Direct Product vs Tensor Product

1. Sep 24, 2015

dman12

Hi,

I am working through a textbook on general relativity and have come across the statement:

"A general (2 0) tensor K, in n dimensions, cannot be written as a direct product of two vectors, A and B, but can be expressed as a sum of many direct products."

Can someone explain to me how this is the case? Am I right in thinking that a (2 0) tensor is the tensor product of two vectors, so how then is the direct product of two vectors different to the tensor product?

Thanks!

2. Sep 24, 2015

andresB

I say that is not correct,

Take a look at quantum mechanic of Cohen Tanoudji page 155.

3. Sep 24, 2015

bcrowell

Staff Emeritus
A direct product of two vectors would mean $A^i B^j$. A tensor product of two vectors would mean the same thing to me (unless you want to consider the inner product $A^i B_i$ as a kind of tensor product as well).

No, this is not true. In four dimensions, a (2,0) tensor has 16 degrees of freedom, while the tensor product of two vectors has only 8. Therefore it is not possible that all (2,0) tensors can be represented as the tensor product of two vectors.

4. Sep 24, 2015

Ravi Mohan

And therefore you need a linear combination of such direct products to build up a general (2,0) tensor. I find Carroll's notes (http://preposterousuniverse.com/grnotes/grnotes-one.pdf [Broken]) quite helpful for the subject.

Last edited by a moderator: May 7, 2017