1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted