The Robertson–Walker metric

To the relativist, cosmology is the task of finding solutions to Einstein's field equations that are consistent with the large-scale matter distribution in the universe. Modern observational cosmology has demonstrated that the real universe is highly symmetric in its large-scale properties, but the evidence for this was not known at the time when Friedmann and Lemaître began their pioneering investigations. Just as Einstein aimed to write down the simplest possible relativistic generalization of the laws of gravity, so cosmological investigation began by considering the simplest possible mass distribution: one whose properties are homogeneous (constant density) and isotropic (the same in all directions).

isotropy implies homogeneity At first sight, one might think that these two terms mean the same thing, but it is possible to construct universes that are homogeneous but anisotropic; the reverse, however, is not possible. Consider an observer who is surrounded by a matter distribution that is perceived to be isotropic; this means not only that the mass density is a function of radius only, but that there can be no preferred axis for other physical attributes such as the velocity field. This has an important consequence if we take the velocity strain tensor ∂vi/∂xj and decompose it into symmetric and antisymmetric parts.