Charged Hayward Black Hole Overview

• Charged Hayward black holes are regular, static, spherically symmetric solutions in General Relativity with a de Sitter core and finite curvature, achieved via nonlinear electrodynamics.
• They exhibit complex horizon structures and observable features, such as black hole shadows and tidal force variations, that depend on the electric or magnetic charge parameters.
• Their thermodynamic behavior, including van der Waals-like criticality and phase transitions, provides insights into microstructure interactions and strong-gravity effects.

A charged Hayward black hole is a regular, static, spherically symmetric black hole solution in General Relativity, supported by nonlinear electrodynamics (NLED). Unlike classic solutions such as Reissner–Nordström, Hayward black holes possess a de Sitter core and are free from curvature singularities at $r=0$. The addition of a magnetic or electric charge introduces a parameter $g$ (or $q_m$, $Q$) into the metric, which governs critical phenomena, microstructure, geodesic properties, and observational features such as tidal forces and black hole shadow. The solution extends naturally to include a cosmological constant and further matter couplings.

1. Metric Structure and Charge Parameters

The static charged Hayward black hole is defined by the Schwarzschild-like line element

$$ds^2 = -f(r) dt^2 + f(r)^{-1} dr^2 + r^2\, (d\theta^2 + \sin^2\theta\, d\phi^2)$$

with the lapse function determined by the choice of charge and NLED model:

Magnetic Charge (Hayward Form)

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

where $M$ is the ADM mass; $g$ characterizes the magnetic charge and regularization scale.

Electric (Ayón–Beato–García Type)

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

with $q$ the electric charge. Both metric functions produce regular black holes with de Sitter behavior at $r\to0$, $f(0) = 1$, and all curvature invariants finite.

Generalizations

With a cosmological constant $\Lambda$, quintessence ($\omega_q$, $\alpha$), and a cloud of strings ($a$), the lapse becomes (Nascimento et al., 4 Nov 2025)

$$f(r) = 1 - a - \frac{2 M r^2}{r^3 + 2\ell^2 M} - \alpha r^{-(3\omega_q+1)} + \frac{Q^2}{r^2} - \frac{\Lambda r^2}{3}$$

The regularity at the core persists for the pure Hayward terms; Maxwell (electric) charge or string clouds reintroduce central singularities unless replaced by NLED.

2. Horizon Structure and Regularity

The horizons are defined by the roots of $f(r)=0$. For the standard charged Hayward model:

• Up to two positive real roots: $r_-$ (inner), $r_+$ (event).
• Extremality is reached when these coalesce ($f(r_e)=0$, $f'(r_e)=0$), giving a critical charge or regularization parameter, e.g. for magnetic charge: $g_{ext} = (16/27)^{1/3} M$ with $r_{ext} = 4M/3$ (Cai et al., 21 Oct 2025). For electric charge: $q_c \approx 0.634\,M$ (Pradhan, 2014).
• With a cosmological constant, up to three horizons can appear (Cauchy, event, cosmological).

The Kretschmann scalar $K = R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma}$ is finite at $r=0$ for pure Hayward or NLED-sourced charges, e.g. $f(r \to 0) \sim 1 - r^2/\ell^2 - (\Lambda r^2)/3$ and $\lim_{r \to 0} K = 24/\ell^4$ for $\Lambda=0$ (Nascimento et al., 4 Nov 2025). Inclusion of a standard Maxwell $Q$ or string cloud $a$ yields a divergence $K \sim Q^2/r^6$ or $K \sim a^2/r^4$.

3. Geodesics, Orbits, and Particle Collisions

Circular Orbits and Effective Potential

For test particles (timelike or null) on the equatorial plane: $$u^t = \frac{E}{f(r)}, \quad u^\phi = \frac{L}{r^2}, \quad \left(\frac{dr}{d\tau}\right)^2 = E^2 - V_{\rm eff}(r)$$ with $$V_{\rm eff}(r) = f(r) \left(1 + \frac{L^2}{r^2}\right)$$. The innermost stable circular orbit (ISCO), marginally bound (MBCO), and photon sphere (CPO) radii are determined by high-order polynomial equations in $r$, dependent on $(M, q, g)$ (Pradhan, 2014).

Orbits Near the Center

The regular core enables geodesics to probe closer to $r=0$ without encountering divergent potentials, in contrast to Kerr, Schwarzschild, or Reissner–Nordström (Nascimento et al., 4 Nov 2025, Cai et al., 21 Oct 2025).

High-Energy Collisions

For extremal configurations, the center-of-mass energy for particle collisions near the horizon diverges: $$E_{\rm CM}\big|_{r\to r_+} = \sqrt{2}m_0 \sqrt{\frac{4r_+^2 + (L_1 - L_2)^2}{2r_+^2}}$$ For $q \to q_c$ (electric), $E_{\rm CM} \to \infty$ due to diverging critical angular momentum (Pradhan, 2014).

4. Thermodynamics and Phase Structure

Basic Thermodynamics

• Hawking temperature $T_H = f'(r_+)/(4\pi)$, could be expressed, for Hayward-AdS with magnetic $g$ (Luo et al., 2023): $$T(r_h,P,g)= \frac{r_h^3 - 2 g^3}{4\pi r_h (r_h^3 + g^3)} + \frac{2P r_h^4}{r_h^3 + g^3}$$
• Entropy: $S = \pi r_+^2$ or, if non-linear electrodynamics modifies the horizon area law, $S = 2\pi (r_+^2/2 - g^3/r_+)$ (Kumara et al., 2020).
• Thermodynamic volume: $V = (4\pi/3)(r_+^3 + g^3)$.
• Heat capacity $C_Q$ diverges at $T_H$'s maximum and stabilizes (becomes positive) in an intermediate domain, indicating a second-order phase transition (Kruglov, 2021).

Criticality and van der Waals Analogue

• The $P$–$r_+$ (or $P$–$v$) equation of state exhibits inflection criticality very similar to the van der Waals fluid, generating a first-order transition below $T_c$ and a continuous second-order critical point at $(T_c, P_c, r_c)$ (Kumara et al., 2020, Luo et al., 2023).
• Critical exponents match mean-field universality: $\alpha=0,\,\beta=1/2,\,\gamma=1,\,\delta=3$.

5. Black Hole Shadow and Observational Characteristics

The shadow cast by the charged Hayward black hole is determined by the photon sphere and the critical impact parameter $b_c = r_{ph}/\sqrt{f(r_{ph})}$:

• The shadow angular radius for an observer at large $D$ is $\theta_{sh} \approx b_c / D$ (Guo et al., 2021).
• As the magnetic charge $g$ increases, both the photon sphere radius $r_{ph}$ and $b_c$ decrease, shrinking the observable shadow diameter $d_{sh}$.
• Example (for $M=1$):

$g$	$r_{ph}$	$b_c$
0.0	3.0000	5.19615
0.2	2.9982	5.19461
0.5	2.9716	5.17169
0.8	2.8748	5.09013

For $g \lesssim 0.5$ the decrease is modest; at larger $g$ the shadow shrinks more rapidly. EHT observations, e.g. for M87*, constrain $g \lesssim 1.7$ at $1\sigma$ (Guo et al., 2021).

The observed intensity profile from accretion flows depends sharply on the nature of the emission (static/infalling sphere, thin disk). The photon ring and lensed ring intensity fractions decrease as $g$ increases, and for infalling spherical accretion the overall emission is $\sim$100$\times$ dimmer than the static case.

6. Microstructure, Ruppeiner Geometry, and Phase Transitions

Ruppeiner geometry encodes the thermodynamic microstructure via curvature scalar $R_N$, with sign signaling dominant interactions:

• $R_N < 0$: net attraction (as in van der Waals or uncharged black holes)
• $R_N > 0$: net repulsion, found in magnetically charged Hayward-AdS only at low temperature and small volume (Kumara et al., 2020)
• The repulsive region appears in certain branches of the small-black-hole phase, while the large-black-hole and coexistence domains are generally attractive, except near the line where $R_N$ changes sign.

The presence of magnetic charge $g$ drives a transition from attractive (bosonic-like) to repulsive (fermionic-like) microstructure, and the critical behavior of $R_N$ directly mimics the behavior in RN-AdS black holes (Kumara et al., 2020, Kumara et al., 2020).

7. Tidal Forces, Geodesic Deviation, and Astrophysical Implications

Tidal strains in the charged Hayward background are finite everywhere and, unlike in Schwarzschild or Reissner–Nordström, can vanish or change sign at specific radii outside the horizon. The tidal tensor in the proper frame contains

$$E_{\hat r \hat r},\quad E_{\hat\theta\hat\theta}=E_{\hat\phi\hat\phi}$$

which approach $-1/l^2$ near the regular center, and exhibit zero crossings as a function of $r$. This structure could affect tidal-disruption events, potentially leaving observational signatures such as points of zero net radial stretching (Junior et al., 2020).

8. Scalarization and Extensions in Nonlinear Theories

Hayward black holes admit spontaneous scalarization in ENM–S models: above a model threshold $\alpha_{\rm th}$, the trivial scalar solution is unstable to growth of "scalar clouds," with the resulting backreacted scalarized black holes labeled by mode number $n=0,1,\dots$ (Cai et al., 21 Oct 2025). The $n=0$ fundamental branch is both dynamically and thermodynamically preferred.

Further generalizations include coupling to rational NLED models (Kruglov, 2021), quintessence and string cloud terms (Nascimento et al., 4 Nov 2025), and AdS extensions supporting van der Waals–like criticality.

9. Summary Table of Key Metric Functions and Properties

Model	$f(r)$ Structure	Core Regular?	Source Lagrangian
Hayward (magnetic)	$1 - \frac{2Mr^2}{r^3+g^3}$	Yes	NLED
Charged (electric)	$$1- \frac{2Mr^2}{(r^2+q^2)^{3/2}} + \frac{q^2r^2}{(r^2+q^2)^2}$$	Yes	NLED
Hayward–AdS	$1- \frac{2Mr^2}{r^3+g^3} + \frac{r^2}{\ell^2}$	Yes	NLED
Hayward–RNED	$1 - 2Gm(r)/r$ with $m(r)$ from RNED	Yes	RNED

10. Physical Implications and Observational Significance

Charged Hayward black holes, via regularized cores, avoid singularities and provide testable deviations from classical GR predictions. Constraints on the magnetic charge parameter $g$ arise from current and future horizon-scale observations, as shadow size and ring structure are sensitive to $g$. Tidal forces, microstructure transitions, and black hole scalarization provide theoretically rich phenomenology for strong-gravity tests and offer a window into quantum-gravitational or nonlinear electromagnetic effects in astrophysical settings.