Reason why $F^{\mu\nu}F_{\mu\nu}$ and $\tilde{F}_{\mu\nu}F^{\mu\nu}$ are Lorentz invariant

Question

I'm trying to think of an intuitive reasoning for why $F^{\mu\nu}F_{\mu\nu}$ and $\tilde{F}_{\mu\nu}F^{\mu\nu}$ are Lorentz invariant. By this I mean that I don't simply want to show that they remain unchanged after actually performing a Lorentz transformation and seeing that I end up with the same expressions, but some sort of 'deeper' understand of why this is so. I just can't really think of why these expressions (written out in vectors like $E^2 - B^2$ and $B \cdot E$ with some constants) would be the same for every inertial observer, while for a space-time interval I can sort of grasp this.

Is there perhaps a good reference someone could point me to?

Answer 1

They're lorentz scalars. Every scalar is lorentz invariant.

Answer 2

$F_{\mu\nu}$ is a Lorentz tensor, easy to see by $\partial_\mu A_\nu - \partial_\nu A_\mu$, which is a 2-form. Contractions of Lorentz tensors are Lorentz tensors. $\tilde{F} = \star F$ is the Hodge dual of $F$, which is also a 2-form, hence a Lorentz tensor, therefore the same applies about its contractions. By these definitions, they are also tensors in curved spacetime.