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The gravitational term is easily evaluated from the Euler-Lagrange equations
for the motion of a massless particle in the background given by [6,8,9]:

| |
(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 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

| |
(16) |

where the differential cross section for Compton scattering is
where *n*_{e} is the free electron number density
and 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
| |
(17) |

where and
and the unprimed
harmonics have argument .Here are the spin-weighted spherical harmonics
[12,13,14,1].

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*Wayne Hu*

*9/9/1997*