Scale factor of special conformal transformation

In summary, the scale factor in a special conformal transformation is a mathematical term that represents the change in distance between points in a given space. It is calculated by taking the derivative of the special conformal transformation function with respect to the transformation parameter and plays a crucial role in understanding the transformation of the space. The scale factor can vary throughout the space, and its value at a point determines the change in distance between points after the transformation has been applied.
  • #1
stegosaurus
2
0

Homework Statement


(From Di Francesco et al, Conformal Field Theory, ex .2)
Derive the scale factor Λ of a special conformal transformation.

Homework 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

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?
 
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  • #2
stegosaurus said:
[...] plugging this and the other similar term in the chain rule gives rise to a very long expression which does not appear to simplify
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
You're right, I made a dumb mistake early on and that prevented me from getting to the final answer! Now I get it.
 
  • #4
$$ \frac{\partial x'^\mu}{\partial x^\nu}=\frac{\delta^\mu_\nu-2 b^\mu x_\nu}{1-2(b\cdot x)+b^2x^2}-\frac{(x^\mu-b^\mu x^2)(-2 b_\nu+2 b^2 x_\nu)}{\left ( 1-2(b\cdot x)+b^2x^2\right )^2} $$
is this expression correct?
 
  • #5
metalvaro18 said:
$$ \frac{\partial x'^\mu}{\partial x^\nu}=\frac{\delta^\mu_\nu-2 b^\mu x_\nu}{1-2(b\cdot x)+b^2x^2}-\frac{(x^\mu-b^\mu x^2)(-2 b_\nu+2 b^2 x_\nu)}{\left ( 1-2(b\cdot x)+b^2x^2\right )^2} $$
is this expression correct?
Afaict, it looks ok to me.
 

1. What is the scale factor in a special conformal transformation?

The scale factor in a special conformal transformation is a mathematical term that represents the change in distance between points in a given space. It is a key component in determining the overall transformation of a space under the special conformal transformation.

2. How is the scale factor calculated in a special conformal transformation?

The scale factor is calculated by taking the derivative of the special conformal transformation function with respect to the transformation parameter. This derivative is then evaluated at the point of interest in the space to determine the scale factor at that particular point.

3. What is the significance of the scale factor in a special conformal transformation?

The scale factor plays a crucial role in understanding how the special conformal transformation affects the geometry of a space. It can provide insight into the overall transformation and can help identify important features of the transformed space.

4. How does the scale factor affect distances in a special conformal transformation?

The scale factor determines the change in distance between points in a space after the special conformal transformation has been applied. If the scale factor is greater than one, distances between points will increase, while a scale factor less than one will result in decreased distances.

5. Can the scale factor vary throughout a space in a special conformal transformation?

Yes, the scale factor can vary throughout a space in a special conformal transformation. This is because the value of the scale factor depends on the point in the space where it is evaluated. Therefore, different points in the space may have different scale factors.

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