---
title: Charged Hayward Black Hole Overview
url: https://www.emergentmind.com/topics/charged-hayward-black-hole
type: topic
---

# Charged Hayward Black Hole Overview

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 [2511.02191]
\[
f(r) = 1 - a - \frac{2 M r^2}{r^3 + 2\ell^2 M} - \alpha r^{-(3\omega_q+1)} + \frac{Q^2}{r^2} - \frac{\Lambda r^2}{3}
\]
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 $r_{ext} = 4M/3$ [2510.18354]. For electric charge: $q_c \approx 0.634\,M$ [1402.2748].
- 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 $\Lambda=0$ [2511.02191]. 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 $(M, q, g)$ [1402.2748].

### 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 [2511.02191, 2510.18354].

### 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 [1402.2748].

## 4. Thermodynamics and Phase Structure

### Basic Thermodynamics
- Hawking temperature $T_H = f'(r_+)/(4\pi)$, could be expressed, for Hayward-AdS with magnetic $g$ [2301.06120]:
\[
T(r_h,P,g)= \frac{r_h^3 - 2 g^3}{4\pi r_h (r_h^3 + g^3)} + \frac{2P r_h^4}{r_h^3 + g^3}
\]
- Entropy: $S = \pi r_+^2$ or, if non-linear electrodynamics modifies the horizon area law, $S = 2\pi (r_+^2/2 - g^3/r_+)$ [2003.00889].
- 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 [2103.14087].

### 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 $(T_c, P_c, r_c)$ [2003.00889, 2301.06120].
- 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 $\theta_{sh} \approx b_c / D$ [2112.11227].
- 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 $1\sigma$ [2112.11227].

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 [2003.10175]
- 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 [2003.00889, 2003.10175].

## 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 [2005.13029].

## 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 $n=0,1,\dots$ [2510.18354]. The $n=0$ fundamental branch is both dynamically and thermodynamically preferred.

Further generalizations include coupling to rational NLED models [2103.14087], quintessence and string cloud terms [2511.02191], 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.

Source: https://www.emergentmind.com/topics/charged-hayward-black-hole