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Lorentz-Einstein transformation

  1. Dec 8, 2006 #1
    Textbooks devoted to the subject present separate derivations for the twoLorentz-Einstein transformations which relate the space-time coordinates of the same event E(x,0,t) and E'(x',0,t')
    x=g(x'+Vt') (1)
    t=g(t'+Vx'/cc) (2)
    Equation (1) can be derived directly from length contraction
    xg^-1=x'+Vt') (3)
    Dividing both sides of (1) by c and taking into account the synchronization of the clocks in the involved inertial reference frames (x/c=t,x'/c=t',t'=x'/c) we obtain
    tg^-1=t'+Vx'/cc (4)
    i.e. (2).
    Did you find that derivation in the literature? Please comment.
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  3. Dec 8, 2006 #2

    Length contraction is a consequence of the LE transforms , not the other way around.
  4. Dec 8, 2006 #3

    Meir Achuz

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    When one thing is derived from another, it is obvious that the derivatiion can be turned around as you do. One could say that GmM/r^2 can be derived from elliptical orbits if one were silly enough. Doing physics usually means presenting a general principle (e.g. Lorentz invariance) and deriving a large number of phenomenological effects from the general principle. Any one of those derived specific results could be used as a starting point if one wanted to engage in sophistry rather than physics. I hope that your derivation is not in the refereed literature.
  5. Dec 8, 2006 #4
    i think that a derivation should start with something probably as you say with a general principle. but that is not enough. the derivations imply supplementary assmptions in order to satisfy the results of thought experiments or other requirements of Nature. do you think that we should start ad-hoc with the LET? physics is a democratic field where each participant could start ad-libitum. i know derivations which start with E=mcc, with the Doppler Effect, with lenght contraction and time dilation (de gustibus...) Concerning the style of your answer it is far from being elegant (do you use sily when you discuss with your students?) and the air of superiority you display is descouraging.
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  6. Dec 8, 2006 #5
    Lorentz Einstein transformation

    Please delete from my message the sentence "(1) can be derived from length contraction" considering that it is derived in a way with which you aggree! What I wanted to underline was that (2) is a direct consequence of (1) if we consider the clock synchronization procedure in the two frames.
  7. Dec 8, 2006 #6
    As to what is fundamental and what is secondary is not always obvious - for example, you can derive an expression of the Gravitational constant G starting with Friedmanns equation for an expanding critical density universe

    While length contraction was the first idea proposed to explain MMx near null result, it may be more comfortable to consider length contraction as a necessary consequence of time dilation - an idea that came a bit later. Since the latter provides a residual record when clocks are compared after they have been brought to rest in the same frame, it may be confusion from a tutorial aspect, to start with length contraction.
  8. Dec 8, 2006 #7

    thank you for your answer. my oppinion is that the best way to the transformation equations is Einstein's clock synchronization procedure. It leads to xx-cctt=x'x'-cct't' which at its turn leads to time dilation which leads to length contraction. time dilation per se is sufficient for deriving the LET. What i wanted to show was that once the LET for the space coordinates is derived in the way we like, the LET transformation for the time coordinates is a direct conseqeunce of the first one if we take into account clock synchronization.
  9. Dec 9, 2006 #8
    You are getting even worse: the Lorentz transform of time cannot be derived from the one for x. What gave you this idea?
  10. Dec 9, 2006 #9

    Could you prove your statement? You do not agree that as a result of clock synchronization in the invoved reference frames x/c=t, x'/c=t' and so
    x/c=g(x'/c+Vt'/c) and finally
    The idea is mine. If it is not correct I am the single guilty! I think we are on the forum to help each other proving our statements.
  11. Dec 9, 2006 #10
    Look at the original derivation:
    From the clock synchronization equation follows the transformation for t.
    From the transformation for t follows the transformation for x.
    This is the natural chain of derivations.
  12. Dec 9, 2006 #11
    Lorentz-Einstein transformations

    Concerning the chain of derivatios we go to C.Moller 'The Theory of Relativity" Oxford at the Clarendon Press 1952 pp.36-41. We see that the derivations of space and time coordinates go in parallel, guessing theirs shapes in order to be in accordance with the invariance of xx-cctt=x'x'-cct't' and so i think that there is not a preference for the succession in which we derive the two transformation equations. Once derived the equations
    lead with some algebra to
    and so the vice-versa to your statement is proved.
    Thank you for your participation to the discussion from which and many readers could learn. I think that is for what we are there.
  13. Dec 9, 2006 #12


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    Review of Einsteins 1905 paper shows that his derivation is based on the properties of light waves as modified by the Light Postulate. His process produces all four relations, x,y,z,t, together. Has anyone disagreed with his LET or with his analysis of the Maxwell equations? The ideas of slow clocks and contraction play no active part in his LET derivation or his Maxwell analysis. It appears that there is still a question whether dilation and contraction actually occur or whether they are connected with the measurement procedure.
  14. Dec 9, 2006 #13

    Meir Achuz

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    I did not say you were silly. I was referring subjunctively to a hypothetical "one". I am very careful in redressing students. I consider you more of an equal. I just wanted somehow to emphasize the importance of going from one fundamental principle, rather than Fitzgerald's patchwork attempt. Now I've insulted him. Anyway, I will not interfere with this thread again.
  15. Dec 9, 2006 #14
    I don't understand your post.

    I do not agree with your attempt to derive the transforms from length contraction. Length contraction is a consequence of the transforms, if you are attempting to start from the Lorentz-FitzGerald contraction that is a very bad idea.

    I do not agree with your attempt to change the original Einstein derivation , i.e. the proper order is clock synchronization equation -> t transform -> x transform

    The original Einstein derivation is excellent and doesn't need any modifications.
  16. Dec 9, 2006 #15

    My intention was not to derive the LET. Einstein did it for all of us. My intention is to show that the direct and the inverse transformations for the space coordinates lead with some simple algebra to the direct and the inverse transformation for the time coordinate and vice-versa. Concentrate please your answer on this side of the problem. In a period when special relativity is a rara avis in the curriculum, I think that Einstein, still alive, would invent a short and transparent enough derivation. I know a nice story which asks if the missionary who was eaten by the natives did his job well? The answer is no, because he did not well his job or he has tried to impose his faith by force. I think the same question aplies in the case of the teacher who starts teaching relativity with four vectors or tensors...
    Thanks for your participation.
  17. Dec 10, 2006 #16
    I would agree Bernhard - it is more important to first obtain the physical connotation rather than start with a mathematical abstraction - we know that Einstein arrived at a result that he did not properly explain and may not have fully comprehended - In part IV of the 1905 paper he first gives an example of two clocks synchronized in one frame then states that when one is moved to the position of the other, there is a peculiar result - the clock that moved will be found to lag behind the one which has not moved - after giving a few more examples which consider both one way and round trip journeys he concludes that a clock at the equator will run slower than a clock at the pole!... but in actuality all clocks run at the same speed - and we know that one inertial frame is no better than another... so he implies something that contradicts the very foundation of SR. If he had said "The moving clock logs less time relative to the clock which is considered to be at rest" it would have generated less debate. The fact is, the moving clock does not run slower - but you do not get this from the equations per se - and that is why it is important to look at these things from perspectives that reveal new insight .. starting with synchronization and length contraction allows you to recover time dilation - and it is easier to see the time difference as being the result of a total accumulation of time in the length contracted frame rather than an alteration of the clock rate.
  18. Dec 10, 2006 #17
    time dilation and length contraction from LET or vice versa

    Please have a look at Asher Peres "Relativistic telemetry" Am.J.Phys. The Author derives the formulas that account for the radar echo, for time dilation, Doppler Effect and for the addition law of velocities without using the LET, considering that using them we obscure the physics behind the mentioned fundamental effects. Personally I learned a lot from the quoted paper which shows how mighty are the two postulates.
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  19. Dec 11, 2006 #18
    An exercise in futility. Not even worth the time finding out what is wrong with such papers.
    Last edited: Dec 11, 2006
  20. Dec 11, 2006 #19
    Why are you assuming it's wrong? It's in a very reputable journal (Am. J. Phys. is about as vanilla as you can get) and written by a well-known physicist.
  21. Dec 11, 2006 #20
    "without using the LET, considering that using them we obscure the physics"

    Really? Either the paper is wrong or Bernhard is misquoting the author. Either way, such papers are not useful for anything, they generally hide some flaw and they bring nothing over the standard derivation. AmJPhys publishes a lot of this stuff because it doesn't publish any original research (by declared charter).
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