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Question on linearity of Lorentz transformations

  1. Jul 12, 2009 #1
    Hello.The way the transformation of coordenates in Special Relativity are ussually derived presuposes linearity or try do demostrate such linearity using wrong arguments. For example some authors state that since linear and uniform motion remains linear and uniform after the transformation this fact imposes linearity, however this simply is not true as is demostrated(for instance) in J. Aharoni's "The Special Theory of Relattivity" where he shows a particular not linear transformation which transforms a uniform motion along a straight line into a similar kind of motion. Other authors simply state that the principle of relativity+homogeinity and isotropy of space-time imposes linearity but don't give details of how this come about.
    Is there a correct,rigorous demostration for linearity from first principles, in which case where can one find it ?
     
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  3. Jul 12, 2009 #2
    Linearity is in fact a consequence of homogeneity.
    Space and time intervals between two points should not depend on a particular point, so in
    [tex]dx'^i = \frac{\partial x'^i}{\partial x^k} dx^k[/tex]
    matrix [tex]\partial x'^i/\partial x^k[/tex] does not depend on x.
     
  4. Jul 12, 2009 #3

    George Jones

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    Try

    Zeeman, E. C. "Causality Implies the Lorentz Group", Journal of Mathematical Physics 5 (4): 490-493; (1964).
     
  5. Jul 12, 2009 #4

    Fredrik

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    I agree. The traditional "derivation" of the Lorentz transformation is mostly BS. It's not really a derivation at all. I've been complaining about that in lots of threads already, so I won't repeat everything here, but consider e.g. the claim that the principle of relativity implies that the group of functions that represent a change of coordinates from one inertial frame to another must be either the Galilei group or the Poincaré group. What we're really doing there is to take an ill-defined statement (the principle of relativity) and interpret it as representing a set of well-defined statements. Then we find out which of the well-defined statements that are consistent with all the other assumptions that we want to make. (One of those assumptions is linearity).

    This is of course a perfectly valid way to find a set of statements that we can take as the axioms of a new theory, but to call it a "derivation" is preposterous.

    No.
     
    Last edited: Jul 12, 2009
  6. Jul 12, 2009 #5
    Noel Doughty, in his book Lagrangian Interaction, refers to a paper by Mariwalla, Uniqueness of classical and relativistic systems, Phys. Lett. A79 143-146.

    "Mariwalla shows that this result [linearity of a boost] may in fact be established without appealing to homogeneity. This follows because there are only three distinct 1-parameter groups acting in one spatial dimension [i.e. along the direction of the boost], and all imply linear transformation equations." (Bracketed comments are mine.)

    He also refers to derivations of the LTs from Am. Journ. Phys. 43 434-437 and 44 271-277.
     
  7. Jul 12, 2009 #6

    atyy

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  8. Jul 12, 2009 #7

    atyy

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    The traditional derivation is not BS if one does not add the conclusion that the Lorentz transformations are the only way that works.

    Edit: Would anyone feel better if instead of "traditional derivation", we said "traditional construction"?
     
    Last edited: Jul 12, 2009
  9. Jul 12, 2009 #8
    Usual linear Lorentz transformations are applicable to coordinates of non-interacting particles (with linear uniform motion). However, they are not valid for coordinates of particles interacting with each other (if this interaction is described by Poincare-invariant Hamiltonian dynamics). This has been proved a long time ago

    Currie, D. G.; Jordan, T. F.; Sudarshan, E. C. G., Relativistic invariance and
    Hamiltonian theories of interacting particles, Rev. Mod. Phys. 35, 350-375
    (1963).

    http://puhep1.princeton.edu/~mcdonald/examples/mechanics/currie_rmp_35_350_63.pdf [Broken]
     
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  10. Jul 12, 2009 #9

    atyy

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    "two particles (not two particles and a field)"
     
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  11. Jul 12, 2009 #10
    Yes, Currie-Jordan-Sudarshan theorem has been proven only for directly interacting particles. If you believe that particles interact via some "field" mediation, then the CJS result does not apply directly. However, the idea of their proof suggests that even in the case of field-mediated interactions Lorentz transformations of particle observables must be non-linear and interaction-dependent.

    The idea is that in any Poincare-invariant interacting theory boosts are represented by interaction-dependent operators in the Hilbert space (or their appropriate analogs in the classical phase space). Therefore, it seems very likely that the action of boosts on particle observables must be interaction-dependent (rather than given by universal linear Lorentz transformations).

    In any case, it seems more logical to derive boost transformations of particle observables from underlying dynamical theory rather than postulate them from the beginning, as has been done in special relativity.
     
  12. Jul 12, 2009 #11
    Thank you for your answers.
    First I want tell quZz that what he said is not correct otherwise the time inteval between two events would be the same for all observers which is what happens in Newtonian physics.
    I see there is a little controversy, Fredrik says there is no such rigorous derivation and ohers cite papers dealing with such derivations. Regretfully I don't have access to papers, I will try the link atyy suggested though
    Can someone tell me what BS stands for?
    By the way I see the discussion went beyound SR
     
  13. Jul 12, 2009 #12

    robphy

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    Along those lines, there are also the theorems of A.D. Alexandrov
    http://books.google.com/books?hl=en...ts=Gx4d8imeP5&sig=YZME-oNHVpD5hf-iwkWFov7Fsp8 [see theorem 1 and its corollary] (1967, with reference to one from 1953).

    Of possible interest are the constructions of http://en.wikipedia.org/wiki/Alfred_Robb" [Broken],
    e.g. http://books.google.com/books?id=vp...YeY2EI&sa=X&oi=book_result&ct=result&resnum=6
    http://www.archive.org/details/theoryoftimespac00robbrich (1914)
     
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  14. Jul 12, 2009 #13
    I'm sorry what I told quZz about why he is wrong is not correct. In fact I would like to know why his argument is incorrect in case it is.
     
  15. Jul 12, 2009 #14
    I think I now know the answer why quZz's argument is wrong, what SR tells us is that only
    [itex]ds^2=g_{ik}dx_idx_k[/itex] have absolute meaning not the [itex]dx_i[/itex] individually, what should be independent of space-time coordenates is [itex]ds^2 \,not\, dx_i[/itex]
     
  16. Jul 12, 2009 #15
    I assume when you wrote that Aharoni's non-linear transformation transforms "a" uniform motion along "a" straight line into a similar kind of motion, that you meant to say it transforms ALL uniform motions along straight lines into similar kinds of motion. This is what is required of a transformation between inertial coordinate systems, so your statement wouldn't make sense unless you replace A with ALL.

    For those of us who don't have access to Aharoni's book, can you state the non-linear transformation that transforms ALL uniform motion in straight lines into similar kinds of motion?
     
  17. Jul 12, 2009 #16
    you may be right perhaps I shouldn't have said "a uniform" pardon my english. The transformation equations are :
    [tex]x_i^{'}=\frac{a_{ik}x_k+b_i}{c_kx_k+d},\quad i=0,1,2,3[/tex]
    where we've used Eintein's convention for repeated indices. Note that the denominator
    does not depend on index "i".
     
  18. Jul 12, 2009 #17
    A derivation is given using the product of ""purely kinematic and non-kinematic transformations"" in:-

    Torretti, Relativity and Geometry (1984). Chapter3 section 3.4--The Lorentz Transformation. Einstein's derivation of 1905.

    Matheinste
     
  19. Jul 13, 2009 #18

    Fredrik

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    I recommend www.urbandictionary.com. :smile:

    There are plenty of rigorous mathematical arguments that end with the Lorentz transformation, but what do they prove, really? They always start with assumptions that are so strong that you might as well have started with the Lorentz transformation right away.

    The "standard" derivation that I'm talking about is the one that starts with Einstein's postulates. You definitely can't prove anything from Einstein's postulates, since they are ill-defined. The biggest problem with them is that they talk about inertial frames as if that concept has been defined already.

    I don't think that works either. What I'd like to see is a clear statement at the start of the "derivation" that says that we're just trying to guess what might be appropriate axioms for a new theory.
     
  20. Jul 13, 2009 #19
    Hi Fredrik,

    I agree with you that existing "derivations" of Lorentz transformations are inadequate. Here are some examples that I am talking about:

    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", Helv. Phys. Acta 70 (1997), 542.
    http://www.arxiv.org/abs/physics/0410062

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

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

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

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

    In my opinion, their major flaw is that they are not applicable to systems of interacting particles. For example, if one uses the 2nd Einstein's postulate (the constancy of the speed of light), then the obtained Lorentz transformations can be logically concluded to be valid for events associated with light pulses only. If one uses the uniformity and linearity of free moving particles, then there is no guarantee that obtained Lorentz transformations will remain valid for interacting particles whose movement in non-uniform and non-linear.

    Fundamentally, we are interested in boost transformations of particle observables (positions, momenta, etc.). In quantum mechanics, such transformations can be unambiguously calculated by applying the operator of boost to particle observables. So, in order to rigorously derive Lorentz transformations one first needs to build a Poincare-invariant dynamical theory of interacting particles. In relativistic quantum mechanics this means construction of 10 Hermitian operators (that correspond to 10 generators of the Poincare group and satisfy commutation relations of the Poincare Lie algebra), which represent total observables of energy, momentum, angular momentum and boost. Then applying the unitary boost operator to observables of individual particles, we can calculate the boost transformation formulas.

    If we follow this prescription, we can immediately realize that universal linear Lorentz transformations cannot be obtained in the interacting case, because it is well-known that the total boost operator must be interaction-dependent. So, Lorentz transformations of special relativity should be regarded as an approximation acceptable only for weakly interacting particles.
     
  21. Jul 13, 2009 #20
    A lot of interesting links =)

    you misunderstood me. I'm not saying that dxi are absolute (have the same values for all observers), that's incorrect. The thing is that the value of dx' depend only on the value of dx but not on x itself, that's what homogeneity is all about. Same dx, different x -> same dx'.
     
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