The Linet-Tian solution with a positive cosmological constant in four and higher dimensions

TL;DR

This paper analyzes the Linet-Tian solution with a positive cosmological constant, revealing its toroidal symmetry, singularities, and higher-dimensional generalizations, thereby clarifying its geometric and physical properties.

Contribution

It provides a detailed analysis of the Linet-Tian solution's geometry, singularities, and extends it to higher dimensions, enhancing understanding of this class of solutions.

Findings

The solution has toroidal symmetry and two curvature singularities.

One singularity can be removed by matching to Einstein static universe.

The solution's properties and higher-dimensional generalizations are explicitly described.

Abstract

The static, apparently cylindrically symmetric vacuum solution of Linet and Tian for the case of a positive cosmological constant $\Lambda$ is shown to have toroidal symmetry and, besides $\Lambda$, to include three arbitrary parameters. It possesses two curvature singularities, of which one can be removed by matching it across a toroidal surface to a corresponding region of the dust-filled Einstein static universe. In four dimensions, this clarifies the geometrical properties, the coordinate ranges and the meaning of the parameters in this solution. Some other properties and limiting cases of this space-time are described. Its generalisation to any higher number of dimensions is also explicitly given.