Why is the volume element a scalar density of weight -1?

[madness]

In Ray d'Inverno's book on general relativity, he defines the volume element in a way which makes it a scalar density of weight -1, meaning it transforms with the inverse of the Jacobian. Every other source I have looked at seems to say it should transform with the Jacobian, making it a scalar density of weight +1. Can anyone clarify this for me?

 

[shoehorn]

I haven't read d'Inverno's book but the conventional way of defining the weight is to say that the weight of any tensorial quantity is the power of $g^{1/2}$ that appears in that quantity. Since the volume element is just $g^{1/2}$, it's a weight 1 quantity.

 

[madness]

The definition of tensor weights as transforming with the square root of the metric should be equivalent to the definition of transforming with the Jacobian, since the metric transforms with the square of the Jacobian. In d'Inverno's book it is defined in terms of the Jacobian.
His definition of a volume element doesn't appear to include the metric, and is actually stated to be a scalar density of weight -1 and not +1 as you have stated. This is my problem, I don't understand where his definition has come from.
He uses this fact to demand that the Lagrangian is a scalar of weight +1, and include the square root of the determinant here.
Can anyone explain this discrepancy with volume definitions?

 

[madness]

Can nobody help with this?

 

[robphy]

Can nobody help with this?

You may need to transcribe the relevant passages here.
It may simply be a confusion in notation or terminology.
(Unfortunately, there are no book previews from Google or Amazon.)

 

[madness]

I'm not sure how to transcribe the mathematics but I can try to explain his argument. He first defines a volume element for an m dimensional subspace of a manifold in a way that makes it an mth rank contravariant tensor tensor. It involves the generalised kronecker delta symbol. He later contracts this with the levi-cevita tensor (which gives the weight of -1). I'm not sure if this is any help at all. My problem is that I don't know why he has chosen any of his definitions (why it is really a volume), and why it differs from other sources.