E.H is a Lorentz invariant, when is it different from 0 ?

lalbatros

From the Lorentz-invariant Faraday tensor F(H,E) two scalar invariants can be constructed:

Inv1 = H²-E²

and

Inv2 = E.H

Thinking at waves, electrostatic fields, magnetostatic fields, I see examples where Inv2 = 0.
It is however easy to arrange an electrostatic field and a magnetostatic field to be parallel, so to get an exemple where Inv2 =/= 0. But this situation is not very interresting, maybe because it is made of static and independent sources.

I would be interrested to know if there are less trivial and more interresting examples of fields where Inv2 =/= 0 ,
if there is a general physical meaning to this condition,
and generally what is the physical importance of this invariant.

Thanks for your comments,

Michel