How Does the Affine Connection Contract in General Relativity?

[coqui82]

Hi everyone!

I have some problems with indices in general relativity. I am now working with the classic textbook by S. Weinberg and in eq. (4.7.4) we find

http://latex.codecogs.com/gif.latex...partial g_{\rho \mu }}{\partial x^{\lambda }}

The question is: where does the last equality come from?
I think that it could come from the comparison between this expression and the same one interchanging μ and ρ. In so doing you would get the same expression except for the last two partial derivatives that would change their sign. Now if you consider (I am not sure if this is right) that http://latex.codecogs.com/gif.latex?\Gamma^{\mu}_{\mu \lambda }=\Gamma ^{\rho }_{\rho \lambda } then it comes straightforwardly that http://latex.codecogs.com/gif.latex...partial g_{\mu \lambda }}{\partial x^{\rho }}
Thanks in advance!

 

[Bill_K]

(You can embed LaTeX code directly in your post by wrapping it with TEX or ITEX.)

Yes, you're correct, the reason the last two terms drop out is that they are antisymmetric in μ and ρ, and we're multiplying by g^μρ which is symmetric.

 

[WannabeNewton]

Does Weinberg actually call the Christoffel symbols an affine connection ?

 

[coqui82]

Thanks a lot Bill! Much more clear now!
And yes, Weinberg uses both terms, although affine connection is a more general one.

 

[George Jones]

Does Weinberg actually call the Christoffel symbols an affine connection ?

What is wrong with this?