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$\det(g)$ 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 coordinates, for example, det$(g)=−r^4sin^2(\theta)$.

A scalar density scales with $J^w$, where $J$ is the Jacobian of the chosen coordinates and $w$ is the weight. (In contrast, a scalar invariant doesn't change under coordinate transformation)

Is there a reason why it's exactly weight 2 and not weight 1 or 3?

I suppose that it has something to do with changes of space and time summing up in the determinant. However, there are three independent spatial directions and if their change would count each as a single it would be rather weight 4 than weight 2.

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This is simply because of the way under which the metric transforms: $$g_{\mu'\nu'}=\frac{\partial x^{\mu}}{\partial x^{\mu'}}\frac{\partial x^{\nu}}{\partial x^{\nu'}}g_{\mu\nu}$$ Taking the determinant, you get $$g(x')=\left|\frac{\partial x^{\mu'}}{\partial x^{\mu}}\right|^{-2}g(x)$$

Hope this helps.

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