Observable Relations in Relativistic Cosmology. II

Recent observations show that the assumption that the universe is isotropic and homogeneous, which forms part of current theories of the “expanding universe”, is probably not satisfied by the actual universe. This paper is an attempt to discover the type of relation which must hold between observable quantities if the theory of general relativity is applicable but if the assumption of isotropy and homogeneity is not made. It is first shown that Hubble's law, or something of a very similar character, must hold to a first approximation, independently of the distribution of matter. In order to relate the apparent recession of the nebulae to the distribution of matter, we must proceed to higher approximations in its relation to distance. With a certain relativistic representation of the universe in our neighbourhood, it is shown how the second approximation in the dependence of the apparent recession on distance can in fact be related to the mean density in our neighbourhood, whether or not the distribution is homogeneous or isotropic. The problem of expressing this relation in terms of quantities which are observable by astronomical methods is discussed, but is not completely solved.