It doesn't happen with photons. Photons themselves always travel exactly at the (vacuum) speed of light, even in water, and they aren't charged particles. Cherenkov radiation happens with charged particles like electrons.

A photon's net speed through a medium is not always the vacuum speed of light.

> For example, the refractive index of water is 1.333, meaning that light travels 1.333 times faster in a vacuum than it does in water.

Light travels at about 0.75c in water.

https://en.wikipedia.org/wiki/Refractive_index

I guess that's relative, depending on which photon is referenced. The corollary is that from the photon POV, not its speed changes, but spacetime is warped, so it sees time in its own frame of reference going faster and/or distances shorter than we.

You are both correct, but on different levels.

>> not its speed changes, but spacetime is warped

No, in water, the spacetime is not significantly warped, certainly not warped enough to slow down light. The different speed of light in a medium compared to vacuum comes from the interaction between the photons and the medium. The photon then, effectively, becomes a hybrid "photon-medium excitation" which is an effective particle with an effective speed lower than the vacuum speed of light.

Maybe not from our point of view, but from the POV of the photon the water is moving with c_0 through the photon, so certainly relativistic effects apply.

Beware the usage of the term "speed of light" here. There is two things: the phase velocity and the group velocity.

I think you mean when an electron (or other charged particle) travels through water.