# Scale factor of special conformal transformation

Tags:
1. Sep 19, 2016

### stegosaurus

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. Sep 19, 2016

### strangerep

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.

3. Sep 20, 2016

### stegosaurus

You're right, I made a dumb mistake early on and that prevented me from getting to the final answer! Now I get it.