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We review a theorem by Hasse and Perlick establishing a result characterizingparallax-free cosmological models via three equivalent properties -- namely theexistence of a redshift potential, the existence of a conformal vector fieldparallel to the observer field, and the vanishing of the shear of the observerfield together with some integrability condition. We are able to provide asimplified proof using Noether's theorem to calculate a conserved quantityalong lightlike geodesics that is connected with the conformal symmetry.Thereby we derive more detailed information about the connection of thekinematical invariants to the redshift isotropy and the connection of conformalvector fields to the causality of spacetime.

On Conformal Vector Fields Parallel to The Observer Field