Undergrad Sum Maxwell Lagrangian 1st Term: Use Minus Signs?

[Gene Naden]

So the first term of the Lagrangian is proportional to ${F_{\mu \nu}}{F^{\mu \nu}}$. I wrote out the matrices for ${F_{\mu \nu}}$ and ${F^{\mu \nu}}$ and multiplied at the terms together and added them up, but some of the terms didn't cancel like they should have. Should I have used minus signs for the fourth (or the first three, depending on the metric convention) components, like when you are raising and lowering indices?

 

[Gene Naden]

F has the electric and magnetic fields in it and ${F_{\mu \nu}}{F^{\mu \nu}}$ is supposed to come out like, depending on the units conventions $\frac{E^2}{c^2}-B^2$

 

[haushofer]

The product is not a mere matrix multiplication. You need to add F_00 F^00 + 2 F_i0 F^i0 + F_ij F^ij, where upper indices are obtained by raising with the Minkowski metric.

 

[Gene Naden]

(F is antisymmettric) So it is something like $2F_{0i}F^{i0} + F_{ij}F^{ij}$ where i runs from 1 to 3 and the metric is diag(-1,1,1,1)?

 

[Gene Naden]

Yes, that works out, but how does one derive the expression $F_{00}F^{00} + 2F_{0i}F^{i0} + F_{ij}F^{ij}$

 

[haushofer]

Yes, F is antisymmetric, but I included the 00 components to be complete ;)

I don't understand your last question; the lagrangian is proposed based on the fact that it is a Lorentz invariant expression containing first order derivatives of A. Written out it spells the sum.

 

[Gene Naden]

Thanks. My last question was not about how to derive the Lagrangian but rather about how to go from $F_{\mu \nu}F^{\mu \nu}$ to $F_{00}F^{00} + 2F_{0i}F^{i0} + F_{ij}F^{ij}$. But maybe that is obvious.

 

[haushofer]

I now see you switched the 0i indices on one of your F's. It should read F_i0 F^i0 or F_0i F^0i, differing a minus sign from your expression.

The way to obtain it is just perform the summation, splitting it into timelike and spacelike components.