I know that Maxwell's equations of electromagnetism are Lorentz invariant in a vacuum. But what about in a generalized medium, e.g. a metal, a rubber, a dielectric, a magnet? I have read it comes down to whether the electric and magnetic polarizations, $M$ and $P$, are themselves Lorentz invariant. (Note: I am ignoring gravity.) My feeling is that they must be, albeit some researchers might use approximations that are not. So can anyone answer my question:
Are Maxwell's equations in a medium Lorentz invariant?
Best Answer
Maxwell is not Lorentz invariant in matter because the material selects a preferred reference frame — the one in which the lump of matter is at rest.
Of course you can make it all look Lorentz covariant by including the local four velocity $u^\mu$ of the piece of matter in the equations for the dielectric constant and the magnetic permeability, and by defining $$ E_\mu = F_{\mu\nu}u^{\nu}, \quad B_\mu = \frac 12 \epsilon_{\mu\nu\sigma\tau} u^\nu F^{\sigma\tau}. $$ to be the ${\bf E}$ and ${\bf B}$ fields in the frame moving with the matter — but that extra $u^\mu$ makes everything rather complicated. It's best to avoid all this unless you really want to to do relativistic fluid/continuum mechanics such as investigating the magnetic field on a neutron star or the accretion disc of a black hole.