Rule of addition of velocities in water Is the speed of light still the same for all inertial observers in water?
If not, what are the rules of addition of velocities in water according to special relativity? 
 A: The invariant speed remains c (the speed of light in vacuum) even if you are in a medium where the light is slowed. All of the relativistic formulas are unchanged. Also, it is possible for particles to travel faster than the speed of light as long as they remain below c. 
A: No, the speed of light in water depends on the observer's speed. If you're at rest relative to the water, you'll see a wave with 4-velocity:
$$ u_{\mu} = \gamma_n(c, 0,0, c/n) = \gamma_n c (1, 0,0,\frac 1 n ) $$
where:
$$ \gamma_n \equiv \frac 1 {\sqrt{1-\frac 1 {n^2}}}$$
Now you can boost it any which way, and you'll find:
$$||v_i'|| = \frac{||u_i||}{\gamma_n} \ne c/n$$
For example, co-linear with the light (boosting with $\beta$ and $\gamma$):
$$u'_{\mu} = \gamma_n c (\gamma-\beta\gamma/n, 0, 0, -\beta\gamma+\gamma/n)$$
$$ u'_{\mu} = \gamma_n\gamma c(1-\beta/n, 0, 0, -\beta+1/n) $$
or
$$ u'_{\mu} = \gamma_n\gamma c[1-\frac{\beta}n](1, 0, 0, \frac{\frac 1 n-\beta}{1-\frac{\beta}n}) $$
which means the 3 velocity is:
$$ v' = \frac c{\gamma_n\gamma} \frac{\frac 1 n-\beta}{(1-\frac{\beta}n)^2} = c \frac {\sqrt{1-\frac 1 {n^2} }}{\gamma} \frac{\frac 1 n-\beta}{(1-\frac{\beta}n)^2} $$
Note what should have been obvious from the beginning: there are (collinear) boosts that can change the direction (and one that zeros it out), which means a priori we should have known the answer is "no".
A: The name "speed of light" is too widespread to change the language now, but a better name might be simply "maximal speed." The maximal speed is the same for all observers, but light does not always travel at the maximal speed. It does in a vacuum, but not in other media like water. 
The rule for adding co-linear velocities is the same in water as it is in a vacuum. The rule is to add the rapidities $\theta$, not the velocities $v$. The relationship is $v=c\,\tanh\theta$, where $c$ is the maximal speed and $\tanh$ is the "hyperbolic tangent" function
$$
   \tanh\theta = \frac{e^\theta-e^{-\theta}}{e^{\theta}+e^{-\theta}}.
$$
The rapidity $\theta$, like the velocity $v$, can be positive or negative. Explicitly, let $A,B,C$ be three objects, and define
\begin{align}
 {v}_{AB} &= \text{ the velocity of $A$ relative to $B$,} \\
 {v}_{BC} &= \text{ the velocity of $B$ relative to $C$,} \\
 {v}_{AC} &= \text{ the velocity of $A$ relative to $C$,}
\end{align}
which are related to the corresponding rapidities by
\begin{align}
 {v}_{AB} &= c\,\tanh\theta_{AB} \\
 {v}_{BC} &= c\,\tanh\theta_{BC} \\
 {v}_{AC} &= c\,\tanh\theta_{AC}.
\end{align}
As usual in special relativity [1], assuming that the velocities are all co-linear, the rule for adding them is:
$$
   \theta_{AC} = \theta_{AB}+\theta_{BC}.
\tag{1}
$$
Again, this rule is the same in any medium. The name "maximal speed" for $c$ is appropriate, because the corresponding rapidity is infinite: 
$$
    \lim_{\theta\rightarrow\infty}\big(c\tanh\theta\big)=c.
$$
Since infinity plus any finite number is still infinity, equation (1) says that the maximal speed $c$ is the same for all observers. This is just as true in water as it is in a vacuum, but light does not travel at this speed in water. The quantity $c$ is the maximal speed, not the speed of light (except in a vacuum).
Since light travels with speed $v<c$ in water, it is possible to outrun light in water. When an aircraft outruns its own sound, it produces a pressure impulse called a "sonic boom." Similarly, when a charged particle outruns light in a medium like water, it produces an electromagnetic impulse, a kind of "optic boom" called Cerenkov radiation (also spelled Cherenkov radiation, which is how it's pronounced in either spelling). Here are two examples:


*

*According to [2], Cerenkov radiation is responsible for the bluish glow of an underwater nuclear reactor. 

*Cerenkov radiation also a useful effect [3]: the opening angle of the resulting cone of light can be used to measure the speed of a particle that is moving faster than light in the given medium (which is typically a gas).

[1] https://en.wikipedia.org/wiki/Rapidity
[2] https://en.wikipedia.org/wiki/Cherenkov_radiation
[3] https://en.wikipedia.org/wiki/Ring-imaging_Cherenkov_detector
