# Local conformal transformation:Coordinate or metrical transformation?

1. Aug 25, 2014

### Johanna222

Hello,

I was wondering what the exact definition of conformal transformations is.

This is a question in the context of Shape Dynamics. In Shape dynamics, time is viewed as a global parameter of the universe, and as such is invariant under spatial coordinate transformation. Part of the diffeomorphism invariance of General Relativity (the diffeomorphisms that mix space and time), is thus not present in the theory, but instead traded for invariance under local spatial conformal transformations (LSCT's).

Interpreting these LCTS's as coordinate transformation ($\vec{x} \mapsto C(x^{\mu})\vec{x}$) leads to a problem:
They should already be part of the diffeomorphism symmetry (of space), giving empty trading.

Are these LCTS's to be interpreted as transformations of the metric, leaving coordinates invariant?

I assume $C(x^{\mu})$ to be positive and differentiable.

2. Aug 28, 2014

### Greg Bernhardt

I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?

3. Aug 29, 2014

### haushofer

I'm confused by the word " local" ; usually the generators are defined by the conformal Killing eqn. Making the generators local again would just give gct's, but I guess you already know that.

I do know that one can obtain GR by gauging the conformal algebra. The " extra" generators (wrt Poincare), generating special conformal transfos and dilatations, give gauge fields which can be solved and gauged away (Stuckelberg) respectively. The action of a conformal scalar then gives the Hilbert action after gaugefixing this scalar field. In the superconformal case this is used to construct matter couplings in supergravity.

Do you have a reference? I'm not so familar with shape dynamics, but am curious :)