On the refractive index transformation laws between inertial frames ...

On the refractive index transformation laws
between inertial frames of reference
Giuseppe Sellaroli
July 17, 2011
Introduction
The following article discusses the transformation laws of the refractive
index of an isotropic, homogeneous, linear medium between the inertial
frame in which the medium is at rest and a laboratory frame, with respect
to which the medium moves with constant velocity.
In the first section the refractive index will be briefly defined from
Maxwell’s equations, and in the second one the actual transformation laws
will be derived, in the framework of Special Relativity; in the third one, as
a simple application, Fresnel formula for the velocity of light in a moving
medium will be obtained from the transformation laws.
Einstein summation convention over repeated indices is utilized in the
article.
1 Refractive index of an isotropic linear medium
Let’s consider a generic medium at rest in an inertial frame of reference,
in absence of free charges and currents; Maxwell’s equations, written in
macroscopic form, read
∇ · D = 0
∇ × E = − ∂B
∂t
∇ · B = 0
∇ × H = ∂D
∂t
(1a)
(1b)
(1c)
(1d)
1

where E is the electric field, B is the magnetic field, D is the electric displacement
field and H is the magnetizing field. D and H are related to E and B through
D = ε 0 E + P
H = 1 µ 0
B − M
(2a)
(2b)
where P and M are electric polarization and magnetization. In a linear and
isotropic medium (such as air or water), P and M are proportional respectively
to E and H, and equations (2) turn into
D = εE
H = 1 µ B
with
ε ≥ ε 0 µ ≥ µ 0 ;
if the medium is also homogeneus, ε and µ do not depend on the space or
time variables.
If we limit ourselves to media that is linear, isotropic and homogeneous,
equations (1) become
∇ · E = 0
∇ × E = − ∂B
∂t
∇ · B = 0
∇ × B = µε ∂E
∂t
which are identical to the microscopic ones, except for the term µε replacing
µ 0 ε 0 ≡ 1 c 2 ; performing the usual calculations on Maxwell’s equations and
discarding the trivial solution
we find the two vector equations 1
E = 0 B = 0
µε ∂E
∂t − ∇2 E = 0
µε ∂B
∂t − ∇2 B = 0
1 These six equations are not independent, since E and B are related to each other through
Maxwell’s equations.
2

describing the propagation of an electromagnetic wave through the medium,
travelling with phase velocity
u = |u| = 1 √ µε
which can be expressed in terms of c as
where
u = c n
√ µε
n :=
µ 0 ε 0
is called the refractive index of the medium, the name deriving from its role
in Snell’s law, which describes refraction of light. Note that, although not
explicitly stated, ε and µ usually depend on the frequency of the wave, thus
the wave phase velocity depends on its frequency (this phenomenon is
called dispersion).
2 Refractive index in moving media
Suppose a linear, isotropic, homogeneous medium is moving with a constant
velocity v with respect to an inertial laboratory frame of reference K.
A monochromatic electromagnetic wave travelling through it, in the inertial
frame K ′ in which the medium is at rest, has, as shown in the previous
section, phase velocity u ′ , whose magnitude u ′ is defined by the medium
refractive index n ′ ; which will be its phase velocity magnitude u, or equivalently
the medium refractive index n, measured by an observer at rest with
respect to K?
First of all, the transformation laws between K and K ′ are given, in the
formalism of Special Relativity, by the Lorentz boost
⎡
⎡
x 0 γ β x γ β y γ β z γ ⎡
⎢x 1
β
⎥
⎣x 2 ⎦ = x γ 1 + (γ − 1) β2 x
β 2
(γ − 1) β xβ y
β 2
(γ − 1) β xβ z
x ′0
β 2
⎢β y γ (γ − 1) β yβ x
β 2
1 + (γ − 1) β2 y
β 2
(γ − 1) β yβ z
⎢x ′1
⎥
⎥ ⎣x ′2 ⎦ (3)
β 2 ⎦
x 3 ⎤
where
⎣
β z γ
(γ − 1) β zβ x
β 2
(γ − 1) β zβ y
β 2
x µ = (ct, x, y, z)
β = (β x , β y , β z ) = v c
γ = ( 1 − β 2) − 1 2
1 + (γ − 1) β2 z
β 2
⎤
x ′3 ⎤
3

Equation (3) can also be written as
x 0 = γ(x ′0 + β · x ′ )
(4a)
x = x ′ + γx ′0 β + γ − 1
β 2 ( β · x ′ ) β (4b)
which will come handy during calculations; transformations (3) and (4) are
valid for any contravariant four-vector
a µ = (a 0 , a)
when substituting it to x µ in the formulas.
Now let’s consider the wave phase in K ′
ψ ′ := ω ′ t ′ − k ′ · x ′
where ω ′ is the wave frequency and k ′ is its wave vector; we postulate 2 this
quantity is a Lorentz scalar, that is, invariant with respect to any Lorentz
transformation, in particular the one beetween K and K ′ :
ψ = ψ ′
If we define the 4-tuple
it’s easy to see that
( ω
)
k µ =
c , k
ψ = k µ x µ
and that for it to satisfy the Lorentz scalar condition for any choice of x µ , k µ has
to be a four-vector. Remembering that
|k ′ | = |ω′ |
c n′ ⇒ k ′ = |ω′ |
c n′ ˆk ′
and applying transformation (4a), we get
ω = γ(ω ′ + n ′ |ω ′ | β · ˆk ′ ) (5)
which describes Doppler effect. If k µ is a four-vector, the scalar product
k µ k µ = ω2
c 2 − ω2
c 2 n2
2 Physical motivation for this postulate lies in the transformation laws of E and B; moreover,
as we will see, this hypothesis leads to the equation describing Doppler effect, which is
experimentally verified.
4

is a Lorentz scalar, which means
ω 2
c 2 − ω2
c 2 n2 = ω′2
c 2 − ω′2
c 2 n′2
n =
√
⇓
(n ′2 − 1) ω′2
ω 2 + 1
Using equation (5) we finally get the transformation law for the refractive
index:
√
(n ′2 − 1) + γ 2 (1 − ω′
|ω
n =
′ | n′ β · ˆk ′ ) 2
∣
∣γ(1 − ω′
|ω ′ | n′ β · ˆk ′ ) ∣
3 An application: Fresnel drag coefficient
Let’s consider the simpler case of a wave travelling in the same direction as
the medium, coinciding with the x-axis direction:
k ′ = (k ′ , 0, 0) β = (β, 0, 0) u ′ = (u ′ , 0, 0)
Moreover, the orientation of the x-axis we will chosen such that u ′ is positive,
which means
n ′ =
c
|u ′ | = c u ′
Remebering that
k ′ = ω′
transformations (4) for k µ become
u ′
v := βc
ω = γω ′ ( 1 + v u ′ )
(
ω 1
u = γω′ u ′ + v )
c 2
Writing ω ′ in terms of ω in the first equation, and substituting it in the
second one we get
( )
ω 1 +
vu ′
u = ω c 2
u ′ + v
which gives the transformation law for the phase velocity 3
u = u′ + v
1 + vu′
c 2
3 Note it is identical to the relativistic velocity addition formula for collinear velocities.
5

In terms of the refractive index, this becomes
( )
u = c 1 +
vn ′
c
n ′ 1 + v
cn ′
which, in the first order approximation in β becomes
u ≈ c (
n ′ + v 1 − 1 )
n ′2
(6)
The term
1 − 1
n ′2
is called Fresnel drag coefficient, and equation (6) was experimentally verified
by the french physicist Hippolyte Fizeau in 1851, half a century before
Special Relativity was formulated. Equation (6) also includes the correction
made by Lorentz in 1895 to the original Fresnel formula, to account for the
effects due to dispersion: in fact, in our derivation
n = n(ω) n ′ = n ′ (ω ′ )
and the formulas automatically include Doppler effect, which changes the
wave frequency between the two frames.
References
[1] J.D. Jackson: Classical Electrodynamics. Wiley, 3rd edition, 1999.
[2] L.D. Landau, E.M. Lifshitz, and L.P. Pitaevskii: Electrodynamics of continuous
media, volume 8 of Course of Theoretical Physics. Pergamon, 2nd edition,
1984.
[3] A. Sommerfeld: Optics, volume 4 of Lectures on Theoretical Physics. Academic
Press, 1954.
[4] C. Wang: Wave four-vector in a moving medium and the Lorentz covariance of
Minkowski’s photon and electromagnetic momentums. ArXiv e-prints, 2011.
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