An exterior for the G\"{o}del spacetime

TL;DR

This paper constructs exterior solutions for G"odel spacetime by matching vacuum, cylindrically symmetric solutions with a dust cylinder, revealing cases with closed timelike curves and similarities to van Stockum's rotating dust solutions.

Contribution

It introduces a method to match G"odel universe exteriors with vacuum solutions, exploring cases with different cylinder radii and the presence of closed timelike curves.

Findings

Existence of three solution cases depending on cylinder radius

Presence of closed timelike curves in some exterior solutions

Similarity between G"odel exteriors and van Stockum's rotating dust spacetime

Abstract

We match the vacuum, stationary, cylindrically symmetric solution of Einstein's field equations with $\Lambda$, in a form recently given by Santos, as an exterior to an infinite cylinder of dust cut out of a G\"{o}del universe. There are three cases, depending on the radius of the cylinder. Closed timelike curves are present in the exteriors of some of the solutions. There is a considerable similarity between the spacetimes investigated here and those of van Stockum referring to an infinite cylinder of rotating dust matched to vacuum, with $\Lambda=0$.