- #1

bdforbes

- 152

- 0

## 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]