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

Understand

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.