Regular AdS3 Black Hole Solutions

• Regular AdS3 black hole solutions are non-singular, asymptotically AdS geometries achieved through modified curvature interactions and additional matter or vector fields.
• They are constructed using regularized Gauss–Bonnet invariants, Lovelock gravity, and nontrivial field couplings that guarantee finite curvature invariants and primary hairs.
• These models exhibit unique thermodynamic features, such as double horizon structures and modified entropy laws, with potential observational signatures in shadow and phase transition analyses.

A regular AdS$_3$ black hole solution refers to a nonsingular, asymptotically anti-de Sitter geometry in three spacetime dimensions, often constructed by modifying the field content or the curvature interactions such that curvature invariants are finite everywhere—including at the would-be central singularity. These solutions typically exhibit unique thermodynamic and horizon structure, with the regularity achieved via mechanisms such as matter profiles with finite core density, nonlinear curvature corrections, or nontrivial coupling to additional field degrees of freedom. Theoretical frameworks for regular AdS$_3$ black holes include bi-vector-tensor theories arising from regularized Gauss-Bonnet invariants, Lovelock gravity with degenerate ground states, and models coupling scalar or vector fields in a manner that enforces global regularity.

1. Regularized Gauss–Bonnet Invariant and Bi-Vector–Tensor Theory

In three dimensions, the usual Gauss–Bonnet term is topological and does not contribute dynamically; however, recent work has shown that nontrivial contributions can be obtained by regularizing the Gauss–Bonnet invariant within the Weyl geometric framework (Alkac et al., 19 Aug 2025). This is achieved by employing two independent Weyl vectors $\mathbf{B}$ and $\mathbf{C}$, leading to an action of the form: $$I = \frac{1}{16\pi G} \int d^3x\, \sqrt{-g}\,\left[R - 2\Lambda_0 + \ell^2 \,(G[\mathbf{B}] - G[\mathbf{C}])\right]$$ with $G[\mathbf{W}]$ encoding coupling between the Ricci curvature and the Weyl vector. The difference $G[\mathbf{B}] - G[\mathbf{C}]$ is nontrivial after a careful $d\to3$ regularization procedure. This action falls within the generalized Proca class, where new vector degrees of freedom supplement the gravitational sector.

The static, rotationally symmetric metric ansatz is: $$ds^2 = -N^2(r)\,f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\theta^2$$ with Weyl vectors taken as linear combinations of $dt$ and $dr$ components, on-shell constrained to null norm. The resulting field equations admit metric solutions of the schematic form: $$f(r) = \frac{r^2(-\Lambda_0 r^2 - m) + 4\ell^2(q_b^2 - q_c^2)}{4\ell^2(q_b-q_c) + r^2}$$ where $m$ is the mass parameter and $q_b, q_c$ are independent "hair" parameters. Proper tuning of these parameters guarantees regularity: for instance, the Ricci scalar near the origin is finite for $q_b > q_c$,

$$R(r\to 0) \to \frac{3(m + q_b + q_c)}{2\ell^2(q_b - q_c)}$$

with higher-order curvature invariants similarly regularized.

2. Primary Hairs, Deformations, and Charged Extensions

These regular AdS$_3$ solutions possess primary hair parameters ($q_b, q_c$), representing degrees of freedom not solely deducible from asymptotic global charges. Deformations of the regularized Gauss–Bonnet theory (i.e. altering the vector coupling strength via a parameter $\sigma$) lead to simple analytic metrics: $$f_{\pm}(r) = -\frac{r^2}{2\sigma \ell^2}\left[1 \pm \sqrt{1 - 4\sigma\ell^2 \Lambda_0 + \frac{4\sigma^2 \ell^4(4q_b^2 - m r^2)}{r^4}}\right]$$ where regularity and asymptotic behavior select the $f_-$ branch and the effective AdS cosmological constant becomes $$\Lambda = [1 - \sqrt{1 - 4\sigma\ell^2 \Lambda_0}]/(2\sigma\ell^2)$$.

Charged regular black holes are constructed by coupling Born–Infeld electrodynamics with the regularized Gauss–Bonnet action. For the static electric field ansatz $A_\mu = \phi(r)\,dt$, the scalar potential is

$$\phi(r) = -q_e\,\log \left(\frac{r + \sqrt{r^2 + q_e^2/b^2}}{C}\right)$$

yielding a regular energy-momentum tensor and modified metric function $f(r)$ that preserves regularity at $r=0$ for admissible choices of $q_b, q_c$.

3. Relation to Other Regular AdS$_3$ Black Hole Frameworks

Mechanisms for singularity resolution in AdS$_3$ black holes span several theoretical frameworks:

• Regularization by matter densities: The introduction of finite core density profiles—as in Hayward-like models or via a zero-point length regularization $r\to\sqrt{r^2+l_0^2}$—yields regular AdS$_3$ black holes with de Sitter cores (Jusufi, 2022, Aros et al., 2019).
• Degenerate Lovelock ground states: Lovelock gravity with finely tuned couplings supporting $n$-fold degenerate AdS vacua allows regular solutions where the Kretschmann scalar is finite at the core (Estrada et al., 11 Feb 2025).
• Scalar fields with nonminimal coupling: Black holes dressed with nonminimally coupled scalar hair accommodate regular metrics given appropriate choices for coupling constants, with analytic solutions possible for special parameter values (Tang et al., 2019).
• Warped or hairy black holes in massive gravity: Minimal massive gravity and bi-metric massive gravity can admit regular or warped AdS$_3$ black hole solutions, with detailed thermodynamic and holographic analyses showing deviations from BTZ thermodynamics (Nam et al., 2018, Chernicoff et al., 2020).

4. Thermodynamic Properties and Phase Structure

Regular AdS$_3$ black holes—whether constructed via regularized Gauss–Bonnet couplings, finite energy densities, or scalar dressing—exhibit distinct thermodynamic behavior. The temperature, entropy (often via Wald’s formula), and heat capacity can differ significantly from singular solutions. Key features include:

• Absence of curvature singularities: All curvature invariants, notably the Ricci and Kretschmann scalars, remain finite for all $r$.
• Modified entropy formulas: The entropy often contains terms reflecting contributions from the regularizing matter or curvature sector, deviating from the area law.
• Multiple horizons and phase transitions: Regular black holes frequently show double horizon structure (Cauchy and event horizons), with phase transitions signaled by divergences in heat capacity and swallow-tail structures in Gibbs free energy plots (Kumar et al., 2023).
• Non-closed inversion curves: In extended thermodynamic analyses (Joule–Thomson expansions), regular AdS black holes display single, non-closed inversion curves; the ratio $T_i^{\min}/T_c$ is universally larger for regular black holes owing to the de Sitter core (Pu et al., 2019).

Thermodynamic profiles can be successfully encoded in observational quantities such as the shadow radius, which positively correlates with the event horizon and serves as a direct probe of phase structure and transitions (Guo et al., 2022).

5. Physical Implications, Observational Connections, and Holography

Regular AdS$_3$ black hole models offer theoretical advantages for quantum gravity and holography. The regularity at the core avoids issues associated with classical singularities, and the presence of additional hairs or deformation parameters enriches the phase space. Theoretically:

• The construction ensures compatibility with AdS/CFT correspondence, with primary hair parameters potentially encoding new degrees of freedom on the boundary CFT.
• Holographic computation of entropy (e.g. via the Cardy formula) takes into account the modified background and hair content, consistent with extended or deformed boundary symmetry algebras.

On the observational front, regular AdS black holes’ connection between horizon, photon sphere, and shadow size can in principle be numerically explored for astrophysical or analog gravity systems (Estrada et al., 11 Feb 2025). It is suggested that future high-resolution shadow observations (such as those performed by the Event Horizon Telescope for M87) may contain signatures distinguishing regularized core structures from singular spacetimes.

6. Comparative Table—Key Regularization Ingredients

Mechanism	Mathematical Origin	Regularity Guarantee
Regularized Gauss–Bonnet	Bi-vector–tensor action (Weyl geometry)	Finite curvature via hair integration
Zero-point length ($l_0$)	$r\to\sqrt{r^2+l_0^2}$ in potentials	Smeared matter/energy density at origin
Lovelock degenerate ground	$n$-fold AdS factorization	Exponential screening of singularity
Scalar field dressing	Nonminimal coupling, self-interactions	Analytical control of core curvature
Born–Infeld electrodynamics	Nonlinear electromagnetic sector	Regular energy-momentum and horizon
Warped/hairy massive gravity	Parameter-tuned massive gravity theories	Modified boundary conditions, smooth core

7. Concluding Remarks

Regular AdS$_3$ black hole solutions unify several avenues for singularity resolution in low-dimensional gravitating systems with negative cosmological constant. Theories constructed from regularized curvature invariants—such as the bi-vector-tensor theory derived from Weyl geometry—provide explicit examples whose horizon structure, thermodynamics, and observables reflect the underlying regularity. The introduction of primary hairs and deformation parameters allows for broad generalizations of the standard BTZ solution, fostering deeper connections to holography and potentially observable gravitational phenomena. The continued development of these frameworks, along with their thermodynamic and observational studies, positions regular AdS$_3$ black hole solutions as versatile probes in quantum gravity and strongly coupled field theory.