Stars as Laboratories for Fundamental Physics - MPP Theory Group

208 Chapter 6
The homogeneous Maxwell equations in Lorentz gauge in an isotropic
medium then have the most general form
(
−K 2 g µν + ∑
a=±,L
)
π a P a
µν Aν = 0. (6.31)
The metric tensor is g = P g + P L + P + + P − where P µν
g
= e µ ge ∗ν
g . Hence
one obtains decoupled wave equations [−ω 2 + k 2 + π a (ω, k)]A a = 0 for
the physical degrees of freedom A a = P a A with a = ±, L. The corresponding
dispersion relation is
−ω 2 + k 2 + π a (ω, k) = 0. (6.32)
It yields the frequency ω k for modes with a given polarization and wave
number. The so-called effective mass is then m 2 eff = π a (ω k , k). This
expression is different for different polarizations and wave numbers,
and may even be negative.
Generally, an isotropic medium is characterized by three different
response functions because the left- and right-handed circular polarization
states may experience different indices of refraction (Nieves and Pal
1989a,b). Such optically active media are not symmetric under a parity
transformation. For example, a sugar solution changes under a spatial
reflection because the sugar molecules have a definite handedness.
If the medium and all relevant interactions are even under parity the
circular polarization states have the same refractive index. Then one
needs to distinguish only between transverse and longitudinal modes;
one defines π T ≡ π + = π − and P T = P + + P − which projects on the
plane transverse to K and U in Minkowski space.
In macroscopic electrodynamics the medium effects are frequently
stated in the form of response functions to applied electric and magnetic
fields instead of a response to A. The displacement induced by an
applied electric field is D = ϵ E with ϵ the dielectric permittivity. Similarly,
the magnetic field is H = µ −1 B for an applied magnetic induction
where µ is the magnetic permeability. For time-varying and/or inhomogeneous
fields these relationships are understood in Fourier space
where the response functions depend on ω and k.
The magnetic field H and the transverse part of D, characterized
by k · D T = 0, do not have independent meaning (Kirzhnits 1987).
Therefore, among other possibilities one may choose H = B, D T =
ϵ T E T , and D L = ϵ L E L . In this case ϵ L ≡ ϵ is the longitudinal and
ϵ T ≡ ϵ L + (1 − µ −1 ) k 2 /ω 2 the transverse dielectric permittivity.