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As we know the speed of light in vacuum is constant for everyone (at least for all inertial frames), i.e., if we run away or toward a beam of light (in vacuum) the speed would be $c$. It doesn't change. But imagine a beam of light traveling in a denser medium (where the speed of light decreases), say, water. Let the speed of light in water be $s$. Suppose a fish is swimming away from that beam of light at velocity $V$. For the fish will the speed of light will be $s-V$ or $s$?

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    $\begingroup$ Possibly helpful: physics.stackexchange.com/questions/155880 and physics.stackexchange.com/questions/197619 $\endgroup$
    – kaylimekay
    Jan 11, 2021 at 6:29
  • $\begingroup$ You mean, C-V or C, not S? If you are moving relative to a bulk of material, and speed of light is less than it is in vacuum, relativistic addition formula will give you different speeds for different moving observers. So, no, speed of light is not te same for all observers in some medium. $\endgroup$
    – Žarko Tomičić
    Jan 11, 2021 at 6:44
  • $\begingroup$ But does the relativistic addition formula hold if you replace c with s? $\endgroup$
    – R. Emery
    Jan 11, 2021 at 7:07
  • $\begingroup$ I don't think that the Einstein's relativistic formula hold if we replace c with s....... $\endgroup$
    – Swayam Jha
    Jan 11, 2021 at 7:35
  • $\begingroup$ Does it hold inside that particular medium? $\endgroup$
    – R. Emery
    Jan 11, 2021 at 8:18

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If we assume linear media, then it is easy enough to transform the E- and D-fields and show that permittivity is not a relativistic invariant.

Thus the refractive index of a medium is not relativistically invariant and observers in different inertial frames measure a different speed of light in a medium.

For example, if the wave motion is parallel to the velocity difference $v$ between the two frames S and S', then $$n' = \frac{n + v/c}{1 + nv/c} .$$ (Shen 2004).

However, note that you can no longer define a single refractive index in S' and it will depend on the direction in which the waves are travelling.

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