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Homework Statement
Prove that
[tex]\Gamma^\mu_{\mu\lambda}=\frac{1}{\sqrt{-g}}\partial_\lambda(\sqrt{-g})[/tex]
where g is the determinant of the metric, and [itex]\Gamma[/itex] are the Christoffel connection coefficients.
The Attempt at a Solution
From the general definition of the coefficients I got:
[tex]\Gamma^\mu_{\mu\lambda}=(1/2)g^{\mu\rho}\partial_\lambda g_{\rho\mu}[/tex]
But I have no idea how to work with the determinant of the metric. I'm not sure if I'm allowed to use this:
det(g)=exp[Tr ln G]
And if I did, would I have to use the GR definition of the trace?
[tex] Tr R = R^\mu_\mu [/tex]
I cleaned it up a little bit with the chain rule:
[tex]\frac{1}{\sqrt{-g}}\partial_\lambda(\sqrt{-g})[/tex]
[tex]=\frac{1}{2g}\partial_\lambda(g)[/tex]