[Paper Review] Revisiting the charged BTZ metric in nonlinear electrodynamics

S. Habib Mazharimousavi, M. Halilsoy+1 more|arXiv (Cornell University)|Jan 27, 2012

Black Holes and Theoretical Physics3 citations

TL;DR

This paper proposes a regular, charged BTZ black hole solution in 2+1 dimensions by replacing linear Maxwell electrodynamics with nonlinear electrodynamics (NED), specifically using a logarithmic Lagrangian. The electric field and potential become regular at r=0, eliminating the singularity present in the linear theory, while the spacetime metric remains singular. A theorem proves that electric black holes necessarily form in 2+1D with circular symmetry and NED, and duality shows magnetic black holes cannot exist in this setting.

ABSTRACT

In contrast to its chargeless version the charged Banados, Taitelboim and Zanelli (BTZ) metric in linear Maxwell electromagnetism is known to be singular at r=0. We show, by employing nonlinear electrodynamics that one obtains charged, extension of the BTZ metric with regular electric field. This we do by choosing a logarithmic Lagrangian for the nonlinear electrodynamics. A Theorem is proved on the existence of electric black holes and combining this results with a duality principle disproves the existence of magnetic black holes in 2+1-dimensions.

Motivation & Objective

• To resolve the r=0 singularity in the charged BTZ black hole solution of linear Maxwell electrodynamics by employing nonlinear electrodynamics (NED).
• To investigate whether regular black hole solutions can be constructed in 2+1 dimensions using NED, particularly focusing on the role of the electric field and metric structure.
• To establish a general theorem proving that any circularly symmetric 2+1D spacetime with a U(1) electric field in NED necessarily forms a black hole.
• To explore the implications of duality in 2+1D NED, particularly the non-existence of magnetic black holes in this framework.

Proposed method

• Adopting a logarithmic NED Lagrangian: $\mathcal{L}(F) = -\frac{1}{\beta^2} \ln(1 + \beta^2 F)$, which reduces to Maxwell theory in the $\beta \to 0$ limit and vanishes in the $\beta \to \infty$ limit.
• Applying a static, circularly symmetric metric ansatz: $ds^2 = -f(r)dt^2 + \frac{dr^2}{f(r)} + r^2 d\varphi^2$, consistent with 2+1D general relativity.
• Deriving the nonlinear Maxwell equation via variation of the action: $d(^\star \mathbf{F} \mathcal{L}_F) = 0$, leading to a conserved electric charge $Q$.
• Solving the Einstein-NED equations to obtain the metric function $f(r)$, incorporating mass $M$, cosmological constant $\Lambda$, and NED parameters.
• Using a duality transformation $Q \to iP$ to generate magnetic solutions from electric ones, and analyzing the resulting field and metric structure.
• Applying a coordinate transformation to remove coordinate singularities and verify the physical nature of the solutions.

Experimental results

Research questions

• RQ1 Can a regular, charged BTZ black hole solution be constructed in 2+1 dimensions using nonlinear electrodynamics instead of linear Maxwell theory?
• RQ2 Does the use of a logarithmic NED Lagrangian eliminate the electric field singularity at $r=0$ while preserving the black hole structure?
• RQ3 Under what conditions does a black hole necessarily form in 2+1D spacetime with a U(1) electric field in NED?
• RQ4 Why do magnetic black hole solutions not exist in 2+1D NED, as implied by duality and the derived theorem?
• RQ5 How does the inclusion of NED affect the metric function $f(r)$ and the behavior of the electric and magnetic fields near $r=0$?

Key findings

• The electric field becomes regular at $r=0$, with a finite maximum value $E_{\text{Max}} = \frac{1}{\sqrt{2}|\beta|}$, eliminating the $1/r$ divergence of the linear theory.
• The electric potential is also regular at $r=0$, with $V(r=0) = -\frac{Q}{2}(1 - \ln(2Q^2\beta^2))$, contrasting with the logarithmic divergence in Maxwell theory.
• The metric function $f(r)$ remains singular at $r=0$ despite the regularity of the electric field, indicating that the spacetime singularity persists.
• A general theorem is proven: any circularly symmetric 2+1D spacetime with a U(1) electric field in NED necessarily admits a black hole solution.
• Using duality, the paper shows that pure magnetic solutions in 2+1D NED do not form black holes, as the corresponding metric lacks an event horizon.
• The logarithmic Lagrangian yields a metric that reduces to the standard charged BTZ solution in the $\beta \to 0$ limit and to the uncharged BTZ in the $\beta \to \infty$ limit.