The Conformal Group SO(4,2) and Robertson-Walker Spacetimes

TL;DR

This paper explores the conformal symmetry of Robertson-Walker spacetimes by deriving conformal Killing vectors from Minkowski spacetime, providing new bases that depend on curvature parameters and include symmetries of special cases.

Contribution

It introduces a local coordinate transformation to express Robertson-Walker metrics as conformally flat and constructs conformal Killing vectors directly from Minkowski spacetime, including bases dependent on curvature.

Findings

Derived conformal Killing vectors for Robertson-Walker spacetimes.

Presented curvature-dependent bases including special symmetric cases.

Compared new bases with previously published ones.

Abstract

The Robertson-Walker spacetimes are conformally flat and so are conformally invariant under the action of the Lie group SO(4,2), the conformal group of Minkowski spacetime. We find a local coordinate transformation allowing the Robertson-Walker metric to be written in a manifestly conformally flat form for all values of the curvature parameter k continuously and use this to obtain the conformal Killing vectors of the Robertson-Walker spacetimes directly from those of the Minkowski spacetime. The map between the Minkowski and Robertson-Walker spacetimes preserves the structure of the Lie algebra so(4,2). Thus the conformal Killing vector basis obtained does not depend upon k, but has the disadvantage that it does not contain explicitly a basis for the Killing vector subalgebra. We present an alternative set of bases that depend (continuously) on k and contain the Killing vector basis as a sub-basis (these are compared with a previously published basis). In particular, bases are presented which include the Killing vectors for all Robertson-Walker spacetimes with additional symmetry, including the Einstein static spacetimes and the de Sitter family of spacetimes, where the basis depends on the Ricci scalar R.