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Derivative of metric and log identity

  1. Feb 29, 2008 #1
    Has anyone seen this identity:

    [tex]g^{ab}\nabla g_{ab}=\nabla ln|g|[/tex]

    I've seen it used, but want to figure out where it comes from.

    Does anyone know a name or have any ideas??
     
  2. jcsd
  3. Feb 29, 2008 #2
    You will have to show the exact formula. The way it's written, it doesnt make sense - covariant derivative of the metric is zero in GR because the connection is chosen 'metric compatible'.
     
  4. Mar 1, 2008 #3
    ok, thanks, I'll check the paper when I get back to my office tomorrow and post.

    Rich
     
  5. Mar 1, 2008 #4

    George Jones

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    I think you mean

    [tex]g^{\alpha \beta} \partial_\mu g_{\alpha \beta} = \partial_\mu \ln \left|g\right|.[/tex]

    See pages 12-13 of Poisson.
     
  6. Mar 1, 2008 #5
    Thanks Greorge. I Don't have that book but I'll see if I can find someone with it. And thanks for pointing out the correction.
     
  7. Mar 1, 2008 #6

    George Jones

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    Lots of have books probably have this, but I won't be able to tell you any others until Monday.

    Maybe Carroll.
     
  8. Mar 1, 2008 #7

    samalkhaiat

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  9. Mar 1, 2008 #8
    I'm not seeing it Sam.

    [tex]\partial_\mu ln|g^{ab}|=\partial_\mu ln[(det.g^{ab})]=\partial_\mu Tr[ln g^{ab} ] =
    \partial_\mu (ln g^{00}+lng^{11}+...)

    [/tex]
     
  10. Mar 1, 2008 #9

    samalkhaiat

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  11. Mar 1, 2008 #10

    robphy

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  12. Mar 1, 2008 #11
     
  13. Mar 1, 2008 #12
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