- #1

Mayan Fung

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- TL;DR Summary
- What is the speed of light in a medium if the medium is also moving at a relativistic speed?

I once naively think that the speed of light is also a constant in a medium in all inertial frames which is not the case. I tried to derive the result yet there is a discrepancy from the results I read in some articles.

For example, from

$$u = \frac{c}{n}(\frac{1+\frac{nv}{c}}{1+\frac{v}{nc}})$$

He derived it with the full set of Maxwell's equations which I think there is another easier way to the solution.

Here's my approach:

Let's say we have a piece of glass with width ##d## and refractive index ##n## moving at a relativistic speed v.

First, we calculate the time, ##t## the light travels through the entire glass in the glass frame. In this frame, the glass is stationary and the speed of light in the medium is ##c/n##. Therefore,

$$t = \frac{d}{c/n} = \frac{n}{c}d$$

The time measured in the lab frame, ##t'##, considering the time dilation effect, is:

$$t'=\gamma t, \gamma = \frac{1}{\sqrt{1-v^2/c^2}}$$

Also, the width of the piece of glass, considering length contraction, is:

$$d' = d/\gamma $$

Therefore, in the lab frame, the light in the glass travels a distance of ##vt' + d/\gamma## to reach the other side of the glass. So, the speed of light in the medium, ##u##, can be found by:

$$\begin{align*}

ut' &= vt' + d/\gamma\\

u &= v + d/\gamma t'\\

u &= v + d/\gamma^2 t\\

u &= v + \frac{c}{n \gamma^2}\\

u &= \frac{c}{n} (1-v^2/c^2) +v

\end{align*}$$

This is different from the results listed in the beginning. I wonder where I made a mistake.

For example, from

**[Link to unpublished paper redacted by the Mentors]**, the author derived the speed of light in a medium of refractive index ##n## moving at a relativistic speed v is$$u = \frac{c}{n}(\frac{1+\frac{nv}{c}}{1+\frac{v}{nc}})$$

He derived it with the full set of Maxwell's equations which I think there is another easier way to the solution.

Here's my approach:

Let's say we have a piece of glass with width ##d## and refractive index ##n## moving at a relativistic speed v.

First, we calculate the time, ##t## the light travels through the entire glass in the glass frame. In this frame, the glass is stationary and the speed of light in the medium is ##c/n##. Therefore,

$$t = \frac{d}{c/n} = \frac{n}{c}d$$

The time measured in the lab frame, ##t'##, considering the time dilation effect, is:

$$t'=\gamma t, \gamma = \frac{1}{\sqrt{1-v^2/c^2}}$$

Also, the width of the piece of glass, considering length contraction, is:

$$d' = d/\gamma $$

Therefore, in the lab frame, the light in the glass travels a distance of ##vt' + d/\gamma## to reach the other side of the glass. So, the speed of light in the medium, ##u##, can be found by:

$$\begin{align*}

ut' &= vt' + d/\gamma\\

u &= v + d/\gamma t'\\

u &= v + d/\gamma^2 t\\

u &= v + \frac{c}{n \gamma^2}\\

u &= \frac{c}{n} (1-v^2/c^2) +v

\end{align*}$$

This is different from the results listed in the beginning. I wonder where I made a mistake.

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