Hayward–AdS Black Hole: Regularization & Thermodynamics

• Hayward–AdS black hole is a regular, static AdS solution derived by coupling Einstein gravity with nonlinear electrodynamics to remove central singularities.
• It exhibits complex thermodynamics with van der Waals-like P-V criticality, distinct Joule–Thomson expansion behavior, and efficient heat-engine cycles.
• Observable signatures like shadow thermodynamics and Ruppeiner geometry provide insights into its microstructure and phase transitions.

The Hayward–AdS black hole is a regular, static, spherically symmetric black-hole solution in anti-de Sitter spacetime, typically obtained by coupling Einstein gravity to nonlinear electrodynamics and by introducing a negative cosmological constant. In its standard four-dimensional form, the line element is commonly written as

$$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2, \qquad f(r)=1+\frac{r^2}{l^2}-\frac{2Mr^2}{r^3+q^3},$$

with a magnetic-type gauge potential $A=Q_m\cos\theta\,d\phi$. The parameter $q$ or $g$, depending on notation, encodes the Hayward regularization and is associated with the nonlinear magnetic sector, while $l$ is the AdS radius and $P=-\Lambda/(8\pi)=3/(8\pi l^2)$ in extended thermodynamics (Guo et al., 2019, Guo et al., 2019, Gogoi et al., 2024). The subject has developed into a broad program spanning black-hole chemistry, Joule–Thomson expansion, heat engines, Ruppeiner microstructure, shadow thermodynamics, Lyapunov diagnostics, and matter-dressed generalizations.

1. Geometric construction and regular character

A widely used realization starts from Einstein gravity coupled to nonlinear electrodynamics, with action

$$S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\left(R-2\Lambda-\mathcal{L}(\mathcal{F})\right),$$

and yields the ordinary Hayward–AdS metric function

$f(r)=1+\frac{r^2}{\ell^2}-\frac{2Mr^2}{r^3+Q^3}.$

Within this framework, $Q$ is the regularization or charge parameter related to the magnetic charge of the nonlinear electrodynamics sector, and the ordinary Hayward–AdS solution is recovered by switching off additional deformations such as quintessence (Belhaj et al., 2022).

The defining geometric feature is regularization of the central region. Several papers describe the Hayward parameter as removing the curvature singularity at $r=0$, replacing the Schwarzschild-type interior by a regular core. In later matter-dressed models this regularity may persist or fail depending on the extra matter content. For a Hayward–AdS black hole surrounded by a fluid of strings, regularity survives only in the window

$A=Q_m\cos\theta\,d\phi$0

and the thermodynamic analysis is then specialized to the regular case $A=Q_m\cos\theta\,d\phi$1 (Nascimento et al., 2024).

The notation is not fully uniform across the literature. Four-dimensional AdS papers use $A=Q_m\cos\theta\,d\phi$2, $A=Q_m\cos\theta\,d\phi$3, or $A=Q_m\cos\theta\,d\phi$4 for the Hayward/nonlinear-magnetic parameter, whereas higher-dimensional studies use a distinct Hayward parameter $A=Q_m\cos\theta\,d\phi$5 in a different family of metrics,

$A=Q_m\cos\theta\,d\phi$6

with $A=Q_m\cos\theta\,d\phi$7 (Abdusattar, 13 Oct 2025). This suggests that “Hayward–AdS black hole” denotes a class of regular AdS geometries rather than a single universally normalized solution.

2. Thermodynamic formulations in AdS

In the extended phase-space formulation, the cosmological constant is treated as pressure and the black-hole mass is interpreted as enthalpy. For the Hayward–AdS solution used in the Joule–Thomson analysis, the horizon radius $A=Q_m\cos\theta\,d\phi$8 is defined by $A=Q_m\cos\theta\,d\phi$9, the entropy obeys the area law $q$0, the thermodynamic volume is

$q$1

and the first law and Smarr relation are

$q$2

Here $q$3 is treated as a thermodynamic variable associated with the nonlinear electromagnetic sector, and $q$4 (Guo et al., 2019).

A separate thermodynamic formulation, used in studies of continuous phase transitions and microstructure, treats the regular Hayward–AdS black hole with non-standard entropy and volume,

$q$5

together with

$q$6

In that formulation the magnetic potential

$q$7

is central and is later used as an order parameter (Kumara et al., 2020).

The literature explicitly notes that these formulations are not always interchangeable. For Hayward–AdS black holes sourced by nonlinear electrodynamics, the temperature $q$8 can differ from the surface-gravity temperature $q$9, and the standard first law must then be corrected by a factor

$g$0

This is emphasized in the fluid-of-strings generalization, where the correction factor bridges the naive thermodynamics and the temperature computed from the metric (Nascimento et al., 2024).

Further modifications arise in 4D Einstein–Gauss–Bonnet gravity, where the first law is written as

$g$1

and the entropy becomes

$g$2

In that setting, the Hayward parameter $g$3 regularizes the geometry while the Gauss–Bonnet coupling $g$4 deforms the gravity sector (Zhang et al., 2021).

3. Equation of state, criticality, and phase structure

A standard equation of state used in the heat-engine literature is

$g$5

which displays an additional $g$6 term induced by the regularization parameter $g$7. In that treatment the critical quantities scale as

$g$8

and $g$9, so isochores coincide with adiabats (Guo et al., 2019).

Other thermodynamic schemes produce a corrected equation of state. In the formulation based on the non-standard entropy and volume,

$l$0

the system undergoes a van der Waals-like phase transition between small and large black holes, with critical values

$l$1

(Kumara et al., 2020). Closely related shadow-thermodynamic work gives

$l$2

again showing explicit scaling with magnetic charge (Luo et al., 2023).

For the ordinary Hayward–AdS solution written in specific-volume form $l$3, the critical ratio is

$l$4

which differs from the standard van der Waals or charged-AdS value $l$5. In the quintessential extension the ratio becomes model dependent, $l$6, and for $l$7 it remains essentially unchanged and equal to $l$8 (Belhaj et al., 2022).

The global phase structure is usually van der Waals-like, but some constrained constructions differ qualitatively from RN–AdS. When a singular parent solution is constrained to produce a Hayward–AdS black hole, the regular solution retains $l$9-$P=-\Lambda/(8\pi)=3/(8\pi l^2)$0 criticality, yet the Gibbs free energy is not swallowtail-like in the RN–AdS sense. For $P=-\Lambda/(8\pi)=3/(8\pi l^2)$1, the $P=-\Lambda/(8\pi)=3/(8\pi l^2)$2-$P=-\Lambda/(8\pi)=3/(8\pi l^2)$3 curves evolve through an “8-shaped” knot, then a “0-like” loop, and finally a “C-shaped” profile; for $P=-\Lambda/(8\pi)=3/(8\pi l^2)$4, the “C-shaped” structure signals a zeroth-order phase transition between small and large black holes (Xia et al., 12 Feb 2026). This suggests that regularization can change not only the ultraviolet geometry but also the topology of the thermodynamic free-energy landscape.

Higher-dimensional Hayward–AdS black holes exhibit the standard mean-field critical exponents

$P=-\Lambda/(8\pi)=3/(8\pi l^2)$5

and the scaling laws of classical critical phenomena. In that family, the Hayward parameter $P=-\Lambda/(8\pi)=3/(8\pi l^2)$6 is explicitly responsible for the emergence of the nontrivial $P=-\Lambda/(8\pi)=3/(8\pi l^2)$7–$P=-\Lambda/(8\pi)=3/(8\pi l^2)$8 criticality, since the $P=-\Lambda/(8\pi)=3/(8\pi l^2)$9 limit reduces to Schwarzschild–AdS and does not exhibit the same phase structure (Abdusattar, 13 Oct 2025).

4. Joule–Thomson expansion and heat-engine cycles

The first dedicated study of Joule–Thomson expansion for the Hayward–AdS black hole treats the process as isenthalpic, with

$$S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\left(R-2\Lambda-\mathcal{L}(\mathcal{F})\right),$$0

Rewriting pressure and temperature as functions of $$S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\left(R-2\Lambda-\mathcal{L}(\mathcal{F})\right),$$1,

$$S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\left(R-2\Lambda-\mathcal{L}(\mathcal{F})\right),$$2

$$S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\left(R-2\Lambda-\mathcal{L}(\mathcal{F})\right),$$3

the paper derives the explicit coefficient

$$S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\left(R-2\Lambda-\mathcal{L}(\mathcal{F})\right),$$4

The inversion curve is determined by $$S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\left(R-2\Lambda-\mathcal{L}(\mathcal{F})\right),$$5. Above it the black hole cools, below it it heats, and unlike the standard van der Waals fluid the Hayward–AdS black hole has only one inversion curve and it is not closed. Increasing $$S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\left(R-2\Lambda-\mathcal{L}(\mathcal{F})\right),$$6 raises the inversion temperature (Guo et al., 2019).

The 4D Einstein–Gauss–Bonnet extension preserves this qualitative picture. It has exactly one positive critical point for any positive $$S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\left(R-2\Lambda-\mathcal{L}(\mathcal{F})\right),$$7 and $$S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\left(R-2\Lambda-\mathcal{L}(\mathcal{F})\right),$$8, and its Joule–Thomson expansion again shows a non-closed inversion curve separating cooling and heating regions. In that model increasing $$S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\left(R-2\Lambda-\mathcal{L}(\mathcal{F})\right),$$9 lowers the inversion temperature at fixed pressure, while increasing $f(r)=1+\frac{r^2}{\ell^2}-\frac{2Mr^2}{r^3+Q^3}.$0 raises it (Zhang et al., 2021).

As a heat engine, the Hayward–AdS black hole is usually studied on a rectangular $f(r)=1+\frac{r^2}{\ell^2}-\frac{2Mr^2}{r^3+Q^3}.$1-$f(r)=1+\frac{r^2}{\ell^2}-\frac{2Mr^2}{r^3+Q^3}.$2 cycle. Because $f(r)=1+\frac{r^2}{\ell^2}-\frac{2Mr^2}{r^3+Q^3}.$3, only the isobaric legs exchange heat, and

$f(r)=1+\frac{r^2}{\ell^2}-\frac{2Mr^2}{r^3+Q^3}.$4

For the ordinary Hayward–AdS black hole, the efficiency increases with pressure, depends nontrivially on entropy, and increases with magnetic charge $f(r)=1+\frac{r^2}{\ell^2}-\frac{2Mr^2}{r^3+Q^3}.$5. The same paper reports that, for the example values used in the plots,

$f(r)=1+\frac{r^2}{\ell^2}-\frac{2Mr^2}{r^3+Q^3}.$6

so the Hayward–AdS engine is more efficient than the Bardeen–AdS engine in that comparison (Guo et al., 2019).

More elaborate matter sectors preserve the heat-engine viewpoint while changing parameter dependence. In the charged Hayward–AdS black hole with a cloud of strings and perfect fluid dark matter, the minimum inversion ratio is

$f(r)=1+\frac{r^2}{\ell^2}-\frac{2Mr^2}{r^3+Q^3}.$7

the parameters $f(r)=1+\frac{r^2}{\ell^2}-\frac{2Mr^2}{r^3+Q^3}.$8 and $f(r)=1+\frac{r^2}{\ell^2}-\frac{2Mr^2}{r^3+Q^3}.$9 contract the cooling region, $Q$0 expands it, and the string-cloud parameter $Q$1 reshapes it non-monotonically. For a representative rectangular cycle the reported efficiencies lie in the range $Q$2–$Q$3, with $Q$4–$Q$5 (Al-Badawi et al., 1 Mar 2026).

5. Order parameters, Ruppeiner geometry, and microstructure

The Hayward–AdS phase transition has been reinterpreted beyond the usual small/large black-hole language. One proposal uses the magnetic potential $Q$6 as a new order parameter and defines

$Q$7

In that picture the transition is between a high-potential phase and a low-potential phase, and a Landau expansion of the Gibbs free energy yields the mean-field exponents

$Q$8

The interpretation given is that the high-potential phase is more ordered and the low-potential phase less ordered (Kumara et al., 2020).

Ruppeiner geometry is then used to diagnose the interaction of effective black-hole molecules. Because $Q$9 in several Hayward–AdS models, the raw Ruppeiner scalar is singular in $r=0$0 coordinates, and the normalized scalar $r=0$1 is used instead. In one four-dimensional analysis, $r=0$2 over most of the phase space, so attractive interactions dominate as in a van der Waals fluid, but a weak repulsive region appears for the small-black-hole phase at low temperature. This is presented as a clear distinction from the ordinary van der Waals fluid, even though the macroscopic phase diagram remains van der Waals-like (Kumara et al., 2020).

The sign structure is model dependent. In the Landau-order-parameter study, charged and uncharged effective molecules are assigned distinct microstructures analogous to boson-gas and fermion-gas behavior, with $r=0$3 interpreted as dominant repulsion and $r=0$4 as dominant attraction (Kumara et al., 2020). In the higher-dimensional family, the normalized curvature again diverges at criticality, but the sign can change along the coexistence region and can become positive in parts of the large-black-hole branch at lower temperatures (Abdusattar, 13 Oct 2025). This suggests that Hayward regularization does not fix a unique microscopic interaction pattern; it constrains a class of regular systems whose reduced thermodynamic geometry remains model sensitive.

6. Shadow thermodynamics and dynamical probes

A major recent direction treats the black-hole shadow as a thermodynamic observable. For the Hayward–AdS black hole, the shadow radius $r=0$5 is found to be a monotonic function of the horizon radius $r=0$6, so it can be used as an equivalent thermodynamic variable. The corresponding $r=0$7 diagrams reproduce the same phase structure as the $r=0$8 diagrams: non-monotonic behavior for $r=0$9, branch merging at $A=Q_m\cos\theta\,d\phi$00, and monotonicity for $A=Q_m\cos\theta\,d\phi$01. Increasing magnetic charge enlarges the shadow radius while lowering the coexistence temperature, and the thermal profile exhibits an N-type temperature change below the critical pressure (Luo et al., 2023).

Quintessential deformations enrich the optical sector further. With dark-energy parameters $A=Q_m\cos\theta\,d\phi$02, the shadow radius increases with $A=Q_m\cos\theta\,d\phi$03, decreasing $A=Q_m\cos\theta\,d\phi$04 tends to enlarge the shadow, and AdS backgrounds lead to smaller shadows than the flat case. In the same analysis, the dark sector reduces the energy-emission rate because it enlarges the shadow while lowering the Hawking temperature in the relevant parameter range (Belhaj et al., 2022).

Dynamical instability of particle orbits provides a complementary diagnostic. For massless and massive particles on unstable circular orbits, the Lyapunov exponent

$A=Q_m\cos\theta\,d\phi$05

is multivalued when the thermodynamic system has multiple branches and single-valued when the phase transition disappears. The discontinuity

$A=Q_m\cos\theta\,d\phi$06

acts as an order parameter, and near the critical point the paper finds

$A=Q_m\cos\theta\,d\phi$07

This directly ties geodesic instability to mean-field thermodynamic criticality (Gogoi et al., 2024).

7. Generalizations, topological structure, and conceptual boundaries

The Hayward–AdS program has expanded into matter-dressed and corrected models. Quintessence introduces a dark-sector moduli space $A=Q_m\cos\theta\,d\phi$08 and changes both criticality and heat-engine performance (Belhaj et al., 2022). A surrounding fluid of strings can preserve regularity only in a restricted parameter range and still yields van der Waals-like criticality in the regular sector (Nascimento et al., 2024). Adding quantum entropy corrections to a Hayward–AdS black hole surrounded by a fluid of strings produces a corrected entropy

$A=Q_m\cos\theta\,d\phi$09

and the reported effect is to stabilize the large-black-hole branch and smooth the swallowtail structure in the Gibbs free energy (Bora et al., 5 Oct 2025).

A distinct extension uses thermodynamic topology. Introducing a generalized Helmholtz free energy and a Duan $A=Q_m\cos\theta\,d\phi$10-mapping current, one assigns a topological charge $A=Q_m\cos\theta\,d\phi$11 to the thermodynamic phase space. In the displayed configuration of the constrained construction, the singular parent black hole has total winding number $A=Q_m\cos\theta\,d\phi$12, while the Hayward–AdS black hole has $A=Q_m\cos\theta\,d\phi$13, placing the regular and singular solutions in different thermodynamic topological classes (Xia et al., 12 Feb 2026).

A recurring source of confusion is the relation between Hayward and Hayward–AdS literature. Several technically important Hayward papers are not AdS studies. The Carathéodory-based analysis of the Hayward thermodynamic manifold is asymptotically flat and explicitly does not introduce a cosmological constant, pressure-volume term, or extended phase space (Fathi et al., 2021). The remnant analysis of Hayward solutions with noncommutativity and rotation is likewise not an AdS construction (Mehdipour et al., 2016), and the accretion–evaporation study of the modified Hayward black hole does not introduce an AdS cosmological constant either (Debnath, 2015). Their relevance to Hayward–AdS lies in thermodynamic method, regular-core phenomenology, and matter-sector modeling rather than in direct AdS black-hole chemistry.

Taken together, the literature presents the Hayward–AdS black hole as a regular AdS black-hole framework whose defining features are nonlinear-magnetic regularization, extended thermodynamic criticality, and strong sensitivity to the chosen thermodynamic ensemble and matter sector. Its thermodynamics is robustly van der Waals-like at the level of critical exponents and coexistence structure, but its inversion curves, free-energy morphology, microstructure diagnostics, and observable shadow signatures depart in systematic ways from both ordinary fluids and singular charged AdS black holes.