special relativity - Mass of particle near light speed in a medium

I am trying to get a common understanding from these two previous questions:

1. Why does the mass of an object increase when its speed approaches that of light?


2. What happens if light/particles exceeded the speed of light for a particular medium (sic)

Does the increase of mass occur only if the particle approaches c (speed of light in a vacuum) or if it simply approaches the speed of light in its current medium? For example, does the mass of charged particles increase during Cherenkov radiation?

Answer

First, your question.
Yes, the increase of mass occurs only when a particle approaches $c$ (speed of light in vacuum).

$c$ is fundamental in Special Relativity, not because it is the speed of photons, but because it is the constant speed in the universe (the only speed invariant to boosts). Just because macroscopic light is transmitted at a lower speed inside a particular medium, that doesn't mean that the fundamental speed of Special Relativity is any different. Even inside mediums where light travels more slowly, all relativistic effects happen when a particle approaches $c$.

Since Cherenkov radiation (CR) is just an effect related to the speed of light in a medium (and not to $c$), it doesn't have anything to do with mass increase either. Though CR and mass increase can happen simultaneously to a particle, they are independent (the first does not imply the second, and vice-versa).

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Second, about the increase of mass.
It has been a historical habit to say that a particle's mass increases as $m=\gamma m_0$ when its velocity approaches $c$. That is not very appropriate. While it may seem convenient to define this relativistic mass, it's not a good habit.

First, because it's confusing to some people. There are physically intuitive ways to explain to a student why time intervals must stretch and why space intervals have to contract, but there's no way at all to explain why a particle's mass should increase.

Second, it's also not accurate. The defined relativistic mass parameter does not sustain the properties you would expect from a mass under close analysis. (I have a reference for this, but the pdf file somehow got corrupted in the last 8 years. I'm looking for a copy.)