Tensor help -- Write out this tensor in a simplified sum

[user1139]

Homework Statement

Write out $$F_{\alpha\beta}F^{\alpha\gamma}$$ in a simplified sum where $$F$$ is the stress tensor and Einstein summation convention is implied.

Relevant Equations

$$F_{\mu\nu}$$ is the usual stress tensor

I managed to write

$$F_{\alpha\beta}F^{\alpha\gamma}=F_{0\beta}F^{0\gamma}+F_{i\beta}F^{i\gamma}$$

where $$i=1,2,3$$ and $$\gamma=0,1,2,3=\beta$$.

How do I proceed?

 

[jedishrfu]

It's like two "for" loops in programming:

0) You have four terms when summing $\alpha$

1) Start with $\gamma$ and expand to four equations for 0,1,2, and 3 with $\beta$ still there.

Repeating with the same expansion with $\beta$, you should now have 16 equations for $F _{\beta}^{\gamma}$.