398 Cosmic microwave background anisotropies
In the orthonormal basis,whereea·eb=δab,one can define the Stokes param-
eters
Q≡ 2 IP 11 =− 2 IP 22 , U≡− 2 IP 12. (9.139)Every direction on the skyncan be entirely characterized by the polar coordinates
θandφ,in terms of which the metric induced on the celestial sphere of unit radius
is
ds^2 =gabdxadxb=dθ^2 +sin^2 θdφ^2. (9.140)In this case, it is convenient to use as a pair of polarization basis vectorseathe
coordinate basis vectorseθandeφ,tangential to the coordinate linesφ=const and
θ=const respectively. Considering the appropriate orthonormal vectorseθ and
ˆeφ≡eφ/
∣∣
eφ∣∣
,the Stokes parameters are
Qθθ≡ 2 IPθθ=− 2 IPφˆφˆ, Uθφ≡− 2 IPθφˆ.Problem 9.19Write down in the original basiseθ,eφthe covariant, contravariant
and mixed components of the polarization tensor in terms of these Stokes parame-
ters.
The reader may question why we work with the polarization tensor and not simply
with the polarization vector. The point is that the polarization tensor multiplied byI,
and consequently the Stokes parameters, are additive for incoherent superposition
of waves and can easily be calculated. The polarization vector is not additive, but
nevertheless the physical interpretation of the polarization pattern is clearest in
terms of this vector. In particular, if the radiation is completely polarized, it is
aligned along the electric field. Note that only the orientation ofpa,and not its
direction, has a physical meaning because the polarization tensor is quadratic in
pa.For partially polarized radiation, the polarization vector points in the direction
of the electric field of the waves which dominate in overall flux, and the magnitude
ofpacharacterizes the excess of the waves with appropriate polarization.
9.10.2 Thomson scattering and polarization
Let us consider the linearly polarized elecromagnetic wave with electric fieldE
scattered by the electron in the directionn(see Figure 9.5). After scattering, the
wave remains completely polarized and its electric field is
E ̃=A((E×n)×n), (9.141)where the coefficientAdoes not depend onEandn.Taking into account that the
polarization basis vectorseaare orthogonal ton,we find that, after scattering, the