[Paper Review] On the stability of black holes with nonlinear electromagnetic fields

Nora Bretón, Santiago Esteban Perez Bergliaffa|arXiv (Cornell University)|Feb 12, 2014

Black Holes and Theoretical Physics49 references3 citations

TL;DR

This paper investigates the thermodynamic and dynamical stability of static, spherically symmetric black holes sourced by nonlinear electromagnetic fields (Born-Infeld, Bronnikov, and Dymnikova solutions). Using specific heat analysis and the Poincaré turning point method, it shows these black holes exhibit a specific heat divergence similar to Reissner-Nordström but remain thermodynamically stable. Dynamical stability is confirmed via perturbation analysis, suggesting a potential equivalence between thermodynamic and dynamical stability for nonvacuum black holes.

ABSTRACT

The stability of three static and spherically symmetric black hole solutions with nonlinear electromagnetism as a source is investigated in three different ways. We show that the specific heat of all the solutions displays an infinite discontinuity with a change of sign, but the turning point method indicates that the solutions are thermodynamically stable (much in the same way as in the case of the Reissner-Nordstrom geometry). We also show that the black holes analyzed here are dynamically stable, thus suggesting that there may be a relation between thermodynamical and dynamical stability for nonvacuum black holes.

Motivation & Objective

• To assess the thermodynamic stability of static, spherically symmetric black holes with nonlinear electromagnetic fields as sources.
• To compare thermodynamic stability criteria (specific heat and Poincaré method) with dynamical stability under gravitational perturbations.
• To investigate whether a relationship exists between thermodynamic and dynamical stability in nonvacuum black hole solutions.
• To derive and apply a general expression for specific heat at constant charge for any charged spherically symmetric black hole.
• To examine horizon structure and thermodynamic behavior across three distinct nonlinear electrodynamics models: Born-Infeld, Bronnikov, and Dymnikova.

Proposed method

• Derives a general expression for specific heat at constant charge valid for any static, spherically symmetric charged black hole using the first law of black hole thermodynamics.
• Applies the Poincaré (turning point) method to analyze thermodynamic stability by examining conjugate variable plots (e.g., mass vs. inverse temperature) for the presence of bifurcations or vertical tangents.
• Performs dynamical stability analysis using established sufficient conditions for stability under gravitational perturbations, based on the sign and convexity of the Lagrangian and its derivatives in the P-frame (for electric case) and dual frame (for magnetic case).
• Analyzes horizon structure and parameter dependence in the three exact solutions: Born-Infeld, Bronnikov, and Dymnikova, using the metric and field equations derived from a general nonlinear electrodynamics Lagrangian.
• Compares results across methods: thermodynamic (specific heat and Poincaré method), and dynamical (perturbation theory and stability criteria from prior works).
• Validated dynamical stability for Born-Infeld black holes via existing literature (Fernando, 2004), and extended stability conditions to regular solutions using the formalism of Moreno et al. (2002).

Experimental results

Research questions

• RQ1 Does the specific heat at constant charge display a divergence in nonlinear electromagnetic black holes, and what does this imply for thermodynamic stability?
• RQ2 Can the Poincaré method detect thermodynamic instability in these black hole solutions, and how does it compare to the behavior in Reissner-Nordström black holes?
• RQ3 Are the Born-Infeld, Bronnikov, and Dymnikova black holes dynamically stable under linear gravitational perturbations?
• RQ4 Is there a consistent relationship between thermodynamic stability (via Poincaré method) and dynamical stability in nonvacuum black hole solutions?
• RQ5 How do the horizon radii and thermodynamic properties evolve with changes in the nonlinear electrodynamics parameters?

Key findings

• The specific heat at constant charge exhibits an infinite discontinuity with a sign change in all three solutions—identical to the Reissner-Nordström case—yet this does not imply thermodynamic instability.
• The Poincaré method reveals no turning points or bifurcations in the mass-inverse temperature diagram, indicating that all three solutions are thermodynamically stable.
• The Born-Infeld black hole is dynamically stable under gravitational perturbations, as confirmed by prior analysis (Fernando, 2004).
• The Bronnikov and Dymnikova regular black holes satisfy the sufficient dynamical stability conditions derived by Moreno et al. (2002), confirming their stability in the electric and magnetic regimes.
• The absence of turning points in the conjugate variable plots suggests a potential equivalence between thermodynamic and dynamical stability for charged, spherically symmetric black holes with nonlinear electromagnetic sources.
• The thermodynamic behavior of the regular solutions remains qualitatively similar to the singular ones as long as the gravitational mass contribution is small, though significant deviations may occur for large mass or extreme parameter regimes.