A Spacetime in Toroidal Coordinates

TL;DR

This paper derives an exact Einstein's field equations solution in toroidal coordinates, describing a spacetime with distinct interior, boundary, and exterior regions, including implications for cosmological constants and anti-de Sitter spacetime.

Contribution

It introduces a novel exact solution in Einstein's equations with a toroidal geometry, incorporating a string equation of state and boundary layers.

Findings

The exterior spacetime is locally isometric to anti-de Sitter space.

The size and mass of the toroidal structure depend on the cosmological constant.

The solution includes an interior string-like region and a boundary layer.

Abstract

We present an exact solution of Einstein's field equations in toroidal coordinates. The solution has three regions: an interior with a string equation of state; an Israel boundary layer; an exterior with constant isotropic pressure and constant density, locally isometric to anti-de Sitter spacetime. The exterior can be a cosmological vacuum with negative cosmological constant. The size and mass of the toroidal loop depend on the size of the cosmological constant.