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 Lorentz transformation 
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Joined: 29 Jan 2010, 19:21
Posts: 11
Post Lorentz transformation
Hello all. I made a similar post to this one on sci.physics.foundations but it seems that I didn't find any interest.
I will try it here, hopefully I can get some comments on this.

I'm fascinated with the Lorentz Transformation and I thought a lot about it.
I have a very simple method to arrive on the Lorentz transformation, and I'm sure this is not new, but I never saw any text dealing with it in this way.
I thought about this after reading Minkowski paper where he introduces the concept of imaginary time, but this is much simpler since it doesn't need any imaginary concept.
I will just introduce in a very directly and simple way, but if there is anyone interested I can surely write a lot more on this.

Let's imagine a simple right triangle
\begin{center}<br />\setlength{\unitlength}{1mm}<br />\begin{picture}(55,25)<br />\put(2,2){\line(1,0){30}}<br />\put(15,0){\scriptsize$b$}<br />\put(2,2){\line(2,0){30}}<br />\put(15,0){\scriptsize$b$}<br />\put(32,2){\line(0,1){15}}<br />\put(34,7){\scriptsize$a$}<br />\put(2,2){\line(2,1){30}}<br />\put(15,12){\scriptsize$c$}<br />\end{picture}<br />\end{center}
We have on this triangle:
Equation (1)
a^2+b^2=c^2 and also
Equation (1a)
a^2= c^2 - b^2

Now let's divide all the sides on the triangle by a, then we would have another triangle similar to the first one:
\begin{center}<br />\setlength{\unitlength}{1mm}<br />\begin{picture}(55,25)<br />\put(2,2){\line(1,0){30}}<br />\put(15,0){\scriptsize$b/a$}<br />\put(2,2){\line(2,0){30}}<br />\put(15,0){\scriptsize$b$}<br />\put(32,2){\line(0,1){15}}<br />\put(34,7){\scriptsize$1$}<br />\put(2,2){\line(2,1){30}}<br />\put(15,12){\scriptsize$c/a$}<br />\end{picture}<br />\end{center}
and we can write
Equation (2)
1 + \frac{b^2}{a^2} =\frac{c^2}{a^2}
that is simply Equation (1) divided by a^2. And if we use the fact that a^2= c^2 - b^2 (Equation 1a) we can rewrite Equation (2) as:
Equation (3):
1 + \frac{b^2}{ c^2 - b^2} =\frac{c^2}{ c^2 - b^2}
And with simple algebric manipulation we can rewrite it as:
Equation (3a):
1 + \frac{b^2}{ b^2(\frac{c^2}{b^2} - 1)} =\frac{c^2}{ c^2(1 - \frac{b^2}{c^2})}
That can be further simplified to:
Equation (3b):
1 + \frac{1}{(\frac{c^2}{b^2} - 1)} =\frac{1}{ (1 - \frac{b^2}{c^2})}

If we call \frac{b}{c}=\beta (so \frac{c}{b}=\frac{1}{\beta}) we can write (3b) as
Equation (3c)
1 + \frac{1}{(\frac{1}{\beta^2} - 1)} =\frac{1}{ (1 - \beta^2)}
And if we simplify a little further (just multiply the second term by \frac{\beta^2}{\beta^2}) we can write
Equation (3d)
1 + \frac{\beta^2}{(1 - \beta^2)} =\frac{1}{ (1 - \beta^2)}

It should be clear that (2) and all the variations of (3) are the samething, so:
\frac{1}{(\frac{1}{\beta^2} - 1)}=\frac{\beta^2}{(1 - \beta^2)} =\frac{b^2}{a^2} \Rightarrow \frac{b}{a} = \frac{\beta}{\sqrt{1 - \beta^2}}
and
\frac{1}{ (1 - \beta^2)}= \frac{c^2}{a^2}\Rightarrow \frac{c}{a} = \frac{1}{\sqrt{1 - \beta^2}}
Notice that \frac{c}{a} = \frac{1}{\sqrt{1 - \beta^2}} is the Lorentz factor (usually called \gamma).
I find this so much simpler that the usual way of looking at Lorentz transformation.
I could write a little more (imagine c constant and b varying), but I think this post is already big enough ;)

Regards,

Rodrigo


13 Apr 2010, 21:23

Joined: 27 Mar 2010, 19:18
Posts: 3
Post Re: Lorentz transformation
Hello Rodrigo!

First what is “sci.physics.foundations”?

Second, the only constructive statement I can see in your post is Pythagorean Theorem, others are 1=1 ;) which is your equation 3, 3d and everything that follows double using of Pythagorean Theorem in one expression (for the same triangle). The last expression for b/a and c/a, is just another way of stating Pythagorean Theorem (it isn’t trivial like 1=1, but isn’t a kind of informative too). I can’t see how it’s related to a Lorentz transforms maybe it’s just a way how you can get \beta and \gamma from the triangle of a body velocity and the speed of light?
In fact using of ict, which is “complex time” (if I’ve got it correctly) isn’t good for a curved space-time and so you’d better not to use it at all. Magic of complex numbers works in another way in space-time (in spinoral approach see R. Penrose, W. Rindler “Spinors and space-time”).

Sorry if I disappointed you a little, but Lorentz transform is a really beautiful thing and it’s worthwhile to think about it anyway.

Regards,
Oleg


14 Apr 2010, 16:26
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Joined: 07 Jun 2008, 08:21
Posts: 235
Post Re: Lorentz transformation
Hi Rodrigo (I never thought I'd correspond with the composer of 'Concierto de Aranjuez',:-))

Essentially you have derived your result from the theorem of Pythagoras:

c^2=a^2+b^2 or
1=a^2/c^2+b^2/c^2\Rightarrow a^2/c^2=1-b^2/c^2=1-\beta^2 if we define \beta=b/c
\Rightarrow c/a=1/\sqrt{1-\beta^2}

and similarly

b^2/a^2=c^2/a^2-1=1/(1-\beta^2)-1=\beta^2/(1-\beta^2)

So really you have just re-arranged the Theorem of Pythagoras.

The Lorentz transformation comes from the theory of Special Relativity. In this theory c represents the speed of light and has nothing to do with Pythagoras.
Regards
Vasco


14 Apr 2010, 16:37

Joined: 29 Jan 2010, 19:21
Posts: 11
Post Re: Lorentz transformation
O.R.S. wrote:
Hello Rodrigo!

First what is “sci.physics.foundations”?

Hi O.R.S., sci.physics.foundations is a newsgroup. You can access it by a news server or using google:
http://groups.google.com/group/sci.phys ... ons/topics
O.R.S. wrote:
Second, the only constructive statement I can see in your post is Pythagorean Theorem, others are 1=1 ;) which is your equation 3, 3d and everything that follows double using of Pythagorean Theorem in one expression (for the same triangle). The last expression for b/a and c/a, is just another way of stating Pythagorean Theorem (it isn’t trivial like 1=1, but isn’t a kind of informative too). I can’t see how it’s related to a Lorentz transforms maybe it’s just a way how you can get \beta and \gamma from the triangle of a body velocity and the speed of light?
In fact using of ict, which is “complex time” (if I’ve got it correctly) isn’t good for a curved space-time and so you’d better not to use it at all. Magic of complex numbers works in another way in space-time (in spinoral approach see R. Penrose, W. Rindler “Spinors and space-time”).

Sorry if I disappointed you a little, but Lorentz transform is a really beautiful thing and it’s worthwhile to think about it anyway.

Regards,
Oleg

You are not disapointing in any way. But if you want to go further and tell some more about your understanding about the Lorentz transform, it would be great.
I agree with you that I wrote some kind of tautology. But I can't see how you don't see the relation with the Lorentz transform. It is the same thing.
Just let c be equal to the magnitude of the speed of light and b equal to the magnitude of the relative speed between the two frames (call it v if you prefer).

Take a look at this paper: http://www.univ-nancy2.fr/DepPhilo/walt ... instd7.pdf , in particular the Appendix (page 35 on the pdf).
By your comments it looks like the Lorentz transformation is a "given". It doesn't need to be understood, just accepted. I don't think this way.

Regards,

Rodrigo


14 Apr 2010, 21:20

Joined: 29 Jan 2010, 19:21
Posts: 11
Post Re: Lorentz transformation
vasco wrote:
Hi Rodrigo (I never thought I'd correspond with the composer of 'Concierto de Aranjuez',:-))

Thanks for your attention vasco. I'm an amateur guitarist. Hopefully I will be able to play Aranjuez some day ;)
Quote:
Essentially you have derived your result from the theorem of Pythagoras:

c^2=a^2+b^2 or
1=a^2/c^2+b^2/c^2\Rightarrow a^2/c^2=1-b^2/c^2=1-\beta^2 if we define \beta=b/c
\Rightarrow c/a=1/\sqrt{1-\beta^2}

and similarly

b^2/a^2=c^2/a^2-1=1/(1-\beta^2)-1=\beta^2/(1-\beta^2)

So really you have just re-arranged the Theorem of Pythagoras.

The Lorentz transformation comes from the theory of Special Relativity. In this theory c represents the speed of light and has nothing to do with Pythagoras.
Regards
Vasco

You got it right. The only observation I would do is that it doesn't "comes from the theory of Special Relativity". It is much more the other way.
Maybe some one can correct me if I'm wrong, but the Lorentz transformation is the way Lorentz explained the failed Michelson-Morley experiment.
Look at this 1895 paper from Lorentz http://www.lawebdefisica.com/arts/lorentz/ .
Einstein used the Lorentz transformation as its main argument for the special relativity. If you look at the minkowski paper (after the special relativity) you will see Minkowski approach to the subject (the link I gave on my reply to O.S.R can give you an idea about it.).
Actually what I did (or tried to do) is to simplify Minkowski approach.
Like O.S.R pointed out, Penrose in his "Spinors and space-time” book approached this same subject in a much more rigorous (and complicated) way. But as far as I can see it is the same tautology that can be solved by a simple Pythagorean theorem.
Keep it simple. Why not?

Rodrigo


14 Apr 2010, 21:38

Joined: 27 Mar 2010, 19:18
Posts: 3
Post Re: Lorentz transformation
Hello Rodrigo.
If mine understanding of the situation is correct you’re trying to find a simple way to derive Lorenz transform. You can’t derive it only by using theorem of Pythagoras merely because theorem of Pythagoras has no relation to time (it only relates spatial lengths of the triangle sides). You can derive it from conservation of Mincowski interval which as you know is c^2t^2-x^2-y^2-z^2. This derivation is presented in L. D. Landau, E. M. Lifshitz “Field Theory”, but I would highly recommend you to take the simplest book possible in special relativity and read the simplest derivation of Lorentz transform. This simplest derivation uses only “classical” Einstein postulates. It derives separately a time slowing and a length contraction and then mixes them into the Lorentz transform. After that you can start thinking about Mincowski space-time, orthogonal transforms in it …

Now what is interesting me in Lorentz transforms is Lorenz invariance which I consider as highly unobvious. It is just an experimental fact, that the influence of the background (absolute space) arises only in second order of time derivative d^2x/dt^2 by producing inertia. This fact is the main fact to define what is an “inertial frame” (“freely moving observer”) and therefore is a key to a special theory of relativity (which states that the laws of nature are the same in any inertial frame). This experimental fact is very well checked, but what if we’ll assume that it’s wrong? What relativity would we have then? What will be Lorentz transform?

Regards,
Oleg


15 Apr 2010, 14:53

Joined: 29 Jan 2010, 19:21
Posts: 11
Post Re: Lorentz transformation
O.R.S. wrote:
Hello Rodrigo.
If mine understanding of the situation is correct you’re trying to find a simple way to derive Lorenz transform. You can’t derive it only by using theorem of Pythagoras merely because theorem of Pythagoras has no relation to time (it only relates spatial lengths of the triangle sides).

Thanks for taking your time and looking into this Oleg.
I don't understand when you say that I "can't" do what I just did. I just did math. It can be right or it can be wrong. I still think it is right, since nobody showed it is wrong.
You say that Pitagoras is not related to time. So I can't use it.
I don't get you. If you want you can substitute c \Rightarrow ct, a  \Rightarrow at and b \Rightarrow bt (or vt if you prefer). You can even use c \Rightarrow f(t), b \Rightarrow g(t), and a \Rightarrow h(t) to make it more general. This won't change the geometry behind it.
Quote:

You can derive it from conservation of Mincowski interval which as you know is c^2t^2-x^2-y^2-z^2. This derivation is presented in L. D. Landau, E. M. Lifshitz “Field Theory”, but I would highly recommend you to take the simplest book possible in special relativity and read the simplest derivation of Lorentz transform. This simplest derivation uses only “classical” Einstein postulates. It derives separately a time slowing and a length contraction and then mixes them into the Lorentz transform. After that you can start thinking about Mincowski space-time, orthogonal transforms in it …

I will search for this reference. Besides Einstein, Minkowski and Penrose. I read Naber, Wheeler and a lot of other references. But it is always good to have other references.
Quote:

Now what is interesting me in Lorentz transforms is Lorenz invariance which I consider as highly unobvious. It is just an experimental fact, that the influence of the background (absolute space) arises only in second order of time derivative d^2x/dt^2 by producing inertia. This fact is the main fact to define what is an “inertial frame” (“freely moving observer”) and therefore is a key to a special theory of relativity (which states that the laws of nature are the same in any inertial frame). This experimental fact is very well checked, but what if we’ll assume that it’s wrong? What relativity would we have then? What will be Lorentz transform?

Regards,
Oleg


This last quote is the one I am more interested. Can you give me more references about this experimental facts? As far as I can get I can't see how to compare the sizes of a moving object with the size of a (inertial) rest object. I couldn't ever find a reference that shows that the Lorentz transformation is absolutey 'real'. It is just a matter of convention (I'm am not saying that it is not real, I'm just playing devil's advocate).
The best (or at least that goes into this point) reference I have is this:
http://www.desy.de/user/projects/Physic ... nrose.html
"People sometimes argue over whether the Lorentz-Fitzgerald contraction is "real" or not. That's a topic for another FAQ entry, but here's a short answer: the contraction can be measured, but the measurement is frame dependent. Whether that makes it "real" or not has more to do with your choice of words than the physics."

Regards,

Rodrigo


15 Apr 2010, 20:23

Joined: 27 Mar 2010, 19:18
Posts: 3
Post Re: Lorentz transformation
Hello Rodrigo.
In the simplest (ideologically, but not the shortest) derivation of Lorenz transform the theorem of Pythagoras actually appears in derivation of time dilation almost in the same way as you stated it here. I think you should look at this derivation and your question will dissolve immediately. This derivation can be found (I think) in every beginners course of SPECIAL relativity and even in popularized books on relativity (like the very old book written by Martin Gardner “Relativity for the million” (it is not the best book as I think, but I just can’t remember others on English)). I think this derivation must be presented in some of an introductory Wheeler’s books on relativity (I just haven’t read them unfortunately).

I don’t know exactly what are the most precise experiments on testing Lorentz invariance today. These experiments are carried out on the accelerators and aimed to check Lorentz invariance of the fundamental laws defining the behavior of elementary particles (usually they are performed as a part of some other interesting experiment at high level of precision). These are not simple things and you should be an expert in particle physics (not only theorist even) to know that.
I don’t think that somebody ever measured Lorenz-Fitzgerald contraction explicitly, but effects of special relativity are actually everyday living thing and everyone deals with them every fraction of a second. Just a few examples.

Electromagnetism is completely relativistic thing.

Another example is a cosmic rays (muons mostly, because they aren’t strongly interacting particles). In average 1 muon comes through 1 decimeter^2 per second. It couldn’t get to Earth's surface if not time dilation it has a lifetime of 1 microsecond by its own clock and therefore can go only to ~ 100 m (c\tau) if not time dilation. These muons are the products of decays of strongly interacting particles at the altitude of ~10 km.

GPS is enough precise to see relativistic effects (influence of general relativity is even more essential in this case so the time on the satellites goes faster (comparing with Erth) not slower).

Accelerators (and detectors) themselves are a great examples of correctness of special relativity. It uses a lot of electromagnetic effects and deals with short living particles (and for people who constructing these machines the whole notion of relativity is a kind of routine, they have a good intuition in it).

I guess I’ve spend enough time on this. It’s time to do Physics, so I probably couldn’t be able to correspond further (if only on something interesting and new).

Regards,
Oleg


16 Apr 2010, 14:15

Joined: 29 Jan 2010, 19:21
Posts: 11
Post Re: Lorentz transformation
From
O.R.S. wrote:
I can’t see how it’s related to a Lorentz transforms

to
O.R.S. wrote:
In the simplest (ideologically, but not the shortest) derivation of Lorenz transform the theorem of Pythagoras actually appears in derivation of time dilation almost in the same way as you stated it here.

Something changed ;)
Sometimes I think that physics are very busy doing Physics and forget to understand the basic stuff. Just think about it. The earth is moving with speed b in one specific direction. The speed of light, doesn't matter where it is coming from (you can look at west, north, up, down), is c. The Lorentz transformation says that there is is only one way to combine these two sides in a right triangle.
But you don't need Lorentz to do that. This is an painfully obvious result from Pitagoras.
And more than that, you don't even need b or c to be constant for this to be true, but if they are constant, then a must also be constant (and equal to a=\sqrt{c^2-b^2}).
But why is it important that c is constant?
Oh well, maybe we are just spinning with a constant speed... (but we are spinning aren't we!).
From one side, this is very basic. A tautology. From the other side "These are not simple things and you should be an expert in particle physics (not only theorist even) to know that.
Well, something sounds strange to me on this, but forget about it. I don't want to take you or nobody else time with this.

Rodrigo


19 Apr 2010, 20:40
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