Black holes in scalar multipolar Universes

TL;DR

This paper constructs scalar multipolar black hole solutions using a generalized Weyl approach, revealing their properties, singularities, and connections to higher-dimensional gravity, extending known scalar and vacuum solutions.

Contribution

It introduces a method to generate scalar multipolar black hole solutions by combining harmonic functions within a generalized Weyl framework, expanding the understanding of scalar-vacuum configurations.

Findings

Scalar multipolar solutions include naked singularities and singular horizons.

Harmonic solutions with decaying multipoles are asymptotically flat.

The scalar-vacuum model maps to five-dimensional pure gravity via Kaluza-Klein oxidization.

Abstract

Recently, a scalar counterpart of the Schwarzschild-Melvin Universe was reported [arXiv:2410.02851]. We show this solution is a special case of a Schwarzschild black hole/mass in a scalar multipolar Universe, that can be constructed algebraically combining known vacuum solutions. This builds on the generalized Weyl construction for scalar-vacuum, that admits two, fully decoupled, independent harmonic functions: one for the gravitational sector and another for the scalar sector. Harmonic solutions with growing multipoles lead to naked singularities (at spatial infinity), including for the scalar counterpart of the Schwarzschild-Melvin; harmonic solutions with decaying (scalar) multipoles are asymptotically flat, but have a singular horizon, in accordance with no-scalar-hair theorems. The scalar-vacuum model can be mapped to a five dimensional pure gravity construction via Kaluza-Klein oxidization and, in this way, to the corresponding $D=5$ generalized Weyl construction.