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Simple Lorentz transformation. Are there objections

  1. Sep 10, 2007 #1
    I find in some textbooks the following generalization of the Galileo transformations
    x=k(x'+vt')
    x'=k(x-vt)
    with the same k because if we transform from I to I' or from I' to I then the distortion factor of lengths or time intervals should be the same.
    Are there objections?
     
  2. jcsd
  3. Sep 10, 2007 #2

    robphy

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    Can you cite specific textbook references?

    Any objections depend on your goal or claim involving these transformations.
    For example,
    should these transformations form a group?
    should the conservation of momentum be preserved under this transformation?
     
  4. Sep 10, 2007 #3
    simple lorentz transformation

    To the first question

    Relativitätstheorie als didaktische Herausforderung
    Journal Naturwissenschaften
    Publisher Springer Berlin / Heidelberg
    ISSN 0028-1042 (Print) 1432-1904 (Online)
    Issue Volume 67, Number 5 / May, 1980
    DOI 10.1007/BF01054529
    Pages 209-215
    I have also a Hungarian version of it. I have it seen in many other places but I do not remember.
    The equations
    x=k(x'+vt')
    x'=k'(x-vt) lead imposing the conditions
    x=ct and x'=ct' to
    t=kt'(1+v/c)
    t'=kt(1-v/c)
    and so
    k=1/sqrt(1-v^2/c^2)
    having in our hands the LT for the space-time coordinates of the same event.
    Is there more to say or ask if the LT satisfy the conditions for which you are asking for.
    Thanks for your help
     
  5. Sep 10, 2007 #4

    robphy

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    So, it looks like "[spatial] symmetry of the observers" and "constancy of the speed of light" leads to equations involving their temporal symmetry for Doppler. This seems like the essence of the Bondi k-calculus (where this k is the Doppler factor, not your "k" which is really [itex]\gamma[/itex] at the end of the day) without the radar experiments or operational definitions.. but in a different order, starting with spatial symmetry.
     
  6. Sep 10, 2007 #5
    k is gamma

    Sorry that is not my k. It is the notation of Roman Sexl and I have respected it as I respect that professor of physics at the University of Viena, But that is not the essence of the problem start at the beginning of the day. The problem is if there are objections? As I see you have not. Thanks for your answer.
     
  7. Sep 12, 2007 #6

    Meir Achuz

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    That is a simple derivation of the LT. It just lacks a bit of explanation of the assumptions and motivation that Prof. Sexl probably provided. With such a simple, straighforward derivation, there is no need to look for more complicated ones.
     
  8. Sep 12, 2007 #7

    robphy

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    In my opinion, it's useful to look for alternative derivations...
    ...particularly those which are pedagogically attractive [possibly building on a particular motivation (e.g. thought experiment, geometrical construction, experimental results, etc...) and which build on the target audience's preparation and ability [which, of course, varies among different target audiences].

    While possibly interesting,... for me, I find purely symbolically-algebraic derivations to be not very effective for teaching relativity... although various "equations" may be obtained in a few steps.
     
  9. Sep 12, 2007 #8
    Lt

    Thanks for your help. My problem is if it is not simpler to avoid thought experiments by simply stating that a distortion in time interval and length takes place without to know start from the begining the formulas which account for them.
    I think that the outcome of teaching is measured by time invested and understanding achieved. My oppinion is: reduce the first increase the second even if an evaluation is not so easy to achieve.
     
  10. Sep 12, 2007 #9
    What I'm I missing here - as Rob points out, you are not really deriving the LT - only Gamma. And that is a one step process from Minkowski via the invariance of the interval!
     
  11. Sep 12, 2007 #10
    LT transformation

    Thanks.
    You are right stating that I derive Gamma but in the following context:
    State that we can add only lengths measured by observers of the same reference frame and that a distorsion in length takes place of the type
    dx=f(V)dx(0) where dx(0) is a proper length, dx the distorted length and f(V) an unknown function of the relative velocity but not of dx(0).
    Then in I
    dx=Vdt+f(V)dx' (1)
    and in I'
    dx'=f(V)dx-Vdt' (2)
    Notice that
    dx/dt=c and dx'/dt'=c. (3)
    Combine (1),(2) and (3) in order to obtain
    f(V)=1/sqrt(1-VV/cc) (4)
    Return to (1) and (2) in order to recover LT.
    Please help me telling where I am wrong.
    The derivation presented above has much in common with a recently presented derivation of the LT in which the function f(V) is known start from the beginning as a result of a thought experiment (the eternal light clock).
     
  12. Sep 13, 2007 #11
    Substantively, I don't see a difference between this derivation and that given by A.P. French at pages 78 and 79 of his 1966 book "Special Relativity." Once you impose the condition of reciprocal symmetry together with x = ct
    and x' = ct' the LT is recovered ...so if you are saying it is not necessary to presuppose anything additional, I would agree
     
  13. Sep 13, 2007 #12
    The Lorentz transformation between the positions and times (x, y, z, t) as measured by an observer "standing still," and the corresponding coordinates and time (x¹, y¹, z¹, t¹) measured inside a "moving" space ship, moving with velocity u are

    x¹ = x - ut/sqrt1 - u²/c²

    y¹ = y,

    z¹ = z,

    t¹ = t - ux/c²/sqrt1 - u²/c²

    There equations relate measurements in two systems, one of which in this instance is rotated relative to the other:

    x¹ = x cos Θ + y sine Θ,

    y¹ = y cos Θ - x sin Θ,

    z¹ = z.
     
  14. Sep 14, 2007 #13
    Lt

    Thanks for the hint. I am collecting "simple derivations" of LT
     
  15. Sep 14, 2007 #14
    Lt


    I have looked in French. He starts as many others do with the "guessed" shape of the transformation as many others do. I think that an approach which starts with the statement that in SR distorsion in length and in time intervals takes place and that we can add and compare only length and time intervals measured in the same inertial reference is better motivated and transparent. But de gustibus nihil disputandum
    Kind regards and thanks for your help.
     
  16. Sep 17, 2007 #15
    These are a few references that I have:

    A. R. Lee, T. M. Kalotas, "Lorentz transformations from the first postulate" Am. J. Phys. 43 (1975), 434

    J.-M. Levy-Leblond, "One more derivation of the Lorentz transformation", Am. J. Phys. 44 (1976), 271

    D. A. Sardelis, "Unified derivation of the Galileo and Lorentz transformations" Eur. J. Phys. 3 (1982), 96

    H. M. Schwartz, "Deduction of the general Lorentz transformations from a set of necessary assumptions", Am. J. Phys. 52 (1984), 346

    J. H. Field, "A new kinematical derivation of the Lorentz transformation and the particle description of light" (1977), Preprint KEK 97-04-145

    R. Polischuk, "Derivation of the Lorentz transformations", http://www.arxiv.org/abs/physics/0110076

    Unfortunately, all these derivations (and your derivation is not an exception) share one weak point. They are normaly performed for events associated with some simple physical systems, like non-interacting particles or freely propagating light rays. For example, a prominent role is often played by the 1st Newton's law (which is valid for non-interacting particles only) which is used to deduce the linearity of transformations. Another example is Einstein's second postulate (the invariance of the speed of light) which can be applied to events associated with light pulses only. There can be no objections against such derivations, and they can be done in a variety of different ways.

    The question that worries me is this: how we can be sure that the same (Lorentz) transformation laws will be valid for events in systems of interacting particles? Do you agree that there is a logical jump when Lorentz transformations derived in non-interacting systems are generalized to all possible physical system and even said to be fundamental properties of space and time, i.e., completely independent on the physical system that is observed?

    Eugene.
     
  17. Sep 17, 2007 #16
    Lt

    Thank you for your help. Can the problem you state be solved in the limits of SR.
     
  18. Sep 17, 2007 #17
    I don't think so. Einstein's special relativity makes the assumption which I find troublesome. It says that transformations of space-time coordinates of events are universal and independent on the type of physical system in which the events occur and on interactions acting in the system. This is an important postulate of SR. Unfortunately, this postulate was never clearly formulated and discussed. Why should we believe in it?

    Eugene.
     
  19. Sep 17, 2007 #18
    Lt

    Sorry but I have no answer to your question.]
    Bernhard
     
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