1
$\begingroup$

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?

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.