Irrotational dust with Div H=0

For irrotational dust the shear tensor is consistently diagonalizable with its covariant time derivative: ${\sigma }_{ab}=0={\dot{\sigma }}_{ab},a\mathrm{\ne }b$, if and only if the divergence of the magnetic part of the Weyl tensor vanishes: $divH=0$. We show here that in that case, the consistency of the Ricci constraints requires that the magnetic part of the Weyl tensor itself vanishes: $H_{ab}=0$.