# Wyman's "other" scalar field solution, Sultana's generalization, and their Brans-Dicke and $R^2$ relatives

## TL;DR

This paper explores Wyman's lesser-known scalar field solutions, their generalizations, and their relations to Brans-Dicke and quadratic gravity theories, revealing their properties, singularities, and physical interpretations.

## Contribution

It provides a detailed analysis of Wyman's solutions, their extensions, and their connections to alternative gravity theories, including explicit mappings and physical insights.

## Key findings

- Wyman's solutions contain central naked singularities.
- Conformal transformations relate these solutions to Brans-Dicke geometries.
- The geometries also solve purely quadratic gravity equations.

## Abstract

Wyman's less known static and spherically symmetric solution of the Einstein-Klein-Gordon equations and its recent generalization for positive cosmological constant are discussed, showing that they contain central naked singularities. By mapping back to the Jordan frame, we obtain the conformal cousins of these geometries that solve the vacuum Brans-Dicke field equations and are time-dependent and spherical. Their physical nature is discussed and the geometry is shown to be also a solution of purely quadratic gravity.