Calculating the invariant E^αβ E_αβ with the metric

[Nusc]

Homework Statement

Calculate the invariant
$$E^{\alpha \beta} E_{\alpha \beta}$$

Homework Equations

The Attempt at a Solution

we apply the metric in this case,
$$E^{\alpha \beta} E_{\alpha \beta} = g_{\alpha n} g_{\beta m} E_{n m} E^{n m}$$

is that even correct?

 

[gabbagabbahey]

we apply the metric in this case,
$$E^{\alpha \beta} E_{\alpha \beta} = g_{\alpha n} g_{\beta m} E_{n m} E^{n m}$$

is that even correct?

No, that equation is non-sensical...on the lefthandside $\alpha$ and $\beta$ are dummy indices (they are being summed over)...on the righthandside they are free indices and you also have $m$ and $n$ appearing 3 times (which is notational nonsense).


I assume you are given either $E^{\alpha \beta}$ or $E_{\alpha \beta}$?