Electromagnetic field invariants
= Electromagnetic field invariants
{title2=$F_{\mu\nu}F^{\mu\nu},\,F_{\mu\nu}\widetilde F^{\mu\nu}$}
{wiki}
For signature $(-,+,+,+)$ and a consistent orientation,
$$
F_{\mu\nu}F^{\mu\nu}
=2\left(B^2-\frac{E^2}{c^2}\right),
\qquad
F_{\mu\nu}\widetilde F^{\mu\nu}
=-\frac4c\mathbf E\cdot\mathbf B.
$$
Overall signs depend on tensor and orientation conventions, while their vanishing does not.