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Prove s^2 = s'^2 using the Lorentz Transformation

  1. Sep 19, 2009 #1
    1. The problem statement, all variables and given/known data

    Using the Lorentz transformation, prove that s2 = s'2


    2. Relevant equations

    s2 = x2 - (c2t2)
    x' = gamma * (x-Beta*ct)
    t' = gamma * (t - Beta*x/c)

    3. The attempt at a solution
    s2 = x2 - (c2t2)
    Therefore
    s'2= x'2 - (c2t'2)

    and x' = gamma * (x-Beta*ct)
    So x'2 = [gamma * (x-Beta*ct)]2
    Using FOIL x'2 = gamma2 (x2 - 2xBeta*ct + Beta2*c2t2)

    and t' = gamma * (t - Beta*x/c)
    So t'2 = [gamma * (t - Beta*x/c)]2
    Using FOIL t'2 = gamma2 (t2 - 2Beta*tx/c + Beta2*x2/c2)

    combining

    s'2 = gamma2 [x2 - 2Beta*cxt + Beta2*c2t2 - c2t2 + 2Beta*cxt - Beta2*x2
    Cancel like terms
    s'2 = gamma2 [x2 + Beta2*c2t2 - c2t2 - Beta2*x2

    This is where I get stuck and cant tell how to progress and make it equal to the original s2 = x2 - (c2t2).

    Any insight would be greatly appreciated.

    Thanks!
     
  2. jcsd
  3. Sep 19, 2009 #2

    gabbagabbahey

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    [itex]x^2-\beta^2x^2-c^2t^2+\beta^2c^2t^2=x^2(1-\beta^2)-c^2t^2(1-\beta^2)=[/itex]______?:wink:
     
  4. Sep 19, 2009 #3
    Thanks, factoring it out like that made things more clear.
    That gives 1-beta2(x2 - c2t2)
    This is what we want except for the factor of 1 - beta2 in the front which I'm still not sure how to get rid of, or account for.
     
  5. Sep 19, 2009 #4

    gabbagabbahey

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    Hint: [itex]\gamma=[/itex]___?
     
  6. Sep 19, 2009 #5
    gamma = (1-Beta2)-1/2.
    But I'm still confused. Possibly because I've been busting my brains on physics for many hours now. Where will the gamma get me? i need to completely remove the 1-beta2 factor dont I?
    I can change it into terms of gamma but then its still there in front.
    gamma = (1-Beta2)-1/2
    thus 1-Beta2) can be called gamma-2 since it is the inverse of gamma2.
    Im still stuck though
     
  7. Sep 19, 2009 #6

    gabbagabbahey

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    Are you forgetting about the gamma^2 you had here:

    ???:wink:
     
  8. Sep 19, 2009 #7
    s'2 = gamma2[1-Beta2(x2-c2t2)
    s'2 = gamma2[Gamma-2(x2-c2t2)
    s'2 = x2-c2t2

    Finally, finished! I feel exhilaration and also relief, I can go to bed now lol. Thank you so much for your help tonight. I can't really pay you back other then by trying my best to help others in the way you have helped me. Thanks a million
     
  9. Sep 19, 2009 #8

    gabbagabbahey

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    You're welcome!...And get some sleep:zzz::smile:
     
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