Axial gravitational quasinormal modes of magnetically charged black holes

The Euler–Heisenberg (EH) Lagrangian, first derived in 1936 [1], serves as a nonlinear extension of quantum electrodynamics (QED) and provides a classical description that goes beyond Maxwell’s theory in strong-field regimes where vacuum polarization effects become significant. By modeling vacuum as a polarizable medium, the EH framework incorporates effective polarization and magnetization responses arising from the virtual charge fluctuations associated with real charges and currents [2]. This theoretical formalism not only refines classical electrodynamics in extreme electromagnetic environments but also forms a cornerstone for studying nonlinear phenomena in astrophysics and cosmology.

Considering these unique features, the first black hole solution in Einstein–Hilbert (EH) gravity—an anisotropic, magnetically charged configuration analogous to the Reissner–Nordström metric but generalized to include dyon-type degrees of freedom—was obtained in 1956 [3]. This framework has also been extended to encompass electrically charged solutions [3, 4], rotating black hole configurations [5, 6], and formulations within various modified theories of gravity [7, 8]. More recently, motivated by developments in string theory and Lovelock gravity, Ref. [9] introduced a novel coupling between the dilaton field and EH electrodynamics, extending the Einstein–Maxwell–dilaton framework. Further studies have investigated various physical phenomena in such spacetimes, including particle dynamics, gravitational lensing effects [10, 11], and the shadow characteristics of magnetically charged black holes [12]. Additionally, Jiang et al. [13] examined the properties of geometrically thin, optically thick accretion disks in these backgrounds, offering insights into the observational signatures of such compact objects.

The recent detection of gravitational waves has opened up new avenues for probing strong gravitational fields in the vicinity of black holes. In particular, during the merger of binary compact objects, the ringdown phase of the emitted gravitational waves can be interpreted as the response of the remnant black hole to perturbations imposed during the coalescence process. At this stage, the final Kerr black hole can be effectively described as being in a perturbed state, with the emitted gravitational radiation characterized by distinct decay timescales. These features are well modeled by the quasinormal modes (QNMs) of the black hole [14, 15], which are unique oscillation frequencies determined solely by the mass and spin of the black hole. Furthermore, QNMs provide a powerful tool for testing the validity of the no-hair conjecture, as deviations from the expected frequency spectrum could indicate violations of the Kerr metric or the presence of exotic compact objects [16−18]. In the context of modified theories of gravity [19−23], QNMs serve as critical probes for constraining alternative gravitational models. The stability of QNMs under the parametric perturbations and deformations of the background spacetime plays a key role in such investigations. However, recent studies have also shown that certain QNM spectra may exhibit instability under small perturbations to the effective potential [24, 25]. Moreover, the behavior of QNMs under various types of perturbations offers valuable insights into the stability properties of the underlying spacetime geometry [26−31].

Motivated by these considerations, this study focuses on the QNMs of the magnetically charged black hole. Note that Cho et al. [32, 33] introduced the asymptotic iteration method (AIM), a systematic iterative technique for deriving the accurate approximations of QNM frequencies. In this study, we employ AIM to numerically solve the perturbation equation. Additionally, we apply the Wentzel–Kramers–Brillouin (WKB) approximation [34−36], a well-established method, to estimate the QNM frequencies of the black holes. By comparing the results from both approaches, we aim to cross-validate and reinforce our findings. Another crucial aspect of black hole perturbation theory is the greybody factor [37−40], which characterizes how the gravitational potential of a black hole modifies the spectrum of emitted radiation. In gravitational wave astronomy, greybody factors and QNMs form a complementary framework for analyzing compact binary mergers: QNMs determine the ringdown phase through their characteristic frequencies and damping times, whereas greybody factors quantify the angular-dependent transmission probability of gravitational waves throughout the inspiral-merger-ringdown process [41]. Although the background geometry remains spherically symmetric, axial perturbations are particularly suitable for analyzing the dynamics of magnetically charged black holes in the string-inspired Euler-Heisenberg framework. Owing to the monopolar structure of the gauge potential, the axial sector exhibits a decoupled evolution governed by a single master equation. This simplification facilitates a clearer computation of QNMs and greybody factors and has been widely adopted in related studies of nonlinear electrodynamics.

Accordingly, we plan to study the QNMs and greybody factor of axial perturbations on the magnetically charged black hole in the string-inspired Euler–Heisenberg theory. The remainder of this paper is organized as follows. In Section II, we review the black hole solution briefly and derive the master equation for axial perturbations. In Section III, we solve the quasinormal frequencies with the AIM and WKB methods and study the effects of the black hole parameters on the QNMs. In Section IV, we investigate the time evolution profiles of the perturbation in Section III. In Section V, we calculate the greybody factor with the WKB method. The conclusions and discussions are given in Section VI.

Motivated by string theory and Lovelock gravity, Bakopoulos et al. [9] have recently introduced an extension of the Einstein–Maxwell–dilaton theory that incorporates a nonlinear Euler–Heisenberg term coupled to the dilaton field. The action describing this model is given by [9]

where R denotes the scalar curvature, ϕ is the scalar field, and $ {\cal{L}}(\phi,F) $ represents the Lagrangian density that governs the interaction between the dilaton and the nonlinear electromagnetic field. This Lagrangian is explicitly defined as

Here, $ f(\phi) $ is a coupling function, $ F^2=F_{\mu\nu}F^{\mu\nu} $, and $ F^4= F_{\mu\nu}F^{\mu\nu}F_{\rho\delta}F^{\rho\delta} $, where $ F_{\mu\nu} $ represents the usual field strength $ F_{\mu\nu}=\partial_\mu A_{\nu}-\partial_{\nu}A_{\mu} $. In the case where $ \alpha=\beta=0 $, the theory reduces to the standard Einstein–Maxwell–dilaton model.

Varying the action (1) with respect to $ g_{\mu\nu} $, ϕ, and $ A_\mu $, we can obtain the three field equations

where $ T_{\mu\nu} $ is an energy-momentum tensor with

With regard to the coupling function $ f(\phi) $, Bakopoulos et al. [9] adopted

and then obtained an exact analytic solution in the presence of magnetic charge and scalar hair. This hyperbolic structure reflects a symmetric dilaton coupling often encountered in string-theoretic effective actions, ensuring that both $ \phi\rightarrow \infty $ and $ \phi\rightarrow -\infty $ regimes are covered in a regular manner. Moreover, this form guarantees a nontrivial contribution from higher-order nonlinear electromagnetic terms and stabilizes the potential structure needed for horizon formation. Defining $ \epsilon=\alpha-\beta $, we obtain the magnetically charged black hole solution [9]

where M and $ Q_m $ are the mass and magnetic charge of this black hole, respectively.

In the limit $ \epsilon=0 $, the solution (9) reduces to the GMGHS or GHS black holes [42, 43], which have been extensively studied [44−47]. Moreover, this solution (9) describes a black hole with a single horizon when $ \epsilon=1 $, whereas for $ \epsilon=-1 $, the black hole horizons can range from two to none. This indicates that the magnetically charged black hole possesses different horizon structures.

To consider the QNMs and greybody factor of axial perturbation on the magnetically charged black hole in the string-inspired Euler–Heisenberg theory, we rewrite the metric (8) and solution (9) into new forms

Here, we focus on the axial perturbation for the magnetically charged black holes. To do so, we assume

where $ \bar{g}_{\mu\nu} $ and $ \bar{A}_{\mu} $ represent the background metric and electromagnetic field, respectively, and $ \delta{g}_{\mu\nu} $ and $ \delta{A}_{\mu} $ denote the corresponding perturbations.

Under the Regge–Wheeler gauge [48], we expand the metric perturbation using tensor spherical harmonics. The axial gravitational field perturbation involves two modes $ h_0(r) $ and $ h_1(r) $, and the perturbed metric is expressed as

where the spherical harmonics can be replaced by Legendre polynomials by setting the azimuthal number $ m=0 $ without loss of generality, i.e., $Y_{lm}|_{m=0}= \sqrt{\dfrac{2l+1}{4\pi}}P_{l}(\cos\theta) $, because the background metric is spherically symmetric.

Following Refs. [49, 50], axial vector perturbation is given by

Substituting the perturbed metric and vector potential ((12), (13), and (14)) into the gravitational field equation (3), the non-zero components of the first-order perturbed gravitational field equation are obtained as

where $ {{\cal{\bar{L}}}}(\phi,F) $denotes the Lagrangian density for black hole background solutions (11) and is expressed as

From Eqs. (15)−(17), one can easily obtain the perturbation function $ u_3(r)=0 $. Moreover, the electromagnetic perturbation function $ u_3(r) $ does not appear in gravitational perturbation equations ((18), (19), and (20)). This indicates that the axial metric perturbation decouples from axial electromagnetic perturbation. In fact, the phenomenon has been recovered in Ref. [51] for magnetic black holes. In addition, we substitute the perturbed metric and vector potential ((12), (13), and (14)) into electromagnetic field equation (5) and then obtain the corresponding perturbed equations. The details are shown in the Appendix, along with the recovery of this decoupling phenomenon. Then, we obtain a single Schrödinger like equation for the perturbed electromagnetic field $ u_3(r) $.

Gravitational perturbations are central to black hole physics because they directly probe spacetime geometry, align with observational priorities in gravitational wave astronomy, and reflect the intrinsic properties of the black hole. In the subsequent section, we only focus on the axial gravitational perturbation. From Eqs. (18), (19), and (20), we can observe that only two of the above three equations are independent. Considering the component $ E_{\theta\varphi} $ (20), we have

Substituting Eq. (22) into the component $ E_{r\varphi} $ (19), we can eliminate $ h_0(r) $ and $ h_0'(r) $, and then obtain a single second order differential equation for $ h_1(r) $. To modify this master equation into the standard Schrödinger form, we further define the function $ \Psi(r) $ with

We assume that the function $ C_0(r) $ takes the following form:

and then obtain the final perturbed equation

where $ r_* $ represents the tortoise coordinate with

and the effective potential $ V(r) $ is

with the background Lagrangian density $ {{\cal{\bar{L}}}}(\phi,F) $; see Eq. (21). Note that the perturbation of the energy-momentum tensor of the electromagnetic part cannot be ignored. A similar phenomenon also appears in Ref. [30].

Note that the solution (11) and potential (27) are invariant under the following rescaling: $ r/M\rightarrow r $, $ Q_m /M\rightarrow Q_m $, and $ \epsilon/ M^2\rightarrow \epsilon $. For a better analysis of the behavior of the potential function, we set $ M=1 $ throughout the paper and leave ϵ and $ Q_m $ free without loss of generality. The effective potentials $ V(r) $ are plotted in Figs. 1, 2, and 3, respectively. The height of the effective potential increases as $ Q_m $ increases in Fig.1. Moreover, the potential increases with an increase in the multipole moment l in Fig. 2. For different $ Q_m $ values, the effective potential exhibits distinct behaviors as the parameter ϵ varies, as shown in Fig. 3. Note that the effective potentials are always positive, indicating that the system is stable under the axial gravitational field perturbation.

Figure 1.  (color online) Effective potential $ V(r) $ for the gravitational perturbation with $ M=1 $ and $ l=2 .$

Figure 2.  (color online) Effective potential $ V(r) $ for the gravitational perturbation with $ M=1 $ and $ Q_m=0.3 .$

Figure 3.  (color online) Effective potential $ V(r) $ for the gravitational perturbation with $ M=1 $ and $ l=2 .$

To accurately determine the QNM frequencies of magnetically charged black holes, we employ two distinct computational approaches: the AIM and WKB approximation method. This dual-methodology approach enables us to perform the cross-validation of our results.

In Refs. [32, 33], the authors applied the AIM to the computation of QNMs. This method has been widely employed in the analysis of black hole perturbations across various spacetime geometries. In the present study, we utilized the AIM to numerically solve the axial gravitational perturbation equation (25), enabling us to determine the QNM frequencies associated with the considered black hole background.

We can rewrite the gravitational perturbation equation (25) in terms of $ u=1-r_+/r $ as

Here, the range of u satisfies $ 0\leqslant u<1 $. The boundary conditions are pure ingoing waves $(\Psi\sim {\rm e}^{{\rm -i}\omega r_*},\; r_*\to -\infty)$at the black hole horizon and pure outgoing waves $(\Psi\sim {\rm e}^{{\rm i}\omega r_*}, r_*\to +\infty)$ at spatial infinity.

To propose an ansatz for Eq. (28), we examine the behavior of the function $ \Psi(u) $ at the horizon $ (u=0) $ and at the boundary $ u=1 $. Near the horizon $ (u=0) $, we have $ A(0)\approx u A'(0) $ and $ B(0)\approx u B'(0) $. Thus, Eq. (28) reduces to

Then, we can obtain the solution

where we must set $ C_2=0 $ to respect the ingoing condition at the black hole horizon.

At infinity $ (u=1) $, the asymptotic form of Eq. (28) can be written as

which has the solution

To impose the outgoing boundary condition, we should set $ D_1=0 $.

From the above solutions at horizon and infinity, we can define the general ansatz for Eq. (28) as

Substituting Eq. (33) into Eq. (28), we have

where

and

Finally, we obtain the functions $ \lambda_0 $ and $ s_0 $ for the gravitational perturbation equation (34).

The WKB method is a well-established approach to solve the black hole perturbation equation in the frequency domain [34−38]. However, the accuracy of the estimated frequencies degrades for $n \geq l $. To ameliorate this issue, the Padé approximation [52] can be used to evaluate QNMs with higher precision. Within this method, the oscillation frequency ω can be determined using the following expression:

where $ n = 0, 1, 2 . . . $ represents the overtone number, $V_0 = V|_{r=r_{\max}}$, and $V_0'' =\dfrac{{\rm d}^2V}{{\rm d}r^2}|_{r=r_{\max}}$. The position $ r_{\max} $ corresponds to the location where the potential function $ V(r) $ attains its highest value. To achieve higher precision in the calculations, correction terms $ \bar{\Lambda}_k $ are introduced. The analytical expressions for these terms, together with the methodology for Padé averaging, are comprehensively presented in Refs. [39, 53].

Table 1 shows the fundamental quasinormal frequencies $ (n=0) $ of black holes with $ \epsilon=0 $ and $ \epsilon=\pm1 $ obtained through the application of the AIM and sixth-order Padé averaged WKB approximation methods for different values of $ Q_m $ with $ M=1 $ and $ l=2 $. Evidently, these fundamental quasinormal frequencies obtained by numerical methods agree well with each other for each branch of these black holes.

$ \epsilon $	l	$ Q_m $	AIM	Padé averaged WKB	$ \Delta_6 $	$\Delta_{\rm AW}$(%)
1	2	0.3	$ 0.3812939-0.0895414\,{\rm i} $	$ 0.381247 - 0.0895259\,{\rm i} $	$ 0.0000272575 $	0.00412788
		0.6	$ 0.4055129-0.0928762\,{\rm i} $	$ 0.405312 - 0.0929958\,{\rm i}$	$ 0.0000877586 $	0.0561951
		0.7	$ 0.4174759-0.0960118\,{\rm i} $	$ 0.41732 - 0.0962066\,{\rm i} $	$ 0.000171463 $	0.0583149
0	2	0.3	$ 0.3813581-0.0894781\,{\rm i}$	$ 0.381338 - 0.0894572\,{\rm i}$	$ 0.0000259841 $	0.00733116
		0.6	$ 0.4071892-0.0912672\,{\rm i} $	$ 0.407187 - 0.0912665\,{\rm i}$	$ 4.24181\times10^{-6} $	0.0006046086
		0.7	$ 0.4214127-0.0922712\,{\rm i}$	$ 0.421409 - 0.0922722\,I{\rm i} $	$ 1.8842\times10^{-6} $	0.000862789
-1	2	0.3	$ 0.3814223-0.0894146\,{\rm i} $	$ 0.381425 - 0.0893987\,{\rm i} $	$ 0.0000502269 $	0.00334693
		0.6	$ 0.4091798-0.0892181\,{\rm i} $	$ 0.40922 - 0.0893678\,{\rm i}$	$ 0.000308716 $	0.0370056
		0.7	$ 0.4261004-0.0874892\,{\rm i} $	$ 0.426335 - 0.0878444 \,{\rm i} $	$ 0.00114384 $	0.0978375

Table 1.  Fundamental QNM frequencies for gravitational field perturbation with $ M=1 .$

The term $\Delta_6 $ in the sixth column of Table 1 serves to quantify the error between two adjacent order approximations, defined as [36]

where $ \omega_7 $ and $ \omega_5 $ represent the QNMs computed using the seventh and fifth-order Padé-averaged WKB methods, respectively. By analyzing these errors, we provide a detailed assessment of the accuracy of our QNM estimates. We further consider the percentage deviation $\Delta_{\rm AW}$ of QNMs obtained via the AIM and Padé-averaged WKB methods. The relative error $\Delta_{\rm AW}$ between the two methods is defined by

Now, we discuss the influence of the magnetic charge $ Q_m $ on the QNM frequencies of the lowest modes $ (n=0) $ in gravitational field perturbation. These fundamental QNM frequencies with respect to different values of $ Q_m $ are plotted in Fig. 4. For different branch solutions, these QNM frequencies exhibit distinct characteristics with an increase in $ Q_m $. The real QNM frequencies for $ \epsilon=0,1 $ increase as $ Q_m $ increases, and the damping rate or decay rate of the perturbed field increases significantly with an increase in $ Q_m $; see Figs. 4(a)−4(d). With regard to $ \epsilon=-1 $ in Figs. 4(e)−4(f)), the real QNM frequencies also increase as $ Q_m $ increases. However, the damping rate increases with the increase in $ Q_m $ and then reaches a maximum decay rate at $ Q_m=0.6 $. As $ Q_m $ surpasses this point, the damping rate begins to decrease.

Figure 4.  (color online) Variation in fundamental QNM frequencies with respect to the magnetic charge $ Q_m $ with $ M=1 .$

The effects of the parameter ϵ on the QNM frequencies for gravitational perturbation can also be investigated. By fixing the magnetic charge $ Q_m= $ 0.3, 0.6, and 0.7, we show the variation in the fundamental QNM frequencies and damping rate with different values of ϵ in Fig. 5. As the parameter ϵ increases, these QNM frequencies decrease for $ Q_m=0.6 $ and 0.7 while remaining almost unchanged for $ Q_m=0.3 $.

Figure 5.  (color online) Variation in fundamental QNM frequencies with respect to ϵ with $ M=1 $ and $ l=2 .$

To study QNM properties via gravitational wave propagation, we analyze the time-domain behavior by converting Eq. (25) into a wave equation, replacing the second-order term with $-\dfrac{{\rm d}^2}{{\rm d}t^2}$. This yields a generalized PDE for the gravitational field

where the potential $ V(r) $ is presented in Eq. (27).

Following the method in Refs. [40, 54−57], we can numerically solve the above time-dependent wave-like equation. Here, the finite difference method is applied for temporal integration, with a Gaussian wave serving as the initial spatial configuration. The radial coordinate is discretized based on the tortoise coordinate transformation

Then, one can further discretize the effective potential into $ V(r(r_*)) = V(j\Delta r_*) = V_j $ and the field into $ \Psi(r, t) = \Psi(j\Delta r_*, i\Delta t) = \Psi_{j,i} $. Subsequently, the wave-like equation [49] is converted to a discretized equation

from which one can isolate $ \Psi_{j,i+1} $ after algebraic operations

The above equation is an iterative equation that can be solved if one gives a Gaussian wave packet $ \Psi_{j,0} $ as the initial perturbation. In our calculations, setting the seed $ r_{i=1}=r_h+10^{-12} $, after imposing the initial condition $ \Psi_{j,i<0} = 0 $, $\Psi_{j,0}=\exp\Big[-\dfrac{(r_{j}-a)^2}{2b^2}\Big]$. We choose the parameters $ a = 40 $ and $ b = 20 $ in the Gaussian profile, and setting $ \dfrac{\Delta t}{\Delta r_*}=\dfrac{0.05}{0.1}=\dfrac{1}{2} $, this iterative equation provides us the evolution of the gravitational field perturbation in the time profile.

Figure 6 shows the temporal profiles for gravitational perturbation while varying the parameter $ Q_m $. These profiles were obtained with an overtone number of $ n=0 $ and a multipole number of $ l=2 $ in both cases. For $ \epsilon=0,1 $ (see Figs. 6(a) and 6(b)), a large $ Q_m $makes the ringing stage of the perturbation waveform more intensive and shorter, which corresponds to a larger value of ${\rm Re}(\omega)$ and absolute value of ${\rm Im}(\omega)$ (see Figs.4(a)−4(d)). In Fig. 6(c) for $ \epsilon=-1 $, the perturbation wave with $ Q_m=0.6 $ exhibits a shorter ringing stage, indicating a larger absolute value of ${\rm Im}(\omega)$. These observations agree well with the results in the frequency domain (Figs. 4(e) and 4(f)). Additionally, as a validation check, we employ the Prony method to fit the fundamental mode, and the results are presented in Table 2.

Figure 6.  (color online) Time evolution for the perturbing gravitational field with $ M=1 $ and $ l=2 .$

$ Q_m $	$ \epsilon=1 $	$ \epsilon=0 $	$ \epsilon=-1 $
0.3	$0.381192-0.090218\,{\rm i}$	$ 0.381057-0.089666\,{\rm i} $	$ 0.382067 - 0.090253\,{\rm i} $
0.6	$ 0.405460-0.092283\,{\rm i} $	$ 0.407387-0.090434\,{\rm i} $	$ 0.409135 - 0.089114\,{\rm i} $
0.7	$ 0.417572-0.096208\,{\rm i} $	$ 0.421273-0.092543\,{\rm i} $	$ 0.425926 - 0.087750\,{\rm i} $

Table 2.  Fundamental QNM frequencies for gravitational field perturbation with $ M=1 $ and $ l=2 .$

In Fig. 7, we show the time domain profiles for gravitational perturbation with different values of ϵ. For $ Q_m=0.3 $, the real and imaginary parts of these QNM frequencies remain almost unchanged as the parameter ϵ varies. This is consistent with the variation in the QNM plots studied in Fig. 5 and potential behavior in Fig. 3(a). For $ Q_m=0.6 $ and 0.7, a large ϵ makes the ringing stage of the perturbation waveform sparser and shorter, which corresponds to a smaller $ {\rm Re}(\omega) $ and larger absolute value of $ {\rm Im}(\omega) $ (see Fig. 5). Moreover, the QNM frequencies with different ϵ are obtained using the Prony method; see Table 3.

Figure 7.  (color online) Time evolution for the perturbing gravitational field with $ M=1 $ and $ l=2 .$

$ Q_m $	$ \epsilon $	WKB method	Prony method
0.3	−10	$0.382084 - 0.0887889 \,{\rm i}$	$ 0.381169 - 0.0889776\,{\rm i} $
	0	$ 0.381338 - 0.0894572\,{\rm i} $	$ 0.381057 - 0.0896657\,{\rm i} $
	10	$ 0.380574 - 0.0902011\,{\rm i}$	$ 0.380790 - 0.0895344\,{\rm i}$
0.6	−5	$ 0.423692 - 0.0835524\,{\rm i}$	$ 0.423566 - 0.0835848\,{\rm i}$
	0	$ 0.407187 - 0.0912665\,{\rm i} $	$ 0.407387 - 0.0904337\,{\rm i}$
	8	$ 0.397065 - 0.102459\,{\rm i} $	$ 0.396758 - 0.1022570\,{\rm i} $
0.7	−2	$ 0.434931 - 0.0853049\,{\rm i} $	$ 0.434850 - 0.0856028\,{\rm i} $
	0	$ 0.421409 - 0.0922722\,{\rm i} $	$ 0.421495 - 0.0923237\,{\rm i}$
	4	$ 0.409369 - 0.105113\,{\rm i} $	$ 0.409109 - 0.1047280\,{\rm i} $

Table 3.  Fundamental QNM frequencies for gravitational field perturbation with $ M=1 $ and $ l=2 .$

In this section, we outline the application of the WKB method to the analysis of the greybody factor, a quantity that provides valuable insight into the transmission properties of the effective potential governing wave propagation in the given spacetime background. We begin by considering the wave equation under boundary conditions that allow for incoming waves originating from spatial infinity. Owing to the symmetry inherent in the scattering process, this configuration is mathematically equivalent to examining the scattering of a wave emanating from the black hole horizon. The appropriate boundary conditions for this scattering scenario are given by

where T is the transmission coefficient, and R is the reflection coefficient.

The square of the amplitude of the wave function at a particular point of spacetime determines the probability of finding it in the given point. The wave incoming toward a regular black hole is partially transmitted and partially reflected by the potential barrier. The greybody factor is defined as the probability of an outgoing wave reaching infinity or an incoming wave being absorbed by the black hole. Therefore, $ |T(\omega)|^2 $ is called the greybody factor, and $ R(\omega) $ and $ T(\omega) $ should satisfy the following relation:

Using the sixth-order WKB method, the reflection and transmission coefficients can be obtained

where K is a parameter that can be obtained by the WKB formula

For more detailed information, refer to reviews such as [37−40] and the references therein.

In Figs. 8−10, we show the behaviors of the greybody factors for gravitational perturbation under varying magnetic charge $ Q_m $, multipole number l, and parameter ϵ. The greybody factors exhibit a noticeable decrease as $ Q_m $ increases (see Fig. 8). This suggests that, when the magnetic charge $ Q_m $ is reduced, the black hole becomes more effective at capturing and interacting with incoming matter or radiation. We also investigate how a change in the multipole number l affects the corresponding behavior of the greybody factors (see Fig. 9). The greybody factors also gradually decrease as l increases, which reveals that the greybody factors are larger for smaller values of l. With regard to the changes in ϵ, the greybody factors gradually increase as ϵ increases (see Fig. 10). This indicates that black holes become less interactive with the surrounding radiation and allow more of the perturbed field to escape. These results are consistent with the effective potentials in Figs. 1−3.

Figure 8.  (color online) Greybody factors for the gravitational field with $ M=1 $ and $ l=2 .$

Figure 10.  (color online) Greybody factors for the gravitational field with $ M=1 $ and $ l=2 .$

Figure 9.  (color online) Greybody factors for the gravitational field with $ M=1 $ and $ l=2 .$

In this study, we examined the axial field perturbations of magnetically charged black holes within the framework of the string-inspired Euler–Heisenberg theory. Under the consideration of axial gravitational perturbations decoupled from axial electromagnetic perturbations, we systematically analyzed the effects of the magnetic charge $ Q_m $, parameter ϵ, and angular quantum number l on the corresponding effective potentials associated with the axial gravitational perturbations.

Then, we adopted the AIM and Padé-averaged sixth order WKB methods to compute QNMs in each scenario. We considered how the QNMs change with the multipole moment l, magnetic charge $ Q_m $, and parameter ϵ. For $ \epsilon=0,1 $, the real part of the QNM frequencies increases as $ Q_m $increases, and the damping rate or decay rate of gravitational waves increases significantly with an increase in $ Q_m $. With regard to $ \epsilon=-1 $, the real part of the QNM frequencies also increases as $ Q_m $increases. However, the damping rate increases with the increase in $ Q_m $ and then reaches a maximum decay rate at $ Q_m=0.6 $. As $ Q_m $surpasses this point, the damping rate slowly begins to decrease. In contrast, these QNM frequencies decrease for $ Q_m=0.6 $ and 0.7 while remaining almost unchanged for $ Q_m=0.3 $ as the parameter ϵ increases. These observations in frequency domains agree well with the results we obtained in the time domains.

Using the sixth-order WKB approximation, we also computed the greybody factor associated with the gravitational field perturbations. The results indicate that the magnetic charge $ Q_m $ has a suppressing effect on the tunneling probability; specifically, a smaller fraction of the perturbed field can penetrate the effective potential barrier as $ Q_m $increases. In contrast, varying the parameter ϵ results in an opposite trend. These observations are consistent with the qualitative behavior of the corresponding effective potential profiles.

Note that Ref. [51] recovered the axial metric perturbation decoupled from axial electromagnetic perturbation for magnetic black holes. Some papers [51, 58, 59] further discussed certain combinations: axial gravitational perturbation coupled with polar electromagnetic perturbation or polar gravitational perturbation coupled with axial electromagnetic perturbation for magnetic regular black holes. We will consider these combinations for magnetically charged black holes in future studies.

We appreciate Guoyang Fu and Zhen-Hao Yang for helpful discussions.

In this appendix, we show the perturbed equation for axial electromagnetic perturbation. Substituting the perturbed metric and vector potential (Eqs. (12), (13), and (14)) into the Maxwell equation (5), we obtain

From these equations, it is evident that the gravitational perturbation functions $ h_0(r) $ and $ h_1(r) $ must vanish and there only exists a single perturbation equation for the perturbed electromagnetic field.

To modify this master equation (A4) into the standard Schrödinger form, we further define the function $ u_3(r) $ with

We assume the function $ C_1(r) $ taking the following form:

and then obtain the final perturbed equation

where $ r_* $ represents the tortoise coordinate with ${\rm d}{r_*} = \dfrac{1}{\sqrt{AB}}{\rm d}r$, and $ V_A(r) $ denotes the effective potential for axial electromagnetic perturbation. Owing to the cumbersome expression of $ V_A(r) $ , its specific form is not provided here.