When do we use tensor densities rather than tensors?

[Phrak]

When do we use tensor densities rather than tensors?

 

[WannabeNewton]

I can't think of definite points where tensor densities are used over tensors but one point is that when integrating some scalar or tensor over a region of an n - manifold the volume element is actually an anti - symmetric tensor density which is constructed through wedge products like differential forms. Another place I have seen tensor densities is in an Einstein Lagrangian $$\L = (-g)^{1/2}R$$ where g is the metric determinant and R is the ricci scalar. If I recall correctly you can always turn a tensor density into a tensor by multiplying the density by $$g^{W/2}$$ where W is the weight of the tensor density you are changing.

If you want check out Appendix B of General Relativity - Wald (he talks about integration and forms but not densities explicitly), Spacetime and Geometry - Carroll (sections 2.8 - 2.10), or Introducing Einstein's Relativity - D'Inverno (sections 7.1 - 7.4).

 

[bcrowell]

For area, volume, and angular momentum:

http://www.lightandmatter.com/html_books/genrel/ch04/ch04.html#Section4.6

 

[nitsuj]

For area, volume, and angular momentum:

http://www.lightandmatter.com/html_books/genrel/ch04/ch04.html#Section4.6

Off topic sorry,

just want to gives props to bcrowell for the free texts. I'm really enjoying them :)

 

[Phrak]

Thanks, bcrowell. That clears it up.