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Some Formulas in Zee's GR Text

  1. Aug 30, 2013 #1
    It seems like Zee lost a trace in his new GR text, but I am sure it is me confusing things.

    First, he establishes:

    [itex]
    log\: det\: M = tr\: log\: M[/itex]

    Then, differentiating:

    [itex]
    (det\: M)^{-1} \: \partial (det\: M)=\partial (tr\: log\: M)=tr(\partial\: log\: M)=tr(M^{-1}\partial M)[/itex]

    Then he applies to the metric, giving:

    [itex]
    \frac{1}{\sqrt{-g}}\partial _{\nu }\sqrt{-g}=\frac{1}{2g}\partial _{\nu }g=\frac{1}{2}\partial _{\nu }log\: g[/itex]

    where g is the determinant of the metric.

    Sure seems like he lost a trace on the very last term. Where I am going wrong here?
     
  2. jcsd
  3. Aug 30, 2013 #2
    In Zee's equation, ##g## means the determinant of the metric tensor. In more explicit notation, the steps are

    ##\frac{1}{\sqrt{-\det g}} \partial_\nu \sqrt{-\det g} = \frac{1}{2 \det g} \partial_\nu (\det g) = \frac{1}{2} \partial_\nu ({\rm tr} \log g) = \frac{1}{2} \partial_\nu (\log \det g)##

    In what I've written here, ##g## is always a matrix and the determinant is kept explicit.
     
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