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Rank 4 Tensor Symmetry Proof

  1. May 3, 2015 #1
    1. The problem statement, all variables and given/known data
    Let Aijkl be a rank 4 square tensor with the following symmetries:
    [tex]
    A_{ijkl} = -A_{jikl}, \qquad A_{ijkl} = - A_{ijlk}, \qquad A_{ijkl} + A_{iklj} + A_{iljk} = 0,
    [/tex]

    Prove that
    [tex]
    A_{ijkl} = A_{klij}
    [/tex]

    2. Relevant equations


    3. The attempt at a solution
    From the first two properties I concluded that:
    [tex]
    A_{iikl} = 0 \qquad A_{ijkk} = 0
    [/tex]

    The last one leaded me to:
    [tex]
    A_{ikli} = -A_{ilik} \qquad A_{ikkj} = -A_{ikjk}
    [/tex]

    However I don't see how this last one may help me.
     
  2. jcsd
  3. May 3, 2015 #2

    Orodruin

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    I suggest not trying to make any contractions and instead just apply the given symmetries, including the last one involving three tensor components.
     
  4. May 3, 2015 #3
    New attempt, got further but still missing something, hope this was what you meant.
    From the third property:
    [tex] A_{ijkl} + A_{iklj} + A_{iljk} = 0 [/tex]
    [tex] A_{klij} + A_{kijl} + A_{kjli} = 0 [/tex]
    Therefore:
    [tex] A_{ijkl} + A_{iklj} + A_{iljk} = A_{klij} + A_{kijl} + A_{kjli} [/tex]
    Since the first two properties refer to switching the first pair or the last pair of indexes, I can write:
    [tex] A_{ijkl} + A_{kijl} + A_{iljk} = A_{klij} + A_{kijl} + A_{kjli} [/tex]
    Leading to
    [tex] A_{ijkl} + A_{iljk} = A_{klij} + A_{kjli} [/tex]
    However I still have one extra term on each side that I can't deal with the same way as before.
     
  5. May 3, 2015 #4

    Orodruin

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    What do you get if you simply do the following renaming of the indices in this equation: ##i \leftrightarrow j##, ##k \leftrightarrow \ell##? Does it remind you of something?
     
  6. May 3, 2015 #5
    That would result in:
    [tex] A_{jilk} + A_{jkil} = A_{lkji} + A_{likj} [/tex]
    The only thing it reminds me is of the third symmetry again, but if I use it I end up with a meaningless result:
    [tex] A_{jlki} = A_{ljik} [/tex]
    Which translates in the first two symmetries.
     
  7. May 3, 2015 #6

    Orodruin

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    Try using the first and second symmetries instead. Also, it will help if you put all of the components on one side and equate to zero.
     
  8. May 3, 2015 #7
    Finally got it! Thanks a lot for the help
     
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