# Confusion on repeated index for Einstein Summation

The rule for Einstein notation is that the same dummy index cannot be repeated twice. However suppose I want to compute Christoeffel symbols: $$\Gamma^{\alpha}_{\beta\gamma} = \frac{1}{2}g^{\alpha\sigma}(\partial_\beta g_{\gamma\sigma}+\partial_{\gamma}g_{\sigma\beta}-\partial_{\sigma}g_{\beta\gamma})$$

Now if my metric is diagonal, then only the terms $$\alpha = \sigma$$ survive, hence we have: $$\Gamma^{\alpha}_{\beta\gamma} = \frac{1}{2}g^{\alpha\alpha}(\partial_\beta g_{\gamma\alpha}+\partial_{\gamma}g_{\alpha\beta}-\partial_{\alpha}g_{\beta\gamma})$$

Of course now the problem is that the index $$\alpha$$ is repeated three times. However, it makes perfect sense to me when I do the computation. Is there some exception to the "not repeated twice" rule?

• If the metric is diagonal and constant, the correct way to write it is $g^{\alpha\beta} = \delta^{\alpha\beta}$... Commented Nov 8, 2020 at 15:52
• @ValterMoretti I was thinking of something more like the Schwarzschild metric Commented Nov 8, 2020 at 15:54

• also $\alpha=\sigma=\lambda=\beta$ so the equation becomes:$\Gamma^{\alpha}_{\alpha\alpha} = \frac{1}{2}g^{\alpha\alpha}(\partial_\alpha g_{\alpha\alpha}+\partial_{\alpha}g_{\alpha\alpha}-\partial_{\alpha}g_{\alpha\alpha})$
• So should it be: $$\Gamma^{\alpha}_{\beta\gamma} = \frac{1}{2}\sum_{\alpha}g^{\alpha\alpha}(\partial_{\beta}g_{\gamma\alpha} + \partial_{\gamma}g_{\alpha\beta} + \partial_{\alpha}g_{\beta\gamma})$$ ? Commented Nov 8, 2020 at 15:42