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Is there a reason why $\det(g)$ is a scalar density of weight 2 and not 1, 3 or 4?
$\det(g)$ is not coordinate-independent - it is a scalar density of weight +2 (or −2 depending on convention) which generally changes across spacetime. In Minkowski space equipped with spherical polar ...
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Is there a relatively quick way to obtain the scale factor on the metric as a response to special conformal transformation on spacetime?
For a special conformal transformation on spacetime positions $$x'^\mu=\frac{x^\mu-x^2b^\mu}{1-2b\cdot x+b^2x^2}$$
I was able to derive the scale factor $$\eta'_{\mu\nu}=\Omega(x)\eta_{\mu\nu}$$by ...
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