Regular Bardeen-AdS Black Hole

• Regular Bardeen-AdS black holes are non-singular, static solutions to Einstein gravity with a de Sitter core achieved by nonlinear electrodynamics.
• They exhibit Van der Waals-like phase transitions and modified thermodynamic behavior in an extended phase space, highlighting unique small/large black hole dynamics.
• Extended models with higher-curvature terms and quantum corrections provide insights into black hole microstructure and observable phenomena such as shadow profiles.

A regular Bardeen-AdS black hole is a non-singular static, spherically symmetric solution to Einstein gravity coupled with a nonlinear electromagnetic field and a cosmological constant, characterized by the absence of curvature singularities (specifically, regularity at $r=0$). Its AdS asymptotics are set by the cosmological constant, with thermodynamics and phase structure that fundamentally differ—in multiple aspects—from those of singular black holes such as Schwarzschild-AdS or Reissner–Nordström-AdS. The regularity is typically achieved by a specific form of the nonlinear electromagnetic source, often interpreted either as a magnetic or electric monopole. When embedded in AdS backgrounds, these black holes are relevant to quantum gravity, black hole chemistry, and holographic duality. Their thermodynamics and critical phenomena display both conventional and distinctive behaviors, including Van der Waals/critical transitions, modified phase diagrams, remnant states, and connections to the black hole shadow.

1. Construction, Regularity, and Nonlinear Electrodynamics

Regular Bardeen-AdS black holes are constructed as static, spherically symmetric solutions to Einstein gravity (potentially with higher-curvature corrections, e.g., Gauss–Bonnet terms) coupled to nonlinear electrodynamics, with line element: $$ds^2 = -f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2\left(d\theta^2 + \sin^2\theta\,d\phi^2\right)$$ The canonical Bardeen lapse function is

$$f(r) = 1 - \frac{2Mr^2}{(r^2 + q^2)^{3/2}} + \frac{r^2}{l^2}$$

where %%%%1%%%% is interpreted as a nonlinear electromagnetic “charge” (typically magnetic, but electric configurations exist (Rodrigues et al., 2018)), and $l^2 = -3/\Lambda$ is the AdS curvature scale.

Distinctively, for parameter values below extremality ($q<q_\mathrm{ext}$), this metric describes a black hole with inner Cauchy and outer event horizons, but curvature scalars are finite everywhere; the $r\to 0$ region realizes a de Sitter core: $f(r) \approx 1 - C r^2 + \dots \qquad (r\to 0)$ Significantly, $q$ may be attributed to either a magnetic or an electric nonlinear electromagnetic field, and the precise form of the required nonlinear Lagrangian can differ between interpretations. In the AdS case, the addition of a negative cosmological constant does not spoil the regularity but modifies the horizon structure and asymptotics.

Nonlinear electrodynamics acts to smear out the singularity by introducing energy–momentum profiles that violate the strong energy condition within a central region ($r < \sqrt{2/3}|q|$), a necessary ingredient for regularity (Rodrigues et al., 2018). This feature remains operative when embedded into AdS space.

2. Thermodynamics in Extended Phase Space and Black Hole Chemistry

The thermodynamics of Bardeen-AdS black holes is conventionally formulated in the extended phase space, where $\Lambda$ is interpreted as thermodynamic pressure, $P = -\Lambda/(8\pi)$, and the black hole mass $M$ as enthalpy rather than internal energy. The first law (for fixed nonlinear charge parameter $q$) is

$dM = T\,dS + V\,dP$

The Hawking temperature, expressed as a function of the event horizon $r_+$, is

$$T = \frac{3r_+^4 + l^2(r_+^2-2q^2)}{4\pi l^2 r_+(q^2 + r_+^2)}$$

The entropy for large black holes reduces to the standard area law: $S = \pi r_+^2$ but incorporates subleading corrections near the regularity scale $q$ (Tzikas, 2018). The thermodynamic volume is

$$V = \frac{4\pi}{3} r_+^3 \left( 1 + \frac{q^2}{r_+^2} \right)^{3/2}$$

and the equation of state links $P$, $T$, and $V$ (Tzikas, 2018): $$P = \frac{12q^2 + \left(6\pi V\right)^{2/3}\left[-1 + 2\pi T\sqrt{(6\pi V)^{2/3}-4q^2}\right]}{2\pi \left[(6\pi V)^{2/3} - 4q^2\right]^2}$$

This formalism leads to "black hole chemistry": the (neutral) Bardeen-AdS black hole undergoes phase behavior closely analogous to the Van der Waals fluid, including a first-order small/large black hole transition (liquid/gas type), critical exponents matching mean-field universality ($\alpha=0$, $\beta=1/2$, $\gamma=1$, $\delta=3$), and a swallowtail in the Gibbs free energy (Tzikas, 2018, Rizwan et al., 2018, Rizwan et al., 2020).

Importantly, for large enough $q$, the phase transitions disappear—mirroring the effect of high charge in Reissner–Nordström–AdS black holes.

3. Critical Phenomena, Phase Structure, and Novel Transitions

Regular Bardeen-AdS black holes exhibit a range of critical behaviors:

• For $q < q_c$, the $T$–$S$ and $P$–$V$ diagrams display multiple branches: small/large black holes separated by an unstable intermediate region, supporting first-order phase transitions. The critical point arises where these branches coalesce and the specific heat $C_P$ diverges, realizing a second-order transition (Tzikas, 2018, Rizwan et al., 2018, Guo et al., 2021).
• The coexistence and spinodal curves have been quantified numerically and display a structure matching van der Waals/Maxwell theory (Rizwan et al., 2020). Order parameters (such as the volume gap $\Delta V_r$ between small and large black hole phases) scale near criticality with exponents in agreement with mean-field predictions.
• Discontinuous and multi-branched temperature–horizon curves emerge, especially in generalized Bardeen-AdS black hole families, leading to additional phases ("tiny black holes") and transitions between non-horizon (regular) backgrounds and black hole states (Hawking–Page analogs) (Wu et al., 29 Jul 2024).
• Under certain constraint-induced reductions of the thermodynamic phase space, the regular Bardeen–AdS black hole’s Gibbs free energy diagram departs from the usual “swallow-tail” and instead exhibits “8-shaped” or “C-shaped” structures, corresponding to first- or zeroth-order transitions between black hole phases (Ma et al., 8 Oct 2025).

4. Extended Theories: Higher Curvature, Massive Gravity, and Quantum Corrections

Bardeen–AdS black holes have been embedded in numerous extensions, with significant impact on both the geometry and the thermodynamics:

• In $D \geq 5$ dimensions, coupling to Einstein–Gauss–Bonnet (EGB) gravity with nonlinear electrodynamics yields exact regular solutions (Bardeen–EGB–AdS black holes), possessing three parameters: mass $M$, (regularizing) charge $e$, and GB coupling $\alpha$. The entropy departs from the area law and involves hypergeometric corrections, and the specific heat exhibits divergences at critical horizon radii signaling phase transitions similar to the Hawking–Page transition (Kumar et al., 2018, Singh et al., 2019, Kumar et al., 2023).
• In de Rham-Gabadadze-Tolley (dRGT) massive gravity, the Bardeen-AdS solution acquires additional mass terms and thermodynamics is modified according to new “gravity parameters,” which shift the critical points and related phase structure (Singh et al., 2022).
• Quantum corrections via a generalized uncertainty principle (GUP) can reinterpret the Bardeen solution as a quantum-corrected Schwarzschild or AdS black hole, tightly relating the minimal length scale (e.g., $r_0$) to quantum gravity parameters. GUP-induced corrections yield logarithmic and inverse area terms in the entropy, modify the Hawking temperature, and can produce black hole remnants (Maluf et al., 2018, Merriam et al., 2021).

5. Microstructure, Thermodynamic Geometry, and Shadow Probes

The microstructure and response functions of the Bardeen–AdS black hole have been explored through thermodynamic geometry:

• Weinhold and Ruppeiner geometries—built from the Hessian of the mass or entropy—yield indicating scalar curvatures: negative curvature suggests attractive microscopic interactions, while positive curvature signals repulsion. Bardeen-AdS black holes display positive curvature (repulsion) for small volumes, a feature absent in standard van der Waals fluids (Guo et al., 2021), but generally show negative curvature (attraction) for large black holes, similar to the Reissner–Nordström–AdS case (Rizwan et al., 2020). Singularities/divergences of the curvature scalars signal phase transitions, and Legendre-invariant Quevedo geometry aligns curvature divergences with those seen in specific heat (Rizwan et al., 2018).
• From a microscopic Landau theory perspective, an order parameter—defined via the potential gap between coexistent phases—enables the analytic extraction of critical exponents (Guo et al., 2021).
• The shadow radius $r_s$ of the Bardeen–AdS black hole, which is the impact parameter of photon orbits forming the black hole shadow, correlates monotonically with the event horizon and can be used as an observable probe of the black hole’s thermodynamic phase structure. The thermodynamic temperature and phase transition order can be rewritten in terms of the shadow radius, implying that astronomical observation of the black hole shadow can, in principle, reveal information about black hole phase transitions and criticality (Guo et al., 2022).

6. Dynamical and Scattering Properties

The dynamical response of Bardeen–AdS black holes, especially their quasinormal modes (QNMs), is strongly influenced by regularity and nonlinear electromagnetic effects:

• Scattering of massless scalar waves demonstrates that Bardeen black holes display interference patterns sensitive to the charge parameter; the cross section can mimic that of Reissner–Nordström black holes but presents distinguishable features for some parameter ranges (Macedo et al., 2015). A combination of partial-wave methods, classical geodesic analysis, and semiclassical glory approximation enables a detailed paper of their response to external perturbations.
• Gravitational QNMs differ for axial and polar perturbations due to the breakdown of isospectrality caused by nonlinear electromagnetic coupling. As the magnetic charge increases, the real and imaginary parts of the QNM frequencies shift, and the deviation between axial and polar spectra (parity splitting) becomes significant (Zhao et al., 2023). AdS asymptotics allow use of methods such as the Horowitz–Hubeny technique for QNM computation.
• The presence of a regular de Sitter core modifies low-energy response properties (e.g., the inversion/critical temperature ratio in the Joule–Thomson expansion is notably higher than that of singular black holes) (Pu et al., 2019).

7. Summary Table: Key Features and Distinctions

Feature	Regular Bardeen–AdS BH	Singular AdS BH (e.g. RN–AdS)
Central structure	de Sitter core, $R<\infty$	curvature singularity, $R\to\infty$
Source	Nonlinear electrodynamics (mag/electric)	Linear E&M (Reissner–Nordström)
Equation of state	Van der Waals–like, with $q$ as minimal length	Van der Waals–like, $Q$ as charge
Phase transitions	First-/second-/zeroth-order, swallowtail and novel “8-”/“C-shaped” structures with constraints	Usual first-order (swallowtail)
Shadow/observable	$r_s$ monotonic with $r_h$, encodes transitions	Similar, but profile can differ near regularity scale
Evaporation endpoint	Stable remnant (at extremality, $T=0$)	Complete evaporation (often $T\to\infty$)
Microstructure geometry	Repulsive for small $V$, attractive for large $V$	Attractive (Ruppeiner)
Isospectrality of QNMs	Broken for high $q$	Typically preserved

In sum, the Bardeen–AdS black hole occupies a unique position in black hole physics, serving as both a laboratory for regular geometries and a key example in AdS/CFT and black hole chemistry. Its thermodynamic and dynamical features, particularly the emergence of novel phase transition structures and microstructural properties, offer broad opportunities for probing gravitational, quantum, and thermodynamic phenomena in the strong-field regime.