Electromagnetic Lagrangian, EoM, Polarisation States

[binbagsss]

Homework Statement

Attached:

Homework Equations

Euler-Lagrange equations to find the EoM

The Attempt at a Solution

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Solution attached:

I follow, up to where the sum over $\mu$ reduces to sum over $\mu=i$ only, why are there no $\mu=0$ terms? I don't understand at all.

Many thanks

 

[TSny]

According to equation (9), what can you say about $F_{00}$? What does this tell you about $F^{00}$?

 

[binbagsss]

According to equation (9), what can you say about $F_{00}$? What does this tell you about $F^{00}$?

oh for a antisymmetric tensor all diagonal entries must be zero? $F_{ii}=0$ as is $F_{00}$?

 

[TSny]

oh for a antisymmetric tensor all diagonal entries must be zero? $F_{ii}=0$ as is $F_{00}$?

Yes. The expression for $F_{\mu \nu}$ in (9) shows that $F$ is antisymmetric. In particular, if you let $\mu = 0$ and $\nu = 0$ in the expression for $F_{\mu \nu}$ in (9) then you see that $F_{00} = 0$.