On Conformal Vector Fields Parallel to The Observer Field

TL;DR

This paper reviews a theorem linking conformal vector fields, redshift potential, and shear in cosmological models, providing a simplified proof and deeper insights into spacetime causality and invariants.

Contribution

It offers a simplified proof of a theorem relating conformal vector fields to cosmological properties using Noether's theorem, enhancing understanding of spacetime symmetries.

Findings

Conformal vector fields relate to redshift isotropy.

Shear vanishing implies parallax-free models.

Conserved quantities along lightlike geodesics are linked to symmetries.

Abstract

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