Putting Maxwell's Equations in Tensor Form. (Carroll Chapter 1 Question 11)

Question

Simply put, if you look at https://en.wikipedia.org/wiki/Electromagnetic_tensor#Significance it says you can go from the traditional four "vector calculus" maxwell equations to two tensor Maxwell equations. Now I was able to convert the first two into the equation $\partial_{\alpha} F^{\alpha\beta} = \mu_0 J^{\beta}$. However, I am struggling to get $\partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha } = 0$ from the last two equations. I am wondering if it easier to work from the more terse antisymmetrization form or if there is some other trick. I guess I'm just concerned that I don't know how to relate the idea of antisymmetrization to the levi-civita tensor which you need to somehow get out of this because in the last two equations you have the curl of $E$ rather than of $B$ which makes getting $F$ out much more difficult because $B$ in terms of $F$ involves the levi-civita symbol and $E$ does not.

Answer 1

[To fix notation: I'll use Greek spacetime indices $\alpha,\beta,\ldots$ running over $0,1,2,3$, and Latin spatial indices $i,j,k$ running over $1,2,3$.]

There are four equations in $\partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha } = 0$, depending on which index is missing. A nice way to rewrite it would be $$ \epsilon^{\alpha\beta\gamma\delta}\partial_\beta F_{\gamma\delta}=0$$ which makes the four equations very clear, depending on how you choose $\alpha$. Here $\epsilon^{\alpha\beta\gamma\delta}$ is the completely antisymmetric tensor, with $\epsilon^{0123}=1$. This is helpful, because once you know where the 0 index is (there must be exactly one to make it nonzero), the rest can be written in terms of the three-dimensional antisymmetric symbol (careful with signs!). For example, $\epsilon^{ij0k}=+\epsilon_{ijk}$. Pick whether $\alpha$ is 0 or spatial, and then, if you need to, split the sums over spacetime indices into (term with 0) + (spatial terms).