- #1
binbagsss
- 1,259
- 11
I'm trying to prove that ##\sqrt{-g}\bigtriangledown_{\mu}v^{\mu}=\partial_{\mu}(\sqrt{-g}v^{\mu}) ##
So i have ##\sqrt{-g}\bigtriangledown_{\mu}v^{\mu}=\sqrt{-g}(\partial_{\mu}v^{mu}+\Gamma^{\mu}_{\mu \alpha}v^{\alpha}) ## by just expanding out the definition of the covariant derivative.
Question
My text next makes the equality :
##\sqrt{-g}(\partial_{\mu}v^{mu}+\Gamma^{\mu}_{\mu \alpha}v^{\alpha}=\sqrt{-g}(\partial_{\mu}v^{mu}+(\partial_{\alpha} ln \sqrt{-g})v^{\alpha}\sqrt{-g}=\partial_{\mu}(v^{\mu}\sqrt{-g}##
I don't understand the last 2 equalities.
Particularity the second to last. I have no idea how you go from the connection term to the ##ln## term, if anyone could provide the identities I need or link me somewhere useful (had a google but couldn't find anything)
I think the last equality has used the product rule.
I'm unsure, if this is correct, of differentiating the ##\sqrt{-g}## to go from the last equality to the second from last, the identities I know that might be of use are:
##Tr (ln M) = In (det M)##
##\partial \sqrt{-g}=\frac{-1}{2}\sqrt{-g}g_{\mu \nu}g^{\nu \mu} ##
Thanks in advance.
So i have ##\sqrt{-g}\bigtriangledown_{\mu}v^{\mu}=\sqrt{-g}(\partial_{\mu}v^{mu}+\Gamma^{\mu}_{\mu \alpha}v^{\alpha}) ## by just expanding out the definition of the covariant derivative.
Question
My text next makes the equality :
##\sqrt{-g}(\partial_{\mu}v^{mu}+\Gamma^{\mu}_{\mu \alpha}v^{\alpha}=\sqrt{-g}(\partial_{\mu}v^{mu}+(\partial_{\alpha} ln \sqrt{-g})v^{\alpha}\sqrt{-g}=\partial_{\mu}(v^{\mu}\sqrt{-g}##
I don't understand the last 2 equalities.
Particularity the second to last. I have no idea how you go from the connection term to the ##ln## term, if anyone could provide the identities I need or link me somewhere useful (had a google but couldn't find anything)
I think the last equality has used the product rule.
I'm unsure, if this is correct, of differentiating the ##\sqrt{-g}## to go from the last equality to the second from last, the identities I know that might be of use are:
##Tr (ln M) = In (det M)##
##\partial \sqrt{-g}=\frac{-1}{2}\sqrt{-g}g_{\mu \nu}g^{\nu \mu} ##
Thanks in advance.