Question about computing Christoffel symbols

Question

I am trying to calculate the Christoffel symbols in polar coordinates, and I am confused on one step. Given that I am here, for example:

$$\Gamma_{r \theta}^{\theta}=\frac{1}{2} g^{\alpha \theta}\left(g_{\alpha r, \theta}+g_{\alpha \theta, r}-g_{r \theta, \alpha}\right)$$

How do I know which of $r$ or $\theta$ to plug into $\alpha$? I know that $g^{r \theta} = 0$ and $g^{\theta \theta} = 1/r^2$, and I know the equation ends up being

$$\Gamma_{r \theta}^{\theta}=\frac{1}{2 r^{2}}\left(g_{\theta r, \theta}+g_{\theta \theta, r}-g_{r \theta, \theta}\right)$$

but why was $\theta$ chosen in place of $\alpha$? It seems that both $\theta$ and $r$ are equally arbitrary choices here since $\Gamma_{r \theta}^{\theta}$ does not contain an $\alpha$ index. How do I know which coordinate to place in this spot generally speaking?

Answer 1

The index $\alpha$ is actually a dummy index. You are using the Einstein summation convention, this means you need to expand the terms, for example $$g^{\alpha \theta} g_{\alpha r ,r} = g^{r \theta} g_{r r,r} + g^{\theta \theta} g_{\theta r,r}.$$ If you do this for all three terms and plug in the metric you should get the right result.