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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 ([itex]\vec{x} \mapsto C(x^{\mu})\vec{x}[/itex]) 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 [itex]C(x^{\mu})[/itex] to be positive and differentiable.

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# Local conformal transformation:Coordinate or metrical transformation?

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