Direct Product vs Tensor Product

[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!

 

[andresB]

Am I right in thinking that a (2 0) tensor is the tensor product of two vectors...?

I say that is not correct,

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

 

[bcrowell]

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).

Am I right in thinking that a (2 0) tensor is the tensor product of two vectors

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.

 

[Ravi Mohan]

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.

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 ) quite helpful for the subject.