Axial Electromagnetic QNMs

• Axial electromagnetic quasinormal modes are fundamental dissipative eigen-oscillations in axially symmetric systems, defined by complex frequencies.
• They are computed using high-order WKB, Prony, and spectral methods to accurately quantify oscillation rates and damping, reflecting underlying spacetime structure.
• These modes serve as probes in black hole spectroscopy and resonator modeling, enabling detection of quantum corrections and modified gravity effects.

Axial electromagnetic quasinormal modes (QNMs) represent the fundamental dissipative eigen-oscillations of electromagnetic fields in axially symmetric systems, most notably black hole geometries and resonators. These modes are characterized by complex frequencies, with the real part specifying the oscillation rate and the imaginary part describing damping (decay). In the context of black holes, axial modes are the odd-parity sector associated with vector spherical harmonics or the spin-weighted harmonics of Maxwell fields. Mathematically, they arise from master Schrödinger-type wave equations subject to outgoing radiation at infinity and ingoing conditions at horizons or singular boundaries. Axial electromagnetic QNMs play a central role in black hole spectroscopy, gravitational wave ringdown analysis, and electromagnetic resonator modeling; they also probe quantum-gravitational and modified-gravity corrections via their sensitivity to spacetime geometry.

1. Governing Wave Equations and Effective Potentials

For a broad class of static, spherically symmetric backgrounds, axial electromagnetic QNMs are governed by a master wave equation of the form

$$\frac{d^2\Psi}{dr_*^2} + \left[\omega^2 - V_{\text{axial}}(r)\right] \Psi = 0$$

where the tortoise coordinate $dr_*/dr = 1/f(r)$ encodes the red-shift structure, and $V_{\text{axial}}(r)$ is determined by the underlying metric and any matter fields or modifications:

• Schwarzschild and Generalizations:

$$V_{\text{axial}}(r) = f(r) \frac{\ell(\ell+1)}{r^2}$$

with $f(r) = 1 - 2M/r$ for Schwarzschild and modifications for quantum corrections (Skvortsova, 2024), dilaton (Blázquez-Salcedo et al., 2019, Brito et al., 2018), electric/magnetic charge, loop quantum cosmology (Skvortsova, 2024), regularization (e.g., Hayward parameter) (Pedraza et al., 2021), and quintessence.

• Nonlinear and Modified Gravity Cases:

Nonlinear electrodynamics generates extra structure in the potential:

$$V_{\rm em}(r) = f(r) \left[\frac{\ell(\ell+1)}{r^2} \left(1+\frac{4Q_m^2L_F(r)}{r^4}\right) - \frac{f{L_F'}^2 - 2L_F(fL_F')'}{L_F^2}\right]$$

as in Plebański-type NLED backgrounds (Fathi et al., 2 Dec 2025). Weyl gravity alters $f(r)$ by a linear term $\gamma r$ (Momennia et al., 2019).

• Kerr Spacetimes:

Axial modes are described by solutions to the Teukolsky angular and radial equations for spin $s = \pm1$ (Staicova et al., 2014).

The universal feature for axial EM modes is the dominance of the angular potential barrier $f(r)\ell(\ell+1)/r^2$ for spin-1 fields and the absence of terms proportional to $dr_*/dr = 1/f(r)$0.

2. Boundary Conditions and QNM Quantization

Axial electromagnetic QNMs are identified as discrete complex eigenfrequencies subject to physically motivated boundary conditions:

• Black Hole Horizons: Purely ingoing waves at the event horizon ($dr_*/dr = 1/f(r)$1), $dr_*/dr = 1/f(r)$2.
• Spatial Infinity: Purely outgoing waves ($dr_*/dr = 1/f(r)$3), $dr_*/dr = 1/f(r)$4 for asymptotically flat backgrounds, or Dirichlet/normalizable vanishing for AdS backgrounds (Fathi et al., 2 Dec 2025).
• Naked Singularities: Ingoing condition at the central singularity (Pathrikar et al., 2 Apr 2025).

These boundary conditions are implemented analytically via expansions, numerically through matching, or—in rotating systems—by exact confluent Heun function prescription at endpoints in the complex plane (Staicova et al., 2014).

3. Computational and Analytical Methods

Several robust approaches have been developed for extracting axial electromagnetic QNMs:

• High-order WKB and Padé Resummation:

Sixth-order WKB with Padé resummation yields accurate spectra, especially for fundamental and overtone modes (Skvortsova, 2024, Pedraza et al., 2021, Momennia et al., 2019, Pathrikar et al., 2 Apr 2025). The WKB formula typically reads:

$dr_*/dr = 1/f(r)$5

where all coefficients are evaluated at the peak of the effective potential.

• Time-domain Integration and Prony Method:

Direct integration on null grids (Gundlach–Price–Pullin scheme) followed by Prony fitting enables extraction from ringdown signals (Skvortsova, 2024, Momennia et al., 2019).

• Continued-Fraction and Direct Shooting:

Leaver's continued-fraction and direct matching of ingoing/outgoing solutions at an internal point (Brito et al., 2018, Blázquez-Salcedo et al., 2019, Blázquez-Salcedo et al., 2020).

• Spectral Methods in NLED/AdS:

Chebyshev–Lobatto discretization of the linear generalized eigenvalue problem for nonlinear electrodynamics in AdS (Fathi et al., 2 Dec 2025).

• Exact Analytic Heun Function Solutions:

Full spectral problem for Kerr: coupled solution of angular and radial Teukolsky equations with boundary conditions via confluent Heun functions (Staicova et al., 2014).

4. Fundamental Properties and Parameter Dependence

Axial electromagnetic QNM frequencies provide probes of spacetime structure, field content, and modifications of general relativity. Principal trends include:

• Effect of Quantum Corrections ($dr_*/dr = 1/f(r)$6 parameter):

In the quantum Oppenheimer–Snyder black hole, increasing $dr_*/dr = 1/f(r)$7 slightly raises the real part of $dr_*/dr = 1/f(r)$8 (oscillation frequency) and significantly reduces the damping rate $dr_*/dr = 1/f(r)$9, up to 10–20% at $V_{\text{axial}}(r)$0 (Skvortsova, 2024).

• Hayward Regularization and Quintessence:

The Hayward parameter $V_{\text{axial}}(r)$1 raises both $V_{\text{axial}}(r)$2 and $V_{\text{axial}}(r)$3 (steeper barrier, faster decay), while quintessence normalization $V_{\text{axial}}(r)$4 reduces both (softening, longer-lived modes) (Pedraza et al., 2021).

• Nonlinear Electrodynamics:

The nonlinearity parameter $V_{\text{axial}}(r)$5 and effective charge $V_{\text{axial}}(r)$6 both raise frequencies and damping rates, with magnetic modes showing systematically less oscillatory and less damped behavior than electric ones. For large $V_{\text{axial}}(r)$7, magnetic modes can become purely imaginary, indicating overdamped evolution in the absence of a trapping barrier (Fathi et al., 2 Dec 2025).

• Dilaton and Scalarization Effects:

Dilaton coupling ($V_{\text{axial}}(r)$8) can induce significant isospectrality breaking, sometimes up to tens of percent for high charge and coupling (Brito et al., 2018, Blázquez-Salcedo et al., 2019, Blázquez-Salcedo et al., 2020).

• Kerr Rotation:

Axial EM QNMs in Kerr spacetimes are sensitive to angular momentum via complex separation constants and Heun function roots, with their real parts typically increasing with spin and azimuthal indices (Staicova et al., 2014).

• Hydrodynamic and Strong-Field Regimes:

In chiral magnetic and Chern–Simons brane spacetimes, the axial electromagnetic QNMs encode chiral magnetic wave propagation, Landau-level dispersion ($V_{\text{axial}}(r)$9 scaling of real part), and damping rates that can vanish in strong anomaly-coupling regimes, producing long-lived modes at large magnetic field (Ammon et al., 2017).

5. Isospectrality, Stability, and Physical Implications

• Isospectrality:

Classical spherical backgrounds (Schwarzschild, RN) feature exact isospectrality between axial and polar electromagnetic modes. Modifications—scalar hair, dilaton, nonlinearities—break this degeneracy, with quantifiable splittings (few–tens percent) depending on model and parameters (Skvortsova, 2024, Brito et al., 2018, Blázquez-Salcedo et al., 2019, Fathi et al., 2 Dec 2025, Blázquez-Salcedo et al., 2020).

• Stability:

All established axial electromagnetic modes for black hole and naked singularity backgrounds have $$V_{\text{axial}}(r) = f(r) \frac{\ell(\ell+1)}{r^2}$$0, ensuring linear stability and decay of perturbations (Pathrikar et al., 2 Apr 2025, Pedraza et al., 2021, Skvortsova, 2024, Blázquez-Salcedo et al., 2020).

• Observational Relevance:

Ringdown times $$V_{\text{axial}}(r) = f(r) \frac{\ell(\ell+1)}{r^2}$$1 pertain directly to electromagnetic (and gravitational) emission signatures. Quantum, dilaton, and NLED modifications produce detectable shifts in frequency and damping, potentially resolvable in future multi-messenger or gravitational-wave events (Skvortsova, 2024, Fathi et al., 2 Dec 2025, Pedraza et al., 2021).

6. Axial Electromagnetic QNMs in Resonator Physics

Beyond gravitation, axial electromagnetic QNMs are essential in modeling axially symmetric dielectric resonators and nanophotonic structures:

• Maxwell Eigenproblem under Axial Symmetry:

$$V_{\text{axial}}(r) = f(r) \frac{\ell(\ell+1)}{r^2}$$2

with m=0 separation in cylindrical coordinates (Kristensen et al., 2019).

• Bi-orthogonal Normalization:

$$V_{\text{axial}}(r) = f(r) \frac{\ell(\ell+1)}{r^2}$$3

guaranteeing modal completeness and orthogonality.

• Response Reconstruction:

$$V_{\text{axial}}(r) = f(r) \frac{\ell(\ell+1)}{r^2}$$4

with explicit field expansions for scattering, Purcell factor, and S-matrix formalism (Kristensen et al., 2019).

7. Tables of Representative Axial Electromagnetic QNM Frequencies

The following tables illustrate generic behavior and dependence on parameters for axial EM QNMs in quantum-corrected, regular, and modified backgrounds (Skvortsova, 2024, Pedraza et al., 2021, Fathi et al., 2 Dec 2025).

Model/Params	ℓ, n	Re(ω)	−Im(ω)
OSC, γ=0.00	1, 0	0.2483	0.0925
OSC, γ=1.50	1, 0	0.2605	0.0858
Hayward + quint c=0.10	1, 0	0.1113	0.0314
NLED β=0.9, Q_e=0.9	2, 0	4.6481	2.6844
NLED β=0.9, Q_m=0.9	2, 0	3.5027	2.0593
JMN-1 M_0=0.7	2, 0	0.4547	0.0944

The data demonstrate monotonic shifts and splittings as γ, c, β, and charges are varied: quantum gravity and regularization typically decrease damping, NLED speeds up and enhances decay, and magnetic/electric modes split in frequency and damping.

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Axial electromagnetic quasinormal modes remain a central probe of spacetime structure, quantum corrections, and field content across both gravitation and electrodynamics. Their calculation—via direct integration, spectral and eigenvalue methods, and exact analytical techniques—enables precise characterization of ringdown signatures, stability analysis, and modal expansion in resonator theory. The sensitivity of axial EM QNMs to geometric and physical modifications makes them indispensable for contemporary theoretical, observational, and experimental investigations in astrophysics and nanophotonics.