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Invarients from the Faraday tensor

  1. Apr 19, 2010 #1
    Hello,

    a full contraction of the faraday tensor with itself can be shown to be

    [tex]F_{\mu\nu}F^{\nu\mu}=2(E^{2}-c^{2}B^{2})[/tex]

    I have done this by calculating 16 terms in the sum i.e. F11F11 + F12F21, and get this answer, but this is very tedious.

    Is there a faster way to show this that I am missing?
     
    Last edited: Apr 19, 2010
  2. jcsd
  3. Apr 19, 2010 #2

    Cyosis

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    Homework Helper

    Yes, by using the fact that the Faraday tensor is antisymmetric. This way you only have to calculate 6 terms.
     
    Last edited: Apr 19, 2010
  4. Apr 19, 2010 #3
    That is a good point. makes it much easier. Thanks
     
  5. Apr 19, 2010 #4

    clem

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    Science Advisor

    It also gets easier if you recognize that the first row is just [tex]-{\vec E}[/tex], and the 3X3 space-like part is just [tex]-{\vec B}[/tex], a bit mixed up.
     
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