# Relativistic effects in non-vacuous media

1. Feb 2, 2005

### freemind

Hello,

I've been pondering the implications of SR in media other than pure vacuum (yes, yes I know such a thing doesn't exist :). More specifically relating to the following thought experiment. If we have a prism of length $$L$$ in the path of a light source, with a wall @ the other end (the prism is btwn the source and the wall), and if the prism is stationary, then the light will take a certain amount of time $$t_s$$ to traverse the distance to the wall (relative to an external observer for whom the source and wall are stationary). Obviously, this time will depend on how much time the light spends inside the prism. Now consider the prism to be moving @ a speed $$v_0$$ toward the wall. The time taken by the light to traverse the same distance shall be $$t_1$$ (again, with respect to the aforementioned FoR). Now, my question is whether $$t_1 < t_0$$ or if $$t_0 < t_1$$. I'm thinking $$t_1 > t_0$$ because the light spends more time in the moving media, hence, the media is effectively "lengthened" because of its speed ("lengthened" only for $$v << c$$ of course; I'm well aware of prism length contraction @ relativistic speeds). Is my reasoning flawed?

Thanks

2. Feb 3, 2005

### pervect

Staff Emeritus
A more direct approach is this

We know that the speed of light in a media with an index of refraction n is c/n

We know that relativistic velocities add by the relativistic velocity addition formula.

v = v1+v2 / (1 + v1 v2 / c^2)

Therfore, we can calculate the speed of light in a moving media

When the media is moving in the same direction as the light, we get

vtot = (v + c/n) / (1 + v/(c*n))

When the media is moving in the opposite direction we get

vtot = (v - c/n) / (1 - v/(c*n))

I haven't worked out the times from the velocities to answer the original question, though.

3. Feb 5, 2005

### freemind

After spending an hour trying to figure out the hyperphysics derivation of the velocity addition formula (hey, it ain't exactly intuitive ya know ), and spending 5 min making the connection to your direct approach, I finally understand it. Thank you.

4. Feb 5, 2005

### pervect

Staff Emeritus
Glad you figured it out - on rereading it my post, I see that my explanation was defintely on the terse side, but it sounds like you got the idea. When you know the speed of light in one frame (the rest frame of the media), you can figure it out in all frames via some very standard formulas.