Metric and Scattering Sources

The gravitational term $\vec{G}$is easily evaluated from the Euler-Lagrange equations for the motion of a massless particle in the background given by $g_{\mu\nu}$[6,8,9]:  

$$\vec{G}[h_{\mu\nu}] = \left({1 \over 2} {n}^i {n}^j \dot h_... ... h_{0i} + {1 \over 2} n^i h_{00\vert i} \,, 0 \, , 0 \right) .$$ (15)

Note gravitational redshift affects different polarization states alike. As should be expected, the modification from the flat space case involves the replacement of ordinary spatial derivatives with covariant ones.

The Compton scattering term $\vec{C}$ was derived in [1,4] in the total angular momentum language. Though the basic result has long been known [10,11], this representation has the virtue of explicitly showing that complications due to the angular and polarization dependence of Compton scattering come simply through the quadrupole moments of the distribution. Here 

$$\begin{array} {rcl}\vec{C}[{\vec {T}}] &=& -\dot\tau \left... ...\bf P}^{(m)}(\hat{n},\hat{n}') \vec{T}(\hat{n}')\, ,\end{array}$$ (16)

where the differential cross section for Compton scattering is $\dot\tau = n_e \sigma_T a$ where n_e is the free electron number density and $\sigma_T$ is the Thomson cross section. The bracketted term in the collision integral describes the isotropization of the photons in the rest frame of the electons. The last term accounts for the angular and polarization dependence of the scattering with 

$$\begin{array} {rcl}{\bf P}^{(m)} = \left( \begin{array} {ccc... ...\displaystyle{\dot a \over a}}\\ \end{array}\right),\end{array}$$ (17)

where $Y_\ell^m {}'\equiv Y_\ell^{m*} (\hat{n}')$ and $\, {}_{s}^{\vphantom{m}} Y_{\ell}^{m}{}'\equiv\, {}_{s}^{\vphantom{m*}} Y_{\ell}^{m*}(\hat{n}')$ and the unprimed harmonics have argument $\hat{n}$.Here $\, {}_{s}^{\vphantom{m}} {Y}_{\ell}^{m}$ are the spin-weighted spherical harmonics [12,13,14,1].