Noncommutative Hayward Black Hole

• The noncommutative Hayward black hole is a singularity-free model that merges the Hayward metric’s de Sitter core with quantum noncommutativity for regularity and modified thermodynamics.
• It employs both Gaussian smearing and Seiberg–Witten deformations to alter horizon structure, evaporation endpoints, quasinormal modes, and gravitational lensing features.
• The framework predicts cold remnant formation and measurable astrophysical corrections, offering insights into high-energy collisions and extra-dimensional scenarios.

A noncommutative @@@@2@@@@ (NH-BH) is a class of regular (singularity-free) black hole solutions in general relativity, formed by combining the Hayward metric—characterized by a de Sitter core and governed by a parameter $g$ or %%%%1%%%%—with noncommutative geometry effects encoded by a scale parameter $\theta$ or $\Theta$. The resulting spacetimes manifest distinct horizon structures, thermodynamics, and phenomenology, including novel regularity conditions, modified evaporation endpoints (black hole remnants), altered quasinormal mode spectra, and new constraints from gravitational lensing and solar-system tests. There are two principal formulations: noncommutative-inspired Hayward black holes, which smear matter sources using Gaussian profiles (Mehdipour et al., 2016), and noncommutative-gauge (Seiberg-Witten–type) deformations of the Hayward solution, which introduce perturbative metric corrections (Heidari et al., 22 Mar 2025). Both frameworks yield non-singular geometries with phenomenology sensitive to the interplay between the noncommutative scale, Hayward core parameter, and rotation.

1. Metric Structure and Noncommutative Deformations

The noncommutative Hayward geometry is constructed by deforming the standard Hayward lapse function $f_{0,g}(r)$ or $f_{0,l}(r)$ through noncommutative effects.

Noncommutative-Inspired (Gaussian Smearing) (Mehdipour et al., 2016)

The line element for the static, spherically symmetric case is

$$ds^2 = f(r)\,dt^2 - \frac{dr^2}{f(r)} - r^2\,d\Omega^2\,,$$

with

$$f(r) = 1 - \frac{2\,m(r)}{r}\,, \qquad m(r) = M_\theta(r)\frac{r^3}{r^3+g^3}\,,$$

where $M_\theta(r)$ is a Gaussian-smeared mass profile,

$$M_\theta(r) = \frac{2M}{\sqrt\pi}\,\gamma\left(\frac{3}{2};\frac{r^2}{4\theta}\right),$$

$\gamma(s;x)$ is the lower incomplete gamma function, $\theta>0$ encodes the noncommutativity scale, and $g>0$ the Hayward core scale.

Noncommutative Gauge (Seiberg-Witten) Framework (Heidari et al., 22 Mar 2025)

The alternative formalism introduces the line element

$$ds^2 = -f_{\Theta,l}(r)\,dt^2 + f_{\Theta,l}(r)^{-1}\,dr^2 + r^2\big(d\vartheta^2 + \sin^2\vartheta\,d\varphi^2 \big),$$

with

$$f_{\Theta,l}(r) = 1 - \frac{2 M r^2}{r^3 + 2 M l^2} + \Theta^2\,\Delta f(r)\,,$$

where $l$ is the Hayward parameter and $\Theta$ the noncommutative parameter. The explicit correction $\Delta f(r)$ at $\mathcal{O}(\Theta^2)$ is constructed from Seiberg–Witten–deformed tetrads.

In both constructions, the geometry is regular everywhere except at specific excluded angular points ($\theta = \pi/2 + n\pi$ for $n\in\mathbb{Z}$) in the gauge-theory approach due to the Kretschmann scalar denominator structure.

2. Horizon Structure, Regularity, and Spacetime Invariants

The event horizon radius $r_h$ is set by the largest real root of $f(r_h)=0$ or $f_{\Theta,l}(r_h)=0$. As the noncommutative parameter increases, $r_h$ shifts mildly. Importantly, the geometry’s regularity is maintained throughout, with the Kretschmann scalar

$$K=R_{\alpha\beta\gamma\delta} R^{\alpha\beta\gamma\delta}$$

remaining finite for all $r$ and angular directions away from excluded points. For $r\to\infty$, all NH-BH solutions approach standard Schwarzschild/Hayward asymptotics.

3. Thermodynamic Properties and Remnant Formation

The Hawking temperature for both static and rotating NH-BH configurations is derived via the surface gravity at the event horizon. For the noncommutative-inspired metric (Mehdipour et al., 2016): $T_H = \frac{1}{4\pi}f'(r_H) \,,$ where $f'(r_H)$ includes the effects of both Gaussian smearing and Hayward regularization.

Similarly, for the gauge-theory construction (Heidari et al., 22 Mar 2025): $$T_H(\Theta,l) = \frac{1}{4\pi}\left.\frac{d f_{\Theta,l}}{dr}\right|_{r=h}\,.$$ In both formulations, the temperature rises, peaks, and then drops to zero as $r_h$ (and the black hole mass) decreases, signaling incomplete evaporation and the emergence of a cold, stable remnant at $T_H=0$, $r_h = r_0$. The remnant mass $M_0$ grows with increasing noncommutative scale ($\theta$ or $\Theta$) and core parameter ($g$ or $l$), and is only mildly sensitive to rotation.

The entropy follows $S = \pi r_h^2$, and the specific heat has sign changes encoding stability properties. Positive heat capacity branches correspond to thermodynamically stable small black holes, while sign changes in $C$ mark Davies phase points.

Thermodynamic quantity	Noncommutative NH-BH	Hayward limit
Hawking temperature	$T_H$ as above; vanishes at minimal $r_0(\theta,g)$	$T_H^0$ as $g,\theta\to0$
Entropy	$S=\pi r_h^2$	$S_{0}=\pi r_h^2$
Remnant mass	$M_0$ increases with $\theta$, $g$ or $\Theta$, $l$	$M_0\to M_{\mathrm{Planck}}$

The threshold energy for black hole formation in high-energy collisions is thus naturally elevated by noncommutative and regularization effects, potentially explaining the absence of black hole signatures at current collider energies.

4. Rotating Solutions and Higher Dimensions

The rotation parameter $a$ is incorporated via the Newman–Janis procedure. The rotating noncommutative Hayward metric in Boyer–Lindquist coordinates reads

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

with

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

where $m(r)$ carries the noncommutative Hayward profile. The ring singularity remains regular due to the finiteness of $m(r \to 0)$, and the thermodynamic endpoint remains a cold remnant, with the remnant radius growing only slightly with $a$ (Mehdipour et al., 2016).

Extensions to $d>4$ spacetime dimensions modify the metric function, remnant mass expression, and threshold energy. The minimal mass for black hole formation, $M_0^{(d)}$, decreases as $d$ increases, reopening the possibility of black hole production at multi-TeV-scale colliders if a sufficient number of extra dimensions exist.

5. Perturbations, Quasinormal Modes, and Shadows

Scalar field perturbations of the NH-BH yield an effective Schrödinger-type equation: $$\frac{d^2\Psi}{dr_*^2} + [\omega^2 - V_{\mathrm{eff}}(r)]\,\Psi = 0,$$ with

$$V_{\mathrm{eff}}(r) = f_{0,l}(r) \left[\frac{\ell(\ell+1)}{r^2} + \frac{1}{r}f'_{0,l}(r)\right] + \Theta^2 \delta V(r) + \mathcal{O}(l^2)\,,$$

and the WKB approach provides the quasinormal frequencies

$$\frac{i(\omega_n^2 - V_0)}{\sqrt{-2V_0''}} + \sum_{j=2}^3\Omega_j = n + \frac{1}{2}.$$

The principal effect of noncommutativity is an increase in both $\operatorname{Re}\,\omega$ and $|\operatorname{Im}\,\omega|$ compared to commutative Hayward black holes, indicating a slightly stronger perturbation barrier.

The photon sphere, $r_\mathrm{ph}$, shrinks with growing $\theta$, $g$, $\Theta$, or $l$. The shadow radius observed at infinity,

$$R_{\mathrm{sh}} = r_\mathrm{ph} \sqrt{\frac{r_\mathrm{ph}^3 + 2M l^2}{r_\mathrm{ph}^3 - 2Mr_\mathrm{ph}^2 + 2M l^2}} + \mathcal{O}(\Theta^2)\,,$$

also exhibits a mild decrease with higher noncommutative or Hayward parameters. Constraints extracted from Event Horizon Telescope data are relatively loose, with $\Theta,\ l \lesssim \mathcal{O}(1M)$.

6. Phenomenological Constraints: Gravitational Lensing and Solar System Tests

Noncommutative corrections to gravitational lensing in the weak-deflection regime are obtained using the Gauss-Bonnet theorem. The leading corrections are

$$\alpha(b; \Theta, l) \simeq \frac{4M}{b} + \frac{3\pi M^2}{4b^2} - \frac{15\pi l^2 M^2}{4b^4} - \frac{8 \Theta^2 M}{3b^3} + \mathcal{O}(l^2\Theta^2),$$

where the $\Theta^2$ term strengthens the bending compared to Schwarzschild.

Precision solar-system data—perihelion precession, light deflection, and Shapiro delay—impose bounds:

• For Mercury: $$-1.5\times 10^{18}\,{\rm m}^2 \leq \Theta^2 \leq 3.0\times 10^{17}\,{\rm m}^2$$, $$-1.9\times 10^{16}\,{\rm m}^2 \leq l^2 \leq 8.6\times 10^{16}\,{\rm m}^2$$.
• For light deflection: $$-1.9\times 10^{13}\,{\rm m}^2 \leq \Theta^2 \leq 9.7\times 10^{13}\,{\rm m}^2$$.
• For Shapiro delay: $$-1.6\times 10^{19}\,{\rm m}^2 \leq \Theta^2 \leq 1.6\times 10^{19}\,{\rm m}^2$$.

A plausible implication is that any deviation from commutative Hayward solutions must be very small on astrophysical scales.

7. Quantum Radiation and Particle Creation

Analysis of Hawking-like quantum radiation, including both bosonic and fermionic modes, with explicit particle creation rates, confirms the existence of a remnant mass at which $T_H \to 0$. These rates are computed from quantum field-theoretic treatments in the NH-BH background up to $\mathcal{O}(\Theta^2)$ (Heidari et al., 22 Mar 2025). This suggests an evaporation scenario where noncommutativity and regularization halt the radiation, preventing total black hole disappearance.

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The noncommutative Hayward black hole paradigm synthesizes UV-regularized black hole geometries with quantum spacetime effects, yielding robust predictions for remnant formation, horizon structure, gravitational lensing, and observable signatures. The different constructive frameworks—Gaussian smearing versus gauge-theory Seiberg–Witten expansion—produce congruent phenomenology with technical distinctions in regularity and parameter space. Experimental constraints currently limit noncommutative and regularization parameters to values indistinguishable from zero at solar-system scales, though they may play a pivotal role near the Planck regime or in models with extra dimensions (Mehdipour et al., 2016, Heidari et al., 22 Mar 2025).