Tensor Manipulation: Is it Legal? | Homework Equations and Attempted Solution

[redstone]

Homework Statement

Wondering if one particular tensor manipulation is "legal"

Homework Equations

$$U^{ij}=\eta_{ab}F^{ia}F^{jb}$$
where F is the electromagnetic tensor, antisymmetric, with zeros on the diagonal

The Attempt at a Solution

So, starting there, is it legal for me to do this?
$$U^{ij}=F^{i}_{b}F^{jb}$$

$$U^{ij}=\eta^{b}_{i}\eta^{ai}F_{ab}F^{ji}$$

$$U^{ij}=\eta^{ab}F_{ab}F^{ji}$$

$$U^{ij}=0$$

since,

$$\eta^{ab}F_{ab}=0$$

that makes my answer seem to trivial to be the real answer, but I don't see the mistake

 

[turin]

... is it legal for me to do this?
...
$$U^{ij}=\eta^{b}_{i}\eta^{ai}F_{ab}F^{ji}$$

If you are using the standard notation, then no. You should never have more than two repeated indices (There are three "i"s).

 

[redstone]

ah, ok, thanks!