christoffel symbols

Qyzren

http://en.wikipedia.org/wiki/Christo...ols#Definition

start with [tex]0=\frac{\partial g_{ik}}{\partial x^l}-g_{mk}\Gamma^m_{il}-g_{im}\Gamma^m_{kl}[/tex]

in wiki it said "By permuting the indices, and resumming, one can solve explicitly for the Christoffel symbols as a function of the metric tensor:"
[tex]\Gamma^i_{kl}=\frac{1}{2}g^{im}(\frac{\partial g_{mk}}{\partial x^l}+\frac{\partial g_{ml}}{\partial x^k}-\frac{\partial g_{kl}}{\partial x^m})[/tex]

but i don't see how they did this step. Can someone please show me?
Thanks you

latentcorpse

well [latex]g_{mk} \Gamma^m_{il}=\Gamma_{kil}[/latex] so you get:

[latex]0=\frac{\partial{g_ik}}{\partial{x^l}}-\Gamma_{kil}-\Gamma_{ikl}[/latex]

i think that wwhat they mean by permuting the indices is just that you can relabel them to create two other equations:
[latex]0=\frac{\partial{g_lk}}{\partial{x^i}}-\Gamma_{kli}-\Gamma_{kil}[/latex]
and
[latex]0=\frac{\partial{g_il}}{\partial{x^k}}-\Gamma_{lki}-\Gamma_{kli}[/latex]

now you know the Christoffel symbol [latex]\Gamma^a_{bc}[/latex] is symmetric in its' lower indices. If you add the first two of those three equations and subtract the last one you get:

[latex]2 \Gamma_{kil}=\frac{\partial{g_ik}}{\partial{x^l}}+\frac{\partial{g_lk}} {\partial{x^i}}-\frac{\partial{g_il}}{\partial{x^k}}[/latex]

and now just bring the 2 across and pull a [latex]g^{im}[/latex] out the front.

this isn't quite right as ive rushed it through... the ideas are right though so try doing it yourself with all the working and seeing where i went wrong.

Qyzren

I do not understand this
"well [tex]g_{mk} \Gamma^m_{il}=\Gamma_{kil}[/tex]"
specifically, how you can have a christoffel symbol with 3 lower indices, how does that work?

should the 2nd equation not be
[tex] 0=\frac{\partial{g_il}}{\partial{x^k}}-\Gamma_{lik}-\Gamma_{ilk} [/tex]
instead of
[tex] 0=\frac{\partial{g_il}}{\partial{x^k}}-\Gamma_{lki}-\Gamma_{kli} [/tex]

and finally how do you pull out a [tex]g^{im} [/tex] factor at the end?

latentcorpse

do you know how the metric [latex]g^{ab}[/latex] operates?

It acts as follows:

[latex]g^{\mu \nu} X_{\nu} = X^{\mu}[/latex] for a vector [latex]X_{\mu}[/latex]
and
[latex]g_{\mu \nu} X^{\nu}=X_{\mu}[/latex] for a [latex]X^{\mu} \in \Lambda^1[/latex] i.e. in the space of one forms.

and so we can manipulate the Christoffel symbol to have as many up and down indices as we like simply by acting on it with a series of metrics.

i pulled out the [latex]g^{im}[/latex] at the end simply so it would be in the same form as the one you quoted in your first post. remember however that when you pull out that factor it will alter the up and down indices accordingly.

hope this helps a bit.

Qyzren

Thanks,
I just did not know it was possible to raise or lower indices for the christoffel symbol.

I've figured it out now though