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In a perturbed metric, gravitational
interactions alter the temperature perturbation .
The redshift properties may be formally derived by employing the
equation of motion for the photon energy where is the 4-velocity of an observer at rest
in the background frame and is the photon 4-momentum.
The Euler-Lagrange
equations of motion for the photon and the requirement
that |*u*^{2}|=1 result in

| |
(45) |

which differs from [20,7] since we take to be the photon propagation direction rather than the viewing
direction of the observer.
The first term is the cosmological reshift due to the expansion
of the spatial metric; it does not affect temperature perturbations
. The second term has a similar origin and is due
to stretching of the spatial metric. The third and fourth term
are the frame dragging and time dilation effects.
Since gravitational redshift affects the different polarization
states alike,

| |
(46) |

in the basis.
We now explicitly evaluate the Boltzmann equation for scalar,
vector, and tensor metric fluctuations of
Eqns. (36)-(38).

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*Wayne Hu*

*9/9/1997*