Vtyjkyuk

I don't quite get the result that I should get.

We have $$Gamma^barmu_baralphabarbeta=frac12g^barmubarnuleft(partial_baralphag_barbetabarnu+partial_barbetag_baralphabarnu-partial_barnug_baralphabarbetaright).$$

Furthermore we find
$$partial_baralphag_barbetabarnu=g_epsilontauleft( x^epsilon_, barbeta baralpha x^tau_,barnu+x^epsilon_, barbeta x^tau_,barnubaralpharight)+x^epsilon_,barbetax^tau_,barnux^lambda_,barbetapartial_lambda g_epsilontau.$$

Here, I've written $x^tau_barnu=partial_barnu x^tau.$

Substituting the partial derivatives of the metric in the Christoffel symbol I get two terms. One is
$$partial_lambda g_epsilontau left(x^epsilon_, barbetax^tau_, barnu x^lambda_,baralpha+x^epsilon_, baralphax^tau_, barnu x^lambda_,barbeta-x^epsilon_, baralphax^tau_, barbeta x^lambda_,barnuright). $$
By renaming the indices I obtain
$$x^epsilon_, barbetax^tau_, barnu x^lambda_,baralphaleft(partial_lambda g_epsilontau+partial_epsilon g_lambdatau-partial_tau g_lambdaepsilonright) .$$

Now, for the second term I get $2g_epsilontaux^epsilon_,barbetabaralphax^tau_,barnu$ after using the interchangeability if the second derivatives, the symmetry of the metric tensor and switching dummy variables in one term.

So, at the end I have $$Gamma^barmu_baralphabarbeta=g^barmubarnu g_epsilontaux^epsilon_,barbetabaralphax^tau_,barnu+frac12g^barmubarnux^epsilon_, barbetax^tau_, barnu x^lambda_,baralphaleft(partial_lambda g_epsilontau+partial_epsilon g_lambdatau-partial_tau g_lambdaepsilonright).$$

However, this deviates what I read in most forums and books. In the first and second term the co- and contravariant metrics should share one dummy variable. Does anyone have a clue?

Starting with $$Gamma^barmu_baralphabarbeta=frac12g^barmubarnuleft(partial_baralphag_barbetabarnu+partial_barbetag_baralphabarnu-partial_barnug_baralphabarbetaright),$$

we have $$partial_baralphag_barbetabarnu=g_epsilontauleft( x^epsilon_, barbeta baralpha x^tau_,barnu+x^epsilon_, barbeta x^tau_,barnubaralpharight)+x^epsilon_,barbetax^tau_,barnux^lambda_,baralphapartial_lambda g_epsilontau,$$

$$partial_barbetag_baralphabarnu=g_epsilontauleft( x^epsilon_, baralpha barbeta x^tau_,barnu+x^epsilon_, baralpha x^tau_,barnubarbetaright)+x^epsilon_,baralphax^tau_,barnux^lambda_,barbetapartial_lambda g_epsilontau,$$

$$partial_barnug_baralphabarbeta=g_epsilontauleft( x^epsilon_, baralpha barnu x^tau_,barbeta+x^epsilon_, baralpha x^tau_,barbetabarnuright)+x^epsilon_,baralphax^tau_,barbetax^lambda_,barnupartial_lambda g_epsilontau.$$

Substitution gives us $$Gamma^barmu_baralphabarbeta
=frac12g^barmubarnug_epsilontauleft(x^epsilon_, barbeta baralpha x^tau_,barnu
+x^epsilon_, barbeta x^tau_,barnubaralpha
+x^epsilon_, baralpha barbeta x^tau_,barnu
+x^epsilon_, baralpha x^tau_,barnubarbeta
-x^epsilon_, baralpha barnu x^tau_,barbeta
-x^epsilon_, baralpha x^tau_,barbetabarnuright)$$
$$+frac12g^barmubarnupartial_lambda g_epsilontauleft(x^epsilon_,barbetax^tau_,barnux^lambda_,baralpha
+x^epsilon_,baralphax^tau_,barnux^lambda_,barbeta
-x^epsilon_,baralphax^tau_,barbetax^lambda_,barnuright).$$

Using the interchangeability of the second derivatives, the first term becomes:
$$frac12g^barmubarnug_epsilontauleft(2x^epsilon_, barbeta baralpha x^tau_,barnu+x^epsilon_, barbeta x^tau_,barnubaralpha-x^epsilon_, baralpha barnu x^tau_,barbetaright)$$
$$=g^barmubarnug_epsilontaux^epsilon_, barbeta baralpha x^tau_,barnu+frac12g^barmubarnug_epsilontaux^epsilon_, barbeta x^tau_,barnubaralpha-frac12g^barmubarnug_epsilontaux^epsilon_, baralpha barnu x^tau_,barbeta.$$

Switching the indices $epsilon,tau$ in the last term cancels both last terms out because of the metric's symmetry. We are then left with: $$g^barmubarnug_epsilontaux^epsilon_, barbeta baralpha x^tau_,barnu
=g^munux^barmu_,mux^barnu_,nug_epsilontaux^epsilon_, barbeta baralpha x^tau_,barnu
=g^munudelta^tau_nu x^barmu_,mug_epsilontaux^epsilon_, barbeta baralpha
=g^mutaug_epsilontaux^barmu_,mux^epsilon_, barbeta baralpha
=delta^mu_epsilonx^barmu_,mux^epsilon_, barbeta baralpha
=x^barmu_,mux^mu_, barbeta baralpha.$$

Now we have obtained the following: $$Gamma^barmu_baralphabarbeta
=x^barmu_,mux^mu_, barbeta baralpha
+frac12g^barmubarnupartial_lambda g_epsilontauleft(x^epsilon_,barbetax^tau_,barnux^lambda_,baralpha
+x^epsilon_,baralphax^tau_,barnux^lambda_,barbeta
-x^epsilon_,baralphax^tau_,barbetax^lambda_,barnuright).$$

Switching the indices $epsilonleftrightarrow lambda$ in the second term of the remaining bracket and $epsilonrightarrowlambdarightarrowtaurightarrowepsilon$ in the third term, we find: $$frac12g^barmubarnupartial_lambda g_epsilontauleft(x^epsilon_,barbetax^tau_,barnux^lambda_,baralpha
+x^epsilon_,baralphax^tau_,barnux^lambda_,barbeta
-x^epsilon_,baralphax^tau_,barbetax^lambda_,barnuright)$$
$$=frac12g^barmubarnux^epsilon_,barbetax^tau_,barnux^lambda_,baralpha
left(partial_lambda g_epsilontau
+partial_epsilon g_lambdatau
-partial_tau g_epsilonlambdaright).$$

As before, we now introduce the barless indices: $$g^munux^barmu_,mux^barnu_,nux^epsilon_,barbetax^tau_,barnux^lambda_,baralpha
=frac12g^munudelta^tau_nu x^barmu_,mu x^epsilon_,barbetax^lambda_,baralpha
=x^barmu_,mu x^epsilon_,barbetax^lambda_,baralpha
cdotfrac12g^mutau.$$

At last, we are left with: $$x^barmu_,mu x^epsilon_,barbetax^lambda_,baralpha
cdotfrac12g^mutauleft(partial_lambda g_epsilontau
+partial_epsilon g_lambdatau
-partial_tau g_epsilonlambdaright)
=x^barmu_,mu x^epsilon_,barbetax^lambda_,baralpha
cdotGamma^mu_lambdaepsilon.$$

Now we have obtained the following: $$Gamma^barmu_baralphabarbeta
=x^barmu_,mux^mu_, baralphabarbeta
+x^barmu_,mux^lambda_,baralphax^epsilon_,barbeta
cdotGamma^mu_lambdaepsilon.$$

This is the sought result.