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If light propagates slower than c in a medium like air, what happens to the way the photons 'experience' time? It is often said that a photon does not 'experience' time since it's simultaneity plane completely overlaps its world-line. But this happens not because it's a photon, but because it travels at c. If photons travel at a subluminal speed, then does it mean that they 'experience' time? Also, what happens to the (rest) mass of a subluminal photon? Does it have some sort of effective mass?

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Photons are massless elementary particles, as defined in the Standard Model, and always travel at speed c in vacuum, when measured locally.

Now there are two ways to explain/interpret on this site why the speed of light itself if slower then c in a medium, and individual photons always travel at speed c when measured locally:

1.

Now in a medium, it is true that light itself travels with speed less then speed c, but the individual photons them selves still travel at speed c inbetween the atoms, in vacuum.

The simplest picture is that light always travels at the speed of light. But in a material it travels at the speed of light until it hits an atom. It is then absorbed and re-emitted in the same direction, which takes a small amount of time. The more this happens, the slower the effective average speed. The denser the material, the more atoms there are in the way.

Why does the light travel slower in denser medium?

It is the wavefront that travels actually at speeds less then c in vacuum.

Maybe an answer of mine will help understand this:

Do photons actually slow down in a medium, or is the speed decrease just apparent?

So the individual photons travel at speed c inbetween the atoms/molecules, and interact with them, and this interaction takes time, and this is why the classical wavefront slows down, and this is proportionate to the density of the medium, because the more atoms/molecules to interact with, the more time it takes.

2.

Light itself propagating through a medium should not be separately handled from the matter/medium it is traveling through.

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.

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

So the answer to your question is that we will have a quantum system, including photons and the particles of the matter/medium, with a new wavefunction describing both.

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