Einstein postulates and the speed of light

[Fredrik]

This paper is really good.

I don't think so. The historical introduction was interesting, but then it gets weird. It doesn't include any proofs. It talks about the curvature of the space of velocities, without even introducing a metric on it. I don't understand this at all. Isn't the space of velocities just an open ball in $\mathbb R^3$, or $\mathbb R^3$ itself?

The paper also claims that the case $\alpha<0$ (where the coordinate transformations are rotations of spacetime) isn't logically inconsistent. I strongly doubt the validity of this claim. In the 1+1-dimensional case, I can prove that the claim is false if I'm allowed to assume that the set of velocities is a subset of ℝ and that 0 is an interior point of that set. If what the author has in mind is that we are not to make that sort of assumptions, then he should have explained a lot more in this section. In particular, how does he make sense of inertial coordinate systems with velocity difference ∞? He makes some weird claim that positive velocities that add up to negative velocities could mean that the third observer is invisible to the first. How does he justify that? With some weird appeal to the undefined curved space of velocities.

The author also keeps suggesting that these things are necessary to see that SR is logically consistent. This is definitely not true. He uses phrases like "logical foundations of the relativity theory" and "proof of the consistency of the foundations". These derivations of the formula for Lorentz transformations have nothing to do with logical consistency. What they tell us is how SR and pre-relativistic physics fit into a larger hierarchy of theories.

The way to see that SR is logically consistent is to note that it's just talking about $\mathbb R^4$ and some functions. These things are defined using set theory, like everything else in a branch of mathematics that's large enough to include everything that's useful and much more. So if SR is logically inconsistent, pretty much all of mathematics would fall with it.

 

[BruceW]

It talks about the curvature of the space of velocities, without even introducing a metric on it. I don't understand this at all. Isn't the space of velocities just an open ball in $\mathbb R^3$, or $\mathbb R^3$ itself?

yeah, it's just R3 itself. But with some curvature. Yes, he doesn't go into much detail. That is one of the problems of this paper. He doesn't seem to go into much detail on anything. Maybe that is because his aim in this paper is mostly just to highlight the work of others.

The paper also claims that the case $\alpha<0$ (where the coordinate transformations are rotations of spacetime) isn't logically inconsistent.

Our universe is alpha positive. But a universe with negative alpha seems consistent to me. Yes, it has some infinite relative velocities, but I don't think that is any worse than our universe which has gamma diverging to infinity when the relative velocity tends to the speed of light. Are you saying something like "what is the point of even considering a negative alpha universe?" If that's what you mean, then I agree with you. I think that is why there is not much talk about negative alpha universes. Also, what do you mean by "where the coordinate transformations are rotations of spacetime"? Do you just mean that in his explanation, he hasn't really said about what happens when we think about transformations such as x'=x+some constant

I strongly doubt the validity of this claim. In the 1+1-dimensional case, I can prove that the claim is false if I'm allowed to assume that the set of velocities is a subset of ℝ and that 0 is an interior point of that set.

How can you prove this?

These derivations of the formula for Lorentz transformations have nothing to do with logical consistency. What they tell us is how SR and pre-relativistic physics fit into a larger hierarchy of theories.

I don't understand really. From the 'postulates', don't the Lorentz transforms logically follow? I guess he is making assumptions about how real numbers work, e.t.c. So I looked on google quickly and found this paper: http://arxiv.org/pdf/1005.0960v2.pdf Which has a different axiomatic explanation, using first-order logic. I suppose this is a way to do it properly. Pretty maths-heavy though. I also remember someone's post in physicsforums which linked to a paper which explained the invariant speed. It was similar to Sergey's paper, but didn't go on about the alpha negative. Also, it was no-where near as maths heavy as the paper I link to in this paragraph, which uses first-order logic. Shame I didn't save the link

 

[Mentz114]

So I looked on google quickly and found this paper: http://arxiv.org/pdf/1005.0960v2.pdf Which has a different axiomatic explanation, using first-order logic. I suppose this is a way to do it properly.

Thanks for finding this. It's blown me away. I never thought my training in formal systems would be useful in physics.

Having axiomatized SR, they extend it to include accelerated observers by adding some axioms. I'm about to read what I hope to be the final definitive resolution of the twin paradox.

 

[Fredrik]

Our universe is alpha positive. But a universe with negative alpha seems consistent to me. Yes, it has some infinite relative velocities, but I don't think that is any worse than our universe which has gamma diverging to infinity when the relative velocity tends to the speed of light. Are you saying something like "what is the point of even considering a negative alpha universe?" If that's what you mean, then I agree with you.

What I mean is this: We're looking for a group G such that each $\Lambda\in G$ is a permutation of $\mathbb R^4$ that changes coordinates from one inertial coordinate system to another. At this stage, we don't have a definition of "inertial coordinate system". Instead, we are trying to find a group that's consistent with some set of assumptions about its members. These assumptions are also supposed to be consistent with our intuition about what an inertial coordinate system is. Otherwise, we can't argue that the result we find has been derived from (a mathematical interpretation of) the principle of relativity. It seems bizarre to me to allow the members of G to take finite-speed world lines to infinite-speed world lines, not just because I expect such groups to be irrelevant to physics, but also because this goes completely against our intuition about inertial coordinate systems.

If we don't allow this, then transformations with velocity c are out, because if $\Lambda$ has velocity c, then $\Lambda^2$ has infinite velocity. This obviously implies that many smaller velocities are out as well. (If $\Lambda\Lambda'$ has a forbidden velocity, then $\Lambda,\Lambda'$ can't both be in G). The details are in the pdf mentioned below.

Also, what do you mean by "where the coordinate transformations are rotations of spacetime"?

In the 1+1-dimensional case, what we find is that there's an $\alpha\in\mathbb R$ such that for each $\Lambda\in G$, there's a $v\in\mathbb R$ such that $1-\alpha^2v>0$ and
$$\Lambda=\frac{1}{\sqrt{1-\alpha v^2}}\begin{pmatrix}1 & -\alpha v\\ -v & 1\end{pmatrix}.$$ If $\alpha=0$, this is a Galilean boost. If $\alpha=1$, this is a Lorentz boost. If $\alpha=-1$, the columns of $\Lambda$ are orthonormal, and this implies that $\Lambda\in\mathrm{SO}(2)$. So $\Lambda$ is a rotation (of spacetime, not space). The rotation angle can be defined by $\theta=\arctan v$.

Do you just mean that in his explanation, he hasn't really said about what happens when we think about transformations such as x'=x+some constant

No, I was only talking about transformations that take 0 to 0.

How can you prove this?

The general idea is described briefly above. The full proof is in the pdf I attached to this post. In version 3 of the pdf, the final step is lemma 12 on page 7, but you should probably start from the beginning.

I don't understand really. From the 'postulates', don't the Lorentz transforms logically follow?

Yes, but what does this have to do with the theory's consistency? My approach to SR starts with the following definitions:

"Minkowski spacetime" is the vector space $\mathbb R^4$ with the Minkowski metric. An "inertial coordinate system" is a permutation of $\mathbb R^4$ that's also an isometry of the metric.​

This is followed by the definition of "proper time", and a statement of the correspondence rule for time measurements. How could this be inconsistent? (Where is the inconsistency? In the axioms for vector spaces? In the axioms for real numbers? In the existence of functions? In the existence of cartesian products? If the answer to any of these is "yes", then ZFC set theory is dead, and it won't be because of special relativity). And more importantly, how could one of these derivations make it less likely that it is?

The way I see it, what the best of these derivations are doing is to answer the following question: Are there any theories of physics in which space and time is represented by $\mathbb R^4$ equipped with global coordinate systems, that are consistent with the principle of relativity and the principle of homogeneity and isotropy of space?

The oversimplified answer is: "Yes, there are exactly two of those". What's interesting here isn't that the derivations show us a way to find the Lorentz transformation formula, but that they rule out all other groups that we might have considered to use as an ingredient in a theory of physics in which spacetime is $\mathbb R^4$ with global coordinate systems.

 

[bcrowell]

So I looked on google quickly and found this paper: http://arxiv.org/pdf/1005.0960v2.pdf

That's interesting. Thanks for posting the link.

In the intro, they go on and on about first-order logic. This seems pretty pointless to me. The distinction between first-order logic and other types of logic doesn't even connect to anything physically observable, so I don't think it's relevant in a physical theory.

It's ugly that they start with explicit coordinates in Minkowski space, when their goal is to develop an axiomatization of GR, which is coordinate-independent. It's also awkward that they start with Minkowski space for their axiomatization of SR and then later have to somehow switch to an arbitrary manifold.

It's essentially just a formalization of Einstein's 1905 axioms. There's nothing deep going on here.

 

[Mentz114]

There's nothing deep going on here.

Depends how tall you are. The brevity and unambiguous language is a change from hand-wavey arguments.

 

[Fredrik]

I'm about to read what I hope to be the final definitive resolution of the twin paradox.

If what you mean by "the twin paradox" is the result that the astronaut twin is younger, then the final definitive resolution is a simple calculation of the proper times of the two curves.

If what you mean by "the twin paradox" is the question of what's wrong with the calculation that finds that the Earth twin is younger, then there are a number of final definitive resolutions, for example the observation that the time dilation formula doesn't apply since the astronaut twin's world line doesn't coincide with the time axis of any inertial coordinate system. I like to supplement this with a diagram that shows the simultaneity lines of the two inertial coordinate systems that are comoving with the astronaut twin before and after the turnaround.

 

[BruceW]

It is an interesting pdf. It says that negative alpha is inconsistent with the group that they have defined. So I guess that a group-theoretic description allowing negative-alpha would need a different group to the one they define. I wonder if there is an easy adaptation to get such a group. Maybe I'll think about it later when I am more awake :)

 

[strangerep]

The historical introduction [of Stepanov's paper] was interesting, but then it gets weird. It doesn't include any proofs.

You might recall that this is the same author I mentioned in a previous thread that wrestled with how to derive the fractional-linear transformations. You might also recall I advised taking the main body of his papers with a grain of salt. His previous papers, which give more detail if you can interpret the poor English are:
http://arxiv.org/abs/astro-ph/9909311 (which is also in Phys Rev D), and
http://arxiv.org/abs/astro-ph/0111306 (which appears not to have been published in a peer-reviewed journal).

It talks about the curvature of the space of velocities, without even introducing a metric on it. I don't understand this at all. Isn't the space of velocities just an open ball in $\mathbb R^3$, or $\mathbb R^3$ itself?

There's quite a lot of literature that works with relativistic velocity space in this way. It's a hyperbolic space (Cayley-Klein-Beltrami projective space). The thing that distinguishes it from an ordinary open ball in $\mathbb R^3$ is the nonlinear velocity addition rule. There's a body of theory going back maybe 100 years or more about how to work with such spaces by tricky mappings.

The paper also claims that the case $\alpha<0$ (where the coordinate transformations are rotations of spacetime) isn't logically inconsistent. I strongly doubt the validity of this claim.

It's true (which can be seen more easily in the context of a dynamical approach which I won't elaborate on here), but the devil is in the detail: one finds that the group of transformations is only defined on a slice of velocity phase space where velocity is 0. I.e., all observers must be at rest relative to each other. Off this slice, the transformation is ill-defined. Hence, for physical reasons, we can discard the possibility $\alpha<0$ because it fails to model realworld situations.

Your "open interval around v=0" hypothesis is (essentially) equivalent to the above: there must be a continuous set of physical velocities containing v=0 or the theory is useless.

 

[Fredrik]

It is an interesting pdf. It says that negative alpha is inconsistent with the group that they have defined. So I guess that a group-theoretic description allowing negative-alpha would need a different group to the one they define. I wonder if there is an easy adaptation to get such a group. Maybe I'll think about it later when I am more awake :)

I could drop assumption 1b from the definition of "linear relativistic group", since it's not used in lemma 3, where the formula for Lorentz/Galilei/SO(2) transformations is found. Then K<0 (i.e. $\alpha<0$) isn't immediately ruled out. However, infinite-speed transformations are still ruled out by assumption 1a, which says that every transformation has a velocity in ℝ, and this makes a lot of velocities "forbidden" in the sense that there's no transformation with that velocity in the group. There is certainly a forbidden velocity in every open interval that contains 0. I haven't thought this through to the end, but I would guess that the set of forbidden velocities is dense in ℝ, but not equal to ℝ-{0}.

So if we want to allow K<0 for some reason, we either have to take the set of allowed velocities to be full of holes, or we allow transformations with infinite speed, i.e. $\Lambda\in G$ such that $(\Lambda^{-1})_{00}=0$. Buuuut...here's something I learned very recently: That would imply that the zero-velocity subgroup of the proper and orthochronous subgroup is not the group of all rotations of space. That sounds undesirable too.

 

[Fredrik]

You might recall that this is the same author I mentioned in a previous thread that wrestled with how to derive the fractional-linear transformations. You might also recall I advised taking the main body of his papers with a grain of salt.

Oops, I had completely forgotten.

There's quite a lot of literature that works with relativistic velocity space in this way. It's a hyperbolic space (Cayley-Klein-Beltrami projective space). The thing that makes distinguishes it from an ordinary open ball in $\mathbb R^3$ is the nonlinear velocity addition rule. There's a body of theory going back maybe 100 years or more about how to work with such spaces by tricky mappings.

OK. Do you know how that space is defined, and how it makes sense to talk about curvature? (If it's a pain to explain it, never mind, I'm just a little bit curious).

 

[strangerep]

Do you know how that space is defined, and how it makes sense to talk about curvature? (If it's a pain to explain it, never mind, I'm just a little bit curious).

Well, yes, it's a bit of a pain -- partly (mainly?) because I don't know that subject in much detail. These Wiki pages might be a start, however:
http://en.wikipedia.org/wiki/Gyrovector
http://en.wikipedia.org/wiki/Beltrami–Klein_model

 

[TrickyDicky]

Stepanov's paper is really weird, it states as its goal to demonstrate that the second postulate is not necessary only to actually proof in sections 5 and 6 that additional assumptions are needed, that ultimately come down to choose a positive alpha by empirical means. This looks suspiciously similar to the second postulate which even though it was introduced as a postulate, it's now closer to an empirical fact that leads us to reject Galilean transformations(alpha=0), and alpha<0 is similarly straightforward to discard empirically.

 

[BruceW]

However, infinite-speed transformations are still ruled out by assumption 1a, which says that every transformation has a velocity in ℝ, and this makes a lot of velocities "forbidden" in the sense that there's no transformation with that velocity in the group... (I skip some of it here)

So if we want to allow K<0 for some reason, we either have to take the set of allowed velocities to be full of holes, or we allow transformations with infinite speed, i.e. $\Lambda\in G$ such that $(\Lambda^{-1})_{00}=0$. Buuuut...here's something I learned very recently: That would imply that the zero-velocity subgroup of the proper and orthochronous subgroup is not the group of all rotations of space. That sounds undesirable to

Yeah, negative alpha is not even consistent with 1a. I think Stepanov wanted to allow these infinite relative velocities, and if this is the case, then negative alpha is not consistent with the "linear relativistic group". If I am following things correctly, am I right in saying that negative alpha is not necessarily a stupid idea, but until someone comes up with a nice group of transformations for it, it is pretty much a useless idea? I wonder if Frank and Rothe (which Stepanov mentions) thought about a group for these transforms... Maybe they were too early in the formulation of SR to be thinking about groups. Anyway, in the standard group formulation of relativity, negative alpha is not consistent, right?

 

[Fredrik]

Yeah, negative alpha is not even consistent with 1a.

I don't see a way to completely rule it out without an assumption like 1b. But of course, the results that most velocities are "forbidden", and that positive velocities can add up to negative ones, are both pretty unappealing.

If I am following things correctly, am I right in saying that negative alpha is not necessarily a stupid idea, but until someone comes up with a nice group of transformations for it, it is pretty much a useless idea?

We would have to do the work to find a subset of ℝ that contains 0, and is closed under the "addition" operation $\oplus$ defined by
$$u\oplus v=\frac{u+v}{1-\frac{uv}{c^2}}.$$ This set must be such that the right-hand side of this formula is well-defined for all u,v in the set, i.e. it can't contain u,v such that uv=c². In particular, it can't contain c or -c. If S is such a set, then
$$\left\{\left.\frac{1}{\sqrt{1+\frac{v^2}{c^2}}} \begin{pmatrix}1 & v/c\\ -v & 1\end{pmatrix}\right|v\in\mathcal S\right\}$$ would be a group that could in principle be used in a theory of physics. But I think any such theory can easily be ruled out by experiments.

Anyway, in the standard group formulation of relativity, negative alpha is not consistent, right?

Special relativity is a theory with a positive alpha, so I'm not sure I understand the question. In my approach, I included assumption 1b (which holds in SR), specifically to rule out negative alphas. It was the simplest assumption I could find that got the job done.

 

[BruceW]

I don't see a way to completely rule it out without an assumption like 1b.

Ah, yeah. I was using the equations derived by Stepanov, then comparing them to 1a and seeing that they are not consistent. I didn't make that clear, sorry.

We would have to do the work to find a subset of ℝ that contains 0, and is closed under the "addition" operation $\oplus$ defined by
$$u\oplus v=\frac{u+v}{1-\frac{uv}{c^2}}.$$

I was thinking of some kind of group which allows the infinite relative velocity transformations. I'm guessing it could be very different to the "linear relativistic group"... Like "who even knows what that would look like" kind of different...

 

[BruceW]

Thanks for finding this. It's blown me away. I never thought my training in formal systems would be useful in physics.

Having axiomatized SR, they extend it to include accelerated observers by adding some axioms. I'm about to read what I hope to be the final definitive resolution of the twin paradox.

Ah, thanks man. To be honest, it was too much maths for me, I skipped over most of it. Am I right in thinking that one of the first theorems they state is that the speed of light is invariant? So I guess this axiomatization would be very far from the idea of having a choice of values for an invariant speed.

 

[Mentz114]

Ah, thanks man. To be honest, it was too much maths for me, I skipped over most of it. Am I right in thinking that one of the first theorems they state is that the speed of light is invariant? So I guess this axiomatization would be very far from the idea of having a choice of values for an invariant speed.

Yes, I think you're correct.

Roughly, the axioms are

1. Defines the domain of the maths. (Looks like the field R⁴)
2. For any inertial observer, the speed of light is the same everywhere and in every direction, and it is finite. Furthermore, it is possible to send out a light signal in any direction.
3. All inertial observers coordinatize the same set of events.

'Sub-axioms'

4a. Any inertial observer sees himself as standing still at the origin of his coordinates.
4b. Any two inertial observers agree as to the spatial distance between two events if these two events are simultaneous for both of them; furthermore, the speed of light is 1 for all observers.

The first theorem derived from these five axioms is : no inertial observer will 'coordinatize' another inertial observer as traveling at v >= c.

The proof is clunky and inelegant, however ( just my opinion). After this they adopt the Poincare group of transformation as the change of coords between inertial frames. But if the Poincare group is adopted as an axiom, the first theorem (and possibly axiom 4b) is redundant and the theory is more elegant.

So, it doesn't live up to my expectations, but still has some interesting points. I may finish reading it sometime.

 

[strangerep]

am I right in saying that negative alpha is not necessarily a stupid idea, but until someone comes up with a nice group of transformations for it, it is pretty much a useless idea?

As I tried to explain briefly in a previous post, there is already a valid transformation group for negative alpha, but it is only well-defined on v=0.
I.e., it only makes mathematical sense in a situation where all observers are at rest relative to each other. Thus, it is indeed "pretty much useless" for physics.

Edit: In more detail...

Consider the velocity addition formulas in Fredrik's treatment for both cases.

1) First the "negative alpha" case. In units where c=1, the velocity addition formula is
$$
u' ~=~ \frac{u + v}{1 - uv} ~=~ \frac{\tan\eta_u + \tan\eta_v}{1 - \tan\eta_u \, \tan\eta_v}
$$where $\eta_u, \eta_v$ can take any real value. (In the +ve alpha case, we'd call them "rapidities".)

We now ask: on what domain of $\eta$ is this formula well-defined? Clearly, if $\eta_v=0$, the denominator is always 1, so that's ok but rather boring since it just means $u'=u$. But now suppose $\eta_v\ne 0$. In that case, for any value of $\eta_v$ I can find a value of $\eta_u$ such that the denominator becomes 0. This is because the range of the tan function is $\pm\infty$. We conclude that this formula is only well-defined on the trivial domain consisting of the single value $v=0$.

2) Now the "positive alpha" case. Staying with units where c=1, the velocity addition formula is
$$
u' ~=~ \frac{u + v}{1 + uv}
~=~ \frac{\tanh\eta_u + \tanh\eta_v}{1 + \tanh\eta_u \, \tanh\eta_v}
$$where, as before, $\eta_u, \eta_v$ can take any real value. In this case, we can validly call them "rapidities". In this case, even if $\tanh\eta_v < 0$, there is no value of $\eta_u$ which makes the denominator 0. This is because the range of the tanh function is bounded by $\pm 1$, and it approaches these limits only asymptotically. Therefore, on the (open) domain of velocities such that $|v|<1$ the transformation is well-defined.

With closer analysis, one can also show that the limit as $v\to 1$ remains sensible -- in that the result of the "addition" always approaches 1 in that limit. So we can complete this to a closed domain in various situations by taking limits carefully.

 

[Fredrik]

There's a sign error in the "positive alpha" case. The velocity addition formula is
$$u'=\frac{u+v}{1+\alpha uv}$$ in both cases. So the denominator is 1+uv when $\alpha=1$, and 1-uv when $\alpha=-1$.

We conclude that this formula is only well-defined on the trivial domain consisting of the single value $v=0$.

I'm not convinced that this is accurate (without an assumption like my 1b). Is {0} really the only subset of ℝ that's closed under the operation $(u,v)\mapsto (u+v)/(1-uv)$? In terms of rapidities, is {0} really the only subset of $(-\pi/2,\pi/2)$ that's closed under the corresponding operation? I think the operation can be defined like this:
$$(\theta,\eta)\mapsto\begin{cases}\theta+\eta &\text{if }-\pi/2<\theta+\eta<\pi/2\\ \theta+\eta-\pi &\text{if }\theta+\eta>\pi/2\\ \theta+\eta+\pi &\text{if }\theta+\eta<-\pi/2\end{cases}$$ Maybe there's a nicer way to state this definition. Edit: Hey, isn't $(-\pi/2,\pi/2)\cap\mathbb Q$ such a set? Two rational rapidities can't add up to a forbidden value like $\pi/2$ (infinite speed) or $\pi/4$ (speed =c=1), because the forbidden rapidities are all irrational.

 

[strangerep]

There's a sign error in the "positive alpha" case.

Corrected, thanks.

I'm not convinced that this is accurate (without an assumption like my 1b). Is {0} really the only subset of ℝ that's closed under the operation $(u,v)\mapsto (u+v)/(1-uv)$?

If the transformations form a group which is continuous and differentiable in a real parameter (i.e., $\eta$ here), we must allow $\eta$ to takes arbitrary real values.
Consider:
$$
e^{\eta K} e^{\eta K} ~=~ e^{2\eta K}
$$ so by composing transformations I can get arbitrarily large rapidities. Any restriction on the values of $\eta$ must also be compatible with arbitrarily many such compositions, else we don't have a group. The only such valid restriction (afaict) is the restriction to $\eta=0$, i.e., a group consisting trivially of the identity and nothing else.

 

[Fredrik]

Corrected, thanks.
If the transformations form a group which is continuous and differentiable in a real parameter (i.e., $\eta$ here), we must allow $\eta$ to takes arbitrary real values.

By this I assume you mean that there's a function that takes each $\Lambda$ in the group to the rapidity of $\Lambda$, and that we require this function to be continuous in some sense, and differentiable in some sense. We could e.g. use the Hilbert-Schmidt norm on the set of matrices (the norm obtained from the inner product $\langle A,B\rangle=\mathrm{Tr}A^TB$) to define a topology on the group. An (equivalent?) alternative is to instead consider the function that takes the 4-tuple of components of the 2×2 matrix $\Lambda$ to the rapidity of $\Lambda$. For this function, we can use the standard definitions of continuity and differentiability from calculus.

There is of course nothing wrong with such assumptions, but I'd like to point out two things. 1. This assumption implies my 1b, and is much stronger than my 1b. 2. This assumption is not one of the statements that turns the principle of relativity into a mathematically precise statement. Rather, this assumption should be thought of as making "boost invariance" mathematically precise. This is when we are talking specifically about rapidities. If the parameter had been a position or an Euler angle, it would have been part of making the principles of "translation invariance" (="spatial homogeneity") or "rotation invariance" (="spatial isotropy") precise.

So I think we have to consider $\alpha<0$ with a non-trivial set of allowed rapidities to be consistent with the principle of relativity, but not consistent with these other principles.

 

[strangerep]

By this I assume you mean [...]

Actually, I was just using the standard notion of "Lie group". No need for anything more elaborate.

1. This assumption implies my 1b, and is much stronger than my 1b.

Certainly one needs the transformation to be well-defined in an open neighbourhood of the identity, else taking derivatives is a problem. So one could alternatively regard what I wrote above as saying that the 1st case is incompatible with such an assumption.

2. This assumption is not one of the statements that turns the principle of relativity into a mathematically precise statement.

That depends on whether one's starting point is dynamical or geometric. If the former, then notions of continuity and differentiability are already there, gratis, since we start with $d^2 x/dt^2 = 0$.

So I think we have to consider $\alpha<0$ with a non-trivial set of allowed rapidities to be consistent with the principle of relativity, but not consistent with these other principles.

Well, in the dynamical approach with Lie symmetries it is not consistent with invariance of the equation $d^2 x/dt^2 = 0$ on a nontrivial range of values of $dx/dt$ .

 

[BruceW]

As I tried to explain briefly in a previous post, there is already a valid transformation group for negative alpha, but it is only well-defined on v=0.
I.e., it only makes mathematical sense in a situation where all observers are at rest relative to each other. Thus, it is indeed "pretty much useless" for physics.

Right. Yeah, that is definitely not what we want. So negative alpha is pretty much useless until someone comes up with a transformation group that can give us all velocities.

 

[Fredrik]

Actually, I was just using the standard notion of "Lie group". No need for anything more elaborate.

OK. But if we're assuming that our group is a Lie group, then we don't need to talk about derivatives to rule out velocity sets like the one I mentioned in the negative alpha case. We only need to use that a manifold is locally homeomorphic to $\mathbb R^n$. And this also rules out the velocity set {0}, because a Lie group can't be a singleton set.

That depends on whether one's starting point is dynamical or geometric. If the former, then notions of continuity and differentiability are already there, gratis, since we start with $d^2 x/dt^2 = 0$.

I don't follow this argument. This implies that the world line of a non-accelerating object is a differentiable curve. But it doesn't seem to imply that we need a Lie group.

 

[strangerep]

But it doesn't seem to imply that we need a Lie group.

If we're considering a family of (zero-acceleration-preserving) mappings of the ambient $(t,x)$ space such that $(t',x') = F(t,x,\xi)$ is continuous (where $\xi$ parameterizes the family), that suggests Lie groups quite readily. :-)

Of course, there may also be discrete groups as well.

 

[Nugatory]

Maxwell's equations include wave propagation. All prior experience with waves suggested a propagation medium. It is easy from our modern standpoint to laugh at aether, but I think it was quite natural for many physicists in the 1800s to suppose all waves must have a material medium to propagate in. Once you assume aether is material, however exotic, it is not necessary to assume that Maxwell's equations, in the 'standard' form, hold only in the aether frame, is a violation of POR.

I agree with this; it's part of the historical perspective that I believe is essential to understanding why Einstein developed the second postulate in the way that he did.