⚛️ high-energy theory

Charged regular AdS $_3$ black holes in a scalar-vector-tensor theory coupled to Born-Infeld electrodynamics

This paper constructs exact, static, circularly symmetric charged regular AdS $_3$ black hole solutions within a scalar-vector-tensor theory coupled to Born-Infeld electrodynamics, demonstrating that the non-linear electrodynamics cures the logarithmic curvature singularity found in the Maxwell-charged counterpart while yielding diverse horizon structures and a singularity-free spacetime suitable for microscopic entropy calculations.

Original authors: Gokhan Alkac

Published 2026-09-29

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Gokhan Alkac

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep in the heart of our universe, gravity acts as the ultimate sculptor, shaping stars and galaxies, but it also holds a dark secret. According to our best current understanding of how the universe works, gravity can crush matter so completely that it creates a point of infinite density called a singularity. This is the center of a black hole, a place where the laws of physics as we know them break down and numbers become meaningless. For decades, scientists have suspected that this breakdown is not a feature of reality, but a sign that our theory of gravity is incomplete. They believe that if we could look closer, perhaps by combining gravity with the strange rules of quantum mechanics, we would find that the center of a black hole is not a broken point, but a smooth, finite region. To test this idea without the impossible complexity of our full three-dimensional universe, physicists often turn to a simpler version of reality: a world with only two dimensions of space and one of time. In this simplified playground, they can build and examine models of black holes to see if the crushing singularity can be avoided.

In this context, a researcher named Gökhan Alkaç has constructed a new, precise model of a black hole that avoids the singularity problem entirely. The work focuses on a specific type of theoretical framework that mixes gravity with other fields, including a scalar field and a vector field, which are types of energy that permeate space. Previous attempts to create a charged black hole in this framework using standard electricity resulted in a failure: the center of the black hole remained singular, with physical quantities blowing up to infinity. The problem was traced to the way the electric field behaved near the center; in the standard model, the electric potential grows in a way that creates a mathematical tear in the fabric of space. Alkaç's breakthrough was to replace this standard description of electricity with a more complex, non-linear version known as Born-Infeld electrodynamics. This older theory, originally proposed to fix similar issues in the early days of particle physics, limits how strong an electric field can become, preventing it from reaching infinite strength.

By weaving this non-linear electricity into the fabric of the black hole model, the researcher found that the singularity disappears. The resulting object is a regular black hole, meaning its core is smooth and finite, with no tears in the geometry of space. The team derived an exact mathematical description of this object, showing that the curvature of space, which measures how much the geometry is bent, remains finite and well-behaved all the way down to the very center. Depending on the specific values chosen for the physical parameters of the model, this black hole can take on different forms. It can exist as a single black hole with one boundary, a double-layered structure with two boundaries, or even a state where the boundaries merge into a single, extreme point. In some configurations, the model describes a smooth, horizonless geometry that looks like a black hole but lacks the event horizon that usually traps light.

The significance of this work extends beyond just finding a pretty solution to a math problem. The primary goal is to use this smooth, singularity-free black hole as a testing ground for understanding the microscopic nature of entropy. Entropy is a measure of disorder, and for black holes, it is a quantity that physicists have long struggled to explain from the bottom up. While we know how to calculate the entropy of a black hole using its size and mass, we do not yet have a complete picture of the tiny quantum states that make up that entropy. Because the new model is free of singularities, it provides a clean, stable background where scientists can attempt to count these underlying quantum states. If successful, this could offer the first microscopic derivation of entropy for a regular black hole, bridging the gap between the smooth geometry of space and the chaotic world of quantum particles. The paper concludes by outlining the next steps, which involve calculating the mass and thermodynamic properties of this new black hole and checking if the microscopic count of states matches the macroscopic entropy, a crucial test for any theory attempting to unify gravity and quantum mechanics.

Technical Summary: Charged Regular AdS $_3$ Black Holes in a Scalar-Vector-Tensor Theory Coupled to Born-Infeld Electrodynamics

Problem Statement
General Relativity (GR) predicts inevitable spacetime singularities at the cores of black holes, leading to divergences in physical observables. While the AdS $_3$ /CFT $_2$ correspondence has provided a microscopic derivation of entropy for the Bañados–Teitelboim–Zanelli (BTZ) black hole, extending this to regular (singularity-free) black holes remains an open challenge. Previous work [23] established that a three-dimensional scalar-vector-tensor theory, derived from regularized Gauss-Bonnet couplings with equal coupling constants ( $\alpha = \beta$ ), yields a static, circularly symmetric regular black hole solution. However, when this theory is coupled to standard linear Maxwell electrodynamics to introduce an electric charge, the resulting solution develops a logarithmic curvature singularity at the origin ( $r=0$ ). The central problem addressed in this paper is the construction of an exact, charged, regular AdS $_3$ black hole solution that resolves this singularity.

Methodology
The author constructs the solution within a three-dimensional scalar-vector-tensor framework obtained by regularizing the Gauss-Bonnet invariant ( $G = R^2 - 4R_{\mu\nu}^2 + R_{\mu\nu\rho\sigma}^2$ ) for dimensions $d < 5$ . The action includes the Einstein-Hilbert term with a cosmological constant, scalar-tensor ( $G_{st}$ ) and vector-tensor ( $G_{vt}$ ) couplings with equal constants ( $\beta$ ), and a non-linear electromagnetic sector.

Theory Setup: The metric is assumed to be static and circularly symmetric ( $ds^2 = -N^2 f dt^2 + dr^2/f + r^2 d\theta^2$ ). The vector field $W$ and scalar field $\phi$ depend only on the radial coordinate $r$ .
Electrodynamics Coupling: Instead of linear Maxwell electrodynamics, the action is coupled to Born-Infeld (BI) electrodynamics, characterized by the Lagrangian $L_{BI} = b^2(1 - \sqrt{1 + \frac{1}{2b^2}F_{\mu\nu}F^{\mu\nu}})$ , where $b$ is the BI parameter. This non-linearity is hypothesized to cure the singularity present in the Maxwell case.
Derivation: A 1D reduced action is derived by inserting the ansatz into the full action. The Euler-Lagrange equations are solved under the simplifying assumptions $N=1$ and $W^2=0$ , which were shown in prior work [24] to yield consistent solutions. The scalar field profile is fixed as $\phi(r) = \log(r/r_0)$ , and the vector field components are determined in terms of the metric function $f(r)$ .
Solution Construction: The equation of motion for the BI gauge field is solved to find the electrostatic potential $\phi(r)$ . This potential is then substituted into the equation for the metric function $f(r)$ , which is integrated to obtain the exact solution.

Key Contributions and Results

Exact Regular Solution: The paper derives an exact metric function $f(r)$ for a charged AdS $_3$ black hole. Unlike the Maxwell-charged counterpart, which exhibits a logarithmic divergence in the Ricci scalar ( $R \sim \log r$ ) at the origin, the BI-coupled solution is shown to be completely regular.
Regularity Verification: The author explicitly demonstrates that the Ricci scalar $R(r)$ remains finite as $r \to 0$ . The regularity holds provided the condition $\beta q > 0$ is met, where $q$ is the vector hair parameter.
Horizon Structures: Through numerical analysis of the metric function with specific parameter sets ( $\Lambda_0 = -1, b = 1, q_e = 1$ $Λ_{0} = - 1, b = 1, q_{e} = 1$ ), the solution is shown to exhibit a rich variety of horizon structures depending on the mass ( $m$ $m$ ), hair ( $q$ $q$ ), and coupling ( $\beta$ $β$ ) parameters:
- Horizonless regular geometries.
- Extremal black holes.
- Black holes with double horizons.
- Black holes with a single horizon.
Limiting Behavior: The solution correctly reduces to the standard Coulomb potential in the weak-field limit where the Born-Infeld parameter $b \to \infty$ . However, the resulting metric function in this limit retains the logarithmic singularity characteristic of the Maxwell-charged case, rather than recovering a regular metric.

Significance and Claims
The paper claims that this work fills a critical gap in the literature by providing the first exact, regular, charged AdS $_3$ black hole solution within this specific scalar-vector-tensor framework. The primary significance lies in its application to black hole thermodynamics and quantum gravity:

Microscopic Entropy: The regular spacetime serves as a testbed for calculating microscopic entropy. By applying the formula derived in [32] (which relates microscopic entropy to the mass of a soliton ground state and electric/magnetic charges), the author suggests this solution allows for the first microscopic derivation of entropy for a regular charged black hole.
Future Directions: The paper outlines a path to verify the consistency of the solution by computing the semiclassical thermodynamic entropy ( $S$ ) and the mass of the corresponding regular soliton ( $M_0$ ) via double Wick rotation, to check if $S_{micro}$ reproduces $S$ . It also notes that constructing rotating versions and exploring other non-linear electrodynamics models are natural extensions.

The author emphasizes that while the mathematical derivation is exact, the physical verification of the entropy correspondence remains a task for future work, positioning this paper as a foundational step toward understanding the quantum mechanical nature of singularity-free spacetimes.

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