26 | | Substituting the Poisson equation into the momentum equation we have |
27 | | |
28 | | [[latex($\frac{d \rho \mathbf{v}}{dt} = (\frac{\nabla^2\phi}{4 \pi G}+\rho_0) \nabla \phi = \nabla \cdot \left [ \frac{1}{4 \pi G} \left ( \nabla \phi \nabla \phi - \frac{1}{2} \left ( \nabla \phi \cdot \nabla \phi + 4 \pi G \rho_0 \phi \right) \right ) \right ]$)]] |
29 | | |
| 26 | Substituting the Poisson equation into the momentum equation we have after some manipulations |
| 27 | |
| 28 | |
| 29 | [[latex($\frac{d \rho \mathbf{v}}{dt} = (\frac{\nabla^2\phi}{4 \pi G}+\rho_0) \nabla \phi = \nabla \cdot \left [ \frac{1}{4 \pi G} \left ( \nabla \phi \nabla \phi - \frac{1}{2} \left ( \nabla \phi \cdot \nabla \phi \right) + \rho_0 \phi \right ) \right ]$)]] |
| 30 | |
| 31 | where we have the momentum flux tensor |
| 32 | |
| 33 | [[latex($\mathbf{T} = \frac{1}{4 \pi G} \left ( \nabla \phi \nabla \phi - \frac{1}{2} \left ( \nabla \phi \cdot \nabla \phi + 4 \pi G \rho_0 \phi \right) \right )$)]] |