Vector Einstein Equations

The vector metric source evolution is similarly constructed from a ``Poisson'' equation

$$\dot V + 2 {\dot a \over a} V = -8\pi G a^2 (p_f \pi^{(1)}_f + \pi_s^{(1)})/k \, ,$$ (58)

and the momentum conservation equation for the stress-energy tensor or Euler equation  

$$\begin{array} {rcl}\displaystyle{}\dot v_f^{(1)}&= &\dot V -... ...\dot a \over a} v_s^{(1)}- {1 \over 2}k\pi_s^{(1)},\end{array}$$

where recall $c_f^2 = \dot p_f / \dot \rho_f$ is the sound speed. Again, the first of these equations remains true for each fluid individually save for momentum exchange terms. For the photons $v_\gamma^{(1)}= \Theta_1^{(1)}$ and $\pi_\gamma^{(1)}= {8 \over 5}\sqrt{3}\Theta_2^{(1)}$.Thus with the photon Euler equation (60) (with $\ell=1$, m=1), the full baryon equation becomes  

$$\dot v^{(1)}_B = \dot V - {\dot a \over a} (v^{(1)}_B - V) + {\dot\tau \over R} (\Theta_1^{(1)}- v^{(1)}_B) \, ,$$ (59)

see Eqn. (67) for details.