# Different speed of light in two inertial frames and the relativity principle

I'm in a frame in which a medium is at rest, and I observe light move at some speed.

Another person observers this medium move at some constant speed, in this case he'll observe a different speed for light.

But the principle of relativity states that the laws of physics have the same form in all inertial systems therefore Maxwells equations should have the same form in both the two frames, which implies that the speed of light should be the same in both frames.

Can anyone please point out my error. Thank you.

• What is this medium of which you speak? Commented Feb 28, 2021 at 17:17
• Special relativity postulates that the speed of light is the same regardless of observer or source motion, so "in this case he'll observe a different speed for light" is incorrect. Commented Feb 28, 2021 at 17:17
• The speed of light in matter is less than c. The index of refraction, n, is used to specify the speed in a medium: Quoted from kleppner and Kolenkow Commented Feb 28, 2021 at 17:29
• @WillO, say water. Speed in water will be v=c/n Commented Feb 28, 2021 at 17:34
• If that's what you mean by "the speed of light", nothing in relativity requires it to be frame-independent. Commented Feb 28, 2021 at 18:03

But the principle of relativity states that the laws of physics have the same form in all inertial systems therefore Maxwells equations should have the same form in both the two frames, which implies that the speed of light should be the same in both frames.

The index of refraction is not the same in different reference frames. In fact, in a reference frame where a transparent medium is moving the index of refraction is anisotropic. It has different values in the directions parallel and anti-parallel to the flow.

• That means Maxwells equations are different in the two frames? Commented Mar 1, 2021 at 4:14
• No, Maxwell’s equations are the same, as is the law for determining the index of refraction of a moving transparent medium.
– Dale
Commented Mar 1, 2021 at 11:49
• So for both the frames the Maxwells equations are the same in the medium. Then how does speed of light vary in the two frames if the speed of light is derived from these equations. If you say the index of refraction changes then the two inertial frames are not equivalent as one of them is a preferred one. Commented Mar 1, 2021 at 16:09
• @Kashmiri see the paper I linked to for details about how the speed of light in the medium varies. It is not correct that one frame is preferred over another. The equation in the linked paper gives the transformation for the index of refraction. This transformation applies to any frame. The same rule gives the index of refraction for any transparent medium in any reference frame.
– Dale
Commented Mar 1, 2021 at 16:19

The principle of relativity, in the form of Einstein's second postulate, says that the speed of light in vacuum is the same in all inertial frames. If you introduce a medium, glass or water or even air, with a refractive index, then you're not in vacuum. The medium has a different motion in different frames, and the basic symmetry is lost.

You might like to look up Fizeau's experiment.

• The principle of relativity does not entail that "the speed of light in vacuum is the same in all inertial frames". Einstein's 1905 two postulates are independent of one another. Commented Feb 28, 2021 at 23:12
• Agreed, as mentioned in my question, what about the Maxwells equations that should remain invarient? Commented Mar 1, 2021 at 4:08
• @PentchoValev we agree really. Whether the postulates are independent is arguable (i.e. I am prepared to argue about it.) I was trying to engage with the language of the question (which has since been edited). The principle of relativity, Einstein's 1st postulate, says the laws of physics are the same in all inertial frames. If you take 'The speed of light is $c$' as one of the laws of physics then the second postulate follows from the first. But for pedagogical purposes it's good to have it made as an explicit statement. Commented Mar 1, 2021 at 11:44
• Maxwell's equations in vacuum are the same. When you introduce the medium and its polarisable molecules, the motion makes it a lot more complicated than the simple $\epsilon_0 \to \epsilon_r \epsilon_0$ substitution we normally deal with. Commented Mar 1, 2021 at 11:48
• Yes the medium is there but it shouldn't matter if the frame is moving or not with respect to the material. Commented Mar 1, 2021 at 16:07

Your error is in the idea that because you are still and can see someone moving, everything that they see is slower. That's not quite the case. Weirdly, the light will move away from the two of you at the same rate. While your friend is moving, the world (from their perspective), is squashed; This conserves the speed of light.