I see that the weyl transformation is gab→Ω(x)gab under which Ricci scalar is not invariant. I am a bit puzzled when conformal transformation is defined as those coordinate transformations that effect the above metric transformation i.e x→x′⟹gμν(x)→g′μν(x′)=Ω(x)gμν(x). but any covariant action is clearly invariant under coordinate transformation? I see that what we mean by weyl transformation is just changing the metric by at a point by a scale factor Ω(x). So my question is why one needs to define these transformations via a coordinate transforms. Is it the case that these two transformations are different things. In flat space time I understand that conformal transformations contain lorentz transformations and lorentz invariant theory is not necessarily invariant under conformal transformations. But in a GR or in a covariant theory effecting weyl transformation via coordinate transformations is going to leave it invariant. Unless we restrict it to just rescaling the metric?
I am really confused pls help.
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