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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 $s$?

Thank you for spending your precious time.

kaylimekay
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Swayam Jha
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1 Answers1

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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.

ProfRob
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