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Tensor Index Notation Question

  1. Jun 7, 2013 #1
    I am just wondering, is there a difference in meaning/definition between the indices of a tensor being right on top of each other

    [tex]A_{\mu }^{\nu }[/tex]

    and being "spaced" as in

    [tex]A{^{\nu }}_{\mu }[/tex]

    I seem to remember that I once read that there is indeed a difference, but I can't remember what it was.

    Thanks in advance.
     
  2. jcsd
  3. Jun 7, 2013 #2

    WannabeNewton

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    Science Advisor

    The spacing is very important if you are contracting with the metric tensor using the abstract index notation. For example if ##A^{a}{}{}_{b}## is a tensor then ##g^{bc}A^{a}{}{}_{c} = A^{ab}## but if we consider ##A_{c}{}{}^{a}## then ##g^{bc}A_{c}{}{}^{a} = A^{ba}## which will not equal ##A^{ab}## unless the tensor is symmetric. The notation ##A^{a}_{b}## makes the contraction with the metric tensor ill-defined in abstract index notation (index free notation is a different story of course).

    The action of the tensor on a covector and a vector will subsequently be ambiguous if all you write down is ##A^{a}_{b}## because ##A_{b}{}{}^{a}v^{b}\omega_{a} = g^{ac}A_{bc}v^{b}\omega_{a} \neq g^{ac}A_{cb}v^{b}\omega_{a} = A^{a}{}{}_{b}v^{b}\omega_{a}## in general, so the spacing is important.
     
  4. Jun 7, 2013 #3

    lurflurf

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    Homework Helper

    You just need a convention for the order.
    $$A{_\mu }^{\nu } \\
    A{^{\nu }}_{\mu } \\
    A_{\mu }^{\nu }$$

    So the third one can be substituted for one of the others as long as you always know which one, or that it does not matter which.
     
  5. Jun 8, 2013 #4
    Non-spaced indices represent symmetric tensor (in respective components).
     
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