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Deriving the Lorentz Transformation from the Homogeneity of Spacetime

  1. Apr 13, 2016 #1
    1. The problem statement, all variables and given/known data
    Show that the isotropy and homogeneity of space-time and equivalence of different inertial frames (first postulate of relativity) require that the most general transformation between the space-time coordinates (x, y, z, t) and (x', y', z', t') is the linear transformation,
    x'=f(v2)x-vf(v2)t; t'=g(v2)t-vh(v2)x; y'=y; z'=z
    and its inverse,
    x=f(v2)x'+vf(v2)t'; t=g(v2)t'+vh(v2)x'; y=y'; z'=z'

    2. Relevant equations


    3. The attempt at a solution
    Now, I know that homogeneity implies that the transformation must be linear in x and t and that the isotropy of space implies that the coefficients can only be functions of the magnitude of the velocity (not the direction) at most. Therefore, I am stuck at the following:
    x'=f(v2)x-vf0(v2)t; t'=g(v2)t-vh(v2)x; y'=y; z'=z
    However, I am having trouble proving that f = f0. The solution I am looking at says that this follows from the homogeneity of space-time but I am having trouble using that fact to prove it.
     
  2. jcsd
  3. Apr 13, 2016 #2

    strangerep

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    Hmm. Which source/solution are you looking at? (Can you post a reference or link?)

    In my derivation of this stuff, the result that ##f = f_0## emerges from the properties of any 1-parameter Lie group (which is what you're deriving here). I.e., 2 successive transformations (in the same direction) with different parameters ##v,v'## must commute. This imposes some constraints on the various functions.
     
  4. Apr 14, 2016 #3
    http://faculty.uml.edu/cbaird/all_homework_solutions/Jackson_11_1_Homework_Solution.pdf
     
  5. Apr 14, 2016 #4

    strangerep

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    Hmm. (Sigh.) Well, I will say that I think there are some unnecessary fudges in that solution. So I'm not sure how I can usefully help you. I could (possibly) show you (a version of) my derivation, but it would be rather different from the solution you've been given.
     
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