Bardeen and Hayward Nonsingular Black Holes

• Bardeen and Hayward nonsingular black holes are regular models that replace the central singularity with a de Sitter core through variable mass functions.
• They integrate quantum gravity insights and effective modifications—such as RG improvement, one-loop corrections, and the generalized uncertainty principle—to reconcile classical and quantum frameworks.
• These proposals predict distinctive thermodynamic, dynamical, and observational signatures, including unique horizon structures, phase transitions, and gravitational lensing effects.

Bardeen and Hayward nonsingular black hole proposals are among the most influential phenomenological models aiming to resolve the curvature singularity present in classical black hole solutions by regularizing the spacetime core—typically via the introduction of a de Sitter patch or effective quantum modifications. These geometries serve as paradigmatic examples in the paper of quantum gravity phenomenology, gravitational collapse, black hole thermodynamics, and the information loss problem.

1. Motivation and Theoretical Construction

Classical black hole solutions such as Schwarzschild and Reissner–Nordström are characterized by curvature singularities at the center, where physical invariants like the Kretschmann scalar diverge. The Bardeen and Hayward spacetimes were introduced as concrete, static, spherically symmetric solutions in which the lapse function $f(r)$ is constructed so as to remain regular for all finite %%%%1%%%%, directly eliminating the singularity at $r = 0$.

Bardeen Metric

The original Bardeen solution introduces a variable mass function,

$$m(r) = M \left( \frac{r^2}{r^2 + g^2} \right)^{3/2},$$

leading to a metric function

$f(r) = 1 - \frac{2m(r)}{r},$

with $g$ interpreted as a magnetic charge in nonlinear electrodynamics. For $r \ll g$, $m(r) \sim r^3$ and $f(r)$ asymptotes locally to de Sitter form, ensuring regularity of curvature invariants.

Hayward Metric

The Hayward solution employs

$m(r) = \frac{m r^{3}}{r^3 + 2 m l^2},$

so that

$f(r) = 1 - \frac{2 m r^2}{r^3 + 2 m l^2}.$

The parameter $l$ sets the scale of regularization and controls the size of the de Sitter core at $r = 0$.

Both solutions are designed to interpolate smoothly between Schwarzschild asymptotics at large $r$ and de Sitter behavior at the center, resulting in bounded curvature invariants everywhere (Saueressig et al., 2015).

2. Quantum Gravity Interpretation and Extensions

The modern interpretation of Bardeen and Hayward black holes links their regular cores to quantum gravitational effects. Several frameworks support this:

• Renormalization group (RG) improved gravity: In asymptotic safety scenarios, Newton’s constant $G$ and the cosmological constant $\Lambda$ become scale-dependent. RG-improved Schwarzschild metrics acquire a variable $G(k(r))$ leading to improved mass functions $M(r)$ of Hayward form, such that

$f(r) = 1 - \frac{2m\, G(k(r))}{r}$

where $k(r) = \xi/d_r(r)$ sets the RG scale by proper radial distance. The central region develops an effective de Sitter patch with a bounded Kretschmann scalar (Saueressig et al., 2015).

• One-loop/corrective approaches: Expanding the improved $f(r)$ at large $r$ recovers the conventional 1-loop quantum corrections to the Newtonian potential (e.g., Donoghue’s correction), providing a bridge between effective field theory and regular geometry construction (Saueressig et al., 2015, Maluf et al., 2018).
• Generalized uncertainty principle (GUP): The Bardeen metric can be mapped to a quantum-corrected Schwarzschild solution where $r_0$, the core size, is related to the GUP deformation parameter $\alpha$ by $r_0 = \alpha l_\text{p}/2$ (Maluf et al., 2018).
• Designer gravity and Lovelock/extended theories: Spherically symmetric, second-order gravity in two dimensions can be engineered to yield any desired regular solution—including the Hayward metric—as its unique vacuum, via suitably chosen Hamiltonian constraints and mass functions (Kunstatter et al., 2015, Bueno et al., 7 Mar 2024).

Higher-dimensional and higher-curvature generalizations further admit Bardeen-like solutions in, e.g., Einstein–Gauss–Bonnet gravity (Singh et al., 2019), and recent work confirms that infinite towers of higher-curvature invariants in pure gravity can also resolve central singularities without recourse to exotic matter (Bueno et al., 7 Mar 2024).

3. Thermodynamic, Dynamical, and Observational Properties

Horizon and Remnant Structure

Bardeen and Hayward solutions typically possess two horizons (event and Cauchy horizons), which merge in the extremal limit (Bonanno et al., 2020). Thermodynamic analyses reveal a non-monotonic temperature profile, with temperature $T_H$ vanishing at both large mass and in the remnant state as the mass approaches a minimal value. This induces the existence of extremal remnants—black holes with zero temperature and finite size—that may play a role in information retention (Mehdipour et al., 2016, Singh et al., 2019).

Thermodynamics and Phase Structure

In AdS backgrounds and extended phase space, Bardeen and Hayward black holes exhibit Van der Waals-type equations of state and critical phenomena. In Bardeen–AdS (BAdS), explicit critical exponents coincide with those of liquid–gas transitions, supporting the analogy between black hole microstructures and classical statistical systems (Tzikas, 2018). For the Hayward–AdS solution (and its extensions by string fluid or other matter), the presence and characteristics of phase transitions are controlled by parameters such as $q$, $l$, $b$, and the equation-of-state parameter $\beta$; the nature of the core (regular vs. singular) is contingent on $\beta$ (Nascimento et al., 30 Nov 2024).

Entanglement Entropy and Information Paradox

The inclusion of regularizing vacuum energy or topological charges modifies the calculation of the Page curve and entanglement entropy via the island formula. In the Bardeen case, entropy and Page time essentially match Schwarzschild, while the Hayward metric—due to its de Sitter core—yields observable differences in the entanglement entropy and a slightly shifted Page time (Luongo et al., 2023).

Dynamical Stability and Mass Inflation

Regularity at the Cauchy horizon strongly depends on the detailed form of the mass function. In the Hayward geometry, the nonlinear dynamical equations driven by the quadratic structure of the mass function tame mass inflation, resulting in at most a weak, integrable (power-law) singularity (Bonanno et al., 2020). In contrast, Bardeen-type solutions, with a linear mass function evolution, cannot generally avoid exponential mass inflation (Bonanno et al., 2020).

Particle Acceleration and Geodesic Structure

Rotating extensions of Bardeen and Hayward metrics have been shown to act as efficient particle accelerators under the Banãdos–Silk–West mechanism. For extremal geometries, the center-of-mass energy of colliding particles diverges due to vanishing denominators in the energy expression, mirroring the dynamics seen in classical Kerr black holes (Pourhassan et al., 2015).

Observable Signatures: Lensing, Shadows, and QNMs

Regular cores subtly influence gravitational lensing in the strong-deflection regime. While the photon sphere (shadow edge) can remain nearly unchanged relative to Schwarzschild, secondary observables—such as the angular separation between images, brightness contrast, time delays, and the spectrum of quasinormal modes (QNMs)—can deviate, providing prospects for observational discrimination (Zhao et al., 2017, Heidari et al., 22 Mar 2025). Parameter constraints on Hayward and Bardeen metrics from EHT shadow data are now being investigated (Heidari et al., 22 Mar 2025).

The universal bounds on minimum orbital periods, $4\pi M \leq T_\text{min} \leq 6\sqrt{3}\pi M$, remain valid for both Bardeen and Hayward metrics, suggesting that the presence of a horizon, rather than a central singularity, underlies these dynamical features (Liu et al., 29 May 2025).

4. Extensions: Noncommutative and Modified Gravity Models

Noncommutative geometry introduces a fundamental minimal length, encoded in a parameter $\theta$, modifying Bardeen and Hayward metrics by replacing local delta distributions with Gaussian profiles. The resulting nonsingular black holes feature larger, colder remnants whose minimal mass $M_0$ and horizon radius $r_0$ scale linearly with magnetic charge $g$ for Bardeen and the relevant scale in Hayward (Mehdipour et al., 2016). Detailed bounds on noncommutative parameters $\Theta$ and Hayward core scale $l$ can be extracted from classical solar system tests, including perihelion precession and light deflection (Heidari et al., 22 Mar 2025).

Further extensions via gravitational decoupling allow for new “hairy” generalizations: starting from seed (regular) Bardeen or Hayward solutions, an anisotropic energy–momentum tensor is added, leading to more general anisotropic metrics. However, arbitrary deformations can spoil central regularity unless the sources are fine-tuned; careful construction can nonetheless yield solutions with preserved de Sitter cores (Misyura et al., 8 May 2024, Nascimento et al., 30 Nov 2024).

5. Critiques, Energy Conditions, and Semiclassical Consistency

Despite their wide adoption, Bardeen and Hayward solutions have faced criticism regarding their energy condition requirements. Specifically, the matter sources needed in GR—often formulated as nonlinear electrodynamics—typically violate the dominant or strong energy conditions somewhere in the spacetime, especially near the core (Maeda, 2021). More restrictive models (e.g., Dymnikova, Fan–Wang) can avoid this issue.

Recent Hamiltonian semiclassical studies, with proper matter backreaction via the expectation values of scalar field operators, demonstrate that Bardeen and Hayward metrics are not solutions to the corresponding semiclassical Einstein equations. The requisite consistency relation between the expectation values of kinetic and momentum contributions cannot be fulfilled with positive definite operators, implying that these metrics do not emerge from self-consistent semiclassical gravity with standard quantum matter (Javed et al., 4 Oct 2025). A plausible implication is that viable regular black hole solutions in semiclassical gravity must differ fundamentally from the phenomenological forms of Bardeen and Hayward.

6. Significance and Broader Impact

Bardeen and Hayward nonsingular black holes serve as archetypes for regular black hole models in gravitational theory and quantum gravity phenomenology. Their construction highlights the interplay between high-energy modifications, matter content (including effective anisotropic fluids and nonlinear electrodynamics), and the preservation of key physical properties—such as bounded curvature, globally defined time, and controlled dynamical instabilities.

They underpin ongoing research into black hole remnants, quantum gravitational singularity avoidance, and the information paradox. Furthermore, as templates for effective geometry in quantum-improved and modified gravity theories, they offer a practical laboratory to test phenomenological consequences—ranging from horizon thermodynamics, phase transitions, and information recovery to observational signatures in lensing and ringdown signals.

Finally, their limitations with respect to energy conditions and semiclassical consistency have sparked the development of new models and analytic techniques, fostering continued refinement in the quest for self-consistent, physically admissible regular black hole solutions.