I am trying to calculate
∂√−g∂gμν,
where g=detgμν. We have
∂√−g∂gμν=−12√−g∂g∂gμν, so the problem becomes how to calculate ∂g∂gμν.
I have used the identity Tr(lnM)=ln(detM) to obtain, applying it with M=gμν and varying it:
δ(Tr(ln(gμν)))=δgg
but then I am stuck. How can I go on? I know the result should be −12gμν√−g
Answer
Use the identity that if M is invertible and δM is "small" compared to M, then we have det In the case of the metric, this implies that -\det(g + \delta g) \approx -\det(g) \left[ 1 + g^{ab} \delta g_{ab} \right] and so \delta (-g) = (-g) g^{ab} \delta g_{ab}.
To complete the calculation you'll then have to relate \delta g^{ab} to \delta g_{ab}, but this should get you on your way. If this isn't a homework problem or the like, let me know and I can expand on this latter part.
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