Conformal Transformation as a diffeomorphism

[radioactive8]

Hello,

I know question surrounding this topic have been asked again before, however the more I search for the answer of my question, the more confused I become since each book, paper and thread uses its own formulation.

I am trying to figure out what is a conformal transformation and as a result what is a Weyl transformation.

For starters, I know that a c.transformation is just a coordinate transformation as are rotations dilatations etc. In addition I understand how the Conformal group arises.

However, the thing that confuses me is the following.

Some books write the conformal transformation as the coordinate transformations which result to:
\begin{align}
g'_{\mu \nu} (x') = e^{-2 \sigma(x)} g_{\mu \nu} (x) \label{1}\\
g'_{\mu \nu} (x') = \frac{\partial x^\alpha}{\partial x'^\mu} \frac{\partial x^\beta}{\partial x'^\nu} g_{\alpha \beta} (x)
\end{align}
where the second equation is due to the conformal transformations being coordinate transformations.

On the other hand, other books and papers write \eqref{1} as:
\begin{equation}
g'_{\mu \nu} (x) = e^{-2 \sigma(x)} g_{\mu \nu} (x)
\end{equation}

My basic question is the following. Does the metric get rescaled in the same point of the manifold as before OR does the rescalling occur for different points?

thanks

 

[samalkhaiat]

On the other hand, other books and papers write \eqref{1} as:
\begin{equation}
g'_{\mu \nu} (x) = e^{-2 \sigma(x)} g_{\mu \nu} (x)
\end{equation}

This is called Weyl rescaling, it effects the (field's) function not the coordinates. To see the connection between Weyl's transformation and the conformal transformations, see

https://www.physicsforums.com/threads/conformal-invariance-klein-gordon-action.630693/post-4057990

 

[radioactive8]

This is called Weyl rescaling, it effects the (field's) function not the coordinates. To see the connection between Weyl's transformation and the conformal transformations, see

https://www.physicsforums.com/threads/conformal-invariance-klein-gordon-action.630693/post-4057990

Believe or not, your answer in the past thread answered almost all my questions. You are a life saver!

I just have two more.

1) So I am trying to imagine all this using Manifolds, maps and etc.

I understand how to imagine a passive (coordinate) transformation which would be simply a map between two different charts containing the same point in a manifold.

I believe that the "active" way to see a coordinate transformation is by taking a map between two identical manifolds, where this map, let's say:
$$
\Phi: M \rightarrow N,
\\ \quad p \rightarrow \phi(p)
$$

connects charts with different coordiante systems on $R^n$, defines a pullback of the metric tensor fieldand the components of the pullback transform like we know.

How can I imagine a Weyl Transformation?

2) How is a metric isometry defined ?

 

[samalkhaiat]

How can I imagine a Weyl Transformation?
2) How is a metric isometry defined ?

Suppose $(M_{1},g_{1})$ and $(M_{2},g_{2})$ are differentiable manifolds of the same dimension. If $F : M_{1} \to M_{2}$ is smooth regular map, then you can pull back $g_{2}$ to a new metric $F^{*}g_{2}$ on $M_{1}$. Now, you have two metrics on the same manifold $M_{1}$, so you can compare them:

a) If $F^{*}g_{2} = g_{1}$, then the map $F$ is called local isometry. The set of all such maps forms a group called the isometry group. An isometry preserves the length of a vector and the angle between vectors.

b) If $F^{*}g_{2} = \Omega^{2} (x) g_{1}$, then $F$ is called conformal. Conformal maps are angle preserving maps. Again, the set of all such maps is a group called the conformal group.

Both, the conformal group and the isometry group, can be realized as groups of coordinate transformations. And this is the difference between them and the scaling group of Weyl. Weyl transformations (just like the gauge transformations) cannot be realized in terms of coordinate transformations.