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A more physical attempt: In general relativity, the metric tensor represents local clock and ruler measurments. If I multiply the metric tensor by a scalar constant, it should be obvious that this is inequivalent (in general) to a set of coordinate transformations, but, at the same time, I'm affecting local clock and ruler measurments (the ratio of the ...

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General relativity is only conformally invariant in two dimensions. This can be proven by making the transformation $g_{ab} \rightarrow \phi g_{ab}$, and seeing what transformation Einstein's equation${}^{1}$ makes. What you will find is that Einstein's equation will MOSTLY transform, but you will get terms proportional to $(d-2)(d-1)$ and derivatives of ...

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Newton's constant is dimensionful. Hence the theory is NOT conformally invariant. In 2 dimensions, newtons constant is dimensionless. But then the apparent conformal symmetry is actually only a REDUNDANCY in the description (sometimes called weyl symmetry).

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