--- title: Rotating Hayward Black Hole url: https://www.emergentmind.com/topics/rotating-hayward-black-hole type: topic --- # Rotating Hayward Black Hole A rotating Hayward black hole 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 Hayward metric. 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 [2006.09869],[1510.08828],[1410.4043],[1604.08584],[1503.08553],[1802.01635],[1703.10154],[2201.02754]. ## 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 [1410.4043],[1510.08828],[1604.08584]. 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 [1410.4043],[1510.08828]. 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 $a_E(g)0$ [1510.08828],[2107.06085]. ## 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 [1410.4043],[2107.06085],[1802.01635]. ## 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 $1\sigma$ [2006.09869]. 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 [2006.09869],[2009.06705],[1802.01635]. ## 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 [1604.08584]. Quantum-gravity–motivated corrections, such as logarithmic entropy and terms from the generalized uncertainty principle (GUP), further modify $T_H$ and $S$ [2201.02754]. 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 [1604.08584]. ## 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 [2107.06085]. 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 [1503.08553],[1506.03443]. 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 $g \lesssim 0.7 M$ [2006.09869]. 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 [1604.08584],[2201.02754]. 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 [1703.10154]. 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. Source: https://www.emergentmind.com/topics/rotating-hayward-black-hole