Rotating Hayward Black Hole

• Rotating Hayward black holes are regular, axisymmetric solutions that generalize Kerr by using a mass function with a regularizing charge g to replace the central singularity with a de Sitter core.
• The metric is derived via methods such as the Newman–Janis algorithm on the static Hayward metric, resulting in modified horizon, ergoregion, and geodesic structures.
• Observable effects include shifted ISCOs, distorted shadows, and enhanced energy extraction processes, informing tests with accretion disks and gravitational wave signatures.

A rotating @@@@2@@@@ is a regular, axisymmetric, asymptotically flat (or de Sitter, in certain generalizations) black hole solution in four-dimensional spacetime that generalizes the classical Kerr metric by introducing a regularizing parameter $g$ (often interpreted as a “magnetic charge” arising from nonlinear electrodynamics). The solution removes the central curvature singularity of Kerr, replacing it with a de Sitter core, and is constructed using techniques such as the Newman–Janis algorithm acting on the original spherically symmetric static @@@@3@@@@. The resulting line element reduces to Kerr for $g\to0$ and to the non-rotating Hayward metric for vanishing spin $a$. The metric is featured in a wide range of studies on regular black holes, black hole shadows, particle acceleration, gravitational wave signatures, and quantum gravity phenomenology (Kumar et al., 2020Lorenzo et al., 20151410.40431604.08584Amir et al., 2015Lamy et al., 2018Gwak, 2017Ali et al., 2022).

1. Metric, Mass Function, and Regularity

The canonical form of the rotating Hayward metric in Boyer–Lindquist coordinates $(t, r, \theta, \phi)$ is

$$ds^2 = -\left(1 - \frac{2\,m(r)r}{\Sigma}\right)dt^2 - \frac{4a\,m(r)r \sin^2\theta}{\Sigma} dt\,d\phi + \frac{\Sigma}{\Delta(r)} dr^2 + \Sigma\, d\theta^2 + \left[r^2+a^2+\frac{2m(r)r a^2\sin^2\theta}{\Sigma}\right] \sin^2\theta\,d\phi^2,$$

where

$$\Sigma(r,\theta) = r^2 + a^2\cos^2\theta, \qquad \Delta(r) = r^2 + a^2 - 2\,m(r)\,r,$$

and the Hayward mass function is

$m(r) = \frac{M r^3}{r^3 + g^3}.$

Here, $M$ is the ADM mass, $a = J/M$ the spin parameter, and $g$ the regularizing charge ($g>0$). For $g=0$, the metric coincides with Kerr; for $a=0$, one recovers the original static Hayward solution (1410.40431510.08828Mehdipour et al., 2016).

The regularity at $r=0$ is ensured by the behavior of $m(r) \sim r^3/g^3$ as $r \to 0$, so that all curvature invariants—such as the Kretschmann scalar—remain finite everywhere, including at the center and the equatorial plane, in contrast to the Kerr singularity (1410.40431510.08828). The matter content supporting the metric can be interpreted as an effective anisotropic fluid violating strong (but not always weak) energy conditions near the core.

2. Horizon Structure, Extremality, and Ergosurfaces

The event horizon(s) are located at real, positive roots $r_+$ of $\Delta(r) = 0$, that is,

$r^2 + a^2 - 2 m(r) r = 0,$

which generically yields two distinct horizons: the outer event ($r_+$) and inner Cauchy ($r_-$) horizons. For arbitrary $g$ these equations must be solved numerically.

Extremality corresponds to the coincidence of the two roots; the double root occurs when, in addition,

$$\Delta(r_{\text{ext}}) = 0, \qquad \Delta'(r_{\text{ext}}) = 0.$$

For fixed $a$, increasing $g$ decreases the critical spin for extremality $a_E(g)<M$ and increases the value of the degenerate horizon $r_H^E$ (Amir et al., 2015). For instance, with $M=1$, $g=0.2$ gives $a_E=0.992$ and $r_H^E=1.015$; for $g=0.6$, $a_E=0.833$ and $r_H^E=1.189$.

The stationary limit (zero of $g_{tt}$, or ergosurface) is determined by $1 - 2 m(r) r / \Sigma = 0$. The region between the outer horizon and the stationary limit is the ergoregion, where negative-energy orbits are possible, enabling energy extraction by the Penrose mechanism. The ergosphere is deformed relative to Kerr, typically being thicker for given $a$ when $g>0$ (Lorenzo et al., 2015Khan et al., 2021).

3. Geodesic Structure and Circular Orbits

The separability of the Hamilton–Jacobi equation is preserved: $$\Sigma \frac{dr}{d\lambda} = \pm \sqrt{\mathcal{R}(r)},\qquad \mathcal{R}(r) = \left[(r^2+a^2)E - aL\right]^2 - \Delta\left[\mathcal{K} + (aE-L)^2 + m_0^2 r^2 \right],$$ where $E$ is the conserved energy, $L$ is the azimuthal angular momentum, $\mathcal{K}$ the Carter constant, and $m_0$ the particle mass (0 for photons). The photon region is determined by the conditions $\mathcal{R}(r_p) = 0$, $\mathcal{R}'(r_p) = 0$.

Innermost stable circular orbits (ISCOs) and photon spheres are modified compared to Kerr, typically with ISCOs shifted inward and higher orbital frequencies for given $a$ as $g$ increases. The analytical structure of geodesic motion is preserved, supporting ray-tracing and accretion disk studies (1410.40432107.06085Lamy et al., 2018).

4. Black Hole Shadows and Observational Signatures

The null geodesic structure defines the boundary of the black hole shadow as seen by a distant observer at $(r_o, \theta_o)$. In celestial plane coordinates,

$$\alpha = -\xi_c\, \csc\theta_o,\qquad \beta = \pm\sqrt{ \eta_c + a^2 \cos^2\theta_o - \xi_c^2 \cot^2\theta_o },$$

where $\xi_c$ and $\eta_c$ are critical impact parameters evaluated at the photon region: $$\xi_c = \frac{[a^2 - 3 r_p^2]\,m(r_p) + r_p[r_p^2 + a^2](1 + m'(r_p))}{ a [m(r_p) + r_p ( -1 + m'(r_p) ) ] },$$

$$\eta_c = -\frac{r_p^3}{ a^2 [ m(r_p) + r_p ( -1 + m'(r_p) ) ]^2 } \left[ r_p^3 + 9 r_p m(r_p)^2 + 2 [2a^2 + r_p^2 + r_p^2 m'(r_p) ] r_p m'(r_p) - 2 m(r_p) (2a^2 + 3r_p^2 + 3 r_p^2 m'(r_p) ) \right].$$

The shadow is computed by mapping $(\xi_c, \eta_c)$ contours to the observer's celestial coordinates.

Systematic bias analysis of shadow observables shows that for $g \lesssim 0.65 M$ the rotating Hayward shadow is indistinguishable from Kerr at present EHT angular resolution ($\sim10\%$ uncertainties). The constraint from the M87* shadow diameter is $g \leq 0.73627M$ at $1\sigma$ (Kumar et al., 2020). Both increasing $g$ and $a$ shrink and distort the shadow, introducing a partial degeneracy. With improved resolution and measurement of higher-order moments, future VLBI facilities may break this degeneracy and provide model discrimination (Kumar et al., 2020He et al., 2020Lamy et al., 2018).

5. Black Hole Thermodynamics and Remnants

Thermodynamic quantities for the outer horizon $r_+$ are given by:

• Surface gravity:

$\kappa = \frac{ \Delta'(r_+) }{ 2(r_+^2 + a^2) },$

• Hawking temperature:

$$T_H = \frac{1}{4\pi(r_+^2 + a^2)} [ 2 r_+ - 2 m(r_+) - 2 r_+ m'(r_+) ],$$

• Entropy:

$S = \pi ( r_+^2 + a^2 ).$

Heat capacity, $C_a = (\partial M / \partial T_H)|_a$, controls local stability (Mehdipour et al., 2016). Quantum-gravity–motivated corrections, such as logarithmic entropy and terms from the generalized uncertainty principle (GUP), further modify $T_H$ and $S$ (Ali et al., 2022).

The evaporation process terminates at the extremal configuration where $T_H=0$, yielding a remnant with minimal mass

$$M_{\rm rem}(a,g) = \frac{ r_{\rm ext}^3 + g^3 }{ 2 r_{\rm ext}^4 } ( r_{\rm ext}^2 + a^2 )$$

and remnant radius $r_{\rm ext}$. Inclusion of noncommutative geometry effects further increases the remnant mass and radius (Mehdipour et al., 2016).

6. Penrose Process, Particle Acceleration, and Dynamical Properties

The regular rotating Hayward black hole supports energy extraction mechanisms analogous to those in Kerr. The maximum efficiency of the Penrose process is

$$\eta_{\max}(a,g) = \frac{1}{2} \left( \sqrt{ r_S / r_H } - 1 \right),$$

with $r_S$ the equatorial static limit and $r_H$ the event horizon. For $g > 0$, $\eta_{\max}$ can exceed the Kerr extremal value (20.7%) due to an enlarged ergoregion (Khan et al., 2021).

In the extremal limit ($a \to a_E(g)$), the metric admits Bañados–Silk–West (BSW) acceleration: infalling particles with properly tuned angular momenta can reach arbitrarily high center-of-mass energy near the horizon, up to Planck scales (Amir et al., 2015Pourhassan et al., 2015).

Table: Comparison of Key Parameters

Quantity	Kerr	Rotating Hayward (generic $g$)
Central singularity	Present	Regular, de Sitter core
Horizon equation	$r^2 + a^2 - 2M r = 0$	$r^2 + a^2 - 2m(r) r = 0$
Extremal spin	$a = M$	$a_{\rm ext} < M,\ \downarrow$ with $g$
Shadow diameter	Maximal ($g=0$)	Decreases with $g$
Remnant mass ($a=0$)	0	$3/2^{5/3} g$

The introduction of $g$ introduces a tunable regularization scale that controls physical deviations from classical Kerr and can be constrained observationally and dynamically.

7. Astrophysical and Quantum Gravity Implications

Rotating Hayward black holes provide regular models for astrophysical black holes avoiding classical singularities. Current EHT observations of supermassive black holes, e.g., M87*, cannot yet discriminate Kerr from Hayward metrics for $g \lesssim 0.7 M$ (Kumar et al., 2020). Future high-resolution EHT or space–VLBI, combined with accretion disk modeling and ringdown analysis, may potentially constrain $g$.

Quantum gravity–motivated corrections (such as noncommutative smearing, GUP, and one-loop modifications to the mass profile) further alter horizon structure, Hawking temperature, and remnants (Mehdipour et al., 2016Ali et al., 2022). These effects may become relevant near the endpoint of black hole evaporation or in the context of TeV-scale black hole formation in extra-dimensional scenarios.

The dynamical merger of two rotating Hayward black holes modifies gravitational wave emission: the upper bound on total emitted radiation is generally increased compared to Kerr, and constraints from GW150914 and GW151226 can be used to place upper limits on $g$ for LIGO-sized events (Gwak, 2017).

Rotating Hayward black holes thus offer a tractable, phenomenologically rich framework to study the interplay of classical no-hair theorems, horizon dynamics, strong-field imaging, and quantum gravity corrections in regular black hole spacetimes.