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Antisymmetrization leads to an identically vanishing tensor

  1. Jul 19, 2009 #1
    This comes from Andersons's Principles of Relativity Physics:

    "Of course, for fifth- or higher-rank tensors antisymmetrization leads to an
    identically vanishing tensor"

    But I don't understand why, even if it's "of course". So can someone show me why?
     
  2. jcsd
  3. Jul 19, 2009 #2

    tiny-tim

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    Hi jason12345! :smile:

    Because any tensor Aabcde has elements which look like Axyztx (or worse), and if A is antisymmetric, then this must be zero. :wink:
     
  4. Jul 19, 2009 #3
    Re: antisymmetrisation

    Antisymmetrising Aabcde means writing it as 1/5!(Aabcde + ... - Abacde -.. ) so that the indices abcde are merely permutated, whereas you have Axyztx which has indices x repeated. Why did you change the labelling from abcde to xyz and then t?

    Thanks for your interest.
     
  5. Jul 19, 2009 #4

    tiny-tim

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    Because the only candidates for the indices for the elements of the matrix are x y z and t (or 1 2 3 and 4, or whatever the four basis elements are) …

    each element of the matrix has to have each of a b c d and e equal to x y z or t. :smile:
     
  6. Jul 20, 2009 #5
    Re: antisymmetrisation

    Thanks, I understand what you're saying now :), although I still say it isn't obvious since Anderson was defining Tensors as general geometrical objects upto this point, it seems, rather than applying them to the space-time manifold.
     
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