# Can speed of light be $c$ in air or other medium?

I know that the speed of light in a vacuum is $c\sim 3\ 10^8\ \mathrm{m/s}$, but I also know that speed of light in a medium (e.g., air) is less than that in vacuum.

Special relativity says that speed of light is same to all observers in any frame of reference. So does this also follow in a medium?

The wavelenght $\lambda$ and frequency $f$ of a wave are related through $$v=\lambda f$$ where $v$ is the velocity of the wave. Shouldn't this mean that different frequencies and wavelengths of light will have different speeds NOT equal to $c$.

How does all this fit with special relativity?

Note: My question is NOT how to move faster than light. I want to know that does special relativity apply between media/mediums.

Note 2: Pay attention to my second question too (about $\lambda$ and $v$ relation)

• Different wavelength in a medium have different speeds – Anubhav Goel Feb 12 '16 at 14:58
• – Carl Witthoft Feb 12 '16 at 15:04
• OP: I edited your question to 1) make it a bit more readable (by using mathjax, click on the edit button to see how I did this) and 2) to remove the "photons" part, which IMHO is pretty irrelevant for your question. If you still think the photons part is important, you can reverse my edit. – AccidentalFourierTransform Feb 12 '16 at 15:32

Light always travels at a (local) velocity of $c$, but light in a medium is not just light, and that's why its velocity can be lower than $c$.

Light is an oscillating electromagnetic field, and when it passes though anything that contains charged particles (i.e. any matter made from electrons and protons) the electric field of the light interacts with those charges. When the light interacts with the charges we have to describe the light/matter system by a new wavefunction that includes all the interacting components. This means the light is not longer purely light - we have a quantum system that mixes up the light with the charged particles. This mixing produces a quasiparticle called a polariton that has a non-zero mass particle so it moves at less than the speed of light.

The frequency of the light can't change, so the reduction in velocity means that the wavelength is decreased.

Finally, in general the interaction of the light with the medium is frequency dependent so the velocity of the light in the medium is frequency dependent. This produces the optical phenomenon called dispersion.

• polariton is an emergent phenomenon suitable for some computations. The OP asks on photon and the principle of the value of c which is not postulated in the billard of a medium – user46925 Feb 12 '16 at 16:49
• @igael: In practice you would only work with polaritons when the coupling is very strong e.g. in a BEC. Nevertheless the principle applies. Once the light starts interacting with a dielectric is is no longer just an EM wave and therefore doesn't travel at $c$. – John Rennie Feb 12 '16 at 16:57

No, all observers do not agree on the speed of a beam of light in a medium.

However, that fact doesn't break special relativity's edict that the laws of physics are the same in all inertial frames of reference. The presence of the medium might make it more convenient to express the laws of physics in an inertial frame of reference in which the medium is at rest, because some values will nicely simply be zero in that frame of reference, but that choice of frame of reference is still just an arbitrary choice.

Even in the presence of a medium, all observers will still agree on the value of the physical constant $c$, and agree on the correctness of any physical law in which $c$ appears. All observers also agree on a particular beam of light's four-velocity $U$ as a geometric object, although they express the components of that four-velocity differently just because they're using different coordinate systems. The components of $U$ are related between two inertial frames of reference by a Lorentz transformation, just like any other four-vector. All observers also agree that the magnitude of $U$ (the square root of $U$'s spacetime interval) is the same constant $c$. And all observers also agree that the relativistic velocity addition formula correctly expresses the relationship between the beam of light's velocity, the velocity of the beam relative to the medium, and the velocity of the medium.

• That clarified my first question, but what about the different frequencies of light? (My second question) – Udit Dey Feb 13 '16 at 8:57

The speed of light in a vacuum (or the constant c) is what is the same to all observers in a reference frame. It is possible to move faster than the speed of light within a medium (see: Cherenkov radiation).

• So does that mean that special relativity applies only in a vacuum? – Udit Dey Feb 12 '16 at 15:30
• Special relativity always applies, but it applies specifically to the constant c. Which light can travel at, because it is massless. – George Feb 12 '16 at 16:00