Photon continuity equation (${\dot{\delta }}_{\gamma }+\frac{4}{3}\nabla \cdot {\mathbf{v}}_{\gamma }-4\dot{\varphi }=0$)

In Newtonian gauge in cosmology, the Free-streaming photon Boltzmann equation is $\dot{\Theta }+\mathbf{e}\cdot \nabla \Theta =\dot{\varphi }-\mathbf{e}\cdot \nabla \psi$. Angular averaging and the photon angular temperature moments give the displayed continuity equation. Its dipole yields the photon Euler equation, with the sign of the anisotropic stress term determined by the stress convention.