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?

• – jacob1729 Sep 24 at 21:45

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.