# Electromagnetic Lorentz invariant apparent violation

We know that the scalar product $$\vec{E}\cdot \vec{B}$$

is a relativistic invariant in any inertial frame of reference. This made me pose the following question:

Suppose we have a charged filament with charge density $$\lambda$$ which moves with a certain constant velocity, producing an equivalent current $$I$$. Thus we have both an electric field $$\vec{E}$$ and a magnetic field $$\vec{B}$$, and the above invariant is non-zero.

Now, let's suppose that we apply a Lorentz transformation and position ourselves in an inertial frame such that the filament looks static, and thus only has a non-zero $$\vec{E}$$ but $$\vec{B}=0$$. This means that the above invariant is zero, which seems to contradict the Lorentz invariance.

Is there something wrong in my reasoning? What's happening here?

• What makes you think the dot product is non-zero? The fields might be perpendicular to each other. The B-field will definitely be perpendicular to the current, but without a diagram I don't know what direction the E-field is. – Bill N Jun 10 at 18:39
• I forgot to include that the fields before were perpendicular, otherwise it would be impossible to find a reference frame were one becomes zero (best case scenario, we could find a frame were they're parallel to each other). But you're right, I forgot that the scalar product is zero in any case, so it doesn't matter. – Charlie Jun 10 at 19:27
• @BillN if you want to elaborate in an answer I will give you the points. – Charlie Jun 10 at 19:32
• actually this result is used to show that if there is a frame where the dot product is $0$ then this product is $0$ in all frames so by the 2nd half of your question it must have been $0$ in the initial frame. – ZeroTheHero Jun 11 at 1:22