# Is classical Snell's law with a moving medium solvable?

• I
Considering classically a light ray in a medium with lightspeed ##c_1## and entering a medium with lightspeed ##c_2## moving with speed ##v## along let say ##y## in the plane of the interface, is it correct to obtain a very complicated formula, having even 3rd power of trigonometric functions of the refracted angles ?

I looked on Google but it's 2D and uses a Lorentz boost wheteas I would consider it classically.

Cryo
Gold Member
I think you will have to use Lorentz boosts. You need this to account for material response. Here is why. In a simplified picture, light induces magnetization ##\vec{M}##, and polarization ##\vec{P}## in the medium it propagates. We usually observe it in a low-energy limit, where the material response is linear so one can introduce electric and magnetic susceptibilities ##\chi_e, \chi_m## and link the polarization/magnetization to electric ##\vec{E}## and magnetic ##\vec{H}##, as follows: ##\vec{P}=\chi_e \vec{E}##, ##\vec{M}=\chi_m \vec{H}##. This then trickles down into your refractive index ##n=\sqrt{\epsilon \mu}=\sqrt{(1+\chi_e)(1+\chi_m)}##.

Now lets return to what, for example, polarization is. It is a volume density of induced electric dipoles. But electric dipole only appears as such if it is stationary. If you have a pair of separated opposite charges (i.e. electric dipole) moving relative to you, you will observe it as electric & magnetic dipoles. This is due to Lorentz boosts. This also applies to polarization and magnetization, but even more so since they are densities (so there are additionnal transformations). Basically if you see a material with polarization moving, it will appear to have both polarization and magnetization. This will affect the susceptibilities and, in turn, the refractive index etc.

So, I think, you have to use Lorentz boosts.

This complicate my problem, I would like to see it as a particle moving with those speeds, and find the quickest path. But in the second medium the classical velocities addition ##\vec{v_2}+\vec{v}## were used instead of the relativistic one.

Ibix
2020 Award
If it's a particle, why is it refracting? It needs to be a wave and needs to be interacting with the medium.

The usual way to do non-relativistic approximation is to Taylor expand the relativistic expression in ##v## and see under what circumstances (if any) you can neglect higher order terms. The last time you asked a similar question PAllen linked to a mathpages page that gives the exact relativistic version of Snell's Law - expand it and see if it matches what you got.

I'll just note that Fizeau's experiment was initially interpreted as supporting partial ether dragging because naive Galilean velocity addition did not match his results. So you may find that the domain of validity of your approximation is extremely narrow.

Cryo
Indeed. So the problem statement should be rather put as a vehicle with different speeds, at a classical level.

Cryo
Gold Member
at a classical level

You keep saying 'classical'='not relativistic', but there is no non-relativistic theory of electromagnetism. You can do full relativistic treatment and then take the limit of low velocity, sure, but you still have to go through relativity first. Maxwell's equations is what is commonly understood to be classical electromagnetism, and they are not invariant under Gallilei transforms, so there is no sane way to do electromagnetism 'non-relativistically'.

Ibix
2020 Award
If you are not considering light, then an example like a lifeguard trying to reach a person in trouble in water is a valid application for this, I think. In that case I presume you can do it in the same way as the usual still water example - fix the end points with respect to their media, parameterise possible paths by the point the lifeguard enters the water, and extremise.

I would expect this to be the low v, extremely high n (in both media), limit of the relativistic formula.

Andy Resnick