Maxwells Equations in a curvilinear coordinate system

Question

I was wondering if anyone would be able to explain to me how the author goes from equation 3 to equations 4(a,b,c) in this paper.

I am confused how to treat the contravariant metric tensor in these equations.

Thanks!

Answer 1

I don't think you "need" physics here. Rather, physics "needs" math. I think the only assumption is TE polarization. That is, that the electric field is transverse to the motion, which mathematically means the tensor $E_j$ is simply $(0,0,E_z)$. I write the equations here for convenience.

$$e^{ijk}\partial_jH_k=-ik_0\mu\sqrt gg^{ij}E_j.$$

You were wondering about equation 4a, so I will try to show how that is found.

First note that $$g^{ij}E_j=g^{iv}E_v+g^{iu}E_u+g^{iz}E_z=0+0+g^{iz}E_z=g^{iz}E_z.$$ This is a contravariant 1-tensor (in the super script $i$).

So, both sides of the equation are 1-tensors in $i$. Let us consider the $i=z$ component. On the left side, we consider

$e^{zjk}\partial_jH_k=\partial_vH_u-\partial_uH_v$, and we are done. That is, equation 4a is

$$\partial_vH_u-\partial_uH_v=-ik_0\mu E_z.$$

I would encourage you to try to make sure you can derive equation 4a yourself. This will require you to develop some more comfort with the notation. Then try tackling the other two equations. Note that for instance on the left side of the Maxwell's equations, we are holding $i$ fixed and summing over all $j$ and $k$. This is the Einstein notation. See here for instance: http://en.wikipedia.org/wiki/Einstein_summation

Note that another component of the above equations, say for $i=v$, we have

$$e^{vuz}\partial_uH_z+e^{vzu}\partial_zH_u=\partial_uH_z-\partial_zH_u=0.$$