Cylindrically and toroidally symmetric solutions with a cosmological constant

TL;DR

This paper introduces cylindrical and toroidal coordinate systems to analyze solutions with a cosmological constant, clarifying their geometric properties and symmetries, including the possibility of matching to Einstein static universe regions.

Contribution

It provides a detailed geometric analysis of Linet-Tian solutions with a cosmological constant using new coordinate systems and explores their properties and higher-dimensional generalizations.

Findings

Toroidal symmetry arises for positive cosmological constant.

One curvature singularity can be removed via matching with Einstein static universe.

The solutions' properties and limits are thoroughly described.

Abstract

Cylindrical-like coordinates for constant-curvature 3-spaces are introduced and discussed. This helps to clarify the geometrical properties, the coordinate ranges and the meaning of free parameters in the static vacuum solution of Linet and Tian. In particular, when the cosmological constant is positive, the spacetimes have toroidal symmetry. One of the two curvature singularities can be removed by matching the Linet-Tian vacuum solution across a toroidal surface to a corresponding region of the dust-filled Einstein static universe. Some other properties and limiting cases of these space-times are also described, together with their generalisation to higher dimensions.