- #1

- 359

- 4

## Main Question or Discussion Point

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?