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Electromagnetic tensor - invariant

  1. Jan 16, 2009 #1
    1. The problem statement, all variables and given/known data

    I have to calculate the invariant: [tex] \tilde{F}^{\mu \nu} \, F_{\mu \nu} [/tex]
    where F is the electromagnetic field tensor and [tex] \tilde{F} [/tex] the dual one.

    2. Relevant equations
    First, the contravariant components of the electromagnetic field tensor are given by:
    [tex]F^{\mu\nu} = \begin{bmatrix} 0 & -E_1 & -E_2 & -E_3 \\ E_1 & 0 & -B_3 & B_2 \\ E_2 & B_3 & 0 & -B_1 \\ E_3 & -B_2 & B_1 & 0 \end{bmatrix}[/tex]

    and the covariant by:
    [tex]F_{\mu\nu} = \begin{bmatrix} 0 & E_1 & E_2 & E_3 \\ -E_1 & 0 & -B_3 & B_2 \\ -E_2 & B_3 & 0 & -B_1 \\ -E_3 & -B_2 & B_1 & 0 \end{bmatrix}[/tex]

    And last but not least, the contravariant components of the dual elm. field tensor:
    [tex]\tilde{F}^{\mu\nu} = \begin{bmatrix} 0 & -B_1 & -B_2 & -B_3 \\ B_1 & 0 & E_3 & -E_2 \\ B_2 & -E_3 & 0 & E_1 \\ B_3 & E_2 & -E_1 & 0 \end{bmatrix}

    3. The attempt at a solution
    First I note:
    [tex]\tilde{F}^{a0} = B^{a} [/tex]

    [tex]\tilde{F}^{ab} = \epsilon^{abi} E_{i}[/tex]

    [tex]F_{a0} = - E_{a} [/tex]

    [tex]F_{ab} = \epsilon_{abc} B^{c}[/tex]

    Now, if I use these relations I obtain the wrong solution:

    [tex] \tilde{F}^{\mu \nu} \, F_{\mu \nu} = 2 \tilde{F}^{a0} F_{a0} + \tilde{F}^{ab} F_{ab} = - 2 B^{a} E_{a} + \epsilon^{abi} E_{i} \epsilon_{abc} B^{c} = 0 [/tex]
    where I used the relation: [tex] \epsilon^{abi} \epsilon_{abc} = 2 \delta^{i}_{c} [/tex]

    Of course, if i simply insert the components explicitly (the "matrix elements") I get the result [tex]\tilde{F}^{\mu \nu} \, F_{\mu \nu} = - 4 \vec{B} \cdot \vec{E} [/tex] and everything is fine.

    I used the following convention: [tex] \epsilon^{0123} = 1 [/tex], latin indices: {1,2,3}, greek indices: {0,1,2,3}

    Further I calculated: [tex] \epsilon_{123} = \eta_{\alpha 1} \, \eta_{\beta 2} \, \eta_{\gamma 3} \, \epsilon^{\alpha \beta \gamma} = - 1 [/tex] where [tex] \eta = diag(1, -1, -1, -1) [/tex] is the metric tensor.

    I think there has to be a wrong sign, but I don't find it. Does anyone have an idea?
  2. jcsd
  3. Jan 16, 2009 #2


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    I think you might want to double check this one :wink:
  4. Jan 17, 2009 #3
    hmmm, I don't see anything wrong here !?

    If I check it I obtain the right elements, for example: [tex] F_{12} = \epsilon_{123} B^{3} = - B^{3} [/tex].
  5. Jan 17, 2009 #4


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    [tex]\epsilon_{123}=+1[/tex] :wink:
  6. Jan 17, 2009 #5


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    This assumes that [tex]\epsilon_{abc}[/tex] transforms as a tensor; but it doesn't. It is a tensor density with weight -1.
  7. Jan 17, 2009 #6
    ok, [tex] \epsilon_{ijk} [/tex] is a pseudotensor, but where is the difference between covariant [tex] \epsilon_{ijk} [/tex] and contravariant [tex] \epsilon^{ijk} [/tex] components?

    In the case with 4 indices there is a difference, for example: [tex] \epsilon^{0123} \neq \epsilon_{0123} [/tex]. Why should it be different with 3 components?
  8. Jan 17, 2009 #7


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    There isn't a difference in the convention that I'm used to seeing.

    I'm assuming that you are using the same convention (the most common convention) that is used here.

    In that case, then [tex] \epsilon^{0123} = \epsilon_{0123} [/tex].

    If you are using some other convention, whereby [tex] \epsilon_{123}=-1 [/tex] , then your problem is in the equation [tex] \epsilon^{abi} \epsilon_{abc} = 2 \delta^{i}_{c}[/tex] instead.
  9. Jan 18, 2009 #8
    i've found something, have a look at this (page 9): http://www.worldscibooks.com/phy_etextbook/6938/6938_chap01.pdf

    Further I found something in another wiki (but it's not in english). There is a remark that in relativity there you have to differ between co- and contravariant indices. It is just convention whether you use [tex] \epsilon^{0123} [/tex]= +1 or -1.

    But all this doesn't explain my problem :confused:
  10. Jan 18, 2009 #9


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    The convention used in the above link is not very common; but if that is the convention you use in your course then you can stick with it.

    According to that convention, [tex]\epsilon_{ijk}=-e^{ijk}[/tex] and [tex]e_{123}=1[/tex]. Therefor, [tex]\epsilon^{abi} \epsilon_{abc} = -2 \delta^{i}_{c}[/tex] in this convention. In addition, epsilon is a tensor in this convention.

    So the question you need to ask yourself is which convention is used in your course text/notes?
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