Bardeen–AdS-Class Black Hole Thermodynamics

• Bardeen–AdS-class black holes are nonsingular, asymptotically AdS solutions with a de Sitter-like core induced by nonlinear electrodynamics and controlled by magnetic charge parameters.
• Their extended thermodynamics interprets black hole mass as enthalpy, revealing van der Waals–type phase transitions and unique Gibbs free energy profiles.
• Key insights also include the classification into Type I and Type II systems, intricate horizon behaviors, and distinctive Joule–Thomson expansion characteristics.

Searching arXiv for recent and foundational papers on Bardeen-AdS-class black holes, thermodynamics, and phase structure. The Bardeen–AdS-class black hole denotes a family of asymptotically anti-de Sitter, nonsingular black-hole solutions supported by nonlinear electrodynamics, together with generalized constructions in which the original Bardeen solution appears as a special case. In the four-dimensional static sector, the metric is typically written as $ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2d\Omega_2^2$, with regularization controlled by a magnetic charge or nonlinear parameter such as $g$, $\beta$, or $q$; in rotating settings it is extended to a Boyer–Lindquist form with rotation parameter $a$ (Li et al., 2018, Chen, 2020, Belhaj et al., 2022). Recent work has also introduced a broader Bardeen–AdS-class construction in which the asymptotic mass and magnetic charge are separated from Lagrangian couplings, yielding two thermodynamic categories, Type I and Type II, with distinct phase structures (Wu et al., 2024, Ma et al., 28 Apr 2026).

1. Geometric definition and regularization mechanism

For the four-dimensional Bardeen–AdS black hole, the metric function is repeatedly given in the form

$$f(r)=1-\frac{2M r^2}{(r^2+g^2)^{3/2}}+\frac{r^2}{\ell^2},$$

with $\Lambda=-3/\ell^2$, where $M$ is the ADM mass and $g$ is the magnetic charge or nonlinear-electrodynamics parameter (Li et al., 2018, Li et al., 2019, Guo et al., 2021). Equivalent notation using $\beta$ or $g$0 is also employed (V. et al., 2019, Wang et al., 2022). In the original static Bardeen–AdS setting, the mass profile is written as

$g$1

or equivalently $g$2 (Li et al., 2018, Chen, 2020).

A recurrent structural point is that the Bardeen solution replaces the central singularity of Reissner–Nordström–AdS by a de Sitter-like core generated by nonlinear electrodynamics (Pu et al., 2019, Wang et al., 2022). One summary states that “near $g$3 the effective stress–energy behaves like a de Sitter core and all curvature invariants remain finite” (Wang et al., 2022). In the rotating Bardeen–AdS geometry, the line element is

$g$4

with

$g$5

and

$g$6

(Chen, 2020, Belhaj et al., 2022).

The generalized Bardeen–AdS-class construction uses the metric

$g$7

with physical mass $g$8 and magnetic charge $g$9, while $\beta$0 are fixed couplings (Wu et al., 2024, Ma et al., 28 Apr 2026). When $\beta$1 and $\beta$2, one recovers the original Bardeen–AdS solution (Ma et al., 28 Apr 2026).

2. Extended thermodynamics and horizon variables

A central convention across the literature is the extended phase space in which

$\beta$3

and the black-hole mass is interpreted as enthalpy $\beta$4 (Pu et al., 2019, Chen, 2020, Guo et al., 2021). For the static Bardeen–AdS black hole, the horizon radius $\beta$5 or $\beta$6 is the largest positive root of $\beta$7, which yields

$\beta$8

(Li et al., 2018, Li et al., 2019, Guo et al., 2021).

Entropy is generally taken to satisfy the area law,

$\beta$9

and several later analyses explicitly emphasize that the correct entropy and thermodynamic volume are the standard ones (Guo et al., 2021). In that treatment,

$q$0

whereas some earlier papers instead obtained

$q$1

or

$q$2

(Pu et al., 2019, Li et al., 2019, Guo et al., 2021). The coexistence of these formulas is part of the thermodynamic ambiguity discussed in the literature.

For the static Bardeen–AdS black hole, the Hawking temperature is commonly written as

$q$3

(Li et al., 2019, Guo et al., 2021). In entropy variables, one convenient enthalpy form is

$q$4

(Pu et al., 2019), with analogous formulas for $q$5 and for the rotating case $q$6 (Chen, 2020).

The extended first law is usually written as

$q$7

for the static case (Guo et al., 2021), and as

$q$8

for the rotating case (Chen, 2020). The corresponding Smarr relation is

$q$9

in the static setting (Pu et al., 2019, Guo et al., 2021), and

$a$0

for rotating Bardeen–AdS black holes (Chen, 2020).

A separate line of work argues that imposing a regularity constraint can break the standard first law. In that framework, the Bardeen–AdS black hole is obtained from a singular “mother” black hole, and “the usual first law of black hole thermodynamics” no longer has the standard independent-variation form (Ma et al., 8 Oct 2025). This suggests that part of the thermodynamic literature is organized around inequivalent conventions rather than a single universally accepted one.

3. Equation of state, criticality, and phase-transition structure

The Bardeen–AdS black hole exhibits van der Waals–type criticality in much of the extended-thermodynamics literature. Using the specific volume $a$1, one finds an equation of state such as

$a$2

or equivalent $a$3 forms (Guo et al., 2021, V. et al., 2019). Critical points satisfy

$a$4

or equivalently the inflection conditions on $a$5 (Li et al., 2018, Guo et al., 2021).

Several papers give explicit critical values. One widely quoted set is

$a$6

with

$a$7

(V. et al., 2019). Another formulation gives

$a$8

(Guo et al., 2021).

The phase structure at fixed pressure is commonly described in terms of small, intermediate, and large black-hole branches. For $a$9, one early analysis found two divergences of $$f(r)=1-\frac{2M r^2}{(r^2+g^2)^{3/2}}+\frac{r^2}{\ell^2},$$0, separating a small stable branch, an unstable intermediate branch, and a large stable branch; at $$f(r)=1-\frac{2M r^2}{(r^2+g^2)^{3/2}}+\frac{r^2}{\ell^2},$$1 the two divergences merge, and for $$f(r)=1-\frac{2M r^2}{(r^2+g^2)^{3/2}}+\frac{r^2}{\ell^2},$$2 only a single stable branch remains (Li et al., 2018). The critical monopole charge is

$$f(r)=1-\frac{2M r^2}{(r^2+g^2)^{3/2}}+\frac{r^2}{\ell^2},$$3

(Li et al., 2018).

In the Gibbs ensemble, the usual account is a swallow-tail below $$f(r)=1-\frac{2M r^2}{(r^2+g^2)^{3/2}}+\frac{r^2}{\ell^2},$$4, signaling a first-order small/large black-hole transition (V. et al., 2019, Guo et al., 2021). However, more recent analyses refine this picture. At fixed pressure, the direct-horizon convention yields four topological classes of raw Gibbs curves: RN–AdS–like, 8-shaped, c-shaped, and single-branch (Wang, 25 May 2026). Three pressure-dependent thresholds are introduced: $$f(r)=1-\frac{2M r^2}{(r^2+g^2)^{3/2}}+\frac{r^2}{\ell^2},$$5 equivalently controlled by $$f(r)=1-\frac{2M r^2}{(r^2+g^2)^{3/2}}+\frac{r^2}{\ell^2},$$6 (Wang, 25 May 2026). The equilibrium Gibbs construction shows that stable small/large coexistence can survive the first topology change, while the c-shaped regime has no stable crossing (Wang, 25 May 2026).

A related 2025 treatment states that the swallow-tail may disappear entirely, being replaced by an “8-shaped” or “c-shaped” structure associated with first-order or zeroth-order transitions under a regularity constraint (Ma et al., 8 Oct 2025). This indicates an active interpretive issue: the Bardeen–AdS thermodynamic phase structure depends sensitively on the adopted thermodynamic state space and constraint structure.

4. Type I and Type II Bardeen–AdS-class systems

The generalized Bardeen–AdS-class system is explicitly classified into Type I and Type II according to whether the Bardeen–AdS black hole itself or a pure Bardeen–AdS spacetime without event horizons is included as a phase state (Wu et al., 2024). In the canonical ensemble with $$f(r)=1-\frac{2M r^2}{(r^2+g^2)^{3/2}}+\frac{r^2}{\ell^2},$$7, the horizon equation becomes

$$f(r)=1-\frac{2M r^2}{(r^2+g^2)^{3/2}}+\frac{r^2}{\ell^2},$$8

(Wu et al., 2024).

For Type I ($$f(r)=1-\frac{2M r^2}{(r^2+g^2)^{3/2}}+\frac{r^2}{\ell^2},$$9), the equation $\Lambda=-3/\ell^2$0 admits at least one root, so the Bardeen–AdS black hole is among the physical states (Wu et al., 2024). The branch structure of $\Lambda=-3/\ell^2$1 is “qualitatively identical to the Reissner–Nordström–AdS case,” with three intersections for $\Lambda=-3/\ell^2$2, and a swallowtail in the Gibbs free energy signaling a first-order small/large transition ending at a second-order critical point (Wu et al., 2024).

For Type II ($\Lambda=-3/\ell^2$3), $\Lambda=-3/\ell^2$4 corresponds to a pure regular AdS spacetime without horizon (Wu et al., 2024). In this case, besides up to four black-hole branches—tiny, small, medium, and large—there is also a vacuum state with free energy

$\Lambda=-3/\ell^2$5

(Wu et al., 2024). As temperature increases from zero, the vacuum undergoes a Hawking–Page-type transition to the smallest black-hole branch at $\Lambda=-3/\ell^2$6, followed at higher temperature by a small-to-large first-order transition (Wu et al., 2024). When $\Lambda=-3/\ell^2$7, the outer-horizon domain splits into disjoint intervals and the $\Lambda=-3/\ell^2$8-$\Lambda=-3/\ell^2$9 curves become discontinuous (Wu et al., 2024).

The stochastic-dynamics analysis of the same class uses the generalized free energy

$M$0

and in the $M$1 frame,

$M$2

(Ma et al., 28 Apr 2026). Extrema satisfy $M$3. Type I admits a small/large-black-hole transition that may pass through a stable, metastable, or unstable regular black hole as an intermediate state, whereas Type II exhibits only vacuum-to-small-black-hole transitions and does not involve any regular-black-hole intermediate state (Ma et al., 28 Apr 2026).

5. Joule–Thomson expansion and throttling phenomena

Joule–Thomson expansion is one of the most developed thermodynamic probes of Bardeen–AdS black holes. The basic coefficient is

$M$4

(Pu et al., 2019, Chen, 2020). For the static Bardeen–AdS black hole, one analytic form is

$M$5

(Li et al., 2019). The denominator vanishes exactly when $M$6, so the divergence point of $M$7 coincides with the zero-temperature point (Li et al., 2019, Chen, 2020).

The inversion curve is defined by $M$8. In the static case the inversion condition may be written as

$M$9

(Pu et al., 2019), or equivalently

$g$0

(Li et al., 2019). Across the Bardeen–AdS literature, the inversion curve is a single lower branch, not a closed curve as in the van der Waals fluid (Pu et al., 2019, Chen, 2020). The rotating case shows the same qualitative feature: “there are only minimum inversion temperature but no maximum inversion temperature” (Chen, 2020).

The minimum inversion temperature occurs at $g$1. One static analysis gives

$g$2

(Pu et al., 2019), while another gives

$g$3

(Li et al., 2019). In the rotating case, the ratio is “a little greater than $g$4” and increases with the nonlinear parameter $g$5; for $g$6, values range from approximately $g$7 at $g$8 to approximately $g$9 at $\beta$0 (Chen, 2020). That work further reports that the ratio $\beta$1 is a monotonically increasing function of $\beta$2, lying between $\beta$3 and $\beta$4 (Chen, 2020).

Isenthalpic curves $\beta$5 at fixed enthalpy intersect the inversion curve at exactly the inversion point (Pu et al., 2019, Li et al., 2019). For the rotating Bardeen–AdS black hole, each isenthalpic curve has one extremum $\beta$6, separating cooling ($\beta$7) from heating ($\beta$8), and cooling exists only when the mass is not less than a certain $\beta$9 (Chen, 2020). The static treatment similarly identifies a minimum mass

$g$00

for the onset of inversion behavior (Li et al., 2019). Below that value, the black hole always heats under expansion (Li et al., 2019).

Quintessence deforms the Joule–Thomson structure. For the regular Bardeen–AdS black hole surrounded by static anisotropic quintessence, the state parameter $g$01 and normalization $g$02 both raise inversion temperatures, and increasing $g$03 raises the isenthalpic-curve maxima (2002.03634).

6. Deformations, probes, and topological characterizations

Several extensions of the Bardeen–AdS family preserve the regular-core logic while changing the thermodynamic and dynamical response.

With quintessence, the metric acquires an additional term $g$04 or $g$05 (Rizwan et al., 2018, Wang et al., 2022, Xie et al., 2024). Thermodynamically, quintessence shifts critical parameters and moves the heat-capacity divergence to lower entropy values (Rizwan et al., 2018). Dynamically, neutral test-particle motion becomes chaotic: Poincaré sections, power spectra, and bifurcation diagrams show that the presence of quintessence creates chaos, and the ISCO radius increases monotonically with the quintessence normalization $g$06 (Xie et al., 2024). A complementary Melnikov analysis of thermal chaos concludes that spatial chaos is always supposed to occur under a tiny spatially periodic perturbation, whereas temporal chaos appears in the unstable spinodal region only when the perturbation amplitude exceeds a critical value $g$07 (Wang et al., 2022).

In Einstein–Gauss–Bonnet gravity, both four-dimensional and higher-dimensional Bardeen–AdS generalizations have been studied (Singh et al., 2020, Kumar et al., 2018, Kumar et al., 2023). The four-dimensional Gauss–Bonnet-corrected Bardeen–AdS black hole exhibits $g$08-$g$09 criticality in the van der Waals universality class, and the Gauss–Bonnet coupling $g$10 lowers the critical temperature and raises the critical volume slightly while leaving $g$11 nearly unchanged (Singh et al., 2020). In the five-dimensional exact Einstein–Gauss–Bonnet solution, the critical exponents are exactly the mean-field values $g$12 (Kumar et al., 2023). In higher-dimensional Einstein–Gauss–Bonnet theory, there is a critical charge $g$13 corresponding to an extremal regular black hole with degenerate horizons, and evaporation leads to a thermodynamically stable remnant (Kumar et al., 2018).

Thermodynamic geometry has been used to probe microstructure. In Weinhold geometry, one study argues that the Bardeen–AdS black hole has standard entropy and volume, van der Waals criticality, and a repulsive interaction in the small-volume state of the microstructure, unlike the van der Waals fluid which has only attractive interaction (Guo et al., 2021). In the quintessence background, Weinhold and Ruppeiner curvatures identify critical behavior but do not diverge at the same point as the specific heat, whereas the Legendre-invariant Quevedo curvature does (Rizwan et al., 2018).

A distinct topological framework uses the generalized off-shell Helmholtz free energy and Duan’s $g$14-mapping. For regular Bardeen–AdS black holes, one finds a single zero of the $g$15-field with positive winding, yielding

$g$16

(Sadeghi et al., 2023). The same topological number persists under quintessence, massive-gravity, and Gauss–Bonnet deformations (Sadeghi et al., 2023). That study interprets $g$17 as the statement that there is a single thermodynamically stable branch in the off-shell free-energy landscape and no coexistence of small/large branches with opposite indices (Sadeghi et al., 2023). This suggests a topological notion of stability distinct from the equilibrium Gibbs constructions discussed above.

Optical probes have also been developed. For rotating Bardeen–AdS black holes, the nonlinear charge $g$18 controls the shadow shape, while dark energy modifies both size and deformation (Belhaj et al., 2022).

7. Conceptual issues and current research directions

Two tensions organize the current literature.

The first concerns the thermodynamic state space. Some works treat the Bardeen–AdS black hole as a straightforward extended thermodynamic system with standard first law, Smarr relation, swallow-tail Gibbs structure, and mean-field critical exponents (V. et al., 2019, Guo et al., 2021). Other works emphasize that the regularity constraint changes the admissible thermodynamic variables, leading to nonstandard Gibbs topology, disappearance of the swallow-tail, or even breakdown of the standard first law (Ma et al., 8 Oct 2025, Wang, 25 May 2026). The Bardeen–AdS-class formulation sharpens this issue by distinguishing Type I and Type II systems and by allowing vacuum states, tiny-black-hole branches, and discontinuous characteristic curves associated with multiple horizons (Wu et al., 2024).

The second concerns the relation between regularity and phase structure. Multiple papers connect the de Sitter-like core to distinctive thermodynamic behavior. For the Joule–Thomson process, one paper explicitly suggests that the unusually large inversion-to-critical-temperature ratio may stem from the repulsive de Sitter core near the origin (Pu et al., 2019). In Weinhold geometry, a repulsive microstructural regime appears at small volume (Guo et al., 2021). This suggests that the regularization mechanism is not merely a short-distance cure of curvature singularities, but also a source of qualitative changes in thermal response.

Across these lines of work, the Bardeen–AdS-class black hole has become a laboratory for comparing several notions of phase behavior: equilibrium Gibbs selection, spinodal topology, Joule–Thomson inversion, topological charge, geometrothermodynamic curvature, and stochastic transition kinetics (Ma et al., 28 Apr 2026, Wang, 25 May 2026). The cumulative result is not a single universal thermodynamic portrait, but a structured family of portraits whose differences track the choice of variables, constraints, and deformations.