On homogeneous and isotropic universe

TL;DR

This paper demonstrates that homogeneity and isotropy of spatial slices do not necessarily extend to the full spacetime metric, providing a new perspective on the symmetry properties of cosmological models.

Contribution

It introduces a simple spacetime metric example showing the distinction between spatial and spacetime symmetries and proves the necessity of a specific metric function being zero under Einstein's equations.

Findings

Homogeneity and isotropy of space do not imply spacetime symmetry.

A second arbitrary function in the metric must be zero if Einstein's equations are satisfied.

Provides a new definition of homogeneous and isotropic spacetime.

Abstract

We give a simple example of spacetime metric, illustrating that homogeneity and isotropy of space slices at all moments of time is not obligatory lifted to a full system of six Killing vector fields in spacetime, thus it cannot be interpreted as a symmetry of a four dimensional metric. The metric depends on two arbitrary and independent functions of time. One of these functions is the usual scale factor. The second function cannot be removed by coordinate transformations. We prove that it must be equal to zero, if the metric satisfies Einstein's equations and the matter energy momentum tensor is homogeneous and isotropic. A new, equivalent, definition of homogeneous and isotropic spacetime is given.