Concerning the 1920 Einstein Derivation of the Lorentz Transformation

[M8M]

I have been reading Lieber's book "The Einstein Theory of Relativity" which describes the derivation of the Lorentz Transformations:

archive.org/details/einsteintheoryof032414mbp -- see pages 39-56

bartleby.com/173/a1.html

The entire derivation makes no sense, whatsoever. Why is it necessary to cover-up the whole problem? You cannot just state that there is a flash of light at the origin and proceed with the mathematics from there. What is the nature of the mysterious light flash at the origin which "stays" with the K and K' frames as they move? I mean, I thought physicists were precise...this is ridiculous.

If somebody thinks they can step-by-step defend and explain either the original Einstein 1920 derivation, or Lieber's butchering of it, please step up.

 

[lalbatros]

M8M,

Your complain about this book is not clearer than the book itself.
The Lorentz transformation can be derived in a few lines and almost no word at all.
The book you are using is so lengthy and complicated that I do not have the patience to even read two pages of it to answer your question.

If you want a simple introductory book on Special Relativity, my best recomendation would be this:

Very Special Relativity: An Illustrated Guide, by Sander Bais

Otherwise derivations satisfying the K-I-S-S principle (according to me) would be:

- There are only two classes of groups relating inertial frames: the Galilean group, and the Lorentz groups based on an arbitrary speed "c". It happens, which is also a consequence, that zero-mass particles also propagates at this speed 'c'. It happens that infinite 'c' brings us back to Galilean group. It appears that the Lorentz transformation is a property of space-time, which was revealed by electromagnetic theory, but 'c' is also the speed of neutrinos which are -a priori- not related in any way to the theory of light.

- The invariance of the equation of light wavefronts: ds²=dx²-c²dt²=0 in K ==> ds'²=dx'²-c²dt'²=0 in K' also lead very directly to the Lorentz transformation. I also tend to believe that the derivation is not interresting and un-pedagogical. It is easy to plug the Lorentz transformation in the above equations and check that it satisfies the requirements. It can easily be accepted (and proved) that there is no other solution.

Finally, note that the original derivation by Einstein dates back to 1905. You can find it there:

http://www.fourmilab.ch/etexts/einstein/specrel/www/


In addition, keep in mind that the Loretnz transformation was derived by Lorentz a few years before the Einstein derivation. There were also several other transformations suggested by other people around the same time. Only the Lorentz transformation satisfied all the requirements of the principle of relativity.

 

[M8M]

M8M,

I also tend to believe that the derivation is not interresting and un-pedagogical. It is easy to plug the Lorentz transformation in the above equations and check that it satisfies the requirements. It can easily be accepted (and proved) that there is no other solution.

Thanks for your response. There is something unsettling about Einsteins derivation. I am starting to think there is more than a grain of truth that the whole business was based on Voigts work...

 

[Mentz114]

Thanks for your response. There is something unsettling about Einsteins derivation. I am starting to think there is more than a grain of truth that the whole business was based on Voigts work...

You'll be alone in that opinion. What have you got against Einstein ?

 

[lalbatros]

Thanks for your response. There is something unsettling about Einsteins derivation. I am starting to think there is more than a grain of truth that the whole business was based on Voigts work...

That's folklore, not physics, sorry.
Even not good history of science.
Start by studying these papers, specially those of Lorentz and Poincaré:

http://www.soso.ch/wissen/hist/SRT/
http://www.soso.ch/wissen/hist/SRT/srt.htm

Try to figure out what the state of physics really was at that time.
Ask yourself if people were supposed to work isolated at that time, or were they also supposed to communicate with each other.
And finally ask yourself if the Einstein1905 paper should be the best way to discuss SR today.

But there are better things to do!

 

[elfmotat]

You do realize that's not Einstein's original derivation, right? That's from a book he wrote in 1920. His original 1905 derivation can be found here: http://www.fourmilab.ch/etexts/einstein/specrel/www/.

 

[M8M]

I think it is safe to say I can believe whatever I want and spout it on this forum as if it is gospel. I see 10 different contradictory opinions on here and they are all supposedly backed by fact.

QED

 

[PAllen]

I think it is safe to say I can believe whatever I want and spout it on this forum as if it is gospel. I see 10 different contradictory opinions on here and they are all supposedly backed by fact.

QED

Yes you can believe what you want. No, you cannot spout anything you want on this forum, as you will soon discover if you try. You should read the forum rules. (Within those rules, there remains plenty of room for differences of informed opinion, interpretation, questions of understanding, etc.)

 

[ghwellsjr]

I have been reading Lieber's book "The Einstein Theory of Relativity" which describes the derivation of the Lorentz Transformations:

archive.org/details/einsteintheoryof032414mbp -- see pages 39-56

bartleby.com/173/a1.html

The entire derivation makes no sense, whatsoever. Why is it necessary to cover-up the whole problem? You cannot just state that there is a flash of light at the origin and proceed with the mathematics from there. What is the nature of the mysterious light flash at the origin which "stays" with the K and K' frames as they move? I mean, I thought physicists were precise...this is ridiculous.

If somebody thinks they can step-by-step defend and explain either the original Einstein 1920 derivation, or Lieber's butchering of it, please step up.

Lieber's explanation of SR and especially his diagrams are hard to learn from but they are not necessarily wrong. It is difficult to represent in a book how two relatively moving observers "watching" the same expanding sphere of light will conclude that they are each in the center of it. This is actually a fact of nature and needs to be explained by a theory so you can't blame the theory for what might seem like a bizarre situation. Let me see if I can help you accept this fact of nature.

First off, if it isn't already obvious to you, no one can directly watch the propagation of light. Once it has left your vicinity, it is invisible to you. You cannot know where it is at any given moment unless you can see it reflected off some object and even then, you cannot know when it hit that object because the light has to make its way back to you before you can see it. So all you know is how long it took for the light to make a round trip. Do you agree with this?

Secondly, the only way that an observer can have any hope of determining the nature of the propagation of the light emanating from a flash that occurred in his vicinity is to surround himself with some mirrors, all placed a particular distance away, in the shape of a sphere and then observe that the reflected light from each mirror arrives back at his vicinity at the same time. Don't you agree that this would be the best indication that he was in the center of the expanding sphere of light, provided that the light was traveling identically in all directions at the same speed?

Now the third point is that if a flash of light occurred when two observers in relative motion exactly passed each other, they would each need to have their own set of mirrors, and if they each observed that the light from all of their own mirrors arrived back at their locations at the same time, even if they were not at the same location when this happened, they could each make the same claim that they were in the center of the expanding sphere of light, couldn't they?

Well, that is exactly what happens when this experiment is performed. So how can we explain this? The explanation that is offered by any theory is that the mirrors are closer together along the direction of motion and that causes the reflected light for each observer to arrive simultaneously. I made an animation to illustrate these points:

https://www.youtube.com/watch?v=dEhvU31YaCw

Does this animation help you understand how both the moving red guy and the stationary green guy can both equally claim to be in the center of the expanding sphere of light or do you need more explanation?

 

[lalbatros]

I think it is safe to say I can believe whatever I want and spout it on this forum as if it is gospel. I see 10 different contradictory opinions on here and they are all supposedly backed by fact.

QED

We see that you are a I-know-it-all beginner.

 

[Mentz114]

- I made an animation to illustrate these points:

Excellent animation.

 

[jtbell]

I agree, that's a nice animation! It would also be nice to have another one that shows what's happening in the red guy's rest frame.

 

[ghwellsjr]

- I made an animation to illustrate these points:

Excellent animation.

Thanks, let's just hope M8M agrees.

I agree, that's a nice animation!

Thanks again.

It would also be nice to have another one that shows what's happening in the red guy's rest frame.

If you want to see it from the red guy's rest frame, just look at it through a mirror and interchange red and green in you mind.

 

[lalbatros]

Thanks, let's just hope M8M agrees.
...

I wouldn't expect anything like that.
He only came on this forum to argue about relativity and Einstein.
This is fashionable in some clubs.

 

[M8M]

Please forgive me for asking such ignorant questions. Perhaps you can take a moment to clarify the following?

I think it is safe to state the following:

The light sphere experiment is a thought experiment, only. There would be serious difficulties in physically conducting such an experiment. First, we are forced to assume that at some instant, t=0, that K and K’ are in the same point O in x,y,z,t space. Obviously, two observers cannot be at the same point at the same time. Additionally, it would pose serious problems to jam the light source S at this already crowded point O.
Next, the spherical mirror in which K’ is centered, by a platform of sorts, would require an entrance opening and exit opening so as to avoid collision with stationary K as it moves past him. This could be resolved by have a circular array of mirrors for K in the x-plane and circular array of mirrors K’ in the y- plane with the diameter of the K circular array being larger. Assuming we can somehow jam K, K’, and S at O at t=0 then we have the following reduction of the problem.

Is the radius of the circular mirror array for K’ shortened in the direction of movement? Would this appear as an ellipse from our privileged frame of reference which watches this video?

This begs the question. How is this any better than using a schematic of the Michelson-Morely experiment as our platform for discussion? Imagine two perpendicular line AB and AC of equal length. Assume AC is on the y-axis and AB is on x-axis with A at the origin. There is an ether wind moving in the direction of the x-axis. Just like the example of a swimmer going upstream and across stream of a river, flowing with a velocity v, with constant velocity c.

We know that t (AC) = 2aβ2/c and t (AB) =2aβ/c. The MM experiment found that t (AC) = t (AB). Lorentz corrected by stating t (AB) had contracted to length by a factor 1/β. Now, Lieber states on pg. 17 that another way of stating this is x’ = β(x-vt) with x’ being the contracted length and time dilation is a consequence of this.

At any rate, a review of the work of Lorentz explains that time dilation came first and that length contraction came second and was “second order” correction. See the theory of "corresponding states". Using the simple diagram of example of MM seen at pg. 11 of the axes being dragged by the ether in page 15. It is not clear how the time dilation is computed. Lieber states that “Now when Lorentz examined other facts, as stated on page 15, he found that equation (4) was quite in harmony with all these facts, but that he was now obliged to introduce a further correction expressed by the equation – for time dilation”. This is leaves us all in the dust.

Why can we not follow and build on the MM experiment and river analogy to build up the time correction and length correction? What is missing? Why is it necessary to jump from example to example? What is the nature of first and second order corrections?

 

[ghwellsjr]

Apparently I misread you. I thought you were having trouble learning Special Relativity and you didn't understand how two co-moving observers could conclude that they were each in the center of the same sphere of light. Isn't that what you were asking about in your opening post?

What is the nature of the mysterious light flash at the origin which "stays" with the K and K' frames as they move?

I thought my discussion and animation would help you understand the answer to your question but now I sense that you already knew all that and more and that you already fully understand Special Relativity, correct?

The purpose of this forum is to teach relativity to those who want to learn and I'm wondering if maybe you know enough to be one of the teachers. It appears that you even know a lot about the history of the development of relativity, too, correct?

 

[M8M]

No. I do not understand any of special relativity.

 

[ghwellsjr]

Then you can't be a teacher. Do you want to learn Special Relativity?

 

[dungonyu]

Is your concern is two equations which is equal to 0 and then they are equated with a factor of some constants? yeah, I also admit that is strange. Just like a=0 and b=0, then arguing a=kb, for some constant k. but, indeed in a mathematical sense, k can be anything!

 

[dungonyu]

I like the derivation in Wikipedia. Starting with : x'= ax + bt and t'=dt + ex, for some constants a, b, d and e. And arguing it's linear, otherwise, there will be acceleration, not an inertial frame. I am satisfied with this derivation. And it uses the two postulates: special case for x=ct and x'=ct' and the principle of relativity.

 

[ghwellsjr]

Is your concern is two equations which is equal to 0 and then they are equated with a factor of some constants? yeah, I also admit that is strange. Just like a=0 and b=0, then arguing a=kb, for some constant k. but, indeed in a mathematical sense, k can be anything!

Can you please point us to the chapter in his 1920 book where he does that?

 

[dungonyu]

Oh, in that little book, as you've said, using a light signal from both orgins of coordinate systems: x=ct; x'=ct'.(in the appendix of deriving Lorentz Transformation) so x-ct=0 and x'-ct'=0. Then Einstein guessed there's some relationship between the two equations, thus x'-ct'=λ(x-ct); and analogously if the light signal is emitted to the negative x,x'-directions, x'+ct'=μ(x+ct); for some constants λ and μ. This step is quite non-sense in a mathematical sense i think.

In the below wikipedia link of Lorentz transformation, there are derivation from different start points. From postulates, and so on. There's also include the Einstein derivation. But still we can omit his approach as we can also derive, from the two postulates, not in Einstein's way, the Lorentz Transformation. It's quite easy to understand as we are pleased that the equations must be linear.
http://en.wikipedia.org/wiki/Lorentz_transformation

 

[ghwellsjr]

Oh, in that little book, as you've said, using a light signal from both orgins of coordinate systems: x=ct; x'=ct'.(in the appendix of deriving Lorentz Transformation) so x-ct=0 and x'-ct'=0. Then Einstein guessed there's some relationship between the two equations, thus x'-ct'=λ(x-ct); and analogously if the light signal is emitted to the negative x,x'-directions, x'+ct'=μ(x+ct); for some constants λ and μ. This step is quite non-sense in a mathematical sense i think.

As you said in post #19, k can be anything which doesn't make it non-sense or invalid. You just have to take more steps to find out what other constraints there are on the value of k. Likewise, in his derivation in the appendix, he takes more steps and gets the answer that is identical to the methods you favor. Doesn't that indicate to you that he followed a correct mathematical procedure?

 

[dungonyu]

urm. Ok, I see your point. And now I understand the arguement: with the disapperance of x-ct, there's also an disapperance of x'-ct'. so there should be some constants times the whole thing but separately such as x-ct=ax'-bct'. and that word "in general" satisfies my worry, since I've been thinking so long that whatever constant it is, the equation is satisfy! And your mentioning of some constrain, ok, i feel better.