Can Snell's Law Be Derived by Passing to a Moving Frame?

[jk22]

Is the relativistic Snell's law : $$\frac {sin\theta_1}{sin\theta_2}=\frac {c_2}{c_1}\sqrt {\frac {c^2-c_2^2}{c^2-c_1^2}} $$ ? OR where could I check this ?

 

[ShayanJ]

Relativistic in what sense?

 

[PAllen]

Relativity would only come into play if media have relative motion along a refraction boundary. In that case, the following link describes the issues:

http://mathpages.com/rr/s2-08/2-08.htm

 

[jk22]

Relativistic in the sense of local time of the photon through the medium not motion of the medium.

 

[Khashishi]

If the medium is not moving, then it's just the standard Snell's law. Light is always relativistic, so the idea of a relativistic Snell's law is redundant.

 

[PAllen]

Relativistic in the sense of local time of the photon through the medium not motion of the medium.

There is no proper time or frame applicable to a photon (or light in general). See our FAQ linked below. Thus, this question is meaningless. The question of refraction across a refraction boundary where one one medium is moving relative to the other is, on the other hand, a very meaningful question for which relativity predicts significant modifications to Snell's law, as noted in my prior post.

https://www.physicsforums.com/threads/rest-frame-of-a-photon.511170/

 

[jk22]

Indeed my question was not very clear :

So it is impossible to find the refraction law for a particle (not forcedly a photon) moving at speed c1 then c2 by passing to the frame 1 and then 2 computing the minimum and then come back to the rest frame of the interface ?

Trying to do the latter I came across : the following formula $$\frac{tan(\theta_1)}{tan(\theta_2)}\sqrt{\frac{1+(1-c_2^2/c^2)tan(\theta_2)^2}{1+(1-c_1^2/c^2)tan(\theta_1)^2}}=\frac{c_1}{c_2}$$

Which does not give Snell's law back. I then thought that it is because it lacks a point like infinite acceleration at the interface. Hence doing this problem would need an accelerated frame with a continuous acceleration and then take the limit of the time of the acceleration to zero ? Then computing the minimal time in the moving frame would need the covariant derivative (concept of General relativity).

So I suppose finding Snell's law by passing to a moving frame is simply impossible due to a to high complexity.

 

[PAllen]

Indeed my question was not very clear :

So it is impossible to find the refraction law for a particle (not forcedly a photon) moving at speed c1 then c2 by passing to the frame 1 and then 2 computing the minimum and then come back to the rest frame of the interface ?

Trying to do the latter I came across : the following formula $$\frac{tan(\theta_1)}{tan(\theta_2)}\sqrt{\frac{1+(1-c_2^2/c^2)tan(\theta_2)^2}{1+(1-c_1^2/c^2)tan(\theta_1)^2}}=\frac{c_1}{c_2}$$

Which does not give Snell's law back. I then thought that it is because it lacks a point like infinite acceleration at the interface. Hence doing this problem would need an accelerated frame with a continuous acceleration and then take the limit of the time of the acceleration to zero ? Then computing the minimal time in the moving frame would need the covariant derivative (concept of General relativity).

So I suppose finding Snell's law by passing to a moving frame is simply impossible due to a to high complexity.

Look at the second derivation in the link I gave earlier (the whole link is well worth reading if you want to understand these issues). It basically does what you are looking for, and it is indeed complex. I am curious why you don't seem to have paid any attention to this link. Note also, as noted in the link, that index of refraction and speed in media are inversely related; thus the whole of the derivations and formulas can be recast in terms of media speed if you like that better than index of refraction.