[Paper Review] Charged black hole solutions of non-linear electrodynamics and generalized gauge field theories

J. Díaz-Alonso, Diego Rubiera-García|arXiv (Cornell University)|Jan 16, 2013

Black Holes and Theoretical Physics1 references4 citations

TL;DR

This paper investigates charged black hole solutions in non-linear electrodynamics (NED) and generalized gauge field theories, focusing on asymptotically anomalous gravitational configurations that deviate from Schwarzschild-like behavior. It classifies electrostatic spherically symmetric black holes based on central and asymptotic field behaviors, identifying six physically admissible models with finite-energy solutions, and demonstrates that certain NED models yield non-Schwarzschild asymptotics, leading to undefined ADM mass and novel horizon structures including single-horizon black holes and extreme black points.

ABSTRACT

We summarize the main features of a class of \emph{asymptotically anomalous} (asymptotically flat, but non Schwarzschild-like) gravitational configurations in models of gravitating non-linear electrodynamics in three space dimensions, whose Lagrangian densities are defined as arbitrary functions of the two field invariants and constrained by several physical admissibility conditions. This class of models and their associated electrostatic spherically symmetric black hole solutions are characterized by the behaviours of the Lagrangian densities around the vacuum and at the boundary of their domain of definition.

Motivation & Objective

• To classify physically admissible non-linear electrodynamics (NED) models that support finite-energy electrostatic spherically symmetric black hole (ESSBH) solutions with non-Schwarzschild asymptotics.
• To analyze the gravitational configurations arising from NED Lagrangians defined as arbitrary functions of field invariants X and Y, under physical admissibility conditions including parity invariance and energy positivity.
• To identify and characterize asymptotically anomalous solutions—where the metric approaches flatness slower than Schwarzschild—resulting in undefined ADM mass and novel horizon structures.
• To extend the analysis to non-Abelian gauge field theories by showing equivalence to the Abelian case under specific field configurations and Lagrangian forms.
• To lay the groundwork for thermodynamic analysis of these configurations, particularly for asymptotically Schwarzschild-like cases, by verifying compliance with the zeroth and first laws of black hole mechanics.

Proposed method

• Formulates the action for generalized non-linear electrodynamics (G-NED) as $ S = S_G + S_{NED} = ∫ d^4x \sqrt{-g} \left[ \frac{R}{16\pi G} - \varphi(X,Y) \right] $, where $ \varphi(X,Y) $ is an arbitrary function of the field invariants $ X = \vec{E}^2 - \vec{H}^2 $, $ Y = 2\vec{E} \cdot \vec{H} $.
• Applies physical admissibility conditions: continuity and regularity of $ \varphi $, parity invariance $ \varphi(X,Y) = \varphi(X,-Y) $, and energy positivity via $ \rho \geq \left( \sqrt{X^2 + Y^2} + X \right) \frac{\partial \varphi}{\partial X} + Y \frac{\partial \varphi}{\partial Y} - \varphi \geq 0 $.
• Derives the first integral for electrostatic spherically symmetric solutions: $ r^2 \left. \frac{\partial \varphi}{\partial X} \right|_{Y=0} E(r) = Q $, linking electric charge Q to the field strength E(r).
• Classifies central and asymptotic field behaviors (A1, A2, UVD and B1–B3, IRD) based on power-law behavior $ E(r) \sim r^p $ at r→0 and $ E(r) \sim r^q $ at r→∞, excluding UVD and IRD cases for finite-energy solutions.
• Solves the Einstein equations for the metric $ ds^2 = \lambda(r) dt^2 - \lambda^{-1}(r) dr^2 - r^2 d\Omega^2 $, with $ \lambda(r,Q,C) = 1 + \frac{C}{r} - \frac{2\varepsilon_{in}(r,Q)}{r} $, where $ \varepsilon_{in}(r,Q) $ is the finite interior energy integral.
• Analyzes horizon structures by finding zeroes of $ \lambda(r) $, identifying configurations such as extreme black holes (EBH), single-horizon black holes, two-horizon black holes, and naked singularities (NS), depending on parameters Q, C, and field behavior.

Experimental results

Research questions

• RQ1 What are the physically admissible non-linear electrodynamics models that support finite-energy electrostatic spherically symmetric black hole solutions with non-Schwarzschild asymptotics?
• RQ2 How do the central and asymptotic field behaviors (A1, A2, UVD, IRD) affect the structure of the gravitational metric and the existence of horizons in these NED models?
• RQ3 What is the nature of the ADM mass in asymptotically anomalous solutions, and why is it undefined in IRD cases?
• RQ4 How do the horizon structures (e.g., single-horizon, extreme, two-horizon) differ between asymptotically anomalous and Schwarzschild-like configurations?
• RQ5 Can the analysis of Abelian NED models be extended to non-Abelian gauge theories, and if so, under what conditions does the solution reduce to the Abelian case?

Key findings

• Six classes of physically admissible NED models support finite-energy electrostatic spherically symmetric black hole solutions after excluding UVD and IRD behaviors, with the remaining six classes forming a complete set of finite-energy configurations.
• Asymptotically anomalous solutions—arising from IRD asymptotic field behavior ($ -1 \leq q < 0 $)—exhibit non-Schwarzschild-like metric decay: $ \lambda(r) - 1 \sim -r^q $ for $ -1 < q < 0 $, and $ \sim \ln r / r $ for $ q = -1 $, leading to undefined ADM mass.
• For A1-IRD and A2-IRD cases, the metric function $ \lambda(r) $ is monotonically increasing and convex, with central singularities that are timelike for $ C > 0 $ and spacelike for $ C \leq 0 $, and critical configurations with $ C = 0 $ exhibit distinct horizon and singularity structures.
• In the A2a case ($ \lambda(0) > 0 $), the critical metric is finite and monotonically increasing with a non-negative slope at the center, leading to timelike naked singularities; in A2b ($ \lambda(0) < 0 $), a single-horizon black hole with spacelike singularity forms.
• For UVD-IRD models, the interior energy integral $ \varepsilon_{in}(r,Q) $ cannot be defined, but the metric is still solved via $ \lambda(r,Q,D) = 1 + \frac{D}{r} - \frac{2\varepsilon(r,Q,0)}{r} $, with horizon radii determined by intersections with $ (r + D)/2 $, yielding EBHs, two-horizon BHs, and NS depending on D.
• The critical case $ C = 0 $ in A1-IRD leads to a metric diverging as $ \lambda(r) \sim 1 + \frac{32\pi Q}{(2-p)(p+1)} r^p $ near r=0, with a single horizon and spacelike singularity for $ C \leq 0 $, and timelike for $ C > 0 $, demonstrating a phase transition in singularity type.