# When is a derivation of the Lorentz transformation good

1. Jun 28, 2007

### bernhard.rothenstein

when is a derivation of the Lorentz transformation "good"

There is an inflation of derivations of the Lorentz transformation from "simple", "fastest"...
When is in your oppinion such a derivation good?
In my oppinion it should fulfill the following conditions:
1. It should be based on the two posulates of special relativity,
2. It should convince that a Lorentz transformation relates the space-time coordinates of an event which detected from two inertial reference frames take place at the same point in space (x,y,z;x',y'z') when the clocks of the two frames located at that point read t an d t' respectively.
3. It should reveal the importance of the clock synchronization in the two frames following the clock synchronization procedure proposed by Einstein.
Would you impose other conditions as well?

2. Jun 28, 2007

### robphy

I feel the answer to this is subjective... and is dependent on the intended audience (and its preparation) and its [long-term] goals (e.g., how it might be used to deduce other results).

An algebraic proof may work well for some audiences. A geometric proof may work better for others. Deduction from experiment would of course be fabulous. Maybe someday, there will be a table-top activity-based derivation that will work for some others.

For historical purposes [i.e. relating to the history of the subject], it should be based on the two postulates. However, I don't feel that, in general, it needs to be based on them. Certainly, in the range of applicability, any alternate sets of starting points together with their inevitable sets of tacit assumptions, should be mathematically equivalent to the two postulates.

I would suggest that a good proof is one formulated with enough precision that misconceptions are avoided as much as possible.

My \$0.02.

3. Jun 28, 2007

### bernhard.rothenstein

good lorentz transformations

thanks. please tell me what is "table-top activity-based derivation". i think that a good derivation should suggest that the LT relate the space-time coordinates of the same event
Regards

4. Jun 28, 2007

### robphy

I am not aware of any for relativity yet... but the comment was making reference to newer pedagogical methods for teaching physics, like
http://physics.dickinson.edu/~abp_web/abp_homepage.html, as well as hopefully forthcoming technological advances in the next few decades or century. [Imagine in the not-too-distant future, a low-cost and [by today's standards] high-precision clock which could be used by a typical student as part of a "radar experiment" done for an introductory physics lab. From a series of experiments, deduce the Lorentz Transformations.]

Implicitly, isn't this what any derivation of the [passive] Lorentz Transformation does?

Last edited: Jun 28, 2007
5. Jun 28, 2007

### bernhard.rothenstein

Implicitly, isn't this what any derivation of the [passive] Lorentz Transformation does?
I have seen many derivations of the LT but I did not found that they do that explicitly.You did?
Thanks

6. Jun 28, 2007

### robphy

Maybe I need an example.
I would think that if one is going to derive something like the Lorentz Transformations, one should provide some motivation for why one is interested in such things [unless it is assumed the reader already knows why].

7. Jun 28, 2007

### bernhard.rothenstein

good lorentz transformation

Have a look please at "A simple derivation of the Lorentz transformation and of the accompanying velocity and acceleration changes"
J.-M. Lévy
Am. J. Phys. 75, 615 (2007) (you can find it on arXiv as well).
The author presents a "simple" derivation of the LT based on the length contraction relativistic effect. Using the "light clock" and deriving LT for events that take place on the overlapped axes the reader does not find out which clocks display the times t and t'. Other authors consider that in order to derive the LT we should consider length contraction and time dilation as well without to mention at all the involved clocks and how are they synchronized.
Thanks for discussing with me a problem with pedagogical importance for others as well.

8. Jul 2, 2007

### meopemuk

In my opinion, a good derivation of Lorentz transformations should take into account that the things that are transformed (e.g., times and positions of events or momenta and energies of particles) are observables measured in real life on real physical systems. Unfortunately, all derivations known to me assume from the beginning that we are looking for some general formulas connecting space-time coordinates of abstract space-time points. It is also assumed without proof that time-positions of all events and energy-momenta of all particles will follow these universal transformation rules independent on the nature of the events and interactions between the particles. I think these assumptions are not well justified, and it is legitimate to ask how accurate they are.

9. Jul 2, 2007

### pervect

Staff Emeritus
The ulitimate "proof" of relativity ultimately lies in experiment. There was a great deal of time spent in noproductive directions until Einstein had the idea of asking the simple question of how time and space transformed.

One he did this, he came up with a simple proposal, which explained the observed experimental facts. It was the fit with observation that made relativity a sucessful theory.

Note that the issue of how energy and momentum transform isn't a separate assumption - it comes as a consequence of their conservation and how space and time trasnsform - it's something that can be derived from the first assumptions.

It is assumed that time and space transform in a uniform manner - i.e. they don't transform one way today, and another way tomorrow, and they don't transform one way in Nebraska and another way in Kansas. While these may be assumptions, they don't seem very objectionable to me.

10. Jul 3, 2007

### meopemuk

I agree, it is impossible to deny successes of special relativity and its perfect agreement with experiment. However, there is always a logical possibility that in some situations Lorentz transformations may not be exact, and that some (very small) corrections should be taken into account. It seems to me that existing derivations of Lorentz transformations leave (at least) one loophole, which can (in principle) lead to such corrections. These derivations often postulate the homogeneity and isotropy of space. This looks like an innocent assumption, especially if it is applied to abstract space-time points or even to non-interacting particles. Indeed, from a given point in empty space all directions are equivalent.

However, this is not so obvious if we consider two particles A and B that interact with each other. If we attempt to derive Lorentz transformations for the position of the particle A from first principles, then we cannot assume that all directions are equivalent. There is a preferred direction that points to the particle B. So, there is a logical possibility that transformations of positions, momenta, etc. of the particle A to a moving reference frame can be slightly different from universal Lorentz transformations of special relativity. The stronger is the A-B interaction the larger deviation from Lorentz formulas one can expect.

Actually, this is not an idle speculation, because there is a well-known theorem, which says that in a system of interacting particles their wordlines can not transform excatly by Lorentz formulas.

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

It seems that non-Lorentz transformation formulas for observables of interacting particles can be in full accord with the principle of relativity and the invariance of the speed of light.

11. Jul 3, 2007

### bernhard.rothenstein

good lorentz

Thanks for the active participation on the thread I have started. Please let me know what do you mean by "space-time coordinates of abstract space-time points. I think that the LT relate the space-time coordinates of events E(x,y,z,t) in I and E'(x',y',z',t') in I' which take place at the same point in space when the clocks C(x,y,t) and C'(x',y',z,) synchronized in I and I' respectively read t and t'. Special conditions are imposed and well specified in the limits of which the LT hold exactly. What is abstract there?
Regards

12. Jul 3, 2007

### meopemuk

My idea is that we cannot measure anything in empty space. We can measure only properties (or observables) of physical systems. For example, we can measure position of a particle. Or we can measure the time and position of an event, such as a collision of two particles. In any case, there should be something material present in order for us to say that there is an event whose coordinates can be measured.

So, my idea is that measurements of space-time coordinates of abstract points in empty space are meaningless. We need to talk about space-time coordinates of physical particles. If there are more than two particles around, then they may interact with each other. If this is so, then it seems logical to assume that transformations of their coordinates to the moving reference frame may depend on these interactions, on distances between particles, etc.

To make this idea more acceptable, let us consider time translations instead of boosts (=transitions to the moving frame). Both time translations and boosts are legimitate transformations of inertial reference frames. They are equal participants in the Poincare group, so it is not so crazy to look for analogies. Now, it doesn't look strange that applying a time translation to an interacting system of particles we get a result that depends on the state of the system and on interactions. It is impossible to describe dynamics of all systems by one universal formula. Different systems evolve in time differently.

I don't think that boosts should be qualitatively different from time translations in this respect. The idea that boost transformations of particle observables may be interaction-dependent looks quite logical to me. So, perhaps it is an approximation to describe boost transformations of particle observables by one universal Lorentz formula that does not take into account what system is measured and what are interactions in this system?

13. Jul 3, 2007

### bernhard.rothenstein

good LT

When I speak about an event I consider that it is generated by a particle or by a light signal. The consequence is that there is a relationship between the space and the time coordinate of an event like E(r,t=r/u) in the case when the event is generated by a tardyon and E(r,t/c) in the case when the event is generated by a light signal.
I would speak abot "ideal" and not "imaginary".
With respect

14. Jul 3, 2007

### country boy

As you say, the Lorentz transformations have been derived in many ways. At this point, I like any derivation that teaches me something new.