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I'm in a frame in which a medium is at rest, and I observe light move at some speed. Now this medium moves at some constant speed, in this case I'll observe a different speed for light.

We can find the speed of light using Maxwell's equations in both the cases by applying those equations in our frame.

I'm curious about what changes in the equations that leads to the change in speed of light in the two cases. The equations being :

$$\begin{cases}\nabla \cdot \mathbf{E}=\frac{1}{\epsilon_{0}} \rho \\ \nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \cdot \mathbf{B}=0 \\ \nabla \times \mathbf{B}=\mu_{0} \mathbf{J}+\mu_{0} \epsilon_{0} \frac{\partial \mathbf{E}}{\partial t}\end{cases}$$

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What changes between frames in the case of a linear medium is not so much Maxwell's equations as the constitutive relations. The displacement field $\mathbf{D}$ and the auxiliary field $\mathbf{H}$ are defined in the rest frame of the medium as $\mathbf{D} = \epsilon \mathbf{E}$ and $\mathbf{H} = \mu^{-1} \mathbf{B}$, where $\mathbf{E}$ and $\mathbf{B}$ are the fields in the rest frame. But in a different rest frame, we will see different, more complicated relationships between $\mathbf{D}', \mathbf{H}', \mathbf{E}'$, and $\mathbf{B}'$. IIRC, in general the speed $\mathbf{u}$ of the medium in the new frame will enter into these constitutive relations; and $\mathbf{D}'$ will generally depend on both $\mathbf{E}'$ and $\mathbf{B}'$, as will $\mathbf{H}'$. The net effect will be to yield a much more complicated set of equations in which wave solutions travel at different speeds in different directions.

It is worth noting that in both the rest frame of the medium and the frame in which the medium is moving, the "vacuum form" of Maxwell's equations (given in the OP) should still hold. However, when viewed this way, the changing polarization and magnetization of the medium gives rise to a time-dependent $\rho$ and $\mathbf{J}$ (the bound charges and the bound currents) in the rest frame of the medium. These bound charges and bound currents would transform when we viewed the wave in a frame in which the medium is moving, in such a way that the new fields and sources $\mathbf{E}'$, $\mathbf{B}'$, $\rho'$, $\mathbf{J}'$ would also satisfy Maxwell's equation in "vacuum form".

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  • $\begingroup$ I believe those are equations valid everywhere not only in vacuum as you said. $\endgroup$
    – Kashmiri
    Commented Feb 27, 2021 at 14:57
  • $\begingroup$ You write $\epsilon$ and $\mu$ as scalars. Pedantically, they are components of a fourth rank Lorentz tensor with the same symmetry as the Riemann tensor. It in general has many components which depend on frequency and wave vector direction. As the medium is unspecified and considered in two different reference frames this may be relevant. $\endgroup$
    – my2cts
    Commented Feb 27, 2021 at 15:04
  • $\begingroup$ @my2cts: I was trying to write an answer without introducing spacetime tensors, since the OP didn't refer to them. But I agree that the tensor approach is the "right" way to think about it. Feel free to write an answer that takes that approach and I'll happily upvote it. :-) $\endgroup$
    – Michael Seifert
    Commented Feb 27, 2021 at 15:23
  • $\begingroup$ @YasirSadiq: True enough. See my edit which addresses this point. $\endgroup$
    – Michael Seifert
    Commented Feb 27, 2021 at 15:31
  • $\begingroup$ @YasirSadiq Correct. However for a medium people tend to use an effective medium approach where the bound charge-current is absorbed in the effective field. $\endgroup$
    – my2cts
    Commented Feb 27, 2021 at 15:49

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