Is This Invariant Calculation Correct?

[Nusc]

Homework Statement

Calculate the invariant
<br /> E^{\alpha \beta} E_{\alpha \beta}<br />

Homework Equations

The Attempt at a Solution

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

is that even correct?

 

[gabbagabbahey]

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

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}?