Hayward Black Hole: A Regular Solution

• Hayward black holes are regular spacetime solutions featuring a de Sitter core that prevents curvature singularities and mimics Schwarzschild behavior at large distances.
• They are defined by a mass parameter M and a regularization scale g, resulting in modified horizon structures and thermodynamic properties, including the formation of stable remnants.
• Extensions with rotation, electric charge, noncommutativity, and higher dimensions enrich the model, offering insights into gravitational lensing, thermodynamic phase transitions, and quantum evaporation.

The Hayward black hole defines a family of static, spherically symmetric, regular spacetimes that interpolate between the Schwarzschild vacuum at large distances and a de Sitter core at short distances, avoiding curvature singularities. Parameterized by a mass $M$ and a "regularization parameter" $g$ (or $\ell$), these geometries are prototypical "regular black holes" in general relativity coupled to nonlinear electrodynamics, admitting generalizations to include dark energy, electric charge, rotation, higher curvature corrections, noncommutative geometry, and extra-dimensional setups. The interplay of horizon structure, thermodynamics, dynamical formation, quantum evaporation, gravitational lensing, and shadow observables makes the Hayward black hole a canonical framework for analyzing singularity resolution, black hole remnants, high-energy collider phenomenology, and tests of Planck-scale gravity.

1. Metric Structure and Regularity

The original Hayward metric is given by

$ds^2 = -N(r) dt^2 + N(r)^{-1} dr^2 + r^2 d\Omega^2$

with

$$N(r) = 1 - \frac{2m(r)}{r}, \quad m(r) = \frac{M r^3}{r^3 + g^3}$$

where $M$ is the ADM mass and $g \equiv \ell$ is a regularizing length scale (Mehdipour et al., 2016). For $r\gg g$, $m(r)\to M$ and $N(r)\to 1-2M/r$, the Schwarzschild limit. For $r\ll g$, $N(r)\to 1 - 2M r^2/g^3$, i.e., a de Sitter core with effective cosmological constant $\Lambda_{\rm eff} = 6M/g^3$. All curvature invariants (such as $R$ and $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$) are finite everywhere: at $r=0$, $K \sim 1/g^4$, $R \sim 1/g^2$.

This regularity persists under modifications: the Hayward metric can be sourced by nonlinear electrodynamics, where the Lagrangian $\mathcal{L}(F)$ is chosen so that Einstein equations admit exactly this $f(r)$ (Halilsoy et al., 2013), and, in the presence of an appropriately designed noncommutative deformation, all curvature singularities remain absent provided $\cos\theta \neq 0$ (Heidari et al., 22 Mar 2025).

2. Horizons, Remnants, and Global Causal Structure

The classical horizon equation is the cubic

$r_H^3 - 2M r_H^2 + g^3 = 0$

with up to two positive real roots: the inner (Cauchy) horizon $r_-$ and the event horizon $r_+$ (Mehdipour et al., 2016). The critical (extremal) configuration occurs when these roots coincide: $r_0 = 2^{1/3} g, \quad M_0 = \frac{3}{2^{5/3}} g$ For $M < M_0$, there is no horizon (horizonless, solitonic object); for $M = M_0$, a degenerate horizon; for $M > M_0$, two distinct horizons.

In the extended metrics—charged (Junior et al., 2020), cloud-of-strings (Nascimento et al., 2023), quintessence (Heydari-Fard, 2022), AdS (Luo et al., 2023, Nascimento et al., 30 Nov 2024), or higher-dimensional (Kumar et al., 2020, Mehdipour et al., 2016)—the characteristic polynomial for horizons can reach quartic or higher order, and extremality criteria and horizon structure depend on the interplay of the additional couplings (e.g., string-fluid parameter $b$, quintessence normalization $c$, cosmological constant, etc.), but always retain the property of degenerating to a single zero-temperature remnant at extremality. For the noncommutative Hayward black hole, horizon positions are shifted (numerically) by the deformation scale $\sqrt{\theta}$ (Mehdipour et al., 2016, Heidari et al., 22 Mar 2025).

The global causal structure resembles Reissner–Nordström but with the central singularity replaced by a regular de Sitter core (cf. conformal diagrams in (Um et al., 2019, Iguchi, 15 Apr 2025)).

3. Thermodynamics, Evaporation, and End-States

The surface gravity at the outer horizon gives

$$T_H = \frac{1}{4\pi} N'(r_+) = \frac{M r_+ (r_+^3 - 2g^3)}{2\pi (r_+^3 + g^3)^2}$$

Entropy follows the standard area law, $S = \pi r_+^2$. For non-extremal Hayward black holes, $T_H$ rises from zero at $r_0$ to a maximum at the Davies point, then falls and diverges as $r_+\rightarrow 0$ in the Schwarzschild limit. However, due to the regular core, evaporation ceases at $M_0$: $T_H(r_0) = 0$, yielding a stable, Planck-scale remnant (Mehdipour et al., 2016, Um et al., 2019).

Thermal fluctuations induce logarithmic corrections to the entropy and other potentials: $S = S_0 - \frac{1}{2}\ln[S_0 T_H^2] + \cdots$ These corrections can restore consistency with the first law and raise the stability bound for the remnant (Pourhassan et al., 2016).

In the noncommutative extension, nonlocal smearing further suppresses $T_H$ in the small-mass regime, increasing both the remnant mass and its radius (Mehdipour et al., 2016, Heidari et al., 22 Mar 2025). For rotating Hayward black holes, remnant properties depend weakly on $g$ but more strongly on the spin parameter $a$ (Mehdipour et al., 2016). In $d$-dimensional scenarios, the remnant mass $M_0$ decreases with $d$, potentially making TeV-scale black holes accessible in large extra dimension theories (Spallucci et al., 2012, Mehdipour et al., 2016).

In the context of black hole formation from collapse, quantum radiation from a shell of matter can be treated within the functional Schrödinger formalism: the unitary evolution and suppression of low-frequency quanta (as $g$ increases) ensures no information loss, and in the extremal limit, Hawking temperature and flux vanish (Um et al., 2019).

4. Rotating and Generalized Hayward Black Holes

The most widely studied rotating extension utilizes the Newman–Janis algorithm. In Boyer–Lindquist-like coordinates: $$ds^2 = -\left(1 - \frac{2 m(r) r}{\Sigma}\right) dt^2 - \frac{4a m(r) r \sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta} dr^2 + \Sigma d\theta^2 + \left(r^2 + a^2 + \frac{2 a^2 m(r) r \sin^2\theta}{\Sigma}\right)\sin^2\theta d\phi^2,$$ with $\Sigma = r^2 + a^2\cos^2\theta$, $\Delta = r^2 - 2 m(r) r + a^2$, and $m(r) = M r^3/(r^3+g^3)$. All curvature invariants remain finite and no geodesics terminate at finite affine parameter if $m(r)$ is suitably defined for $r < 0$ (Lamy et al., 2018).

The horizon structure is determined by the real roots of $\Delta(r)$, with parameter regimes yielding (i) a two-horizon regular black hole, or (ii) a horizonless regular (traversable) wormhole (Lamy et al., 2018, Amir et al., 2015). The spin parameter $a$ at extremality decreases as $g$ increases, with $r_H$ increasing (Amir et al., 2015).

Hayward black holes admit Bañados–Silk–West (BSW) acceleration: in the extremal limit, the center-of-mass energy for infalling particle collisions diverges for particles with critical angular momentum. Non-extremal cases yield finite but $g$-dependent $E_{\rm CM}^{\rm max}$ (Amir et al., 2015).

These geometries exhibit millimeter-wave images and shadow structures nearly indistinguishable from Kerr at present EHT resolutions (few-percent corrections for $g \sim M$), but differences in ISCO frequencies, shadow shift, and the potential for higher $E_{\rm rad}$ in binary mergers may enable discrimination with future high-precision data (Lamy et al., 2018, Gwak, 2017).

5. Observational Signatures: Shadows, Lensing, and Constraints

The photon sphere and shadow radius are determined by circular null geodesics, with critical impact parameter $b_c = r_{\rm ph}/\sqrt{f(r_{\rm ph})}$ (Heydari-Fard, 2022, Zhao et al., 2017). For Hayward black holes, the shadow size is only mildly decreased relative to Schwarzschild—by a few percent for $l \ll M$. Additional fields, such as quintessence (dark energy with equation-of-state parameter $\omega$), substantially affect the shadow size and ring brightness: quintessence corrections dominate over the Hayward core in modifying shadow observables (Heydari-Fard, 2022).

Strong-deflection gravitational lensing observables (angular separations, brightness ratios, time delays) require resolution beyond current VLBI—$\mathcal{O}(0.1\,\mu{\rm as})$—to distinguish Hayward signatures from Schwarzschild (Zhao et al., 2017). Shadow and lensing features are further modified in noncommutative Hayward models: both $l$ and noncommutativity $\Theta$ shrink the shadow and enhance gravitational deflection (Heidari et al., 22 Mar 2025).

Constraints from EHT observations of M87* and Sgr A* yield upper bounds on deviations of the dark energy parameter $c$ and, in principle, can be used to restrict the $(l,\Theta)$ parameter space of regular black holes (Heydari-Fard, 2022, Heidari et al., 22 Mar 2025).

Numerical estimates for Sgr A* in the modified Hayward case yield photon-sphere angles $\theta_\infty \sim 25$–$37\,\mu$as, within EHT reach. However, sub-nanoarcsecond imaging and high-cadence timing would be required to exploit the discriminating power of lensing observables (Zhao et al., 2017).

6. Thermodynamic Phase Structure and Extended Scenarios

In AdS, the Hayward–AdS black hole displays a van der Waals-like phase structure, with critical radius and temperature

$$r_c \sim g, \quad T_c \sim 1/g, \quad P_c \sim 1/g^2$$

and characteristic swallow-tail behavior in the Gibbs free energy, signifying a first-order small/large black hole transition (Luo et al., 2023, Nascimento et al., 30 Nov 2024). The shadow radius $R_{\rm sh}$ is a monotonic function of horizon radius $r_h$, mapping geometric optically observable features directly to thermodynamic parameters and phase transitions (Luo et al., 2023).

For Hayward black holes surrounded by anisotropic string fluids or quintessence, the properties of the critical point and heat capacity are affected by the new parameters ($b$, $\beta$, $c$, $\omega$), but regularity at $r=0$ is preserved only for restricted parameter regions (e.g., $-1\leq\beta<0$ for string fluids (Nascimento et al., 30 Nov 2024), $a=0$ for string clouds (Nascimento et al., 2023)).

In five-dimensional EGB gravity with nonlinear electrodynamics, the Hayward solution generalizes to possess two horizons, curvature regularity, non-area-law entropy, and a well-characterized second-order (Davies-type) phase transition at a critical radius $r_C$ (Kumar et al., 2020).

7. Information, Gravitational Entropy, and Quantum Gravity Phenomenology

The Hayward black hole provides a test bed for gravitational entropy definitions beyond Bekenstein–Hawking. The Weyl/Kretschmann entropy ratio $P^2=W/K$ quantifies the reduction in gravitational degrees of freedom due to regularization, and vanishes in the extremal limit at $r_+=r_-$, reflecting the loss of tidal degrees of freedom in the de Sitter core (Iguchi, 15 Apr 2025).

In the context of quantum gravity and TeV-scale collider phenomenology, the existence of a minimal length $l_0$ translates into a black hole production threshold $\sqrt{s_0}=3/(4G_N l_0)$ and a geometric cross section $\sigma(\sqrt{s}) \to \pi r_+^2$, with exponential suppression for $\sqrt{s}<\sqrt{s_0}$, providing an explanation for the absence of black holes at current LHC energies; the minimal remnant mass is $M_0=3/(4G_N l_0)$ (Spallucci et al., 2012).

Information loss is avoided due to the absence of a singularity: geodesics can traverse the regular de Sitter core and re-emerge, and the semiclassical dynamics of Hawking radiation remain unitary (Nojiri et al., 2023, Um et al., 2019).

In mergers, extremal Hayward black holes pose subtleties: the final state of two merging extremals is again an extremal with vanishing entropy, appearing to violate the usual second law, unless additional degrees of freedom (e.g. angular momentum, higher-order corrections) are included to lift the system out of the extremal manifold (Fathi et al., 2021).

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Table: Key Modifications and Their Consequences

Extension	Key Parameters	Regularity & Global Structure
Nonlinear Electrodynamics	$g$	Regular at $r=0$, de Sitter core
Electric Charge	$Q, l$	Two horizons if $Q^2 < Q_{\rm crit}^2(l)$
Rotation	$a, g$	Regular everywhere, extremal at $a_E(g)$
Noncommutativity	$\Theta, l$	Regularity for $\cos\theta\neq 0$
Quintessence	$c, \omega, l$	Three horizons, shadow dominantly $c,\omega$
Cloud of Strings	$a, \ell$	Singularity at $a\neq 0$
AdS/Fluid of Strings	$q, b, \beta$	Regular for $-1\leq \beta<0$
Scalar–EGB Gravity	$\lambda$	Regular, ghost-free for specific $f(\xi)$

• (Mehdipour et al., 2016, Lamy et al., 2018, Um et al., 2019, Halilsoy et al., 2013, Nascimento et al., 2023, Nascimento et al., 30 Nov 2024, Luo et al., 2023, Heydari-Fard, 2022, Kumar et al., 2020, Zhao et al., 2017, Amir et al., 2015, Iguchi, 15 Apr 2025, Pourhassan et al., 2016, Spallucci et al., 2012, Heidari et al., 22 Mar 2025, Fathi et al., 2021, Nojiri et al., 2023, Junior et al., 2020, Gwak, 2017)

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Summary

Hayward black holes represent a robust paradigm for regular, horizon-bearing solutions that avoid spacetime singularities and admit a cold remnant at the endpoint of Hawking evaporation. Their generalized forms—incorporating charge, rotation, noncommutativity, accretion, or higher-curvature corrections—preserve global regularity subject to constraints on the parameters. The rich phase structure, shadow observables, and implications for information loss, gravitational entropy, and collider phenomenology render the Hayward class foundational in the landscape of regular black hole models.