Local Conformal Transformations:Coordinate or metric transformations?

In summary, local conformal transformations preserve angles and change the scale of distances on a local level, while global conformal transformations preserve angles and distances across the entire space. Examples of local conformal transformations include translations, rotations, and dilations, and they have applications in mathematics, physics, image processing, and computer graphics. Local conformal transformations are a subset of conformal symmetry, which is a type of symmetry that preserves angles and scales, but not necessarily distances.
  • #1
Johanna222
2
0
Hello,

I'm wondering what the exact definition of a local conformal transformation is, in the context of General Relativity (/Shape Dynamics)

To be more precise:
1. Are local conformal transformations coordinate transformations or scalar transformations of the metric?
2. If they are coordinate transformations, are they of the form [itex]\vec{x} \mapsto C(x^{\mu})\vec{x}[/itex], with [itex]C(x^{\mu})[/itex] a differentiable function?

Good evening to you all!
 
Last edited:
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  • #2
I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?
 

Related to Local Conformal Transformations:Coordinate or metric transformations?

1. What are local conformal transformations?

Local conformal transformations refer to coordinate or metric transformations that preserve angles and change the scale of distances locally. In other words, the transformation only affects points in a small neighborhood, rather than the entire space.

2. How are local conformal transformations different from global conformal transformations?

Global conformal transformations preserve angles and distances across the entire space, whereas local conformal transformations only preserve angles and change distances on a small scale. Global conformal transformations are also rigid, meaning that the shape and size of objects do not change, while local conformal transformations can change the shape and size of objects locally.

3. What are some examples of local conformal transformations?

Some examples of local conformal transformations include translations, rotations, and dilations. These transformations can be applied to specific points or regions in a space, rather than the entire space.

4. What are the applications of local conformal transformations?

Local conformal transformations have many applications in mathematics and physics. They are used in differential geometry to study the curvature of surfaces, and in general relativity to describe the geometry of spacetime. They are also useful in image processing and computer graphics for transforming and manipulating images.

5. How do local conformal transformations relate to conformal symmetry?

Conformal symmetry is a type of symmetry where a transformation preserves angles and scales, but not necessarily distances. Local conformal transformations are a subset of conformal symmetry, as they only preserve angles and change scales on a local level. In other words, local conformal transformations are a type of local conformal symmetry.

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