Relativistic invaraiance of a simple electrodynamic quantity

[ak99]

Homework Statement

Under what conditions is the quantity $$E^2+B^2$$ relativistically invariant?

Homework Equations

$$E\cdot B$$ and $$E^2-c^2B^2$$ are invariant.

(Also Lorentz transformation laws for E and B which I won't type here.)

The Attempt at a Solution

I think you can just subtract multiples $$E^2-c^2B^2$$ from the given quantity to show that $$E^2$$ and $$B^2$$ must also be separately invariant if the given quantity is invariant, which I think happens only if E and B are both zero. But I have been told this is wrong by someone.

 

[ak99]

(Technical point: You may set c=1 in the above expressions.)

 

[pam]

I don't think there are any such conditions, except if each is zero.

 

[olgranpappy]

Homework Statement

Under what conditions is the quantity $$E^2+B^2$$ relativistically invariant?

Homework Equations

$$E\cdot B$$ and $$E^2-c^2B^2$$ are invariant.

(Also Lorentz transformation laws for E and B which I won't type here.)

The Attempt at a Solution

I think you can just subtract multiples $$E^2-c^2B^2$$ from the given quantity to show that $$E^2$$ and $$B^2$$ must also be separately invariant if the given quantity is invariant, which I think happens only if E and B are both zero. But I have been told this is wrong by someone.

E^2 + B^2 is not invariant in general since it is the energy-density of the field and thus transforms as the 00 component of a rank 2 tensor... but... how about rotations?

 

[pam]

E^2 + B^2 is not invariant in general since it is the energy-density of the field and thus transforms as the 00 component of a rank 2 tensor... but... how about rotations?

It is invariant under rotations.

 

[olgranpappy]

true.