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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?

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

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The Einstein summation rule is true for tensor-equations. Once you assume a form for the metric (diagonality), the equation you get is no longer a true tensor-equation (it is only true in some coordinate-systems). This is the reason why you need to write the summation by-hand from this point on.

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  • $\begingroup$ 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})$ $\endgroup$
    – JEB
    Commented Nov 8, 2020 at 15:29
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    $\begingroup$ So should it be: $$ \Gamma^{\alpha}_{\beta\gamma} = \frac{1}{2}\sum_{\alpha}g^{\alpha\alpha}(\partial_{\beta}g_{\gamma\alpha} + \partial_{\gamma}g_{\alpha\beta} + \partial_{\alpha}g_{\beta\gamma}) $$ ? $\endgroup$
    – D. Soul
    Commented Nov 8, 2020 at 15:42

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