conformal transformation (CT)

[xiaomaclever]

CTs are not a change of coordinates but an actual change of the geometry, right? In principle, we can change the Minkowski spacetime into Riemannian one even Riemann-Cartan one by some kind of CT. In the Riemann-Cartan spacetime there is torsion while it is torsion-free for Minkowski spacetime. So how do we understand the problem? Can we consider the torsion coming from the CT and torsion is not an intrinsic geometrical quantity ? Thanks for any reply!

[adamwho]

CTs are not a change of coordinates but an actual change of the geometry, right? In principle, we can change the Minkowski space-time into Riemann one even Riemann-Cartan one by some kind of CT. In the Riemann-Cartesian space-time there is torsion while it is torsion-free for Minkowski space time. So how do we understand the problem? Can we consider the torsion coming from the CT and torsion is not an intrinsic geometrical quantity ? Thanks for any reply!

1. It is a change of geometry not just coordinates.

2. Minkowski space has nothing to do with conformal mapping.

3. I have no idea what you are talking about when you mention torsion.

[Phrak]

I'm not sure what you're looking for. But consider a coordinate system where the coordinates system XY is normal at all points, though not necessarily orthonormal. The off-diagonal elements of the metric are zero for all points (x,y).

\hat{e}_i \hat{e}_j = 0, \ when \ i \neq j

In a conformal transform of coordinates, XY --> UV the off diagonal elements in the UV basis remain zero.