Mathematical Proof For Lorenz Coordinate Transformation

[DrorH]

Hi, I have a pretty in depth understanding of special relativity. Recently I have been searching for mathematical proof of the Lorenz transformation.
I found some information about it, but to tell the truth I didn't understand much of it.
Maybe one of you guys can shed some light on the proof or can link me to an article or a book?

Thanks in advance, Dror.

 

[robphy]

Hi, I have a pretty in depth understanding of special relativy. Recently I have been searching for mathematical proof of the Lorenz transformation.
I found some info about it, but to tell the truth I didn't understand much of it.
Maybe one of you guys can shed some light on the proof or can link me to an article or a book?

Thanks in advance, Dror.

What "info about it" did you find? Can you post a reference?

Can you elaborate of the "depth" of your "understanding of special relativ[it]y" so that appropriate alternative references can be offered? For example, I'll rattle off some terms in no particular order to see what you are familiar and unfamiliar with: interval, time-dilation, doppler-effect, radar-measurements, 4-velocity, rapidity, rotation, vector-space, spacetime diagram, metric, vector, tensor...

 

[DrorH]

First, sorry for typing errors which are now fixed. Heh.

I am well familiar with space-time interval as an invariant size, time dilation as a result of Lorenz time transformation, doppler-effect relaltive formula as a result of Lorenz transformations, I am in the Israeli navy so I have a very good understanding of radars if that is what you meant, I am not familiar with the term 4- velocity, I don't know rapidity as a SR term, rotation duh, familiar with vector space, familiar with Minkowski space-time diagram, metric as in meter system?, vector duh, and I have no idea what tensor is.

Sigh! :)
I have a book which breifly describes the math behind it. Tell me if you want me to copy it to the message with much effort.

 

[cristo]

A google search finds the following proof of the Lorentz transformation. It's quite mathematical, but its similar to the proof I was taught when learning the subject.

http://farside.ph.utexas.edu/teaching/em/lectures/node109.html

That ok?

 

[DrorH]

Thanks a lot.
At first glance I understand close to nothing. I'll try my best.. hehe.
I wonder if einstein found the transformations through math or did he find it intuitivly. From my knowledge he found those transformation without even knowing that they were also made by Lorenz as a technical tool to correct a "calculation error" rather then natural fact as they are used in SR. Because after all these transformations make rational sence. Their proof makes less sense though.. :)
I'll appreciate if you try to explain it with more depth on the mathematical tools used which I am obviously not familiar with. I'll understand if it is too much of a hassle.

 

[robphy]

Well... here is a transcription of Einstein's paper:
http://www.fourmilab.ch/etexts/einstein/specrel/www/

A Lorentz Boost Transformation is analogous to a Euclidean rotation. So, if you can follow the mathematics of the derivation of a rotation, the Lorentz Boost shouldn't be too difficult... mathematically.

There is a neat (and arguably more physically motivated) mathematical proof using the (doppler-effect) k-calculus and radar measurements... but I don't have the time to write it up right now.

 

[DrorH]

I will appreciate it if you write it up when and if you find the time.
Thanks a lot for the help.

 

[bernhard.rothenstein]

Hi, I have a pretty in depth understanding of special relativity. Recently I have been searching for mathematical proof of the Lorenz transformation.
I found some information about it, but to tell the truth I didn't understand much of it.
Maybe one of you guys can shed some light on the proof or can link me to an article or a book?

Thanks in advance, Dror.

please have a critical look at


arXiv.org > physics > physics/0607048


abstract

Following an approach proposed by Rosser for deriving the transformation equations of volume charge density and current density we derive the transformation equations for the space-time coordinates of the same event, for the mass and the momentum of the same particle and for the electric and the magnetic field generated by the same distribution of electric charges.
the best things a physicist can offer to another one are information and constructive criticism

 

[lalbatros]

I wonder if einstein found the transformations through math or did he find it intuitivly.

The Lorentz transformation was known already at least ten years before Einstein's special relativity.
The interpretation was a problem, as well as the reconcilition with the other part of physics, specially Newtonian mechanics.

Einstein, solved all these problems by understanding that spacetime was involved and that classical physics had to be modified. Since 1905, spacetime has to be discovered and studied experimentally just like any physical object. This led Einstein to the further steps of general relativity.

Michel

peace and freedom

 

[bernhard.rothenstein]

LET derivation: what was first the chicken or the egg?

Well... here is a transcription of Einstein's paper:
http://www.fourmilab.ch/etexts/einstein/specrel/www/

A Lorentz Boost Transformation is analogous to a Euclidean rotation. So, if you can follow the mathematics of the derivation of a rotation, the Lorentz Boost shouldn't be too difficult... mathematically.

There is a neat (and arguably more physically motivated) mathematical proof using the (doppler-effect) k-calculus and radar measurements... but I don't have the time to write it up right now.

I am glad seeing that you aggree with other ways to derive the LET, illustrating the way in which the relativistic postulates should be applied. As I see, here on the Forum, there are physicists who become angree when they here about them.
There is an old, nonsolved problem: What was first the egg or the hen? I think that the relativistic postulate is compulsory as clock synchronization and definition of the concept of same event are. The first results are time dilation and length contraction which lead to the addition law of relativistic velocities or two the LET. With them in hands and with the concept of proper physical quantity the way to all the relativistic transformaion equations in all the fields of physics is paved avoiding paradoxes. Of course hundred years of special relativity generated many derivations with other starting points. Far from me to blame them because i learned a lot from them.
sine ira et studio

 

[bernhard.rothenstein]

help

A google search finds the following proof of the Lorentz transformation. It's quite mathematical, but its similar to the proof I was taught when learning the subject.

http://farside.ph.utexas.edu/teaching/em/lectures/node109.html

That ok?

please let me know if all parts of the course are available?

 

[DrorH]

please have a critical look at


arXiv.org > physics > physics/0607048


abstract

Following an approach proposed by Rosser for deriving the transformation equations of volume charge density and current density we derive the transformation equations for the space-time coordinates of the same event, for the mass and the momentum of the same particle and for the electric and the magnetic field generated by the same distribution of electric charges.
the best things a physicist can offer to another one are information and constructive criticism

I found 4 articles that corespond to that number but only one of them has anything to do with math and it's about "Compactness of the solution operator to d-bar in weighted L^2 - spaces"...
Couldn't find what you were reffering to.

 

[bernhard.rothenstein]

arxiv paper

I found 4 articles that corespond to that number but only one of them has anything to do with math and it's about "Compactness of the solution operator to d-bar in weighted L^2 - spaces"...
Couldn't find what you were reffering to.

go google and give it to look for
physics/0607048
the first answer will be
Special relativity from a single scenario following the same strategy. If you have some problem give me an e-mail address and I will forward you the paper and some others devoted to the same problem.

 

[cristo]

please let me know if all parts of the course are available?

I just googled this site- it's nothing to do with me personally! It's from a course on classical electromagnetism, and the rest of the course is here http://farside.ph.utexas.edu/teaching/em/lectures/

 

[DrorH]

go google and give it to look for
physics/0607048
the first answer will be
Special relativity from a single scenario following the same strategy. If you have some problem give me an e-mail address and I will forward you the paper and some others devoted to the same problem.

This article reveals a way of reaching the Lorenz transformation without actually using it. But it does a huge intuitive leap by incorperating the size v^2/C^2 without any mathematical analysis. You can't really reach the conclusion without being familiar with the proper Transformation. Get my drift?
From my understanding there is a mathematical way of revealing this particular size, and others, with no physical intuition. Am I wrong?

 

[robphy]

I am glad seeing that you aggree with other ways to derive the LET, illustrating the way in which the relativistic postulates should be applied. As I see, here on the Forum, there are physicists who become angree when they here about them.
There is an old, nonsolved problem: What was first the egg or the hen? I think that the relativistic postulate is compulsory as clock synchronization and definition of the concept of same event are. The first results are time dilation and length contraction which lead to the addition law of relativistic velocities or two the LET. With them in hands and with the concept of proper physical quantity the way to all the relativistic transformaion equations in all the fields of physics is paved avoiding paradoxes. Of course hundred years of special relativity generated many derivations with other starting points. Far from me to blame them because i learned a lot from them.
sine ira et studio

You may recall an earlier 2005 post of mine
https://www.physicsforums.com/showpost.php?p=694535&postcount=8
which features a chart (as an attachment) from
"Spacetime and Electromagnetism" by J.R. Lucas, P.E. Hodgson (https://www.amazon.com/exec/obidos/tg...60834?v=glance&tag=pfamazon01-20 ) where they try to diagram the various approaches to obtain the Lorentz Transformations.

 

[bernhard.rothenstein]

This article reveals a way of reaching the Lorenz transformation without actually using it.I have soome problems understanding your statement. Of course we reach something without using it! But it does a huge intuitive leap by incorperating the size v^2/C^2 without any mathematical analysis. The big leap was done by those who derive the time dilation formula, the length contraction forumula and the addition law of relativistic velocities. I only uise theirs results. You can't really reach the conclusion without being familiar with the proper Transformation. Get my drift?
From my understanding there is a mathematical way of revealing this particular size, and others, with no physical intuition. Am I wrong?

Before to go further with our discussion please have a look at the paper by Asher Peres "Relativistic telemtry" published in Am.J.Phys. who made among others the big jump showing where from vv/cc arrises.
pleas do not hesitate to put further questions. i learn a lot from them.