I am trying to calculate
∂√−g∂gμν,
where g=detgμν. We have
∂√−g∂gμν=−12√−g∂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(M+δM)=det(M)det(1+M−1δM)≈det(M)[1+tr(M−1δM)].
To complete the calculation you'll then have to relate δgab to δgab, 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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