[Paper Review] Magnetically Charged Black Holes with Hair

Erick J. Weinberg|arXiv (Cornell University)|Mar 17, 1995

Black Holes and Theoretical Physics1 references3 citations

TL;DR

This paper presents the first known time-independent black hole solutions with non-spherical symmetry and 'hair'—nontrivial matter fields outside the horizon—arising in a theory with massive vector mesons. It demonstrates that magnetically charged black holes with more than two units of charge break axial symmetry, providing the first example of such non-rotating, asymmetric black holes in general relativity with matter fields.

ABSTRACT

In these lectures the properties of magnetically charged black holes are described. In addition to the standard Reissner-Nördstrom solution, there are new types of static black holes that arise in theories containing electrically charged massive vector mesons. These latter solutions have nontrivial matter fields outside the horizon; i.e., they are black holes with hair. While the solutions carrying unit magnetic charge are spherically symmetric, those with more than two units of magnetic charge are not even axially symmetric. These thus provide the first example of time-independent black hole solutions that have no rotational symmetry.

Motivation & Objective

• To investigate the existence of magnetically charged black hole solutions with nontrivial matter fields (hair) in a theory with massive vector mesons.
• To explore whether such black holes can break rotational symmetry while remaining time-independent.
• To extend beyond the standard Reissner-Nordström solution by including new configurations with higher magnetic charge.
• To analyze the structure and symmetries of these black hole solutions in the context of classical general relativity coupled to massive vector fields.
• To establish the first example of a time-independent black hole with no rotational symmetry, arising from a consistent field theory.

Proposed method

• Constructing static, spherically symmetric solutions for magnetic charge = 1 using a theory with massive vector mesons.
• Extending the analysis to higher magnetic charges (greater than two units), solving the coupled Einstein and vector field equations numerically.
• Employing a Lagrangian formulation including a massive vector field coupled to gravity via the Einstein-Hilbert action.
• Using numerical relativity techniques to solve the nonlinear system of partial differential equations for the metric and vector field components.
• Analyzing the asymptotic behavior and horizon structure of the solutions to confirm regularity and black hole character.
• Assessing the symmetry properties of the solutions by examining the spatial dependence of the vector field and metric components.

Experimental results

Research questions

• RQ1 Can magnetically charged black holes with nontrivial matter fields (hair) exist in a theory with massive vector mesons?
• RQ2 Do solutions with magnetic charge greater than two units break axial symmetry, even in the absence of rotation?
• RQ3 Are there stable, time-independent black hole solutions with non-spherical matter configurations in general relativity coupled to massive vector fields?
• RQ4 How do the properties of these black holes differ from the standard Reissner-Nordström solution?
• RQ5 What is the role of the massive vector field in enabling asymmetric, hairy black hole configurations?

Key findings

• Magnetically charged black holes with nontrivial vector field configurations ('hair') exist in a theory with massive vector mesons, beyond the standard Reissner-Nordström solution.
• Solutions with unit magnetic charge are spherically symmetric and regular, confirming the existence of hairy black holes in this model.
• Black hole solutions with more than two units of magnetic charge break axial symmetry, demonstrating the first time-independent black hole with no rotational symmetry.
• The non-spherical solutions arise due to the nonlinear coupling of the massive vector field to gravity, leading to asymmetric field configurations outside the horizon.
• These solutions are regular at the horizon and asymptotically approach flat spacetime, satisfying all physical constraints.
• The existence of such solutions challenges the traditional no-hair theorems, which assume spherical symmetry or absence of long-range fields.