Geodesics and Global Properties of the Liouville Solution in General Relativity with a Scalar Field

TL;DR

This paper presents a family of exact solutions in General Relativity with a scalar field, revealing their global geodesic structure and implications for cosmological models with accelerated expansion.

Contribution

It introduces a new class of solutions using the Liouville metric with exponential scalar potential, analyzing their global properties and geodesic completeness.

Findings

Solutions include models with naked singularities and accelerated expansion.

All geodesics are explicitly determined and shown to be either complete or ending at singularities.

The solutions extend Friedmann universes smoothly through zero scale factor.

Abstract

One parameter family of exact solutions in General Relativity with a scalar field has been found using the Liouville metric. The scalar field potential has exponential form. This model is interesting, because, in particular, the solution corresponding to the naked singularity provides smooth extension of the Friedmann universe with accelerated expansion through the zero of the scale factor back in time. All geodesics are found explicitly. Their analysis shows that the Liouville solutions are global ones: every geodesic is either continued to infinite value of the canonical parameter in both directions or ends up at the singularity at its finite value.