Saturday, 15 October 2016

homework and exercises - Variation of square root of determinant of metric, deltag



I am trying to calculate



ggμν,


where g=detgμν. We have


ggμν=12gggμν,

so the problem becomes how to calculate ggμν.


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 12gμνg



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+M1δM)det(M)[1+tr(M1δM)].

In the case of the metric, this implies that det(g+δg)det(g)[1+gabδgab]
and so δ(g)=(g)gabδgab.


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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