Question

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)=−r4sin2(θ).

A scalar density scales with Jw, 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.

Answer 1

The weight of det(g) in spherical polar coordinates is exactly 2 because there are only three independent spatial directions, causing the weight to double. If each direction counted as a single, the weight would be 4 instead.

Step-by-step explanation:

Det(g) is a scalar density of weight +2 (or −2) which generally changes across spacetime. The weight of det(g) in spherical polar coordinates is exactly 2 because there are only three independent spatial directions, causing the weight to double. If each direction counted as a single, the weight would be 4 instead.

The weight of a scalar density scales with the Jacobian of the chosen coordinates. In the case of det(g) in Minkowski space equipped with spherical polar coordinates, it is exactly weight 2 because there are only three independent spatial directions. If each direction would count as a single, it would have been weight 4 instead of weight 2.