Killing vector fields and a homogeneous isotropic universe

TL;DR

This paper reviews key theorems on Killing vector fields, provides a detailed proof of the metric form for homogeneous isotropic spacetimes, and discusses examples and definitions related to constant-curvature spaces.

Contribution

It offers a comprehensive proof of the general metric form for homogeneous isotropic space-times and clarifies the distinction with non-homogeneous constant-curvature spaces.

Findings

Detailed proof of the metric form for homogeneous isotropic space-times

Example of a space with constant-curvature spatial sections that is not homogeneous or isotropic

Alternative definition of homogeneous and isotropic space-time via embedded manifolds

Abstract

Some basic theorems on Killing vector fields are reviewed. In particular, the topic of a constant-curvature space is examined. A detailed proof is given for a theorem describing the most general form of the metric of a homogeneous isotropic space-time. Although this theorem can be considered to be commonly known, its complete proof is difficult to find in the literature. An example metric is presented such that all its spatial cross sections correspond to constant-curvature spaces, but it is not homogeneous and isotropic as a whole. An equivalent definition of a homogeneous and isotropic space-time in terms of embedded manifolds is also given.