Refraction vs Special Relativity

Question

Special relativity postulates that the speed of light (while traveling through vaccum) is a constant ($c = 3 \times 10^8 m/s$). But, when light is refracted and change the medium in which it's traveling, its speed changes. The question is:

If the new medium has $n=2$, for example, then the speed of light in this medium is $v=1.5 \times 10^8 m/s$. Does this new speed become a limit to the speed of objects in this medium? And the Lorentz transformations still unchanged or they'll use this new speed too?

Answer 1

Does this new speed become a limit to the speed of objects in this medium?

Not in the same sense that the speed of light is a limit in vacuum. There is a well-known phenomenon which is caused by a charged object moving through a medium faster than the speed of light in that medium: Cherenkov radiation.

Now, this radiation emits energy, so a charged particle moving faster through a medium with a speed greater than $c/n$ will rapidly lose energy until its speed falls below this threshold. In this sense, $c/n$ is a kind of limit for charged particles moving through the medium; one wouldn't expect to find charged particles traveling long distances at a speed higher than this. But it's not a fundamental limit the way $c$ is a limit in vacuum; neutral particles could conceivably travel at speeds greater than $c/n$.

And [are] the Lorentz transformations still unchanged or they'll use this new speed too?

The index of refraction of a medium can be thought of as arising from the interaction between an external EM wave and the skillions of tiny dipoles inside the the medium. Importantly, the assumption that's usually made in this derivation is that the dipoles are at rest. This means that the index of refraction is implicitly defined in a particular reference frame. We therefore shouldn't expect Lorentz symmetry and Lorentz transformations to transfer over to a medium in the "naïve" way, where we just replace $c$ with $c/n$ everywhere and call it a day; in a medium, we wouldn't expect there to be symmetry between inertial reference frames.

(Actually working out how Lorentz symmetry extends to linear media is a complicated business. See this answer from a related question for more information on the subject.)

Answer 2

Massless particles like photons move in vacuum with the maximum speed allowed in the hybrrboluc geometry of spacetime. Here geometry is primary while the fact that light moves with this speed is a consequence. We can clearly see this distinction in a refractive medium where light moves slower, but the spacetime geometry is essentially unchanged.

To answer your question, in $n=2$, light slows down twice, but the maximum speed remains unchanged and the Lorentz transformation remains unchanged as well. For example, if neutrinos travel at $0.99c$ in vacuum, they would pass through this medium still at $0.99c$.

The asymptotically local maximum speed in the local frame of the observer, in spacetime of any smooth curvature, is always 1 light-second per second. Based on this constant we define 1 meter as exactly 1/299,792,456 part of 1 lught-second.