On the existence of conformally coupled scalar field hair for black holes in (anti-)de Sitter space

TL;DR

This paper investigates the existence of scalar field hair around black holes in (anti-)de Sitter space, finding that hair exists only in the negative cosmological constant case under certain conditions.

Contribution

It demonstrates the conditions under which conformally coupled scalar field hair can exist for black holes in (anti-)de Sitter space, extending previous no-hair results.

Findings

No scalar hair in positive cosmological constant case.

Stable scalar hair exists in negative cosmological constant case for small scalar masses.

The existence of hair depends on the scalar field mass and cosmological constant.

Abstract

The Einstein-conformally coupled scalar field system is studied in the presence of a cosmological constant. We consider a massless or massive scalar field with no additional self-interaction, and spherically symmetric black hole geometries. When the cosmological constant is positive, no scalar hair can exist and the only solution is the Schwarzschild-de Sitter black hole. When the cosmological constant is negative, stable scalar field hair exists provided the mass of the scalar field is not too large.