I would like to have your opinion about the following fact.
Is seems to me that the fact that a 2-form in Minkowski spacetime cannot be either self-dual of anti-self-dual implies that the electric and the magnetic field cannot be the same (i.e. equal magnitude, equal direction) in the same point in space
Does this look obvious to you, or do you want me to explain myself?
I think you'd need to explain that a little bit. Is it something you've concluded from considering duality of the Faraday 2-form and how this interchanges the electric and magnetic fields?
I'm of course supposing that the correct way to formalize the electromagnetic field in Minkowski spacetime is through the Faraday 2-form.
"The correct way" meaning the most fundamental one: define the electromagnetic quadripotential as a four-vector field, transform it into a 1-form, then exterior-derive it.
Hm, it does not seem obvious to me that this should be true. Is it something you're only considering for vacuum (so that the dual 2-form and the Faraday 2-form both obey the same differential equations)?
It doesn't depend upon the dynamical equations. It is a straightforward fact about 2-forms in Minkowski spacetime. I show you the calculations (they really are two equations).
Let us say that we want the electric field $\vec{E}$ to be equal to the magnetic field $\vec{B}$ in some point of space.
Gosh, don't equations work in the chatroom?
Do you know whether there is a way of writing equations in here?
No, I don't think so. Consider F = E_1 dx^0 \wedge dx^1 - B_1 dx^2 \wedge dx^3. Then the dual is E_1 dx^2 \wedge dx^3 + B_1 dx^0 \wedge dx^1. There's a sign change on one of the components, but not both.
(It's precisely for that reason that there are no self-dual or anti-self-dual 2-forms.)
Yes. I was asking myself whether mine was an error of calculation or a bad definition. The dual operator is defined in the same way in Minkowski spacetime as in a Riemanniang manifold, isn't it?
