Dyonic Kerr-Newman-Melvin-Swirling Spacetime

• Dyonic Kerr-Newman-Melvin-Swirling spacetime is an exact six-parameter solution to the Einstein–Maxwell equations that models a rotating black hole with electric, magnetic, and swirling charges.
• It unifies Kerr–Newman, Melvin, and swirling metrics via solution-generating techniques, ensuring global regularity through specific algebraic parameter constraints.
• The geometry features complex horizon structures, deformed ergoregions, and nonintegrable, chaotic geodesic dynamics alongside a universal near-horizon conformal symmetry.

The dyonic Kerr–Newman–Melvin–swirling spacetime is an exact, six-parameter family of solutions to the Einstein–Maxwell field equations, modeling a rotating black hole with both electric and magnetic (dyonic) charges embedded within a rotating, magnetized cosmological background. This geometry generalizes and unifies the well-known Kerr–Newman, Melvin, and swirling metrics, introducing intricate coupling between black hole and background parameters. The geometry is completely regular—free of curvature and conical singularities—subject to explicit algebraic relations among parameters, and features rich horizon, ergoregion, and dynamical structure, including nonintegrable and chaotic particle motion.

1. Metric Structure and Field Content

The dyonic Kerr–Newman–Melvin–swirling metric is presented in Boyer–Lindquist–type coordinates $(t,r,\theta,\varphi)$, and depends on six parameters: mass $M$, spin $a$, electric charge $Q$, magnetic charge $H$, Melvin magnetic field strength $B$, and swirl (universe rotation) parameter $\jmath$ (Pinto et al., 10 Mar 2025).

The full solution is (Eq. (3.1)): $$\boxed{ \begin{aligned} ds^2 &= F(r,\theta)\,\Bigl[ -\frac{\Delta(r)}{\Sigma(r,\theta)}\,dt^2 + \frac{dr^2}{\Delta(r)} + d\theta^2\Bigr] + \frac{\Sigma(r,\theta)\,\sin^2\theta}{F(r,\theta)}\Bigl[d\varphi - \frac{\Omega(r,\theta)}{\Sigma(r,\theta)}\,dt\Bigr]^2, \ A &= \frac{A_0(r,\theta)}{\Sigma(r,\theta)}\,dt + \frac{A_3(r,\theta)}{F(r,\theta)}\Bigl[d\varphi - \frac{\Omega(r,\theta)}{\Sigma(r,\theta)}\,dt\Bigr]. \end{aligned} }$$ Key structural functions: $$\begin{aligned} Z^2 &= Q^2 + H^2, \ \Delta(r) &= r^2 - 2Mr + Z^2 + a^2, \ \lambda(r) &= 2Mr - Z^2, \ \Sigma(r,\theta) &= (r^2 + a^2)^2 - a^2\Delta(r)\sin^2\theta, \ R^2 &= r^2 + a^2\cos^2\theta, \ \Xi(r,\theta) &= (r^2 + a^2)\sin^2\theta + Z^2\cos^2\theta. \end{aligned}$$

The nontrivial functions parameterizing the metric and potential are expanded in $(B,\jmath)$: $$\boxed{ \begin{aligned} F &= R^2 + 2B\phi_{(0)} + \tfrac{B^2}{2}\phi_{(1)} + \tfrac{B^3}{2}\phi_{(2)} + [\jmath^2 + \tfrac{B^4}{16}]\phi_{(3)} + 2\jmath B\phi_{(4)} + \jmath F_{(1)}, \ \Omega &= a\lambda + 2B\chi_{(0)} + \tfrac{B^2}{2}\chi_{(1)} + \tfrac{B^3}{2}\chi_{(2)} + [\jmath^2 + \tfrac{B^4}{16}]\chi_{(3)} + 2\jmath B\chi_{(4)} + \jmath\Omega_{(1)}, \ A_0 &= \chi_{(0)} + \tfrac{B}{2}\chi_{(1)} + \tfrac{3B^2}{4}\chi_{(2)} + \tfrac{B^3}{8}\chi_{(3)} + \jmath\chi_{(4)}, \ A_3 &= \phi_{(0)} + \tfrac{B}{2}\phi_{(1)} + \tfrac{3B^2}{4}\phi_{(2)} + \tfrac{B^3}{8}\phi_{(3)} + \jmath\phi_{(4)}. \end{aligned} }$$ The auxiliary polynomials $\{\chi_{(i)},\phi_{(i)},F_{(1)},\Omega_{(1)}\}$ depend on $(r,\theta;a,Q,H,M)$ and are explicitly given in [(Pinto et al., 10 Mar 2025), Sec. 3.2].

2. Global Structure: Horizons, Extremality, and Ergoregion

Horizons and Extremality

Killing horizons are located at roots of $\Delta(r)=0$: $$\boxed{ \Delta(r_\pm)=0 \ \Longrightarrow\ r_\pm = M \pm \sqrt{M^2-a^2-Z^2} }$$ Extremal black holes satisfy $M^2 = a^2 + Z^2$, i.e., $r_+ = r_- = M$. Coordinate singularities also occur at $\Sigma=0$ and $F=0$, but only $\Delta=0$ corresponds to true Killing horizons.

Ergoregion

The ergoregion, defined by $g_{tt} > 0$, is encoded as: $$g_{tt}=g_{(1)}(r,\theta)(1-\cos^2\theta) + g_{(2)}(r,\theta)\Delta(r)$$ The “doughnut” ergoregion, bounded by $g_{tt}=0$, is topologically similar to the standard Kerr–Newman case but is deformed by $B$ and $\jmath$ (Pinto et al., 10 Mar 2025). Notably, the ergoregion can extend to infinity along the rotation axis for $g_{(2)} < 0$, a feature not present in the standard Kerr family.

3. Regularity, Conical Defect Removal, and Curvature Invariants

Global regularity requires explicit parameter constraints to eliminate conical singularities (defects in the azimuthal coordinate $\varphi$) and Dirac strings associated with the magnetic charge.

• Conical singularities are removed by imposing [(Pinto et al., 10 Mar 2025), Eq. (3.20)]:

$$\delta_0=\delta_\pi=2\pi \ \Longrightarrow\ \jmath=-\frac{B\,H\,(4+B^2 Z^2)}{16\,a\,M -4\,B\,Q\,Z^2}$$

followed by a rescaling $\phi\to(2\pi/\delta\varphi)\phi$.

• Dirac strings are eliminated via [(Pinto et al., 10 Mar 2025), Eqs. (3.24)-(3.25)]:

$$\jmath=\frac{H\,(4+3B^2 Z^2)}{4\,Q\,Z^2},\quad a = \frac{Q\,B^3\,Z^4}{2M(4+3B^2Z^2)}.$$

A corresponding gauge shift in $A_\varphi$ and $\phi$ rescaling completes the defect removal.

• Curvature invariants. The Kretschmann scalar

$K = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$

decays as $r^{-6}$ for $B=\jmath=0$, but as $r^{-12}$ if either $B$ or $\jmath$ is nonzero; at $r=\cos\theta=0$ (“center”), $K \propto r^{-8}$ unless one imposes parameter values incompatible with conical regularity constraints. The regularized solution is free of curvature singularities on and outside the horizon.

4. Thermodynamical and Near-Horizon Properties

Despite the presence of external fields, the deformations $B$ and $\jmath$ leave the thermodynamic quantities associated with the outer horizon invariant (Pinto et al., 10 Mar 2025):

$$\boxed{ T_H = \frac{\kappa_+}{2\pi},\quad \kappa_+ = \frac{r_+-r_-}{2(r_+^2 + a^2)},\quad \mathcal{A}_+ = 4\pi(r_+^2+a^2),\quad S_{\rm BH} = \pi(r_+^2+a^2) }$$

For extremal black holes, the near-horizon geometry is a universal $SL(2,\mathbb R)\times U(1)$ warped-twisted product of $AdS_2\times S^2$: $$\boxed{ d\hat s^2 = \Gamma(\theta)\left[ -\hat r^2 d\hat t^2 + \frac{d\hat r^2}{\hat r^2} + \alpha^2(\theta)d\theta^2 + \gamma^2(\theta)(d\hat\varphi + \tilde\kappa \hat r d\hat t)^2\right] }$$ with explicit gauge fields. The Kerr/CFT correspondence yields a microscopic derivation of the Bekenstein–Hawking entropy: $$c_L = 3\tilde\kappa\int_0^\pi \Gamma\,\alpha\,\gamma\,d\theta=-6\kappa,\quad T_L = -\frac{M^2+a^2}{2\pi\kappa},\quad S_{\rm CFT} = \frac{\pi^2}{3}c_L\,T_L = S_{\rm BH}$$ Thus, the Cardy formula exactly reproduces the macroscopic Bekenstein–Hawking entropy.

5. Chaotic Geodesic Dynamics and Nonintegrability

Test particle motion in the dyonic Kerr–Newman–Melvin–swirling spacetime is governed by the Lagrangian: $$\mathcal{L} = \frac{\mu}{2}g_{\alpha\beta}\dot x^\alpha\dot x^\beta + qA_\alpha\dot x^\alpha,$$ with conserved energy $E = -p_t$ and axial angular momentum $L = p_\varphi$.

Due to the nontrivial $r$–$\theta$ dependence of the metric and electromagnetic potential (in particular, nonvanishing $B$ and $\jmath$), the Hamilton–Jacobi equation ceases to separate, leading to generically nonintegrable, chaotic dynamics (Cao et al., 11 Nov 2025).

Standard diagnostics for chaos were employed:

• Poincaré sections: Regular orbits yield KAM tori; chaos manifests as fuzziness and destruction of tori.
• Fast Lyapunov indicator (FLI): Algebraic growth signals regularity; exponential growth signals chaos.
• Recurrence analysis: Long diagonals in recurrence plots denote regularity; broken diagonals/absence of structure indicates chaos.
• Bifurcation diagrams and basins of attraction reveal period-doubling, fractal basin boundaries, and windows of chaotic escape/binding.

Quantitative findings:

• Increasing $B$ or $\jmath$ increases both the number and region of chaotic orbits.
• Increasing $Q$, $H$, or $a$ decreases the chaotic region and raises its threshold.
• The critical value $j_c$ for chaos onset is approximately $6.1\times10^{-6}$ for representative parameters; for $B$, the threshold is $B_c\sim2\times10^{-6}$ (at $j=10^{-6}$).

The regularity conditions that remove conical and Dirac string singularities do not eliminate chaos, although they reduce the space of independent continuous parameters.

6. Construction via Solution-Generating Techniques

The solution is constructed via Ehlers–Harrison transformations of the Ernst formalism (Pinto, 15 Jul 2024). Starting from the Lewis–Weyl–Papapetrou ansatz, the procedure involves:

• Defining complex Ernst potentials $E=f-|\Phi|^2 + ih$ and $\Phi = A_t + i\psi$.
• Solving the coupled Ernst equations.
• Successive application of:
  • Ehlers transformations (introducing the swirl parameter $\jmath$) and
  • Harrison transformations (introducing the Melvin field $B$).

The explicit metric and Maxwell fields are reconstructed by back-substituting into the seed ansatz; the full component functions are provided in [(Pinto, 15 Jul 2024), Eqs. (5.13)–(5.14)] and [(Pinto et al., 10 Mar 2025), Sec. 3.2].

The swirling parameter $\jmath$ generically mixes electric and magnetic charges (dyonicity) and allows, in conjunction with $B$, the removal of all conical and string singularities for nonzero $(a,Q,H,B)$.

7. Physical Implications, Limits, and Uniqueness

Subcases and Limits

Special cases of the general solution recover known metrics by setting parameters to zero:

• $B=\jmath=0$: Standard Kerr–Newman black hole.
• $B\neq0,\,\jmath=0$: Kerr–Newman–Melvin (magnetized) black hole.
• $\jmath\neq0,\,B=0$: Swirling–Kerr–Newman.
• $M=a=Q=H=0$: Pure Melvin–swirling cosmology.

Supersymmetry and BPS Properties

No covariantly constant Killing spinors (i.e., no supersymmetric solutions) exist for $B$ or $\jmath$ nonzero in $N=2$, $d=4$ supergravity (Pinto, 15 Jul 2024). Only the standard extremal BPS Kerr–Newman (in AdS) is supersymmetric.

Astrophysical and Theoretical Significance

The swirling–Melvin interaction drastically modifies the near-horizon and asymptotic geometry, enhancing curvature falloff ($r^{-12}$) and deforming the ergoregion. The nonintegrability and chaos in geodesic motion suggest observable consequences for charged particle dynamics and radiation near strongly magnetized, rotating astrophysical objects. The universal near-horizon conformal structure ensures the validity of the Kerr/CFT correspondence and its entropy accounting.

A plausible implication is that, in realistic settings where rotating black holes interact with external magnetized, swirling cosmological environments, intricate dynamical phenomena—such as chaotic scattering, fractal basin boundaries for particle capture/escape, and characteristic near-horizon conformal symmetry—could provide indirect probes of such backgrounds in astrophysical observations.