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Ok, simple question about index notation. If I have this:-

$$ \delta^\mu_\eta (\partial_\mu g_{\eta\nu}) $$

Where $\delta^\mu_\eta$ is the Kronecker delta, does this become:-

$$ \partial_\eta g_{\mu\nu} $$

Or can I just switch one of the indicies and get this:-

$$ \partial_\mu g_{\mu\nu} $$

I'm guessing they both can't be correct because in the first all indexes are free and the second $\mu$ is a dummy index.

rgvcorley
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  • If you're using Einstein notation, then you are summing over $\mu$, so you get $\partial_\eta g_{\eta\nu}$. Note that you wrote $\partial_\eta g_{\mu\nu}$, but I can't see why you turned $g_{\eta\nu}$ into $g_{\mu\nu}$. – A.P. Apr 20 '13 at 09:26
  • Ok, but are we not also summing over $\eta$? Why doesn't $\delta^\mu_\eta g_{\eta\nu} = g_{\mu\nu}$? – rgvcorley Apr 20 '13 at 15:04
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    You're not; in Einstein summation, you sum over indices that are repeated in the sense that they appear both as an upper index and as a lower index. Hence, in your expression, you sum over $\mu$, which appears as an upper index in $\delta_\eta^\mu$ and as a lower index in $\partial_\mu g_{\eta\nu}$, but not over $\eta$, which only appears as a lower index. – Branimir Ćaćić Apr 20 '13 at 23:55

1 Answers1

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Even though the Kronecker delta is sometimes treated like an index substitution symbol, it actually defined as $$ \delta_j^k = \begin{cases} 1, & j=k\\ 0, & j\neq k \end{cases} $$ Now consider what happens when summing over one of those indices $$ \sum_\mu \delta^\mu_\eta (\partial_\mu g_{\eta\nu})=\partial_\eta g_{\eta\nu} $$ since this is the only non-zero summand. If we sum over the other index, then $$ \sum_\eta\sum_\mu \delta^\mu_\eta (\partial_\mu g_{\eta\nu})=\sum_\eta \partial_\eta g_{\eta\nu}=\partial_1 g_{1\nu}+\partial_2 g_{2\nu}+\dotsb $$ which is the same result you would get for summing over $\eta$ and then over $\mu$. Observe that in both cases this is different from $\partial_\eta g_{\mu\nu}$.

Finally, as I alluded to in my comment and as Branimir Ćaćić made explicit, in Einstein notation you sum only over those indices that appear both as an upper and a lower index. The other indices are considered constant throughout the summation.

Daniel Fischer
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A.P.
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