Light moves through a slowly moving medium with refractive index n. That medium moves with a speed v in parallel with the direction of light. What speed will be measured for that light by a rest observer?
If we have a medium with refractive index n, the speed of light relative to that medium is c/n.
Using relativistic addition of velocities, we get for the rest observer:
V
=
c
/
n
+
v
1
+
v
/
n
c
{\displaystyle V={\frac {c/n+v}{1+v/nc}}}
But as
v
≪
c
{\displaystyle v\ll c}
expand that expression in terms of
v
/
c
{\displaystyle v/c}
V
=
c
1
/
n
+
(
v
/
c
)
1
+
(
v
/
c
)
/
n
{\displaystyle V=c{\frac {1/n+(v/c)}{1+(v/c)/n}}}
V
=
c
1
/
n
1
+
c
1
⋅
(
1
)
−
1
/
n
⋅
(
1
/
n
)
1
v
/
c
+
.
.
.
≈
c
n
+
c
(
1
−
1
n
2
)
v
/
c
=
c
n
+
v
(
1
−
1
n
2
)
{\displaystyle {\begin{aligned}V&=c{\frac {1/n}{1}}+c{\frac {1\cdot (1)-1/n\cdot (1/n)}{1}}v/c+...\\&\approx {\frac {c}{n}}+c\left(1-{\frac {1}{n^{2}}}\right)v/c\\&={\frac {c}{n}}+v\left(1-{\frac {1}{n^{2}}}\right)\\\end{aligned}}}
The factor
(
1
−
1
n
2
)
{\displaystyle \left(1-{\frac {1}{n^{2}}}\right)}
v
′
=
V
+
v
/
Γ
2
{\displaystyle v'=V+v/\Gamma ^{2}}
