Inhomogeneous generalization of Einstein’s static universe with Sasakian space

Abstract

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 the inhomogeneous number density of particles, are a generalization of Einstein’s static universe. For some examples of explicit solutions, we discuss non-linear density contrast and deviation of the metric functions.