21
(1949), 392). So, when boosts are applied to particle observables, we are bound to find that the corresponding transformations are non-linear and interaction-dependent. It then follows that (linear and interaction-independent) Lorentz transformations of special relativity cannot be exactly valid in interacting systems, and that one should expect interaction-dependent corrections to Einstein's time dilation and length contraction formulas.

 

[Ich]

@facenian:

The transformation you provided does map straight lines to straight lines, but not uniform motion along straight lines to uniform motion along straight lines.

 

[facenian]

@facenian:

The transformation you provided does map straight lines to straight lines, but not uniform motion along straight lines to uniform motion along straight lines.

Yes, it does. Calculate carefully de folowing
V_{\alpha}^{'}=\frac{dx^{'}_\alpha}{dx^{'}_0} =\frac{\frac{dx^{'}_\alpha}{dx_0}}{\frac{dx^{'}_0}{dx_0}},\quad \alpha=1,2,3 \quad, and \, under\, the\, assumption\, \frac{dx_\alpha}{dx_0}=V_\alpha=const

 

[Al68]

You definitely can't prove anything from Einstein's postulates, since they are ill-defined. The biggest problem with them is that they talk about inertial frames as if that concept has been defined already.

Wasn't he referring to the same concept of "inertial frame" as that used for a couple hundred years in Newtonian physics?

ie, a reference frame in which Newton's first law is valid, which according to Newton was valid in any reference frame "neither rotating nor accelerating relative to the fixed stars."

It seems obvious to me that Einstein was specifically referring to the same concept of an inertial frame in his postulates.

 

[Ich]

You're right - of course, since straight lines in spacetime represent uniform motion.
So this statement alone does not require linearity. We need homogeneity, too.

 

[Fredrik]

Wasn't he referring to the same concept of "inertial frame" as that used for a couple hundred years in Newtonian physics?

ie, a reference frame in which Newton's first law is valid, which according to Newton was valid in any reference frame "neither rotating nor accelerating relative to the fixed stars."

It seems obvious to me that Einstein was specifically referring to the same concept of an inertial frame in his postulates.

Would you also define the real numbers as "numbers with the properties that people who don't know math or physics expect distances to have"? The above is obviously not a definition. It doesn't tell us which functions are inertial frames.

 

[Al68]

Would you also define the real numbers as "numbers with the properties that people who don't know math or physics expect distances to have"? The above is obviously not a definition. It doesn't tell us which functions are inertial frames.

I don't see what you're saying here. If F=ma, then the reference frame is inertial. Otherwise it's not. Sounds well defined to me.

 

[Fredrik]

It's not. If it was, then an inertial frame in SR would be the same thing as an inertial frame in Newtonian mechanics. It clearly isn't. A coordinate system is a function from (an open subset of) spacetime into \mathbb R^4. Inertial frames are coordinate systems. If x and y are coordinate systems, then x\circ y^{-1} represents a coordinate change. I'm not sure if it's a standard term, but I call these functions "transition functions". In Newtonian mechanics, the set of transition functions associated with inertial frames is the Galilei group. In SR, it's the Poincaré group. So the transition functions are clearly not the same in both theories, and therefore the inertial frames aren't either.

 

[facenian]

Fredrik, I'm not sure whether you last said is all right because the transition functions are different in both theories doesn't mean the concept of inertial frame are necesarily different.
I think fron a phisical stand point an inertial frame may be defined as one where the law of inertia holds or perhaps can be defined,more abstracly, like Landau does as I mentioned in an erlier post.

 

[Fredrik]

What I said is definitely correct. In both theories you can take spacetime to be \mathbb R^4 and let the identity map on \mathbb R^4 (i.e. the function I defined by I(x)=x for all x) be one of the inertial frames. This choice makes the set of transition functions identical to the set of inertial frames.

 

[facenian]

What I said is definitely correct. In both theories you can take spacetime to be \mathbb R^4 and let the identity map on \mathbb R^4 (i.e. the function I defined by I(x)=x for all x) be one of the inertial frames. This choice makes the set of transition functions identical to the set of inertial frames.

I see your definiton is mathematically correct and this leads to defferent inertial frames whether you us LT transformations or Galiei transformation.
I don't know if in this contex the physical content an inertial frame is in order.

 

[Al68]

I don't see what you're saying here. If F=ma, then the reference frame is inertial. Otherwise it's not. Sounds well defined to me.

It's not. If it was, then an inertial frame in SR would be the same thing as an inertial frame in Newtonian mechanics. It clearly isn't. A coordinate system is a function from (an open subset of) spacetime into \mathbb R^4. Inertial frames are coordinate systems. If x and y are coordinate systems, then x\circ y^{-1} represents a coordinate change. I'm not sure if it's a standard term, but I call these functions "transition functions". In Newtonian mechanics, the set of transition functions associated with inertial frames is the Galilei group. In SR, it's the Poincaré group. So the transition functions are clearly not the same in both theories, and therefore the inertial frames aren't either.

I was referring to Newtonian mechanics, since the issue was how well defined an inertial frame was prior to Einstein's postulates in 1905. There was no SR inertial frame definition prior to 1905.

I didn't say the Newtonian definition was correct in SR, I said it seemed to be well defined in Newtonian physics..

 

[Fredrik]

I was referring to Newtonian mechanics, since the issue was how well defined an inertial frame was prior to Einstein's postulates in 1905. There was no SR inertial frame definition prior to 1905.

I didn't say the Newtonian definition was correct in SR, I said it seemed to be well defined in Newtonian physics..

Huh. Why would it be relevant that there's an older theory in which the same term is used to mean something different?

 

[quZz]

This is very interesting, Fredrik! Though don't quite understand you =) but never mind... The question: is there a homogeneity of spacetime in an inertial system in SR? in Newtonian physics?

 

[Al68]

Huh. Why would it be relevant that there's an older theory in which the same term is used to mean something different?

Because Einstein was clearly referring to a Newtonian inertial frame in his 1905 paper.

 

[Fredrik]

Because Einstein was clearly referring to a Newtonian inertial frame in his 1905 paper.

I really hope he wasn't, because his postulates are false if he was.

If he was referring to an undefined concept, the task of "deriving" something from the postulates can be interpreted as the task of finding out which definitions are consistent with other assumptions that also seem natural. I haven't been able to find any other way to make sense of Einstein's "postulates" and the attempts to "derive" things from them.

 

[atyy]

I really hope he wasn't, because his postulates are false if he was.

Yeah, he was. I've read that it's ok if one says an inertial frame is one in which Newtonian laws hold at low velocities - maybe an article by Ohanian in the American Journal of Physics - didn't study it, so can't reproduce the reasoning here.

 

[atyy]

I don't know what's up with so(4).

http://books.google.com/books?id=PE...oNXYDA&sa=X&oi=book_result&ct=result&resnum=3

 

[Al68]

Because Einstein was clearly referring to a Newtonian inertial frame in his 1905 paper.

I really hope he wasn't, because his postulates are false if he was.

Clearly, he was, although he didn't use the phrase "inertial frame", he used the phrase "frames of reference for which the equations of mechanics hold good" and "system of co-ordinates in which the equations of Newtonian mechanics hold good."

How does that make his postulates false?

 

[Fredrik]

Clearly, he was, although he didn't use the phrase "inertial frame", he used the phrase "frames of reference for which the equations of mechanics hold good" and "system of co-ordinates in which the equations of Newtonian mechanics hold good."

How does that make his postulates false?

See #34 and #36. The set of inertial frames in Newtonian mechanics can be identified with the set of Galilei transformations, but not with the set of Poincaré transformations. The first postulate is consistent with both the Galilei group and the Poincaré group, but the second postulate rules out the former.

I don't doubt that you're right about what Einstein said, but he was incredibly sloppy.

 

[Al68]

See #34 and #36. The set of inertial frames in Newtonian mechanics can be identified with the set of Galilei transformations, but not with the set of Poincaré transformations. The first postulate is consistent with both the Galilei group and the Poincaré group, but the second postulate rules out the former.

I think that was his point, that in Newtonian physics, the postulates were contradictory. His derivations were the result of modifying Newtonian physics to be consistent with the postulates.

I don't doubt that you're right about what Einstein said, but he was incredibly sloppy.

I think he preferred the phrase, "not too pedantic". After all, how exact should an inertial frame be defined in a paper that uses a rail car as an example of one?

 

[Fredrik]

I think that was his point, that in Newtonian physics, the postulates were contradictory. His derivations were the result of modifying Newtonian physics to be consistent with the postulates.

And one of my points is that if you start with a set of assumptions and end up with a contradiction, you have only proved that your theory is inconsistent. You certainly haven't derived a new theory.

That's why I'm saying that the only way to make sense of the "derivation" is to interpret the "postulates" as ill-defined statements, and the "derivation" as finding out which of the corresponding well-defined statements are consistent with the other assumptions we want to make.

I think he preferred the phrase, "not too pedantic". After all, how exact should an inertial frame be defined in a paper that uses a rail car as an example of one?

I don't have a problem with the fact that the first paper ever written on SR is "not too pedantic". I just don't think that's a good reason for us do the same. It's not even too difficult to talk about SR in a way that makes sense, so we have no excuse. I think it's absurd that professors still give students the impression that SR is defined by Einstein's postulates, and that the rest of the theory can be "derived" from the "postulates". You really can't derive anything from them, and they can't be taken as the definition of SR.

 

[dx]

It is possible to have well defined postulates from which the linearity of Lorentz transformations follows.

Let V be a four dimensional vector space. An inertial frame is a map ψ from the set of events into V which satisfies the following postulates:

1. The world lines of free particles are straight lines.
2. Clock rates are uniform, i.e. intervals measured by clocks agree with the linear structure of V.

That there exist such frames is an experimental question and has nothing to do with the mathematical structure of SR. The ideas of 'free partice' and 'clock' are primitive notions which are not defined within the theory.

Given two such inertial frames ψ and ψ', it is easy to see that the transformation between them given by ψ^-1ψ' : V → V is a linear transformation.

(Note that the Lorentz behavior of clocks represented by the Lorentz metric dt² - dx² - dy² - dz² is not the only one compatible with these postulates)

 

[meopemuk]

1. The world lines of free particles are straight lines.
2. Clock rates are uniform, i.e. intervals measured by clocks agree with the linear structure of V.

These two postulates apply only to free particles. So, you need also a third postulate:

3. Events with interacting particles (e.g., their worldlines) transform by the same formulas as events with free particles.

Then, according to the Currie-Jordan-Sudarshan theorem, your theory must be interaction-free.