Writing tensors in a different way?

[Physicist]

Hi all,

I have 2 tensors of rank 2. I want to write their product in a way else than a matrix.

Or let's say, for example: how can I write the electic field in a form of matrix (tensor)?

Thanks

 

[dextercioby]

What kind of a product?A simple tensor product,or a contracted tensor product...?Please,for our illumination,post the product of tensors in component form.

The electric field is a 3-vector and can be put under the form of a column:
$$(\vec{E})^{i}=\left ( \begin{array}{c}E^{1}\\E^{2}\\E^{3}\end{array}\right )$$

Daniel.

 

[Physicist]

I have a proof to do, starting from the covariant & contravariant field tensors (which are 4 X 4 matrices) & ending with E^2 & B^2.

I couldn't know where did those bold E & B come from? I mean how to transform the calculations from dealing with matrices to the bold symbols?

I think the E represents the electric field tensor. How is it written in form of a 4 X 4 matrix? I found different forms in different sites & couldn't know which one is right.

I hope I'm clear now.

Thanks

 

[dextercioby]

Components of E & B are elements of the em tensor $$\hat{F}$$...When u consider operations with theis tensor,u're making operation with the fields as well.E.g.The Lagrangian (density) of the em field is:
$$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$$... (in Heaviside-Lorentz units)

Consider all the terms in the summation & u'll end up with something ~(E^{2}-B^{2})...

Daniel.

 

[Physicist]

Consider all the terms in the summation

How??

OK I've done the following:

http://physicist.jeeran.com/untitled.JPG

I noticed some notes about the elements of the resultant matrix but still couldn't complete!

I asked about the E & B to try to get them from this matrix.

Can you help?

just a hint please, I wanted to do it myself but I'm stuck at that point since few days

Thanks

 

[dextercioby]

Nope,the double contraction MUST BE A LORENTZ SCALAR.That matrix form is highly useless.

Your calculus is included in any standard book on electrodynamics,as Jackson,maybe...
I told you what to do:consider that sum and you'll get your answer.

Daniel.

 

[Physicist]

Still couldn't understand how to do as you said (consider that sum and you'll get your answer).

I've got the book of Jackson, he went through it briefly & didn't explain the mathematical steps.

Can anyone please do it step by step with explaining in details? because I'm somehow new to tensors.

I will be thankful.

 

[dextercioby]

Do what,step by step...?The summation...?You can't add 16 terms...?

Daniel.

 

[Physicist]

OF COURSE I CAN!

But I didn't understand what do you mean? to add what?

Do you mean I have to add the 16 terms in the matrix?! What would that equals?

 

[jcsd]

add the terma together, you can't add scalars to a matrix.

edited to add you need to go back to your textbook and see exactly what $F^{\mu}^{\nu}F_{\mu}_{\nu}$ means.

 

[dextercioby]

He knows what $$F^{\mu\nu}F_{\mu\nu}$$ means.And that should equal the lagrangian density,what else...?

Daniel.

 

[jcsd]

Yes, but he seems unsure what the noataion represents mathematically.

 

[dextercioby]

There are 16 terms in all,4 of which are 0.So the problem is even simpler.

Daniel.

 

[jcsd]

One simpler (to me) way of looking at it is that $F^{\mu}^{\nu}$ are the compents of a vector in the (16 dimensional) vector space of tensors of type (2,0) and $F_{\mu}_{\nu}$ are the compoents of it's dual vector, so $F^{\mu}^{\nu}F_{\mu}_{\nu}$ is it's square norm.

 

[Physicist]

If I know the answer I wouldn't ask!

I didn't want you to give me the answer directly, I really wanted to understand because I tried reading in many books & sites but still didn't understand it. I didn't have any course in tensors & now I need to deal with it in a research.

If it looks simple for you dextercioby, it's not for me & that's why I asked!

Thanks anyways.


jcsd, you are right (unsure what the noataion represents mathematically).

(add the terma together)

Do you mean that I should add the terms in the resultant matrix? What would the result represent?

Thank you.

 

[dextercioby]

What matrix are you talking about...?

Daniel.

 

[Physicist]

The matrix that results from the multipication (see reply #5).

 

[quantumdude]

Physicist,

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

$$F^{ \mu \nu } F_{ \mu \nu } = \sum _ { \mu = 1} ^ {4} \sum _ { \nu = 1} ^ {4} F^{\mu \nu} F_{\mu \nu}$$

That is the Einstein summation convention. So, you let the indices $\mu$ and $\nu$ 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,

 

[quantumdude]

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

I have a proof to do, starting from the covariant & contravariant field tensors (which are 4 X 4 matrices) & ending with E^2 & B^2.

I couldn't know where did those bold E & B come from? I mean how to transform the calculations from dealing with matrices to the bold symbols?

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

$$\mathbf {A} ^2= \mathbf {A} \cdot \mathbf {A}= A_x^2+A_y^2+A_z^2$$

The other thing you're missing is this issue of matrix multiplication. $F^{\mu \nu }F_{\mu \nu }$ does not mean that you are supposed to multiply the matrix representations of $F$ 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:

$$F^{\mu \nu } F_{\nu \lambda }$$

 

[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 $F^{\mu}_{\alpha}F^{\alpha}_{\nu} = F^{\mu}_{\nu}$ is the kind of operation that phsyicist is doing.

 

[Physicist]

Thank you all..

The other thing you're missing is this issue of matrix multiplication. $F^{\mu \nu }F_{\mu \nu }$ does not mean that you are supposed to multiply the matrix representations of $F$ 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:

$$F^{\mu \nu } F_{\nu \lambda }$$

That was the missing point.

Thanks a lot