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Construction of Regular Black Holes in General Relativity

Zhong-Ying Fan, Xiaobao Wang

TL;DR

The paper develops a general, analytically tractable method to construct exact, regular black hole solutions in General Relativity coupled to nonlinear electrodynamics, using a mass-function f=1-2m(r)/r and a L(F) that supports magnetic or electric charges. By selecting appropriate mass profiles, three explicit magnetically charged regular classes (Bardeen, Hayward, and a new class) plus a generic case are obtained, with regularity requiring μ≥3 and a two-parameter structure tied to a magnetic charge; the framework is extended to electric charges, Legendre-transformed Lagrangians, and thermodynamics, including a first law, Smarr relations, and entropy products. The authors further generalize to asymptotically anti-de Sitter spacetimes, deriving AdS BH solutions and an extended phase-space thermodynamics with pressure-like variables, enabling exploration of critical behavior. Overall, the work provides a unified, constructive approach to regular black holes in GR with nonlinear electrodynamics and analyzes their global, thermodynamic, and AdS properties.

Abstract

We present a general procedure for constructing exact black hole solutions with electric or magnetic charges in General Relativity coupled to a nonlinear electrodynamics. We obtain a variety of two-parameter family spherically symmetric black hole solutions. In particular, the singularity at the central of the space-time can be cancelled in the parameters space and the black hole solutions become regular everywhere in the space-time. We study the global properties of the solutions and derive the first law of thermodynamics. We also generalize the procedure to include a cosmological constant and construct regular black hole solutions that are asymptotic to anti-de Sitter space-time.

Construction of Regular Black Holes in General Relativity

TL;DR

The paper develops a general, analytically tractable method to construct exact, regular black hole solutions in General Relativity coupled to nonlinear electrodynamics, using a mass-function f=1-2m(r)/r and a L(F) that supports magnetic or electric charges. By selecting appropriate mass profiles, three explicit magnetically charged regular classes (Bardeen, Hayward, and a new class) plus a generic case are obtained, with regularity requiring μ≥3 and a two-parameter structure tied to a magnetic charge; the framework is extended to electric charges, Legendre-transformed Lagrangians, and thermodynamics, including a first law, Smarr relations, and entropy products. The authors further generalize to asymptotically anti-de Sitter spacetimes, deriving AdS BH solutions and an extended phase-space thermodynamics with pressure-like variables, enabling exploration of critical behavior. Overall, the work provides a unified, constructive approach to regular black holes in GR with nonlinear electrodynamics and analyzes their global, thermodynamic, and AdS properties.

Abstract

We present a general procedure for constructing exact black hole solutions with electric or magnetic charges in General Relativity coupled to a nonlinear electrodynamics. We obtain a variety of two-parameter family spherically symmetric black hole solutions. In particular, the singularity at the central of the space-time can be cancelled in the parameters space and the black hole solutions become regular everywhere in the space-time. We study the global properties of the solutions and derive the first law of thermodynamics. We also generalize the procedure to include a cosmological constant and construct regular black hole solutions that are asymptotic to anti-de Sitter space-time.

Paper Structure

This paper contains 13 sections, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Einstein gravity coupled to a non-linear electrodynamics
  3. Asymptotically flat black holes with magnetic charges
  4. Case 1: Bardeen class
  5. Case 2: Hayward class
  6. Case 3: a new class
  7. Generic case
  8. Asymptotically flat black holes with electric charges
  9. The first law of thermodynamics
  10. Derivation of the first law
  11. Entropy product formulae
  12. Asymptotically anti-de Sitter black holes
  13. Conclusion

Figures (1)

  • Figure 1: The metric function $f(r)$ for Bardeen class solution with zero Schwarzschild mass. Along the vertical axis, the $\mu$'s value decreases from top to bottom. For solid lines $\mu=5\,,3$ and for dashed lines $\mu=2\,,1$. Some parameters have been set as $\alpha=1/2\,,q=1$. The dotted line corresponds to a Schwarzschild black hole with $M=2$.