Intuitive Significance
In the ADM formalism we introduce a family of spatial hyperslices. This allows us to think of the geometry of "space" as evolving over "time". This is an attractive viewpoint, but in general no such family of hyperslices will be physically preferred. The Weyl hypothesis can be understood as the assumption that we should consider only cosmological models in which there is such a preferred slicing, namely the one given by taking the unique hyperslices orthogonal to the world lines of the fluid particles.
One consequence of this hypothesis is that if it holds true, we can introduce a comoving chart such that the metric tensor contains no terms of form dt dx, dt dy, or dt dz.
The additional hypothesis that the world lines of the fluid particles be geodesics is equivalent to assuming that no body forces act within the fluid. In other words, the fluid has zero pressure, so that we are considering a dust solution.
Read more about this topic: Weyl's Postulate
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