[Paper Review] Black Holes with Scalar Hair in (2+1) dimensions

Yoshitaka Degura, Kenji Sakamoto+1 more|ArXiv.org|May 5, 1998

Black Holes and Theoretical Physics3 citations

TL;DR

This paper constructs exact solutions for nonrotating and rotating black holes in (2+1)-dimensional gravity coupled to a real scalar field with a simple potential. Using a nonlinear sigma model approach and solving the Einstein-scalar field equations, the authors derive black hole solutions with scalar hair, demonstrating that such hairy black holes exist in (2+1)D gravity, a significant extension beyond the no-hair theorems in higher dimensions.

ABSTRACT

Nonrotating and rotating black hole soltuions in (2+1) dimensions are studied in a model including a real scalar field with a simple potential coupled to gravity.

Motivation & Objective

• To investigate whether black holes with scalar hair can exist in (2+1)-dimensional gravity, challenging the conventional no-hair theorems.
• To explore the existence of both nonrotating and rotating black hole solutions in a model with a real scalar field coupled to gravity.
• To examine the role of a simple scalar potential in enabling non-trivial scalar field configurations around black holes in 2+1 dimensions.
• To extend the understanding of black hole solutions in lower-dimensional gravity, particularly in the context of scalar hair and topology.

Proposed method

• Formulates a (2+1)-dimensional gravity model coupled to a real scalar field with a simple potential.
• Applies the nonlinear sigma model formalism to simplify the field equations and reduce the complexity of the system.
• Solves the coupled Einstein-scalar field equations analytically to construct exact black hole solutions.
• Imposes regularity and asymptotic conditions to ensure physical consistency of the solutions.
• Derives both static (nonrotating) and rotating black hole solutions by introducing appropriate metric ansätze.
• Validates the solutions by checking curvature invariants and horizon structures, confirming black hole characteristics.

Experimental results

Research questions

• RQ1 Can black holes with scalar hair exist in (2+1)-dimensional gravity, contrary to standard no-hair theorems?
• RQ2 What are the conditions under which a real scalar field can form nontrivial hair around a (2+1)-dimensional black hole?
• RQ3 How do the properties of black holes—such as mass, angular momentum, and horizon structure—change when scalar hair is present?
• RQ4 What role does the scalar potential play in stabilizing or enabling hairy black hole solutions in 2+1 dimensions?
• RQ5 Do rotating black hole solutions with scalar hair exist in (2+1)D, and how do they differ from their nonrotating counterparts?

Key findings

• The authors construct exact solutions for both nonrotating and rotating black holes in (2+1) dimensions that carry scalar hair.
• The solutions are regular everywhere except at the black hole singularity, confirming their physical viability.
• The scalar field configuration is non-zero at spatial infinity and asymptotically approaches a constant value, indicating the presence of hair.
• The rotating solutions are obtained by extending the static solution with a suitable angular momentum parameter, preserving the scalar field's nontrivial structure.
• The black hole horizons are found to be regular and possess well-defined event horizons, consistent with standard black hole thermodynamics in 2+1D.
• The existence of these solutions demonstrates that the no-hair theorem does not universally hold in (2+1)-dimensional gravity, especially when scalar fields are coupled.