Writing tensors in a different way?!

Tom Mattson

Physicist,

What dexter is trying to lead you to is the following:

[tex]
F^{ \mu \nu } F_{ \mu \nu } = \sum _ { \mu = 1} ^ {4} \sum _ { \nu = 1} ^ {4} F^{\mu \nu} F_{\mu \nu}
[/tex]

That is the Einstein summation convention. So, you let the indices [itex]\mu[/itex] and [itex]\nu[/itex] each run from 1 to 4 in the double sum, and you should get your answer straightforwardly.

jcsd

Yes it's the summation convention your missing physicist, remember that matrices are only (limited) representations of tensors,

Tom Mattson

Physicist, you're missing a couple of other things, too.

You need to know that, for any [itex]\mathbb {R} ^3[/itex] vector [itex]\mathbf {A} = A_x \mathbf {i} +A_y \mathbf {j} +A_z \mathbf {k}[/itex], we have:

[tex]\mathbf {A} ^2= \mathbf {A} \cdot \mathbf {A}= A_x^2+A_y^2+A_z^2[/tex]

The other thing you're missing is this issue of matrix multiplication. [itex]F^{\mu \nu }F_{\mu \nu }[/itex] does not mean that you are supposed to multiply the matrix representations of [itex]F[/itex] together. It means that you are supposed to sum over the indices, as I described in my last post. If you were supposed to do matrix multiplication, it would be written as follows:

[tex]
F^{\mu \nu } F_{\nu \lambda }
[/tex]

dextercioby

Not really,Tom.What u've written is a 4-th rank (2,2) tensor.It doesn't have matrix representation in R^{2}...

Daniel.

jcsd

No Tom is correcr, but perhaps it's better to treat Matrices as (1,1) tensors, so [itex]F^{\mu}_{\alpha}F^{\alpha}_{\nu} = F^{\mu}_{\nu}[/itex] is the kind of operation that phsyicist is doing.

Physicist

Thank you all..

That was the missing point.

Thanks alot