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?

  • $\begingroup$ 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. $\endgroup$ – Bill N Jun 10 at 18:39
  • $\begingroup$ 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. $\endgroup$ – Charlie Jun 10 at 19:27
  • $\begingroup$ @BillN if you want to elaborate in an answer I will give you the points. $\endgroup$ – Charlie Jun 10 at 19:32
  • $\begingroup$ 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. $\endgroup$ – ZeroTheHero Jun 11 at 1:22

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