# Is the speed of light in all media independent of reference frame?

Let's use the Lorentz transformation to calculate this. Let S be a reference frame, in which a coordinate system $$(t,x)$$ is used. Let S$$'$$ be a frame moving in the $$x$$ direction relative to S at speed $$v$$. The Lorentz transformation asserts that if event A is at $$(t,x)$$ then its coordinates in S$$'$$ are given by $$\begin{eqnarray} t' &=& \gamma(t - v x / c^2) \\ x' &=& \gamma(x - v t) \end{eqnarray}$$ where $$\gamma = (1-v^2/c^2)^{-1/2}$$. Now consider a pulse of light that starts from $$(0,0)$$ and propagates at the speed $$c/n$$ relative to S. For example, there could be some water at rest in S, and $$n$$ is the refractive index of this water. At time $$t$$ in S, such a pulse will have reached the event $$(t,x) = (t, ct/n)$$. The coordinates of this event relative to S$$'$$ are given by $$\begin{eqnarray} t' &=& \gamma(t - v (ct/n) / c^2) \\ x' &=& \gamma((ct/n) - v t) \end{eqnarray}$$ Now the coordinates of the starting point are $$(0,0)$$ in both frames, so we can find the speed of this pulse relative to S$$'$$ by using the distance traveled divided by the time elapsed: $$\mbox{speed relative to S}' = \frac{x' - 0}{t' - 0} = \frac{c/n - v}{1 - v (c/n) / c^2} = \frac{c/n - v}{1 - v / n c}.$$ If you are familiar with the formula for addition of velocities, you could find this same result by applying it. We now find that when $$n=1$$ the speed relative to S$$'$$ is equal to $$c$$, but when $$n \ne 1$$ the speed relative to S$$'$$ is not equal to $$c/n$$.

The above formula gives the speed that will be measured by your detector number 3, if we take it that the detector works in the usual way by measuring distances and times in its own rest frame. The result for detector number 2 will be $$\frac{c/n + v}{1 + v / n c}.$$

Another question that arises is the speed of the light relative to the water. That is just $$c/n$$. As soon as one says "relative to the water" then to calculate it you must use the rest frame of the water. End of story. But someone might ask instead, what is the rate of change of the distance between the light pulse and something floating in the water? If, relative to some given frame, the water is flowing at speed $$w$$ and the light is moving at speed $$u$$, then the answer to this question is $$u-w$$.

The media itself constitutes a preferred reference frame in which the speed of light is $$v=c/n$$. The speed of light in a reference frame that moves relative to the media will change anisotropically.

The formula for adding velocities is $$\frac {v_1 + v_2}{1+\frac{v_1v_2}{c^2}}$$. Plugging $$c$$ in for $$v_1$$ yields $$c$$ for any $$v_2$$. Plugging in a value of $$v_1$$ other than $$c$$ and a nonzero value for $$v_2$$ results in a value other than $$v_1$$.

In other words, anything observed to be travelling at $$c$$ in one reference frame will be observed to be travelling at $$c$$ in all reference frames. Anything observed to be travelling at a velocity other than $$c$$ in one reference frame will be observed to be travelling at other velocities (and by "other", I mean different from its observed velocity in the first reference frame, not just different from $$c$$) in other reference frames.

In particular, anything travelling at a velocity other than $$c$$ has a rest reference frame in which it will be observed to have zero velocity. If Detector 3 in your diagram is travelling at $$\frac c n$$, then the light wave will appear to be a standing wave.