Local properties and global structure of McVittie spacetimes with non-flat FLRW backgrounds

TL;DR

This paper explores the global structure of McVittie spacetimes with non-flat FLRW backgrounds, revealing their geometric properties, singularities, and black hole features across different spatial curvatures.

Contribution

It extends the definition of McVittie spacetimes to closed FLRW backgrounds and analyzes their global properties, including singularities and black hole regions, for the first time.

Findings

Closed case spacetimes are confined between minimum and maximum area radii.

Radial null geodesics in open universes originate at the Big Bang or a hypersurface.

Black holes are common in eternally expanding open McVittie spacetimes.

Abstract

McVittie spacetimes embed the vacuum Schwarzschild(-(anti) de Sitter) spacetime in an isotropic FLRW background universe. We study the global structure of McVittie spacetimes with spatially non-flat FLRW backgrounds. This requires the extension of the definition of such spacetimes, previously given only for the flat and open cases, to the closed case. We revisit this definition and show how it gives rise to a unique spacetime (given the FLRW background, the mass parameter $M$ and the cosmological constant $\Lambda$) in the open and flat cases. In the closed case, an additional free function of the cosmic time arises. We derive some basic results on the metric, curvature and matter content of McVittie spacetimes and derive a representation of the line element that makes the study of their global properties possible. In the closed case (independently of the free function mentioned above), the spacetime is confined (at each instant of time) to a region bounded by a minimum and a maximum area radius, and is bounded either to the future or to the past by a scalar curvature singularity. This allowed region only exists when the background scale factor is above a certain minimum. In the open case, radial null geodesics originate in finite affine time in the past at a boundary formed by the union of the Big Bang singularity of the FLRW background and a non-singular hypersurface of varying causal character. Furthermore, in the case of eternally expanding open universes, we show that black holes are ubiquitous: ingoing radial null geodesics extend in finite affine time to a hypersurface that forms the boundary of the region from which photons can escape to future null infinity. We revisit the black hole interpretation of McVittie spacetimes in the spatially flat case, and show that this interpretation holds also in the case of a vanishing cosmological constant, contrary to a previous claim of ours.