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 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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