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Rasing and lowering indicies

  1. Jul 12, 2007 #1
    In GR the metric can raise or lower indices, but which index will it raise in this case:

    [itex] \delta^{ij} \partial_i \xi_j [/itex]

    is it,

    [itex] \partial^j \xi_j [/itex]

    or,

    [itex] \partial_i \xi^i [/itex]

    Or are these equal?
     
  2. jcsd
  3. Jul 12, 2007 #2

    nrqed

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    Didn't you actually mean [itex] g^{ij} \partial_i \xi_j [/itex]??
    In that case, yes, it is equal to both the expressions you gave, which are equal to one another.
     
  4. Jul 12, 2007 #3
    Brilliant, thanks very much! :smile:
     
  5. Jul 13, 2007 #4

    HallsofIvy

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    Note that int both
    [itex] \partial^j \xi_j [/itex]
    and
    [itex] \partial_i \xi^i [/itex]
    you will be doing a further contract. What you are really saying is that the order in which the contractions are done does not matter.
    [itex] \delta^{ij} \partial_i \xi_j [/itex]
    is a scalar quantity.
     
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