Tensor transformations for change of coordinate system

[aeson25]

In school I've always learned that tensor transformations took the form of:

$$\mathbf{Q'}=\mathbf{M} \times \mathbf{Q} \times \mathbf{M}^T$$

However, in all the recent papers I've been reading. They've been doing the transformation as:

$$\mathbf{Q'}= \frac {\mathbf{M} \times \mathbf{Q} \times \mathbf{M}^T}{det(\mathbf{M})}$$

Where Q is the tensor in question and M is the transformation matrix and M^T is the transpose of M.

Does anyone know why the difference?

 

[Ben Niehoff]

Such on object is properly called a "tensor density". Specifically, it is a "tensor density of weight -1", since the determinant appears with the power -1.

 

[aeson25]

I can't seem to find a clear definition of "tensor density" online. How does this differ (or provide an advantage) (practically, not mathematically) from a regular coordinate transformation? FYI, I'm trying to follow the transformation of a anisotropic density (actual matter density) in a paper.

 

[Ben Niehoff]

Did you even try Googling it? For your claim that you "can't find it", I have serious doubts.

 

[aeson25]

Of course I tried googling it. I didn't say I didn't find ANY definitions, I said I couldn't find a CLEAR definition which included information pertaining to the entirety of the second sentence of my last post with an emphasis on the parenthetical parts. The online definitions basically say its a coordinate transformation with a weight to it based on a Jacobian determinant. Big whoop, those definitions tell me nothing about why it's used over a normal transformations. Why use a weight of -1 rather than 1000?