Speed of Light Logic: Flaw Explained

[Silviu]

Hello! My question is: If you travel in a material at the same speed as the speed of light (I assumed this is possible due to the existence of Cherenkov radiation) you will see the light basically stationary so you see a constant electric field which means no magnetic field. But light has both so this is impossible. What is the flaw in my logic? Thank you!

 

[Dale]

you will see the light basically stationary so you see a constant electric field which means no magnetic field.

This doesn't sound right. Can you show your work?

 

[Silviu]

This doesn't sound right. Can you show your work?

You move at the same speed as the light. Let's assume that you see the electric field to have a value $E_0$. In your frame that value will not change, as the speed of the electric field, is the same speed as the light (obviously). You can imagine you "sit" on the electric field on one of the crests - maximum amplitude - so as you move at the same speed as the electric field, you conclude that there is no oscillations of the electric field (you don't feel like going up and down on the wave). So if you see no variation of the electric field and laws of physics must hold in your frame of reference, there is no source for a magnetic field. Is there something wrong with this?

 

[PeterDonis]

What is the flaw in my logic?

You're ignoring the medium. If the light is traveling in a medium, then the light by itself does not satisfy Maxwell's Equations (which is what you are using to deduce that there is no magnetic field if the electric field is stationary). Only the total system, light plus medium, satisfies Maxwell's Equations. (And in general, dividing things up into "light" plus "medium" is arbitrary in such a case.)

 

[vanhees71]

Well, yes you have in-medium electrodynamics, and it can be formulated in a covariant way (already Minkowski did this). The usual in-medium edynamics you learn in the EM lecture is linear-response theory close to equilibrium of the matter. It's very clear, how the fields $(\vec{E},\vec{B})$ (or $F_{\mu \nu}$ in the four-vector formalism) transform under Lorentz transformations, and indeed they transform as the microscopic fields do. This already tells you that the Lorentz boosted fields of the usual plane-wave solutions in the rest-frame of the medium cannot be a static field, but are still plena-wave solutions.

 

[Dale]

Is there something wrong with this?

Yes, you didn't show your work, just some hand waving.

Showing your work would be to take a plane wave solution in a medium, boost both the field and the medium, and show that the result behaves as described.

This is along the lines of what @vanhees71 suggests above, and I think his description of the result sounds correct.