# A direct derivation of the speed transformation formula?

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1. Nov 28, 2014

### Sugdub

In SR the speed transformation formula (in response to a change of inertial frame of reference) is usually derived from the Lorentz transformation of space and time coordinates. I would like to find a direct derivation starting from the existence of a maximum speed limit (c) in respect to any IRF, avoiding to go through the transformation of coordinates as an intermediate step. Any hint?
Thanks.

2. Nov 28, 2014

### robphy

3. Nov 29, 2014

### m4r35n357

It is well worth reading the whole of the development described here: http://mathpages.com/rr/s1-07/1-07.htm

But the bit that is relevant to your question starts about two thirds of the way down: "A more persuasive argument for a finite upper bound on speeds . . .", culminating in equation 8.

4. Nov 30, 2014

### Sugdub

Thanks for your suggestion. I went through this book and noted the section on “Velocity Composition” p. 104-105. By combining the invariance of the propagation speed of radar signals (c) and the time dilation formula established in the previous sections, one can effectively derive the speed transformation formula. However the discussion is based on thought experiments, it is therefore similar to Einstein's derivation in 1905. Not neglecting its historical value, there are several drawbacks to this approach insofar it is difficult to control the list and the actual contribution of various secondary assumptions or hypotheses. On top of setting the principle of relativity and the existence of a maximum speed limit, one must postulate that there exists a useable signal actually propagating at the maximum speed limit (light, radar pulses), so that the thought experiment looks feasible in principle. The reflexion of the signal on a distant object must be assumed instantaneous, as well as occurring at mid-time between emission and reception events. Furthermore, the derivation assumes that time and length measurements are performed whereas the validity of SR is expected to hold even in the absence of any measurement or observation activity.

This is not what I'm looking for. It is quite puzzling that both initial principles of SR relate to velocities and that they must be converted into statements about experimental measurements of time or length, whilst involving assumptions about non-observable events (e.g. the reflexion of a radar signal) in order to derive intermediate equations which get converted back into the speed transformation formula (which is ultimately independent from the time dilation/length contraction factor). I'm looking for a direct “axiomatic” derivation of this formula starting from a non-ambiguous mathematical transcription of the initial principles.

5. Nov 30, 2014

### pervect

Staff Emeritus
The existence of a signal that travels exactly at the universal speed limit is very helpful. However, one could consider that the "signal" was just throwing electrons or even baseballs, if one could throw the baseball / electron either at the universal speed limit (which isn't possible), or close enough to the universal speed limit that the difference didn't matter (which is). But it would complicate things to do the later, i.e. if you tried to keep track of how much under the universal speed limit the electrons/baseballs/ generalized signals were going.

[add]I think I can sharpen this up a bit. What you have to be able to do, experimentally, is find the limiting earliest time a signal would arrive if it did travel at the "speed limit", based on measurements you can actually carry out with signals (which are presumed to exist) which may in practice travel at less than the universal speed limit.

Not really, you can handle this by having the signal received and re-transmitted rather than reflected. You do need to assume that you can make any delay that occurs between reception and re-transmission "short" - if there is a significant delay between reception and re-transmission, it again complicates the numbers. I think you also need to imagine that the "receiver" and "transmitter" are point objects, or at least small.

[add]Since the signals in most cases will be arriving periodically, I don't think it's particularly onerous to imagine that you can arrange things so that you phase-lock the periodic signals you receive to the periodic sigals you send. You know you've succeeded when the signal you send and the signal you receive are both present at the same point at the same time, within experimental error. Again, we see the need for imagining that our signals are small in spatial extent.

I'd have to agree that the Bondi's approach isn't a formal mathematical one. On the plus side, it gives one a very "physical" feel for how one might actually go about measuring lengths and times using familiar instruments.

Last edited: Nov 30, 2014
6. Dec 2, 2014

### Fantasist

Have a look at Levy-Leblond's derivation of the Lorentz transformation ( https://fenix.tecnico.ulisboa.pt/downloadFile/3779571248372/Levy-Leblond_(76).pdf.) . From invariance principles applicable to coordinate transformations in general (that is without involving the invariance of the speed of light), he formally obtains as solutions 4 different types of transformations, two of which are only physically acceptable, the Galilei transformation and the Lorentz transformation. The Lorentz transformation can then be singled out by assuming the existence of a limiting speed c.

Last edited: Dec 2, 2014
7. Dec 2, 2014

### Sugdub

Thanks for your input. The derivation of the speed transformation formula (equation 8) follows the standard method and does not match my expectations (see my previous input).
However the discussion which follows this derivation seems to address the relevant issue: a direct derivation of the speed transformation formula. Unfortunately I don't understand the way the problem is presented: “One approach is to begin with three speeds u,v,w representing the pairwise speeds between three co-linear particles, and to seek a composition law of the form Q(u,v,w) = 0 relating these speeds...Q(u,v,w)=Auvw+Buv+Cuw+Dvw+Eu+Fv+Gw+H ...Treating the speeds symmetrically requires B = C = D and E = F = G”.
Although three collinear velocities are effectively on stage, they relate to two inertial frames of reference and they are not mutually independent: for example v is the velocity of a particle in IRF-1, w is the velocity of the same particle in IRF-2 and u is the velocity of IRF-1 in respect to IRF-2. Hence the symmetry of the problem at stake differs from what is described (their respective role is not equivalent) and I don't understand either the rationale for the requirement for linearity for all three variables. May be someone can explain this to me.
The author seems to confuse the law of composition of speeds which relate to the same IRF (the addition of speeds still holds in SR as well as in the Galilean case), with the law of transformation of speeds in response to a change of IRF (the addition which holds in the galilean case must be replaced with a new speed transformation formula).

Interestingly the next section in the book seems to indicate that the problem I'm raising has been addressed by Minkowski and others but it is not clear whether a clean and simple solution has been found.

8. Dec 2, 2014

### m4r35n357

Ah OK, the way I read it seemed to be what you were looking for, in the sense that he derives both the galilean and lorentzian speed addition formulas "from scratch", and shows that a speed limit distinguishes the two (with the Lorentz transform just being an intermediate step). I think the Q(u,v,w) stuff is his own personal musings on the matter. That said, I counted three IRFs in there so I think I'm ready to admit that I don't understand your question and I'll shut up now ;)

9. Dec 2, 2014

### robphy

10. Dec 2, 2014

### Staff: Mentor

Sugdub, it seems like everyone is guessing what you want and providing answers to questions you are not asking. I think you need to reformulate the question. Please list what you want to allow as givens and what you want to prove and the specific steps or types of reasoning that you want to exclude.

11. Dec 3, 2014

### Sugdub

Good suggestion. I'll try again.
In SR, two initial principles (principle of relativity of motion, existence of a maximum limit for a speed related to an IRF) … plus a few assumptions... enable to derive in a first instance the Lorentz transformation of the 4-coordinates of any event in response to a change of the IRF. Taking stock of the Lorentz transformation, one can then derive a set of theorems such as the invariance of the space-time interval between two events through a change of the IRF, the length contraction and time dilation formulas, the transformation formula for speeds in response to a change of the IRF.

I wish to find a direct derivation of the speed transformation formula which:
• proceeds on the basis of the same initial principles above plus a few “assumptions” to be clarified,
• does not deal with space and time coordinates (x,t), but deals directly with constraints on the composition of speeds such as f(u,0)=f(0,u)=u f(u,1)=f(1,u)=1 f(u,-u)=0 f(u,v)<=1
• does not involve any kind of thought experiment (neither based on measurements of length or time, nor based upon the imagined propagation of a signal).
Since the target speed transformation formula relates to the velocities which are dealt with by the initial principles, and since an indirect proof of this formula exists on the basis of the coordinates of imagined events relevant to some thought experiments, I think it should be possible to find a direct proof of the speed transformation formula in compliance with the above requirements.
Thanks.

Note: for example, the discussion at the end of the chapter 1.7 Staircase Wit in http://mathpages.com/rr/s1-07/1-07.htm looks promising insofar it matches the constraints above. However I do not understand how the equation kuvw+u+v+w=0 is arrived at.

12. Dec 3, 2014

### pervect

Staff Emeritus
If all you demand is that your velocity composition law satisfies the group axioms of associativity, closure, invertibility, and the existence of an identity element, any such composition law over a finite interval has a mapping / rescaling to a simple addition over real numbers, according to http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/relativity.pdf

In special relativity, this rescaled law is that the [correction] inverse hyperbolic tangents, also known as rapidities, add.

So you will need more than the fact that velocity addition is a group in an open interval of the real numbers to get SR. I think. Without more structure you know some rescaled function of velocities add, but you won't know which function that is. Equivalently, you can specify some arbitrary invertible mapping from your open interval [-c,c] to the real numbers R, and using this arbitrary function, map from [-c,c] to R, add the resulting reals, then pulls back via the inverse map to your open interval to generate a general composition law that satisfies the group axioms, but will not be SR UNLESS you choose your arbitrary function to be the [correction] inverse hyperbolic tangent.

Last edited: Dec 4, 2014
13. Dec 3, 2014

### robphy

I think you mean that "rapidity" is the "inverse hyperbolic tangent of the velocity" (that is, tanh(rapidity)=velocity).
These are additive since "rapidity" is the angle between worldlines (or an arc length on a unit hyperbola) in Minkowski geometry.
Of course, hyperbolic-tangents are not additive... their composition law resembles the formula for tanh(A+B).

By the way, the additivity of rapidity corresponds to the multiplicativity of exp(rapidity) ( that is, exp(A+B)=exp(A)exp(B) ).
exp(rapidity) and exp(-rapidity) are special because these are precisely the eigenvalues of a boost, with lightlike eigenvectors.
In the language of Bondi's k-calculus, k=exp(rapidity)=DopplerFactor=sqrt((1+v)/(1-v))...
and the multiplicativity of the k-factors is equivalent to the velocity-composition-formula.

14. Dec 4, 2014

### pervect

Staff Emeritus
Yes, sorry for any confusion. Insert irrelevant excuses here...

It's interesting (and mentioned by the paper as an example of the main theorem) that there is a transformation that transforms the group structure of multiplication over a suitable interval into the addition of real numbers. The needed mapping is of course logarithmic, adding logarithims gives the logarithm of the product. So it's not surprising and indeed expected that the additive quantity is the log of the multiplicative quantity.

If I understood the OP correctly he wasn't so interested in k-calculus, as he didn't want to have to introduce coordinates, just consider what we could learn from the velocity composition law. My conclusion is that there are (unsurprisingly) an infinte number of velocity composition laws that obey the group axioms, only one of which is the relativistic velocity composition law.

It's not clear if the OP has more structure to add to the velocity composition laws than the group axioms, but it's unclear how one might go from saying that "there is some function of velocities that adds" to "rapidities add" without considerably more structure than the OP was proposing.

Once one introduces coordinates, and demands that there be some linear transformation between between coordinates that represent a change-in-frames, i.e. a Lorentz boost or an addition of velocity, it becomes obvious (once explained, at least) the requirement that a point on a light cone through the coordinate origin be mapped to another point on a light cone through the coordinate origin implies that the coordinates of the points on the lightcone are eigenvectors of said linear transformation.

If we let the velocity of light be c, and consider a 2-d space (t,x), then points on the light cone are all a scalar multiple of (0,c).

And the definition of an eigenvector of a linear transformation is that it maps vectors to scalar multiples of a vector. Since the points on the lightcone are all scalar multiples of the same vector, that vector, representing the velocity of light, must be an eigenvector of the linear transformation, as must any vector on the light cone.

The rest follows as you explained, the k-values are the eigenvalues corresponding to the eigenvectors.

But this requires the introduction of coordinates, something the OP was resisting (at least from what I read).

Last edited: Dec 4, 2014
15. Dec 12, 2014

### Sugdub

Very interesting. What kind of additional logical “structure” could be missing? For me there are two options. Either some additional hypotheses/assumptions are missing, or the available hypotheses are not fully “exploited”.

On the one hand we have a set of postulates, axioms, hypotheses which in the standard SR derivation context lead to the speed transformation formula via the Lorentz transformation of coordinates. In the new derivation context, the key principles are still on. However we have replaced the existence of an invariant speed for the propagation of light rays with the existence of a maximum speed limit for any physical object. Also we won't use the hypothesis whereby there exists a physical object (the light) which propagates at c, and neither the definition whereby my time is what gets measured by a clock collocated with me ... We don't need such hypotheses any longer since we are now excluding the perspective of deriving the speed transformation formula via thought experiments based on light rays and time measurements. But so far we only removed the hypotheses which made the standard SR formalism dependent on a pre-existing model of the world (at least in respect to the propagation of light rays). As long as we believe that SR should provide an a-priori formal structure which any physics theory (outside gravitation) will have to fit into, the absence of these specific hypotheses should not hinder the feasibility of a direct derivation of the speed transformation formula. If not, it could mean that SR has a specific debt toward electromagnetism, or whichever loophole one may imagine.

For me it is likely that the available hypotheses must get combined in a different, more powerful way. I'll keep thinking about it. Thanks.

16. Dec 12, 2014

### Staff: Mentor

Such as combining them to derive the Lorentz transform perhaps?

17. Dec 17, 2014

### Alkel

I have a question about the Lorentz transform. Why is c used as the unit speed? Is it not just as possible to use 10*c, or the speed of sound for that matter?

18. Dec 17, 2014

### Staff: Mentor

Because that's the unit speed that we measure experimentally. If you plug a different unit speed into the Lorentz transform, you will get answers that don't agree with experiment.

19. Dec 17, 2014

### Alkel

Well that is fine if we are only looking at light. The experiment uses light as the measurement and as the receiver. If, for example, we used a ball and this ball could travel at the any other speed greater or less than the speed of light the equation would follow for Vball. Would it not?

20. Dec 17, 2014

### PAllen

No,it wouldn't, unless the ball's speed were an invariant speed. In our world, a ball can't move faster than c. A different way of saying this that has nothing to do with light is the following:

A suppose a rocket moving at speed v relative to a lab, fires a ball in its forward direction at speed u relative to itself. What speed does the lab measure for the ball?

The prediction from Newtonian mechanics, that people believed was exact before the 20th century, is that the lab would measure u+v.

The prediction from relativity, that we have verified by experiment, is that the ball's speed measured by the lab is: (u+v)/(1 + (u*v/c2))

Note, this means that if the rocket were moving at .9c relative to the lab,and fired a ball at .9c relative to it, the lab measures this ball going .9945c NOT 1.8c.