clj4 said:
Sure, if you do your calculations correctly. This is what you were asked all along. And, if you want to be thorough , find the error in Kirshner as well. It is only two papers, that's all.
Only two papers. I will hold you to that.
Okay, I spent time on the Gagnon paper. The error is fairly obvious, but I wasted hours brushing up on this and that in order to feel confident enough to state it here.
The error is this: electrodynamics cannot be formulated with Maxwell's equations alone. It is Maxwell's equations (how the fields interact and are produced by the sources) _AND_ Lorentz's force law (how the fields act back on the sources). Gagnon used the transformed Maxwell's equations from reference 9 without using the transformed Lorentz force law. Reference 9 doesn't calculate it, so they must have (incorrectly) assumed that it retained the same form. This is incorrect.
Lorentz force law:
K^\mu = q n_\nu F^{\mu \nu}
where K is the Minkowski force, q the charge, n the proper velocity, F the field tensor.
Reference 9 chose the components of the covarient field tensor to define the electric and magnetic fields (instead of the contravarient field tensor, which is why the two source dependent Maxwell's equations come out horrid while the non-source dependent ones come out fairly clean). So we need to rewrite the equation to depend on that, as well as depend on the contravarient proper velocity (corresponds to the physical velocity as opposed to the covarient proper velocity).
K^\mu = q (g_{\nu a} n^a) (g^{\mu b}F_{b c}g^{c\nu})
Rearranging and noting that g^{c\nu} g_{\nu a} = \delta^c_{\ a} we have:
K^\mu = q n^a g^{\mu b}F_{b a}
Let's move to another frame and see how the dependence of the force on the fields and the velocity changes. (I'll use a bar to denote quantities in this other frame.)
Of course we still have \bar{K}^\mu = q \bar{n}^a \bar{g}^{\mu b}\bar{F}_{b a} but this will correspond to the same dependence on the velocity and fields ONLY if g^{\mu b}=\bar{g}^{\mu b}. In special relativity, the metric is frame independent, so the force law maintains the same form (as expected). However, this is not true for GGT. In GGT the metric is frame dependent and thus the Lorentz force law changes form when we change frames (the metric is worked out in reference 9, so you can calculate the horrid form of the Lorentz force law using GGT if you so wish).
How does this affect Gagnon's paper? It means the boundary conditions they invoke when solving for the fields in the wave-guide are not correct. So their calculations are flawed right at the beginning.
EDIT: changed kronecker delta symbol for clarity
