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Scale factor of special conformal transformation

  1. Sep 19, 2016 #1
    1. The problem statement, all variables and given/known data
    (From Di Francesco et al, Conformal Field Theory, ex .2)
    Derive the scale factor Λ of a special conformal transformation.

    2. Relevant equations
    The special conformal transformation can be written as

    x'μ = (xμ-bμ x^2)/(1-2 b.x + b^2 x^2)

    and I need to show that the metric transforms as

    g'μν = Λ(x) gμν

    3. The attempt at a solution
    My attempt was to differentiate the transformation law in order to then use the chain rule (the derivatives are intended as partial):

    gσλ=dx'μ/dxσ dx'ν/dxλ g'μν

    For a particular partial derivative I get:
    dx'μ/dxν = (δμν-2bμxν)/(1-2 b.x + b^2 x^2)- (xμ-bμ x^2)(-2 bν+2b^2 x ν)/(1-2 b.x + b^2 x^2)^2

    however plugging this and the other similar term in the chain rule gives rise to a very long expression which does not appear to simplify (I've checked it does in 1D, if all quantities were scalars).
    Am I doing something wrong, or am I just missing something?
     
  2. jcsd
  3. Sep 19, 2016 #2

    strangerep

    User Avatar
    Science Advisor

    Oh c'mon, it's not all that "long". (It might seem less intimidating if you used latex and \frac ...)

    The combined numerator ends up as a sum of 4 terms, and you can contract some of indices with ##g##.

    You'll have to post the whole expression before we can help you figure out what's wrong.
     
  4. Sep 20, 2016 #3
    You're right, I made a dumb mistake early on and that prevented me from getting to the final answer! Now I get it.
     
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