Inhomogeneous Generalization of Einstein's Static Universe with Sasakian Space

TL;DR

This paper presents exact inhomogeneous solutions to Einstein's equations using Sasaki metrics, generalizing Einstein's static universe by incorporating arbitrary inhomogeneous particle densities and analyzing their properties.

Contribution

It introduces a new class of static inhomogeneous solutions with arbitrary density functions, extending the Einstein static universe framework.

Findings

Solutions exhibit non-linear density contrast

Metric functions show deviations from homogeneous models

Explicit examples demonstrate the inhomogeneity effects

Abstract

We construct exact static inhomogeneous solutions to Einstein's equations with counter flow of particle fluid and a positive cosmological constant by using the Sasaki metrics on three-dimensional spaces. The solutions, which admit an arbitrary function that denotes inhomogeneous number density of particles, are a generalization of Einstein's static universe. On some examples of explicit solutions, we discuss non-linear density contrast and deviation of the metric functions.