On spacetime models with an isotropic Hubble law

We consider a Lorentzian manifold (,g) with an observer field (timelike vector field) V. Along each lightlike geodesic the redshift z and the angular diameter distance D are then well defined functions. (Instead of the angular diameter distance one may equivalently use the corrected or uncorrected luminosity distance.) (,g,V) is said to admit an isotropic Hubble law at p if all past-oriented lightlike geodesics issuing from p yield the same z-D relation. For infinitesimally short lightlike geodesics (i.e. to within a linear approximation with respect to the variable D) it is well known that an isotropic Hubble law holds at all points p if and only if V is freely falling and shear-free. Here we derive the necessary and sufficient conditions for an isotropic Hubble law beyond the linear regime. To that end we expand the z-D relation in a Taylor series (Kristian-Sachs series) and we investigate the validity of an isotropic Hubble law order by order. In particular, we prove that (,g,V) admits an isotropic Hubble law of third order at every point p if and only if (,g,V) is either redshift-free or a Robertson-Walker model (locally around every point p).