Einstein postulates and the speed of light

[MHD93]

Hello

Some authors claim that Einstein's second postulate (constant speed of light) simply emerges from the first one (or more precisely, its converse contradicts the first postulate).
Serway Modern Physics:

Note that postulate 2, the principle of the constancy of the speed of
light, is consistent with postulate 1: If the speed of light was not the same in
all inertial frames but was c in only one, it would be possible to distinguish
between inertial frames, and one could identify a preferred, absolute frame
in contradiction to postulate 1.

Now, is that true? And if yes, what's so special about light than other object (ex, sound waves) to consider its speed a law of nature (note that their argument, that is, the first postulate indicates the second, doesn't involve experiments)

Thanks

 

[ghwellsjr]

Hello

Some authors claim that Einstein's second postulate (constant speed of light) simply emerges from the first one (or more precisely, its converse contradicts the first postulate).
Serway Modern Physics:

Note that postulate 2, the principle of the constancy of the speed of light, is consistent with postulate 1: If the speed of light was not the same in all inertial frames but was c in only one, it would be possible to distinguish between inertial frames, and one could identify a preferred, absolute frame in contradiction to postulate 1.

Now, is that true? And if yes, what's so special about light than other object (ex, sound waves) to consider its speed a law of nature (note that their argument, that is, the first postulate indicates the second, doesn't involve experiments)

Thanks

No, it's not true. Whoever wrote that statement doesn't understand Einstein's Special Relativity. That person thinks that it is possible to measure or observe or be aware of the speed of light that is talked about in postulate 2. But if you look at what Einstein wrote in his 1905 paper, he said in the introduction:

...light is always propagated in empty space with a definite velocity c...

And in section 2 he said:

Any ray of light moves in the “stationary” system of co-ordinates with the determined velocity c...

It is clear that he is talking about the One-Way Speed of Light which cannot be measured.

In section one, he discusses how to measure the two-way speed of light and he says:

In agreement with experience we further assume the quantity

to be a universal constant—the velocity of light in empty space.

"Experience" there means experiment-something that can be measured.

If you read these first two sections of his paper, you will see that what he is doing is assigning the unknowable one-way speed of light to the measurable value of the two-way speed of light. And he can do this because he postulates without proof that light propagates in any Inertial Reference Frame at c. Prior to that, the Lorentz viewpoint was that light propagated at c only in the absolute ether rest state.

 

[Fredrik]

Some authors claim that Einstein's second postulate (constant speed of light) simply emerges from the first one

The first postulate is just a loosely stated idea, so you can't derive mathematical statements from it. But you can write down a mathematical statement that people would agree is an acceptable way to make the first postulate (and something else, like isotropy of space) mathematically precise, and then you can try to derive things from that. What you will find is that there's an invariant speed, i.e. that there are lines in spacetime that represent motion that's assigned the same speed by all inertial coordinate systems. However, you will not find the value of the invariant speed this way. It can be any positive real number, and it can also be infinite.

In this context, the significance of the second postulate is only that it tells us that the invariant speed is finite.

(or more precisely, its converse contradicts the first postulate).

Converse? Do you mean its negation?

Now, is that true? And if yes, what's so special about light than other object (ex, sound waves) to consider its speed a law of nature (note that their argument, that is, the first postulate indicates the second, doesn't involve experiments)

Sound waves don't have the same speed in all inertial coordinate systems. Electromagnetic waves do.

It's a consequence of relativistic quantum mechanics that for any elementary particle, the value of $-E^2+\vec p^2$ (in units such that the invariant speed is 1) is the same in all inertial coordinate systems. That value is one of the things that distinguishes one particle species from another. It turns out that a theory that involves particles for which that value is 0 makes extremely accurate predictions about experiments involving light. Those particles are called photons, and it can be shown that the equality $-E^2+\vec p^2=0$ implies that they move at the invariant speed.

 

[Meir Achuz]

c is a parameter in Maxwell's equations. Electromagnetism would be different in each Lorentz frame if c were not the same. c must be constant to satisfy the first postulate.
c was measured in 1856 by the ratio of magnetic to electric phenomena with no relation to the speed of light.
It turns out that c determines the speed of light, which thus must be invariant if c is.
E's seond postulate necessarily follows from the first.

 

[Fredrik]

c is a parameter in Maxwell's equations. Electromagnetism would be different in each Lorentz frame if c were not the same. c must be constant to satisfy the first postulate.
c was measured in 1856 by the ratio of magnetic to electric phenomena with no relation to the speed of light.
It turns out that c determines the speed of light, which thus must be invariant if c is.
E's seond postulate necessarily follows from the first.

You mean from the first and Maxwell's equations?

 

[bcrowell]

Now, is that true?

I agree with it, but you'll find many people who wouldn't.

And if yes, what's so special about light than other object (ex, sound waves) to consider its speed a law of nature

The speed of light is predicted to have a unique value by Maxwell's equations, which are laws of physics. The speed of baseballs isn't predicted to have a unique value by Newton's laws.

I wouldn't get too hung up on Einstein's 1905 axiomatization. It's really not optimal from a modern point of view. In 1905, electromagnetism was the only known fundamental field. From a modern point of view, it doesn't make sense to give it a special role. Here's something with a more modern outlook: http://arxiv.org/abs/physics/0302045

 

[Nugatory]

c is a parameter in Maxwell's equations. Electromagnetism would be different in each Lorentz frame if c were not the same. c must be constant to satisfy the first postulate.
c was measured in 1856 by the ratio of magnetic to electric phenomena with no relation to the speed of light.
It turns out that c determines the speed of light, which thus must be invariant if c is.
E's second postulate necessarily follows from the first.

I must confess that I find this to be the most natural way of developing SR: PoR plus Maxwell, doubling down on the first postulate.

However, the historical development of the theory is important here. If it were really as clear as all that, then shortly after Maxwell the entire physics community would have realized that they couldn't have all three of the PoR, Galilean transforms, and Maxwell's equations, and would have started looking for an alternative to the Galilean transforms. And that's not even close to what happened... So apparently this "most natural" way of developing SR wasn't at all natural in historical context.

Thus, even if the second postulate is in some sense redundant, at the time it was essential to the argument. It granted permission to abandon the ether and Galilean transforms. Now, a century later, we're altogether comfortable without ether and Galilean transforms and the second postulate sounds to the modern ear more like "Oh, and I really mean that first postulate". If we no longer feel the need for the second postulate, that just goes to show how convincing the theory has become over the past century.

 

[PAllen]

You mean from the first and Maxwell's equations?

and that Maxwell's equations hold in all inertial frames, unchanged. This is not required by the principle of relativity if you believe that an exotic material aether picks a preferred frame in which Maxwell's equations hold in their standard form. It is not necessary to consider it a violation of the principle of relativity to be able to detect motion relative to (exotic) matter (any more than the ability to detect motion relative to CMB isotropy is a violation of any relativity principle).

Thus, even with the type of modern arguments Frederik (and Bcrowell) mention, you need something else, take your pick:

- that there is a finite invariant speed
- that Maxwell's equations are interpreted without reference to aether, or that aether is undetectable
- that light speed is constant in all inertial frames

you've got to pick something more.

 

[Nugatory]

Thus, even with the type of modern arguments Frederik (and Bcrowell) mention, you need something else, take your pick:

- that there is a finite invariant speed
- that Maxwell's equations are interpreted without reference to aether, or that aether is undetectable
- that light speed is constant in all inertial frames

The third is a restatement of the second, it it not? And also a more specific form of the first?

But it's the second that is the most interesting, because it's so odd that it's needed. Why, if we must state that Maxwell's equations are to be interpreted without reference to aether, do we not also need to state that they are to be interpreted without reference to the breath of angels, or the giant tortoise that supports the entire universe, or ...?

That's a rhetorical question, of course, and it's why I believe that the second postulate is best understood in historical context. Einstein was developing his thinking in an era that accepted (almost without question) the existence of the aether, so "no aether" was a new and important and challenging idea. Today it's as easy to just never introduce the notion of aether in the first place, and then we don't need a postulate to get rid of it.

 

[Fredrik]

But it's the second that is the most interesting, because it's so odd that it's needed. Why, if we must state that Maxwell's equations are to be interpreted without reference to aether, do we not also need to state that they are to be interpreted without reference to the breath of angels, or the giant tortoise that supports the entire universe, or ...?

I prefer the first option on the list: We assume that the invariant speed that's predicted by (mathematical interpretations of) the principle of relativity, translation invariance and rotation invariance, is finite rather than infinite.

When we make the second assumption about "light", it sounds like SR is just an ingredient in a theory of electrodynamics, rather than a framework in which both classical and quantum theories of particles and fields can be defined. If we write down a theory of electrodynamics in this framework, it's automatically independent of ether, angels and tortoises.

 

[PAllen]

The third is a restatement of the second, it it not? And also a more specific form of the first?

Not really. The third can be taken independently of Maxwell's equations. The first says nothing about light. If you assume the first you must empirically determine that light is the invariant speed.

But it's the second that is the most interesting, because it's so odd that it's needed. Why, if we must state that Maxwell's equations are to be interpreted without reference to aether, do we not also need to state that they are to be interpreted without reference to the breath of angels, or the giant tortoise that supports the entire universe, or ...?

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.

That's a rhetorical question, of course, and it's why I believe that the second postulate is best understood in historical context. Einstein was developing his thinking in an era that accepted (almost without question) the existence of the aether, so "no aether" was a new and important and challenging idea. Today it's as easy to just never introduce the notion of aether in the first place, and then we don't need a postulate to get rid of it.

If you want to treat POR as part of space and time symmetries and nothing else, you still need at least (1), as Fredrik has noted.

 

[ghwellsjr]

I wouldn't get too hung up on Einstein's 1905 axiomatization. It's really not optimal from a modern point of view. In 1905, electromagnetism was the only known fundamental field. From a modern point of view, it doesn't make sense to give it a special role. Here's something with a more modern outlook: http://arxiv.org/abs/physics/0302045

When that paper invokes the "isotropy of space" (between equations 6 and 7), aren't they invoking Einstein's second postulate, except they aren't limiting it to the speed of light? Is that your point, not that Einstein's second postulate was included in the first but that it should be stated in a more general context that also includes light?

 

[PAllen]

When that paper invokes the "isotropy of space" (between equations 6 and 7), aren't they invoking Einstein's second postulate, except they aren't limiting it to the speed of light? Is that your point, not that Einstein's second postulate was included in the first but that it should be stated in a more general context that also includes light?

I think that is a valid way of looking at it.

 

[Fredrik]

When that paper invokes the "isotropy of space" (between equations 6 and 7), aren't they invoking Einstein's second postulate, except they aren't limiting it to the speed of light?

What makes you say that? The assumption they're making is that for each inertial coordinate system S, there's an inertial coordinate system T with the property that for all events p, if S assigns it the coordinates (t(p),x(p)), then T assigns it the coordinates (t(p),-x(p)).

This doesn't seem to have any obvious connection to a statement about an invariant speed.

 

[Fredrik]

I have a couple of other issues with Pal's argument. To get from (1) and (2) to (3) and (4), he's using that for all transformations f, if the velocity of f is v, then the velocity of f^-1 is -v. This doesn't follow from the principle of relativity alone.

An informal argument for it would look something like this: Consider two guns built according to identical specifications, in gun factories that are identical except for their velocities and orientation in space. (They are oriented in opposite directions, so the guns will be aimed in opposite directions). Now let's get rid of the factories and keep only the guns. Suppose that they meet at some event, which is assigned coordinates (0,0) by their comoving inertial coordinate systems. Suppose also that they're both fired at (or near) that event, and that the specifications are such that the bullet from gun A will end up comoving with gun B. Then the principle of relativity and principle of isotropy (which in 1+1 dimensions means reflection invariance, not rotation invariance) demand that the bullet from gun B will end up comoving with gun A. To be more precise, the principle of relativity suggests that guns according to identical specifications must fire bullets at the same speed relative to the gun, and the principle of isotropy suggests that the speed of the bullets won't depend on how the gun factory was oriented.

I don't see an informal argument that doesn't rely on reflection invariance. An alternative to this is to introduce a function f that takes the velocity of S' in S to the velocity of S in S', and make a technical assumption about its properties. The principle of relativity strongly suggests that $f\circ f$ is the identity map, but this doesn't imply that f(v)=f(-v) for all v unless we assume continuity or something. The assumptions must of course also imply that f is not itself the identity map. This point was (I think) first argued by Berzi & Gorini (http://physics.sharif.ir/~sperel/paper1.pdf). The argument can be found in Giulini's The rich structure of Minkowski space as well. I actually haven't studied the details myself, because I was trying to find an approach that doesn't require ugly technical assumptions.

Another issue with the paper is that the argument that rules out K<0 is pretty weak. He uses strong words like "not self-consistent", but the results he derives from the assumption K<0 are just "unexpected". There's no clear statement of what mathematical statement it contradicts. He says that "we want A_v to reduce to unity when v=0", but desire is of course irrelevant. This could be interpreted as an assumption that we're dealing with a connected topological group, but then why doesn't he say that he's making an assumption like that?

I recently tried to work out my own version of this argument. I was able to derive a genuine contradiction from the assumption that K<0, with only one simple technical assumption: 0 is an interior point in the set of velocities. Unfortunately I didn't realize until after I was done that, just like Pal and many others before him, I too had used reflection invariance right at the beginning.

I now think that there is no "nothing but relativity" argument of the sort I was hoping to find (no ugly technical assumptions, no use of reflection invariance), at least not for the 1+1-dimensional case. There is however a theorem for the 3+1-dimensional case that looks really awesome. Unfortunately, you have to go to a university library to read the proof, and it's a rather horrible exercise in matrix multiplication. Gorini's theorem can be stated like this: Let G be a subgroup of GL(ℝ⁴) such that the subgroup of G that takes the 0 axis to itself is
$$\left\{\left.\begin{pmatrix}1 & 0^T\\ 0 & R\end{pmatrix}\right| \,R\in\operatorname{SO}(3) \right\}.$$ Then either G is the group of Galilean boosts (which has invariant speed +∞), or there's a c>0 such that
$$G=\left\{\Lambda\in\operatorname{GL}(\mathbb R^4)\left| \Lambda^T\eta_c\Lambda=\eta_c\right.\right\},$$ where
$$\eta_c=\begin{pmatrix}-c & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{pmatrix}.$$ This is of course the "Lorentz" group with invariant speed c. (The actual "Lorentz group" has invariant speed 1, but this doesn't have any deeper significance. It's just a choice of units).

Those zeroes in my notation for the rotation subgroup denote the 3×1 matrix with all zeroes. The notation should be interpreted as a short way of writing a 4×4 matrix whose components are numbers, not as a 2×2 matrix whose components are matrices. I like to number coordinates and the rows and columns of these matrices from 0 to 3, so a transformation that takes the 0 axis to itself is a $\Lambda\in G$ with $\Lambda_{i0}=0$ for all $i\in\{1,2,3\}$.

 

[harrylin]

Hello

Some authors claim that Einstein's second postulate (constant speed of light) simply emerges from the first one (or more precisely, its converse contradicts the first postulate). [..]
Thanks

Hi. It's quite the contrary: the second postulate is (at first sight) "apparently irreconcilable" with the first postulate.
- http://www.fourmilab.ch/etexts/einstein/specrel/www/

 

[PAllen]

What makes you say that? The assumption they're making is that for each inertial coordinate system S, there's an inertial coordinate system T with the property that for all events p, if S assigns it the coordinates (t(p),x(p)), then T assigns it the coordinates (t(p),-x(p)).

This doesn't seem to have any obvious connection to a statement about an invariant speed.

I think the point is that while two way speed is independently measurable, one way speed is not; thus the assumption of invariant one way speed is equivalent to assuming isotropy for light (Edwards frames are anisotropic). The newer derivations can be interpreted as replacing this special assumption about light with a generally assumed symmetry of space/time.

 

[bcrowell]

When that paper invokes the "isotropy of space" (between equations 6 and 7), aren't they invoking Einstein's second postulate, except they aren't limiting it to the speed of light? Is that your point, not that Einstein's second postulate was included in the first but that it should be stated in a more general context that also includes light?

No, isotropy of space simply means that space (not spacetime) is the same in all directions, i.e., that the laws of physics are invariant under spatial rotations.

The type of axiomatization used by Pal actually doesn't refer, even implicitly, to any dynamical laws of physics such as Newton's laws or Maxwell's equations. Unlike Einstein's 1905 axiomatization, it's purely geometrical. Geometrically, the laws imply that either (a) spacetime is Galilean, or (b) spacetime is Lorentzian with some frame-invariant velocity c. In a universe where electromagnetism didn't exist, this axiomatization would still make sense, and we would still have a frame-invariant velocity c.

 

[bcrowell]

Another issue with the paper is that the argument that rules out K<0 is pretty weak. He uses strong words like "not self-consistent", but the results he derives from the assumption K<0 are just "unexpected". There's no clear statement of what mathematical statement it contradicts.

I agree that this is a flaw in his presentation. The way I prefer to present this is that there is a separate axiom asserting causality, i.e., there exist events P and Q such that Q is later than P in all frames of reference. Since K<0 describes a rotation in the x-t plane, it allows us to do a 180-degree rotation that reverses the time-ordering of any two events, and that violates the axiom.

I now think that there is no "nothing but relativity" argument of the sort I was hoping to find (no ugly technical assumptions, no use of reflection invariance), at least not for the 1+1-dimensional case.

I don't see what's objectionable about those as assumptions. If you're interested in seeing other presentations of this flavor, try:

W.v.Ignatowsky, Phys. Zeits. 11 (1911) 972

P. Frank, H. Rothe, Ann Phys (Lepizig) 34 (1911) 825.

L.A. Pars, Philos. Mag., 42 (1921) 249

Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51

David Mermin, American Journal of Physics, 52 (1984) 119

Gannett, "Nothing but Relativity, Redux," http://arxiv.org/abs/1005.2062

Morin, Introduction to Classical Mechanics, Cambridge, 1st ed., 2008, Appendix I

I don't think Pal's is the best treatment in this style, but I do think that this general approach is the only sensible one in the 21st century. It's ridiculous that people are still slavishly following Einstein's 1905 axiomatization, which is just philosophically wrong from a modern point of view. The reason I always point people to Pal's treatment is that it's conveniently available on arxiv.

 

[Fredrik]

I agree that this is a flaw in his presentation. The way I prefer to present this is that there is a separate axiom asserting causality, i.e., there exist events P and Q such that Q is later than P in all frames of reference. Since K<0 describes a rotation in the x-t plane, it allows us to do a 180-degree rotation that reverses the time-ordering of any two events, and that violates the axiom.

The approach I used was roughly this: I called the group of coordinate transformations G. I assumed that there's an ε>0 such that for all v in (-ε,ε), there's a $\Lambda\in G$ with velocity v. This seemed like the "smallest" technical assumption one can possibly make that leads to good things. Then I proved that if K<0, the following statements are true.

1. There's no proper $\Lambda\in G$ with velocity c. (If there is, then the 00 component of $\Lambda^2$ would be 0, and this means that the velocity of $(\Lambda^2)^{-1}$ is infinite, contradicting my assumption that every transformation has a velocity in ℝ).
2. For each v in (-ε,ε), there's a proper $\Lambda\in G$ with velocity v.
3. There's a proper $\Lambda\in G$ and a positive integer n such that the velocity of $\Lambda^n$ is c. (Since $\Lambda^n$ is proper too, this contradicts item 1 on this list).

I don't see what's objectionable about those as assumptions.

I find assumptions that can be motivated by physical principles more appealing than assumptions that are introduced just to ensure that the theory is simple enough to work with. But I guess there's no logical reason for this. It's just a matter of taste. I also find assumptions that involve advanced math uglier than assumptions that involve simple math.

If our goal is only to find out what a transformation between inertial coordinate systems looks like, then there is nothing objectionable about assuming reflection invariance. My goal was however more ambitious. I wanted to show that if G is the group of transformations, then there's a K≥0 such that G is equal to one of exactly five groups. The number is five because there are exactly five inequivalent answers to the question of which of these three matrices are members of G:
$$P=\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix},\qquad T=\begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix},\qquad -I=\begin{pmatrix}-1 & 0\\ 0 & -1\end{pmatrix}.$$ After a lot of work, I was finally able to prove this. This result would (in my opinion) have been really beautiful if we could interpret the statements $P\in G$ and $P\notin G$ respectively as "space is reflection invariant" and "space is not reflection invariant". I find this interpretation objectionable because the best way to justify my starting assumptions is to use reflection invariance. I was really frustrated when I figured this out. This is the reason why I haven't said that everyone should read my proof instead of Pal's. If anyone wants to see it, there's a pdf attached to this post.

If you're interested in seeing other presentations of this flavor, try:

W.v.Ignatowsky, Phys. Zeits. 11 (1911) 972

P. Frank, H. Rothe, Ann Phys (Lepizig) 34 (1911) 825.

L.A. Pars, Philos. Mag., 42 (1921) 249

Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51

David Mermin, American Journal of Physics, 52 (1984) 119

Gannett, "Nothing but Relativity, Redux," http://arxiv.org/abs/1005.2062

Morin, Introduction to Classical Mechanics, Cambridge, 1st ed., 2008, Appendix I

Thanks for the tips. I have heard of most of those, but I haven't studied any of them. Most of these are probably difficult to access for most of us.

I don't think Pal's is the best treatment in this style, but I do think that this general approach is the only sensible one in the 21st century. It's ridiculous that people are still slavishly following Einstein's 1905 axiomatization, which is just philosophically wrong from a modern point of view.

I agree, but "wrong" may be too strong a word. If we remove the references to "light" and just say that there's a finite invariant speed, we have made the assumptions pretty without changing the mathematics, and now the only problem is that the assumptions are stronger than they need to be. This may be undesirable, but it has the advantage that it makes the calculations significantly easier.

The reason I always point people to Pal's treatment is that it's conveniently available on arxiv.

It's a good article, so it makes sense to link to it.

 

[harrylin]

[...] It's ridiculous that people are still slavishly following Einstein's 1905 axiomatization, which is just philosophically wrong from a modern point of view. [..]

I suspect that that "ridiculous" thing is non-existent, as I don't know people who "slavishly" follow such things. For many people it will be as it is for me, a matter of precise definitions of what the axioms are of which theory. "SR" happens to be a definition that was made by Einstein. Also, it is rare in modern physics to subject a theory to philosophical arguments. One merely subjects a theory to experimental verification, as philosophy is a matter of personal opinion.

 

[Fredrik]

I suspect that that "ridiculous" thing is non-existent, as I don't know people who "slavishly" follow such things. For many people it will be as it is for me, a matter of precise definitions of what the axioms are of which theory. "SR" happens to be a definition that was made by Einstein. Also, it is rare in modern physics to subject a theory to philosophical arguments. One merely subjects a theory to experimental verification, as philosophy is a matter of personal opinion.

I don't consider Einstein's postulates to be the definition of special relativity. Every theory is defined by its mathematics and its correspondence rules (i.e. assumptions about how to interpret the mathematics as predictions about results of experiments). The postulates aren't even mathematical statements. They can be interpreted as mathematical statements, but this isn't even a good way to specify the mathematics.

I would say that the mathematics of SR is defined by the definitions of mathematical terms like "Minkowski spacetime" and "proper time". Together with the correspondence rule for time measurements, this defines a framework in which both classical and quantum theories of matter can be defined. I would define "special relativity" as this framework, not as a specific theory of physics. When we e.g. explain the twin paradox by drawing a bunch of lines in a spacetime diagram, what we're using is a special relativistic theory of a finite number of non-interacting classical particles. But it's much easier to just say that we're using "special relativity".

In my opinion, a "derivation" of the Lorentz transformation based on (a mathematical interpretation of) Einstein's postulates is only of historical interest. This derivation is a good way for someone who knows Newtonian mechanics and Maxwell's equations (or the result of the Michelson-Morley experiment) to find a mathematical structure that might be appropriate to use in a new theory of space, time and motion, or a theory of electrodynamics without ether. But once we've found it, it makes more sense to build the definition around that structure than around the postulates.

Even Gorini's theorem, which I find really awesome, doesn't really tell us anything about SR. What it tells us is how Galilean spacetime and Minkowski spacetime fit into a larger hierarchy of mathematical models of space and time.

 

[rjbeery]

C is defined in terms of distance by time, yet all manner of spatial measurements ultimately rely upon c. If you look at the current NIST definition of a meter it explicitly does so, but prior to this it is easy to see that even our concept of any arbitrary measuring stick would depend on the propagation rate of force carriers. It's a self-referential system. IOW, unless the rate of clocking changed locally (which would contradict SR's first postulate), we have no choice but to find c locally invariant, therefore SR's second postulate is redundant.

 

[harrylin]

I don't consider Einstein's postulates to be the definition of special relativity. Every theory is defined by its mathematics and its correspondence rules (i.e. assumptions about how to interpret the mathematics as predictions about results of experiments). The postulates aren't even mathematical statements. They can be interpreted as mathematical statements, but this isn't even a good way to specify the mathematics.

It should be a good way to specify the physics...

[..]I would define "special relativity" as this framework, not as a specific theory of physics.

It appears to me that the framework that you call SR is in fact known as Minkowski spacetime. That is of course a very suited framework for applying SR, as it was conceived for that very purpose.

[..] In my opinion, a "derivation" of the Lorentz transformation based on (a mathematical interpretation of) Einstein's postulates is only of historical interest. This derivation is a good way for someone who knows Newtonian mechanics and Maxwell's equations (or the result of the Michelson-Morley experiment) to find a mathematical structure that might be appropriate to use in a new theory of space, time and motion, or a theory of electrodynamics without ether. But once we've found it, it makes more sense to build the definition around that structure than around the postulates. [..]

Well of course we don't need to relate to the postulates when we use the LT. However, they follow from each other and "historical interest" can be very useful for a good and logical understanding of the foundations of a theory. In many discussions here I found that people who asked questions about SR did so because they didn't understand how to correctly apply the transformations; while if they understood the postulates, they probably would understand how to correctly apply the LT. Regretfully Serway's book as cited in the OP implies a misunderstanding of the second postulate and I find that particularly unhelpful.

 

[Fredrik]

It should be a good way to specify the physics...

The physics is in the correspondence rules, and the postulates don't say anything about those.

It appears to me that the framework that you call SR is in fact known as Minkowski spacetime.

I would be OK with that definition too, but I chose to also include the correspondence rule that all theories in this framework have in common. This is the one that says that clocks measure proper time. To be more precise: For all events A,B such that both are on the world line of a clock, and B is in the chronological future of A, the number displayed by the clock at B, minus the number displayed by the clock at A, is the proper time of the segment of the clock's world line from A to B.

...if they understood the postulates, they probably would understand how to correctly apply the LT.

I'm somewhat skeptical of this.

 

[harrylin]

The physics is in the correspondence rules, and the postulates don't say anything about those. [...]

The second postulate is a physics statement (which apparently Serway completely ignores!): it implies wave propagation as opposed to ballistic emission theory. It's the second postulate that bases SR on Maxwell's EM theory instead of for example the theory of Ritz.

 

[Fredrik]

The second postulate is a physics statement (which apparently Serway completely ignores!): it implies wave propagation as opposed to ballistic emission theory. It's the second postulate that bases SR on Maxwell's EM theory instead of for example the theory of Ritz.

I've been thinking of the second postulate as a statement about the existence of a finite invariant speed, but I guess that's just because I've never not been under the influence of the modern way of thinking about SR. Einstein very likely thought of it as a statement about electrodynamics. So it makes sense to interpret it as saying that the new theory has the same correspondence rules as the ether theory of electrodynamics. I guess that means that I have to retract my statement that the postulates don't say anything about physics. In the future I will just say that they are an ugly way to do physics. We can certainly do better in this century.

 

[harrylin]

I've been thinking of the second postulate as a statement about the existence of a finite invariant speed, but I guess that's just because I've never not been under the influence of the modern way of thinking about SR. Einstein very likely thought of it as a statement about electrodynamics. [..]

Right (consider the title of his paper). The postulates merely served as inputs based on observations to solve the puzzle; they were sufficient input to find the correct transformation rules for the New Mechanics. But I agree that with that New Mechanics in place, the existence of an invariant speed is what really characterizes it. BTW, that in this New Mechanics "the velocity of light would become an impassable limit" was already remarked by Poincare in 1904; it's not new insight.

 

[BruceW]

http://synset.com/pdf/100_en.pdf

This paper is really good. It gives the standard 'axiomatisation' of SR. Essentially, the axioms of SR are the same as for classical mechanics, except we drop the postulate that simultaneity is absolute, and instead take up the postulate that the speed of light is invariant.

What is really interesting is that if we drop the postulate that simultaneity is relative and don't use the postulate that the speed of light is invariant. We get that there is some invariant speed 'alpha'. Newtonian mechanics comes about when we assume 1/alpha^2 is zero, and SR comes about when we say alpha equals the speed of light.

Edit: In other words, from other postulates, there must be some invariant speed (except in the case 1/alpha^2=0, which is classical mechanics). To establish SR, we need the other postulates AND a postulate to say that the speed of light in vacuum is this invariant speed.

 

[rbj]

what's so special about light than other object (ex, sound waves) to consider its speed a law of nature[?]

the speed of light is the speed of the EM interaction. it's one of these fundamental forces, like gravity, and i think what Einstein (and some other physicists) are saying is that the fundamental nature of these fundamental forces is that they are not instantaneous; that when cause happens at some place, effect occurs somewhere else at a later moment as observed by some observer that is at an equal distance from both cause and effect. the time difference is directly proportional to the distance between cause and effect and the constant of proportionality is, according the the units we use to measure time and distance, equal to \frac{1}{c}. and it's the same, whether the interaction is EM, gravity, or the strong nuclear force. this parameter is really just a manifestation of the units we use to measure things. the only thing that nature insists on is that it is a real, positive, and finite value. in Planck units or many other systems of natural units, this value is simply 1. and it's 1 for any inertial observer because there is no reason why some inertial observers should be preferred as "stationary" while other inertial observers are "moving".

 

[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.