Constructing a family of conformally flat scalar field models

TL;DR

The paper introduces a geometrical method to solve non-minimally coupled scalar field equations in spherically symmetric, conformally flat spacetimes, simplifying the problem to a single equation dependent on a specific coordinate, and provides explicit solutions including Anti-de Sitter cases.

Contribution

It presents a complete classification of conformally flat scalar field spacetimes with gradient conformal vector fields and reduces the field equations to a single solvable equation.

Findings

Full set of Petrov type O spacetimes with scalar fields determined

Scalar field equations reduced to a single equation in variable w

Explicit solutions including Anti-de Sitter and scalar bubble configurations

Abstract

Using purely geometrical methods we present a mechanism to solve the scalar field equations of motion (non-minimally coupled with gravity) in a spherically symmetric background. We found that the \emph{full }set of spacetimes, which are of Petrov type O (conformally flat) and admit a \emph{gradient} Conformal Vector Field, can be determined completely. It is shown that the full group of scalar field equations reduced to a \emph{single} equation that depends only on the distance $w=r^{2}-t^{2}$ leaving the metric function (equivalently the functional form of the scalar field or the potential) freely chosen. Depending on the structure of the metric or the potential $V$ (as a function of $\phi $) a solution can be found either analytically or via numerical integration. We provide physically sound examples and prove that (Anti)-de Sitter fits this scheme. We also reconstruct a recently found solution \cite{Strumia:2022kez} representing an expanding scalar bubble with metric that has a singularity and corresponds to what is termed as Anti-de Sitter crunch.