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Diagonal connection problem

  1. Sep 22, 2007 #1
    1. The problem statement, all variables and given/known data
    I am trying to show that the connection [tex] \Gamma^a_{bc} [/tex] is equal to 0 when the metric g_ab is diagonal. Will the formula
    [tex] \Gamma^a_{bc} = 1/2 g^{ad}(\partial_bg_{dc} + \partial_cg_{bd} - \partial_dg_{bc}) [/tex] be of use? How can I manipulate that equation and use the fact that the metric is diagonal?


    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Sep 22, 2007
  2. jcsd
  3. Sep 22, 2007 #2

    Dick

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    Why do you think the connection is zero if the metric is diagonal? It's not even true. Look at spherical coordinates.
     
  4. Sep 22, 2007 #3
    It comes right out of my book!
     

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  5. Sep 23, 2007 #4

    Hurkyl

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    No it doesn't; your book doesn't ask you to prove all of the components are zero: just some of them.
     
  6. Sep 23, 2007 #5
    OK. So it only wants me to prove that the off-diagonal components are zero (that statement in parentheses was pretty important). Anyway, I still have the same two questions as in the first post.
     
  7. Sep 23, 2007 #6

    dextercioby

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    If [itex] \Gamma [/itex] is meant to be the Riemann - Christoffel (or the metric) connection, then, yes, that's the formula you should use.
     
  8. Sep 23, 2007 #7
    Yes. It is the Christoffel symbol of the second kind. So am allowed to do this:

    [tex] \Gamma^a_{bc} = 1/2 (\partial_bg^{ad}g_{dc} + \partial_cg^{ad}g_{bd} - \partial_dg^{ad}g_{bc}) [/tex]
    ?

    Then the first term becomes the Kronecker delta, I think.

    If not, how should I use the fact that the metric is diagonal?
     
  9. Sep 23, 2007 #8

    Hurkyl

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    Why do you think, for all d, that gad is a constant with respect to the b, c, and d-th variables?



    Any way you can imagine. You could substitute zero for the off-diagonal terms. You could decompose the metric into a linear combination of simpler tensors. Et cetera.
     
    Last edited: Sep 23, 2007
  10. Sep 23, 2007 #9
    I see. You just replace d with a.
     
  11. Sep 23, 2007 #10
    Now, can someone explain to me where in the world this natural log comes from at the bottom of attached page of the Hobson book?
     

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  12. Sep 23, 2007 #11

    Hurkyl

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    Did you try differentiating it?
     
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