Confusion regarding conformal transformations

[nigelscott]

I am confused about conformal transformations on Riemannian manifolds. Here's what I have so far.

1. Under a conformal transformation the metric changes by:

g' -> Ω²g

2. Under a Weyl transformation the metric changes by:

g' -> exp(-2f)g

3. Any 2D Riemann manifold is locally conformally flat and the metric can be defined in terms of isothermal coordinates.

g = exp(f)(du² + dv² )

Where u and v are Euclidean.

How are these all tied together? There appears to be a commonality in form but the multiplying functions are different. Where does the exponential come from?

 

[nigelscott]

OK, I have modified my original questions. I think (hope) I now have a better understanding. Perhaps somebody could critique this. However, I am still a little confused about the relationship between λ and Ω and the difference in exponents. Thanks.


Conformal Manifolds:

A manifold, M, is said to be conformal if two Riemannian metrics, g and g', on the smooth manifold M are related by the equation:

g'_μν = Ω²g_μν

Where Ω is a real-valued smooth function defined on the manifold. Metrics obeying this equation are said to be conformally equivalent.

This equation represents a general coordinate transformation, x -> x', such that x' = f(x) has the following effect on the metric:

g'_μν = (∂x^ρ/∂x'^μ)(∂x^σ/∂x'^ν)g_ρσ

Under these conditions both the metric and the Weyl tensor are invariant meaning:

g_μν(x)dx^μdx^ν = g'_μν(x)dx'^μdx'^ν or ds² = ds'².

and

C_abcd = C'_abcd

Conformal Transformations:

A conformal transformation can be defined as a subset or special case of the above general coordinate transformation which acts as a rescaling of the metric x -> x by a scale factor, λ . This rescaling of the metric is often referred to as WEYL transformation and the 2 terms are often used interchangeably, although they are in fact different things. Therefore:

g_μν = λg_μν

From above:

g_μν = (∂x'^ρ/∂x^μ)(∂'x^σ/∂x^ν)g'_ρσ

So we can write:

(∂x'^ρ/∂x^μ)(∂'x^σ/∂x^ν)g'_ρσ = λg_μν

or

g_μν = λ^-1{(∂x'^ρ/∂x^μ)(∂'x^σ/∂x^ν)g'_ρσ}

In other words any coordinate transformation such that g'_μν = λg_μν can be 'undone' by a Weyl transformation. The Weyl transformation takes us to a coordinate system where the metric has the same form as the one we started with, but the points have all been moved around and pushed closer together or farther apart depending on the scale factor. Therefore, under a Weyl transformation:

ds² -> λds²

The Weyl transformation changes the proper distances at each point by a scalar factor λ; and, therefore, has no directional meaning. It is a local change of scale which preserves the angles between all lines. It is not a coordinate transformation on the space or spacetime.

Conformal Flatness:

A necessary condition for a Riemannian manifold to be conformally flat is that the WEYL TENSOR vanish. The interpretation of this is that if a manifold (M,g) has a neighborhood (U,h) at each point such that g'_μν = η'_μν = Ω²η_μν it is locally conformally flat. Under these circumstances the Riemannian metric is conformal to the Euclidean metric and can be described by a local coordinate system in which the metric ds² satisfies:

ds² = Ω²(dx₁² + dx₂² + ... dx_n²)

These are referred to as ISOTHERMAL COORDINATES.

 

[haushofer]

The multiplying functions can be defined as you want and are conventional. In 2d, a metric has 3 components, so gaugefixing your 2 gct's leaves you with one function for the metric. A conformal metric can the be used to write the metric in the form you give.

Hope this helps.

 

[nigelscott]

Thanks. I think I am almost there. By convention Ω = exp(2f). Both the CT and the WT are applicable to d ≥ 2 but for d = 2, f = f(z) and is holomorphic. The scale factor becomes |df/dz|². Correct?

 

[samalkhaiat]

I am confused about conformal transformations on Riemannian manifolds. Here's what I have so far.

1. Under a conformal transformation the metric changes by:

g' -> Ω²g

2. Under a Weyl transformation the metric changes by:

g' -> exp(-2f)g

3. Any 2D Riemann manifold is locally conformally flat and the metric can be defined in terms of isothermal coordinates.

g = exp(f)(du² + dv² )

Where u and v are Euclidean.

How are these all tied together? There appears to be a commonality in form but the multiplying functions are different. Where does the exponential come from?

You may need to look at this post
Conformal Invariance Klein Gordon Action