# Question with Einstein Convention

Quick question, lets say I have the following term

$$a_{\mu}b^{\mu}c_{\nu}d^{\nu}\tag{1}$$

I have repeated indices over $a$ and $b$ so their components are summed over, and I have repeated indices over $c$ and $d$ so their components are summed over. However, since the indices are repeated, I may relabel them:

$$a_{\mu}b^{\mu}c_{\mu}d^{\mu} \tag{2}$$

This suggests that I can now sum the components of $b$ and $c$ and likewise with $a$ and $d$, etc. however I don't think this is valid and I just wanted to confirm. More specifically, as a general rule you should keep repeated indices unique to other repeated indices so confusions such as this don't occur. Can anyone confirm?

• Think about this analogy: in the double integral $\iint xy\ dx\ dy$, $x$ and $y$ are dummy variables, so you may relabel them. Does this mean you can change $y$ to $x$? Feb 13, 2017 at 8:48

Yes, your conclusion is correct. The initial expression evaluates to just $(a \cdot b) (c \cdot d)$ while the rewriting of it to $a_{\mu}b^{\mu} c_{\mu} d^{\mu}$ is then meaningless (In Einstein summation convention you have an upper and lower repeated index and indices to be summed over come in pairs).