Gravitational Redshift

In a perturbed metric, gravitational interactions alter the temperature perturbation $\Theta$. The redshift properties may be formally derived by employing the equation of motion for the photon energy $p \equiv -u^\mu p_\mu$where $u^\mu$ is the 4-velocity of an observer at rest in the background frame and $p^\mu$ is the photon 4-momentum. The Euler-Lagrange equations of motion for the photon and the requirement that |u²|=1 result in  

$${\dot{p}\over p} = -\dot{a \over a} - {1 \over 2} {n}^i {n}... ...j} - {n}^i \dot h_{0i} - {1 \over 2} n^i \nabla_i h_{00} \, ,$$ (45)

which differs from [20,7] since we take $\hat{n}$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 $\delta T/T$. 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,  

$$\vec{G}[h_{\mu\nu}] = \left({1 \over 2} {n}^i {n}^j \dot h_... ..._{0i} + {1 \over 2} n^i \nabla_i h_{00} \,, 0 \, , 0 \right) ,$$ (46)

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