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Quick Question about the Christoffel Symbol of the Second Kind

  1. Aug 11, 2012 #1
    The Christoffel symbol is as followed:

    [tex]{\Gamma ^{m}}_{ab}=\frac{1}{2}g^{mk}(g_{ak,b}+g_{bk,a}-g_{ab,k})[/tex]

    where [itex]k[/itex] is a dummy index. What values is it summed over? If I had to guess I'd say 0 to 3, but it seems somewhat counter intuitive.

    Does the Christoffel symbol become

    [tex]{\Gamma ^{m}}_{ab}=\frac{1}{2}([g^{m0}(g_{a0,b}+g_{b0,a}-g_{ab,0})]+[g^{m1}(g_{a1,b}+g_{b1,a}-g_{ab,1})]+[g^{m2}(g_{a2,b}+g_{b2,a}-g_{ab,2})]+[g^{m3}(g_{a3,b}+g_{b3,a}-g_{ab,3})])[/tex]

    ?
     
  2. jcsd
  3. Aug 11, 2012 #2

    PeterDonis

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    Staff: Mentor

    You've got it right, at least for the "standard" case in relativity of a 4-D spacetime. The general rule is that summed indexes range over the full range of indexes in whatever manifold you are working with.
     
  4. Aug 11, 2012 #3
    Why is this counter intuitive? It simply follows from the einstein summnation notation.
     
  5. Aug 11, 2012 #4
    I understand the Einstein summation notation. I found it somewhat counter intuitive seeing as how it was a dummy index that didn't seem to "be" the 4 coordinates of space-time in any equation or tensor, etc. but it makes more sense the way PeterDonis said it... it's on a 4-dimensional manifold...
     
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