On the generalization of McVittie's model for an inhomogeneity in a cosmological spacetime

TL;DR

This paper analyzes the limitations and generalizations of McVittie's spacetime model, revealing geometric constraints that restrict the types of inhomogeneities it can represent in cosmological settings.

Contribution

It clarifies the geometric constraints, specifically spatial Ricci-isotropy, that limit the generalizations of McVittie's solution and corrects misconceptions in recent literature.

Findings

Spatial Ricci-isotropy constrains solution generalizations.

Certain generalized solutions do not include Schwarzschild geometries.

Contradicts some recent claims about solution classes.

Abstract

McVittie's spacetime is a spherically symmetric solution to Einstein's equation with an energy-momentum tensor of a perfect fluid. It describes the external field of a single quasi-isolated object with vanishing electric charge and angular momentum in an environment that asymptotically tends to a Friedmann--Lemaitre--Robertson--Walker universe. We critically discuss some recently proposed generalizations of this solution, in which radial matter accretion as well as heat currents are allowed. We clarify the hitherto unexplained constraints between these two generalizing aspects as being due to a geometric property, here called spatial Ricci-isotropy, which forces solutions covered by the McVittie ansatz to be rather special. We also clarify other aspects of these solutions, like whether they include geometries which are in the same conformal equivalence class as the exterior Schwarzschild solution, which leads us to contradict some of the statements in the recent literature.