# Special Relativity, refractive index and catching up with a wave

Einstein was partially motivated by the following: With Maxwell's equations, a plane wave is a sinusoidal wave that varies in space in time and moving with speed $c$. These variations are linked by Maxwell's equations. What would happen if you could travel along with a plane wave at the speed c? You would observe fields that would be fixed in space and this would contradict Maxwell's equations. (See http://www.pitt.edu/~jdnorton/Goodies/Chasing_the_light/).

However, for the case of light moving in a medium with refractive index $n > 1$, the (phase) speed is $c/n < c$. In principle it is possible for an observer to travel at this speed. If they were to do so, would they note time independent fields?

• Is this not just a circular argument as a plane wave is a massive simplification which requires infinite space and time. Edit also the wave does not really "travel" slower in a microscopic level, refractive index is used to combine a hell of a lot of properties and really only explains macroscopic effects. – MJC Jun 13 '17 at 14:38

## 2 Answers

Not quite, although if an observer were to move at the group speed would see a time independent intensity distribution although there would still be time variations within the pulse. Maxwell's equations in a medium are not Lorentz covariant, simply because there is material medium. To understand this, imagine an extreme case where the the wave propagates through a stratified medium; the refractive indices will be time varying for an observer moving normal to the layers. Or, from another standpoint: imagine the medium as a lattice of atoms. As you move relative to it, the Lorentz Fitzgerald contraction means that you see a lattice with its period in the direction of motion shrunken, whereas the lattice periods normal to the motion are unchanged. Put simply, an isotropic material becomes anisotropic, as the material's optical density in the direction of motion is increased.

A peculiar aspect to this problem is that group velocities add according to the wonted relativistic 3-velocity addition law, but phase velocities do not; there is a more complicated transformation for the latter. If you could magically color the pulse so that different surfaces of constant phase were given their own different colors, you'd see a stationary pulse with colors moving though it. However, even if group and phase speeds are the same (nondispersive medium), if one could see microscopically, one could see charges shuttling backward and forth so that the magnetic fields are not the simple magnetostatic fields that we would get if we imagined the charges in the medium to be moving by. There are electric and magnetic fields somewhat like what one would see if one could see the fields in a resonant cavity.

These conclusions are fairly messy to reach, but the basic line of reasoning is that the wave's phase field must be a Lorentz scalar. From this postulate, one works as described in:

Kirk T. McDonald, "Index of Refraction of a Moving Medium"

to conclude the results above.

I think that the fields will be time-dependent according to an observer moving with the wave, because the phase of the wave is invariant, and so

$\omega(t-x/v)=\omega'(t'-x'/v')$

where $v=c/n$ is the phase velocity of the wave. (There is a chapter on the transformation of waves in Moller's book The Theory of Relativity).

Another way to come to the same conclusion is to note that the EM fields obey wave equations in both frames (according to the principles of relativity) and so the solution must be time-dependent in both frames. A third way to come to the same conclusion is to realize that even if the fields were static in the wave frame they would still have spatial variation, and hence a curl, so therefore E and B fields would be induced.

I think you've probably already worked this out and what you're really asking for is a physical reason why this is true (because it's so weird!). I've racked my brains and the only thing that helps is to consider the idea that the fields themselves are measurable only via the Lorentz force, and if the Lorentz force is wiggling a particle about in the lab frame (due to the time-dependence of the wave in the lab frame) you'd actually hope the particle would also be wiggling about (albeit with gamma corrections) in a frame moving with the wave...so at least the idea that a wave is always wiggling even if you catch up with it does not change the actual qualitative behavior of matter.