Purely radiative irrotational dust spacetimes
  We consider irrotational dust spacetimes in the full non-linear regime which are "purely radiative" in the sense that the gravitational field satisfies the covariant transverse conditions div(H) = div(E) = 0. Within this family we show that the Bianchi class A spatially homogeneous dust models are uniquely characterised by the condition that $H$ is diagonal in the shear-eigenframe. 
N Van den Bergh, B Bastiaensen, H R Karimian and L Wylleman