Spacetimes with a transitive similarity group

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Abstract
Spacetimes admitting a four-dimensional transitive similarity group are studied. It is shown that when the homogeneous hypersurfaces (which necessarily exist) are spacelike (spatially homogeneous case) then the connection components in a Lorentz frame adapted to the spatially homogeneous slicing are proportional to t-1 where t is the proper time of the homogeneous slices. An analogous statement holds when those slices are timelike except that t is a spacelike coordinate in that case. The transitively self-similar universes correspond exactly to the exact power law solutions in Wainwright's terminology, (1984). They are also closely connected to the regularised form of the Einstein field equations used in the qualitative approach to spatially homogeneous cosmology. In fact the transitively self-similar models are in one-one correspondence and those critical points of the regularised system which lie in the physical region of that system.

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