Cylindrically Symmetric Scalar Field and its Lyapunov stability in General Relativity

TL;DR

This paper presents an exact cylindrically symmetric scalar field solution in general relativity with a cosmological constant, analyzing its properties, energy conditions, and stability, extending known solutions like Levi-Civita and de-Sitter spaces.

Contribution

It introduces a new exact solution for a massless scalar field with a cosmological constant, generalizing previous cylindrical solutions and analyzing their stability and energy conditions.

Findings

The solution generalizes Levi-Civita and de-Sitter spacetimes.

Conditions under which the energy conditions are satisfied are identified.

The focusing theorem's validity for this solution is demonstrated.

Abstract

In this paper we found an Exact solution for massless scalar field with cosmological constant. This exact solution generalized the Levi-Civita vacuum solution\cite{8} to a massless scalar field,with a cosmological constant term. This solution in the absence of the Cosmological constant recovers the spacetime of a massless scalar field with cylindrical symmetry (\emph{Buchdahl metric}\cite{2}). Also if the scalar field disappears, the spacetime is a representation of de-Sitter space. We prove that the form of the metric's function which was purposed in \cite{1} is valid even if we assume a general form. Too we show that in which conditions this solution satisfies energy conditions. Finally the validity of focusing theorem is proved.