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?
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?
via Physics Forums RSS Feed http://www.physicsforums.com/showthread.php?t=707860&goto=newpost
0 commentaires:
Enregistrer un commentaire