Invariant quantities in the EM field

[Mentz114]

I understand that the quantities

$$E^2 - B^2$$

$$\vec{E} \cdot \vec{B}$$

(the dot is vector inner product).
where E and B are the electric and magnetic components of an EM wave,
are invariant under Lorentz/Poincare transformations.
Can someone explain the physical significance of this ? Is either quantity related to the velocity of light ( or the invariance of the velocity of light ) ?

The second expression must be zero at all times surely ?

 

[dextercioby]

I understand that the quantities

$$E^2 - B^2$$

$$\vec{E} \cdot \vec{B}$$

(the dot is vector inner product).
where E and B are the electric and magnetic components of an EM wave,
are invariant under Lorentz/Poincare transformations.
Can someone explain the physical significance of this ? Is either quantity related to the velocity of light ( or the invariance of the velocity of light ) ?

The second expression must be zero at all times surely ?

Not necessarily wave. An EM wave is just a particular case of a radiated EM field. That's why the scalar product is not always 0, because the radiated EM field is not always a wave.

There's not too much physical significance of the invariants, just that the first one is good for a lagrangian density since it leads to field equations second order in time.

Daniel.

 

[Mentz114]

Thanks, Daniel.

I didn't know there are solutions to Maxwells equations other than the EM wave.

It's hard getting my head around the idea that the E and B fields 'mix' like space and time, when boosted.