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Writing tensors in a different way?

  1. Mar 3, 2005 #1
    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
     
  2. jcsd
  3. Mar 3, 2005 #2

    dextercioby

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    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:
    [tex] (\vec{E})^{i}=\left ( \begin{array}{c}E^{1}\\E^{2}\\E^{3}\end{array}\right ) [/tex]

    Daniel.
     
  4. Mar 3, 2005 #3
    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
     
  5. Mar 3, 2005 #4

    dextercioby

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    Components of E & B are elements of the em tensor [tex] \hat{F}[/tex]...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:
    [tex] \mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} [/tex]... (in Heaviside-Lorentz units)

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

    Daniel.
     
  6. Mar 3, 2005 #5
    How??

    OK I've done the following:

    Click here please

    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 :smile: but I'm stuck at that point since few days :frown:

    Thanks :smile:
     
    Last edited: Mar 3, 2005
  7. Mar 3, 2005 #6

    dextercioby

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    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.
     
  8. Mar 9, 2005 #7
    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.
     
  9. Mar 9, 2005 #8

    dextercioby

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

    Daniel.
     
  10. Mar 9, 2005 #9
    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?
     
  11. Mar 9, 2005 #10

    jcsd

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    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 [itex]F^{\mu}^{\nu}F_{\mu}_{\nu}[/itex] means.
     
    Last edited: Mar 9, 2005
  12. Mar 9, 2005 #11

    dextercioby

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

    Daniel.
     
  13. Mar 9, 2005 #12

    jcsd

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    Yes, but he seems unsure what the noataion represents mathematically.
     
  14. Mar 9, 2005 #13

    dextercioby

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

    Daniel.
     
    Last edited: Mar 9, 2005
  15. Mar 9, 2005 #14

    jcsd

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    One simpler (to me) way of looking at it is that [itex]F^{\mu}^{\nu}[/itex] are the compents of a vector in the (16 dimensional) vector space of tensors of type (2,0) and [itex]F_{\mu}_{\nu}[/itex] are the compoents of it's dual vector, so [itex]F^{\mu}^{\nu}F_{\mu}_{\nu}[/itex] is it's square norm.
     
    Last edited: Mar 9, 2005
  16. Mar 9, 2005 #15
    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 becuase 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.
     
  17. Mar 9, 2005 #16

    dextercioby

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    What matrix are you talking about...?

    Daniel.
     
  18. Mar 14, 2005 #17
    The matrix that results from the multipication (see reply #5).
     
  19. Mar 14, 2005 #18

    Tom Mattson

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    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.
     
  20. Mar 14, 2005 #19

    jcsd

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    Yes it's the summation convention your missing physicist, remember that matrices are only (limited) representations of tensors,
     
  21. Mar 14, 2005 #20

    Tom Mattson

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    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]
     
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