Exploring the Enigma of Particle Dynamics and Plasma Lensing Using Einstein–Euler–Heisenberg Black Hole Geometry

Abstract

The unified Einstein–Euler–Heisenberg theory is utilized to investigate the particle motion and weak gravitational lensing characteristics of black holes. This black hole solution is developed using spherically symmetric possessing electric and magnetic charges. Quantum electrodynamics corrections reveal a screening effect for BH electric charges and paramagnetic impacts on magnetic charges. We analyzed the motion of massive as well as massless particles by studying their effective potential, event horizon, photon orbit and inner circular orbit. It was demonstrated that magnetic and electric fields of spherically symmetric black holes have significant impact. Then, we also delve to study the weak gravitational lensing phenomenon. A comprehensive approach was employed to investigate this phenomenon and explore the angle of deflection of light rays near magnetically and electrically charged black holes.

1. Introduction

General Relativity (GR) was introduced in 1915, and has been examined by several experiments and observations. Recent findings of gravitational waves in space [1] and the shadow of M87 [2] offer an intense field domain test of GR, whereas solar system tests [3] are considered weak field regime tests. However, General Relativity (GR) falls short in understanding the singularity that occurs during the collapse associated with the acceleration of cosmic expansion, the rotation curves of galaxies, and alignment with the theory of quantum fields. To address such shortfalls, one can either employ other gravity theories or break GR’s symmetry. For several years, researchers in physics have been captivated by the electromagnetic interactions with nonlinear behavior of the Reissner’s solution to Einstein and Maxwell’s equations. Gravitational Born–Infeld (BI) theory [4] is the widely recognized example. In the 1930s, gravitational nonlinear electrodynamics was researched [5,6]. The revelation that string theory and D-brane physics generate Lagrangians resembling both Born–Infeld theories (Abelian and non-Abelian) in the environment of their regime low energy (as referenced in [7,8,9]) has sparked a renewed fascination with these nonlinear operations. There exist black hole (BH) solutions that are static, spherically symmetric, and asymptotically flat (as discussed in [10,11]).

Black holes in Born–Infeld that have a universal constant have been researched in many aspects [12,13]. In recent decades, various nonlinear electrodynamics theories have been recommended to clarify static and spherically symmetric structures, including those with a gauge invariant’s general function [14,15,16,17], a logarithmic Maxwell invariant’s function [18], and a nonlinear Langrangian [19], which is generalized and can lead one to the Lagrangian of BI, and is the limit of the weak-field of the Lagrangian of the Einstein–Euler–Heisenberg theory [20]. Within Ref. [21], the gravity of BHs (spherically symmetric and static) was examined alongside electrodynamics, which is a nonlinear limiting of the weakfield of the Lagrangian of the Heisenberg–Euler–Schwinger model. Attempts to acquire BH solutions that are spherically symmetric and regularly (singularity free) static using gravitational electrodynamics that is nonlinear have been studied [22,23,24,25,26]. References [27,28] provide detailed explanations of the unique characteristics associated with these solutions.

In various works [29,30,31,32], researchers have explored the extension of spherically symmetric BHs to higher dimensions using a nonlinear Lagrangian that focuses on the power of the Maxwell invariants. These studies delve into the generalization of such BHs in a broader context. In the rotating version of Einstein–Born–Infeld model, black branes and black strings [26,33,34] have been studied. An effective Lagrangian Heisenberg–Euler–Schwinger model for nonlinear electromagnetic fields was developed. Heisenberg and Euler implemented the theory of a Dirac electron–positron for the first time [20]. Schwinger recasted the nonperturbative single loop effective Lagrangian in quantum electrodynamics (QED) [35]. When the strength of electric fields is greater, the value of threshold energy of vacuum $E_c={\displaystyle \frac{m^2{c}^3}{eh}}$ can be reduced by the spontaneous formation of pairs of electron and positron [20,35,36]. For decades, researchers have become fascinated by the creation of pairs of electrons and positrons from the vacuum of quantum electrodynamics (QED) and the polarization of the vacuum by an outside electromagnetic field (Refs. [37,38]). QED is basic framework that describes electromagnetic interactions and has been demonstrated through experiments. It is crucial to investigate QED implications in physics of BH.

The structure of this paper is as: In Section 2, we study the review of Einstein–Euler–Heisenberg BH. Section 3 analyses the motion of massive as well as massless particles, including it is effective potential, applying the unified Einstein–Euler–Heisenberg theory. The focus of this section includes examining structure of horizon, and orbits of photons, and radius of the inner circular orbit (ISCO) of massive particles orbiting around the BHs enriched with electric and magnetic charges. It is demonstrated that magnetic and electric fields of spherically symmetric BH do not vanish. Then, Section 4 delves deeper into the gravitational weak lensing phenomenon. A comprehensive approach is employed to investigate this phenomenon and explore the deflection angle of light rays near magnetically and electrically charged BH. In the last section, we conclude our findings of the study.

2. Black Holes in Einstein–Euler–Heisenberg (EEH) Theory

The EEH theory’s action incorporates the Einstein–Hilbert action with an added term that corresponds to a forcible electromagnetic field. This field is characterized by its invariants F and G, which are functions $F_{\mu \nu }$ and $F^{\mu \nu }$. The EEH theory’s action describes dynamics of the electromagnetic and gravitational fields. This is an integral over the spacetime of Lagrangian density that is a function of fields and their derivatives. EEH action from [39] is expressed as

$$S={\displaystyle \frac{1}{4\pi }}\int d^{4}x\sqrt{-g}\left({\displaystyle \frac{1}{4}}R-\mathcal{L}(F,G)\right),$$

(1)

where $L(F,G)$ refers to the functional of $F={\displaystyle \frac{1}{4}}{F}_{\mu \nu }{F}^{\mu \nu }$ and $G={\displaystyle \frac{1}{4}}{F}^{\mu \nu }{F}_{\mu \nu }^{*}$ with strength of electromagnetic field $F_{\mu \nu }$ and $F_{\mu \nu }^{*}={\displaystyle \frac{1}{2}}{ϵ}_{\mu \nu \sigma \rho }{F}^{\sigma \rho }$. The Levi-Civita tensor satisfies ${ϵ}_{\mu \nu \sigma \rho }{ϵ}^{\mu \nu \sigma \rho }=-4!$. As single-loop corrections to quantum electrodynamics (QED), the Lagrangian of Euler–Heisenberg is

$$\mathcal{L}(F,G)=-F+{\displaystyle \frac{1}{2}}{F}^2+{\displaystyle \frac{7a}{8}}{G}^2,$$

(2)

where a is the coupling constant [20,21]. Two frameworks are important nonlinear electrodynamics. One is the P framework constructed by the tensor $P_{\mu \nu }$, and the other is framework F constructed by the electromagnetic field tensor $F_{\mu \nu }$. P is defined by

$$P_{\mu \nu }=(1-aF)F_{\mu \nu }-{\displaystyle \frac{7a}{4}}{F}_{\mu \nu }^{*}G.$$

(3)

Einstein–Euler–Heisenberg theory predicts the spherically symmetric spacetime given as follows in spherical polar coordinates [21]:

$$d{s}^2=-f\left(r\right)d{t}^2+{\displaystyle \frac{d{r}^2}{f\left(r\right)}}+r^2(d{\theta }^2+{sin}^2\theta d{\Phi }^2),$$

(4)

where

$$f\left(r\right)=1-{\displaystyle \frac{2GM}{r}}+{\displaystyle \frac{G{Q}^2}{4\pi r^2}}+{\displaystyle \frac{G{Q}_m^2}{4\pi r^2}}.$$

(5)

where Q is electric charge and $Q_m$ is magnetic charge. Using the Einstein–Euler–Heisenberg BH solution taking $f\left(r\right)=0$, we can evaluate the event horizon of BH, as shown in Figure 1. One can notice that the event horizon decreases by increasing the electric and magnetic charge.

3. Particle Dynamics Around Black Hole in EEH Theory

Within this section, we investigated the dynamics of massless and massive particles orbiting within framework of EEH BH.

3.1. Massive Particle Motions Orbiting Black Holes in EEH Theory

First, we investigate the movement of massive particles typically orbiting around the Einstein–Euler–Heisenberg BH. To determine the orbits of a test particle with mass m, consider the Lagrangian of the EEH BH in the following form:

$${\mathcal{L}}^{\prime }={\displaystyle \frac{1}{2}}{g}_{\mu \nu }{u}^{\mu }{u}^{\nu },u^{\mu }={\displaystyle \frac{d{x}^{\mu }}{d\tau }}.$$

(6)

where the affine parameter is abbreviated by $\tau$, four velocity and coordinates of the test particle are expressed by $u^{\mu }$ and $x^{\mu }$, respectively. Also, we can write energy “$\epsilon$” and angular momentum “$\mathcal{L}$” for the test particle as

$$\epsilon ={\displaystyle \frac{\partial L^{\prime }}{\partial {\mu }^{\prime }}}=-f\left(r\right){\displaystyle \frac{dt}{d\tau }},$$

(7)

$$\mathcal{L}={\displaystyle \frac{\partial L^{\prime }}{\partial {\mu }^{\varphi }}}=r^2{sin}^2\theta {\displaystyle \frac{d\theta }{d\tau }},$$

(8)

By substituting the expressions in Equation (8) and using the condition $g_{\mu \nu }{u}^{\mu }{u}^{\nu }=-ϵ$ one may obtain an expression representing the test particle’s motion in the following structure:

$${\displaystyle \frac{d\theta }{d\tau }}={\displaystyle \frac{1}{r^2}}\sqrt{\kappa -{\displaystyle \frac{{\mathcal{L}}^2}{{sin}^2\theta }}},$$

(9)

$${\displaystyle \frac{d\Phi }{d\tau }}={\displaystyle \frac{{\mathcal{L}}^2}{r^2{sin}^2\theta }},$$

(10)

$${\displaystyle \frac{dt}{d\tau }}={\displaystyle \frac{\epsilon }{f\left(r\right)}},$$

(11)

where $\kappa$ represents the Carter constant, and the parameter $\epsilon$ is defined as follows:

$$ϵ=\left\{\begin{array}{cc}1,\hfill & \mathrm{for}\mathrm{timelike}\mathrm{geodesic}\hfill \\ 0,\hfill & \mathrm{for}\mathrm{null}\mathrm{geodesics}\hfill \\ -1,\hfill & \mathrm{for}\mathrm{spacelike}\mathrm{geodesics}\hfill \end{array}\right.$$

(12)

For the purpose of simplicity, we consider the velocity of particle’s is in the equatorial plan that meets the criteria ${\displaystyle \frac{d\theta }{d\tau }}=0$ and $\theta ={\displaystyle \frac{\pi }{2}}$. In this particular case, $\kappa ={\mathcal{L}}^2$, and equation representing radial motion will be in the following form:

$${\left({\displaystyle \frac{dr}{dt}}\right)}^2={\epsilon }^2-v_{\mathrm{eff}}\left(r\right)={\epsilon }^2-f\left(r\right)\left(1+{\displaystyle \frac{{\mathcal{L}}^2}{r^2}}\right),$$

(13)

$$V_{\mathrm{eff}}=f\left(r\right)\left(1+{\displaystyle \frac{{\mathcal{L}}^2}{r^2}}\right).$$

(14)

The relationship of radial coordinates and effective potential for massive test particles near BH for various values of Q and $Q_m$ parameters is represented in Figure 2. It can be noted that these parameters have a potential impact on the effective potential. Graphs show that with an increase in values of parameter Q, stable circular orbits start moving towards BH’s center, while parameter $Q_m$ also has the same effect.

To calculate the circular motion of the neutral particle around BH, we apply the ahead-stated conditions $\dot{r}$ = 0 and $\stackrel{\dot{}\dot{}}{r}$ = 0. Application of these conditions will give formulas for $\mathcal{L}$ and $ϵ$ for test particles, given as

$${\mathcal{L}}^2={\displaystyle \frac{r^2(G{Q}^2-4GM\pi r+G{Q}_m^2)}{2(G{Q}^2+2\pi r(-3GM+r)G{Q}_m^2)}},$$

(15)

and

$${\epsilon }^2={\displaystyle \frac{{(G{Q}^2+4\pi r(-2GM+r)+G{Q}_m^2)}^2}{8\pi r^2(G{Q}^2+2\pi r(-3GM+r)+G{Q}_m^2)}}.$$

(16)

Plotting these conservative quantities can provide the readers with an additional information, as can be seen in Figure 3 and Figure 4. These graphs show that the increase in parameters Q and $Q_m$ shifts the radial coordinates towards the center.

Now, we will examine the $ISCO$ radius $r_{\mathrm{ISCO}}$. To find the $ISCO$ radius, we will utilize the following conditions:

$$\left\{\begin{array}{c}{V}_{\mathrm{eff}}^{\prime }=0,\hfill \\ V_{\mathrm{eff}}^{″}=0,\hfill \end{array}\right.$$

(17)

With the complicated nature of effective potential, we cannot acquire a precise analytical expression for $r_{ISCO}$. However, we can plot $ISCO$ radius directly without explicitly solving it, as shown in Figure 5. Based on Figure 5, one can obtain information about dependence of $ISCO$ radius for different values of Q and $Q_m$ EEH BH. $ISCO$ radius falls as the parameters Q and $Q_m$ increase.

3.2. Motion of Massless Particles (Photon)

In this section, we analyze the motion of photon (massless) particles around EEH BH spacetime. Using Lagrangian (6), we put $ϵ=0$ in Equation (12) to derive the equation of photon motion orbiting the BH and at equatorial plane, which can be expressed as

$${\dot{r}}^2={\epsilon }^2-f\left(r\right){\displaystyle \frac{{\mathcal{L}}^2}{r^2}},$$

(18)

$$\dot{\varphi }={\displaystyle \frac{\mathcal{L}}{r^2}},$$

(19)

$$\dot{t}={\displaystyle \frac{\epsilon }{f\left(r\right)}},$$

(20)

Utilizing Equation (18), we will have an expression for effective potential $V_{\mathrm{eff}}$ for photon’s radial motion,

$$V_{\mathrm{eff}}=f\left(r\right){\displaystyle \frac{{\mathcal{L}}^2}{r^2}},$$

(21)

Figure 6, displays the relation between effective potential and radial coordinates for different values of electric charge Q and magnetic charge $Q_m$. Graphs reveal that the Q and $Q_m$ parameters cause photon orbits to move towards the center of BH.

The photon’s orbit radius $r_{ph}$ orbiting around BH is the result of the solution to the following equation:

$$V_{\mathrm{eff}}^{\prime }=0,$$

(22)

which will give us expression for $r_{ph}$, written as:

$$r_{ph}={\displaystyle \frac{2GM\pi -\sqrt{\pi }\sqrt{-G{Q}^2+4G{M}^2\pi -G{\left(Q_m\right)}^2}}{2\pi }}.$$

(23)

The relationship between photon orbit’s radius $r_{ph}$ and the EEH BH parameters Q and $Q_m$ is graphed in Figure 7. An increase in parameters Q and $Q_m$ causes a decrease in the photon’s orbit radius.

4. Weak Gravitational Lensing in Einstein–Euler–Heisenberg Theory

This part examines the optical properties of EEH BH utilizing the weak gravitational lensing phenomenon. We use following notation of metric tensor for approximation of weak field [40,41]:

$$g_{\alpha \beta }={\eta }_{\alpha \beta }+h_{\alpha \beta },$$

(24)

where $h_{\alpha \beta }$ and ${\eta }_{\alpha \beta }$ are expressions for perturbation gravity field and Minkowshki spacetime describing EEH theory, respectively. One must have to consider the following properties for ${\eta }_{\alpha \beta }$ and $h_{\alpha \beta }$:

$${\eta }_{\alpha \beta }=diag(-1,1,1,1)$$

$$under{x}^{\alpha }\to \infty ,h_{\alpha \beta }\to 0,h_{\alpha \beta }<<1,$$

$$h^{\alpha \beta }=h_{\alpha \beta },g^{\alpha \beta }={\eta }^{\alpha \beta }-h^{\alpha \beta }.$$

(25)

In this part, we studied the optical properties of EEH BH using weak gravitational lensing effects. For weak field approximation, we consider the metric tensor’s notation as used in references [31,35]. Utilizing the basic equations, we will obtain the expression for angle of deflection for EEH theory, given as follows [31]:

$${\alpha }_b={\int }_0^{\infty }{\displaystyle \frac{b}{r}}\left({\partial }_r\left(h_{33}\right)+{\displaystyle \frac{1}{1-\frac{w_e^2}{w}}}{\partial }_r\left(h_{\theta \theta }\right)-{\displaystyle \frac{k_e}{w^2-w_e^2}}\left({\partial }_{r}N\right)\right)dz,$$

(26)

where ${\omega }_{\theta }$ and $\omega$ are quantities representing plasma frequencies and photon frequencies, respectively. One can rewrite the line element from Equation (4) in the following form:

$$\begin{array}{cc}\hfill d{s}^2=& d{s}_{\theta }+\left({\displaystyle \frac{R_{s}G}{r}}-{\displaystyle \frac{G{Q}^2}{4\pi r^2}}-{\displaystyle \frac{G{Q}_m^2}{4\pi r^2}}\right)d{t}^2\hfill \\ \hfill & +\left({\displaystyle \frac{R_{s}G}{r}}-{\displaystyle \frac{G{Q}^2}{4\pi r^2}}-{\displaystyle \frac{G{Q}_m^2}{4\pi r^2}}\right)d{r}^2,\hfill \end{array}$$

(27)

where

$$d{s}_{\theta }=-d{t}^2+d{r}^2+(d{\theta }^2+sin{\left[{\theta }^{\prime }\right]}^{2}d{\Phi }^2)$$

$$h_{\theta \theta }={\displaystyle \frac{G}{r}}\left(R_s-{\displaystyle \frac{1}{4\pi r}}(Q^2-Q_m^2)\right)$$

(28)

$$h_{ij}={\displaystyle \frac{G}{r}}\left(R_s-{\displaystyle \frac{1}{4\pi r^2}}(Q^2-Q_m^2)\right)n_i{n}_j,$$

(29)

$$h_{33}={\displaystyle \frac{G}{r}}\left(R_s-{\displaystyle \frac{1}{4\pi r^2}}(Q^2-Q_m^2)\right){cos}^2\chi ,$$

(30)

where ${cos}^2\chi =z^2/(b^2+z^2)$ and $r^2=b^2+z^2$. The derivatives of $h_{00}$ and $h_{33}$ by radial coordinate are defined as:

$${\displaystyle \frac{d}{dr}}\left(h_{\theta \theta }\right)={\displaystyle \frac{G{Q}^2}{2\pi r^3}}+{\displaystyle \frac{G{Q}_m^2}{2\pi r^3}}-{\displaystyle \frac{G{R}_s}{r^2}},$$

(31)

$${\displaystyle \frac{d}{dr}}\left(h_{33}\right)={\displaystyle \frac{Z^2(G{Q}^2+G{Q}_m^2-3G\pi r{R}_s)}{\pi r^5}},$$

(32)

and one can write the mathematical form for angle of deflection [42] as:

$$\widehat{\alpha }=\widehat{{\alpha }_1}+\widehat{{\alpha }_2}+\widehat{{\alpha }_3},$$

(33)

with

$$\widehat{{\alpha }_1}={\displaystyle \frac{1}{2}}{\int }_{-\infty }^{\infty }{\displaystyle \frac{b}{r}}{\displaystyle \frac{d{h}_{33}}{dr}}dz,$$

$$\widehat{{\alpha }_2}={\displaystyle \frac{1}{2}}{\int }_{-\infty }^{\infty }{\displaystyle \frac{b}{r}}{\displaystyle \frac{1}{1-\frac{w_e^2}{w}}}{\displaystyle \frac{d{h}_{00}}{dr}}dz,$$

$$\widehat{{\alpha }_3}={\displaystyle \frac{1}{2}}{\int }_{-\infty }^{\infty }{\displaystyle \frac{b}{r}}\left(-{\displaystyle \frac{K_e}{w^2-w_e^2}}\right){\displaystyle \frac{dN}{dr}}dz,$$

(34)

Our objective is to investigate the angle of deflection for different distributions of plasma density.

4.1. Case 1: In Presence of Uniform Plasma

The deflection angle of EEH BH with uniform plasma may be expressed as [42]:

$${\alpha }_{\mathrm{uni}}={\alpha }_{\mathrm{uni}1}+{\alpha }_{\mathrm{uni}2}+{\alpha }_{\mathrm{uni}3}$$

(35)

Using Equations (30), (33), and (34), one will be able to find the expression for the deflection angle in the presence of a homogeneous plasma around BH:

$${\alpha }_{\mathrm{uni}}={\displaystyle \frac{G{Q}^2}{16b^2}}+{\displaystyle \frac{G{Q}_m^2}{16b^2}}-{\displaystyle \frac{G{R}_s}{b}}+\left({\displaystyle \frac{G{Q}^2}{8b^2}}+{\displaystyle \frac{G{Q}_m^2}{8b^2}}-{\displaystyle \frac{G{R}_s}{b}}\right){\displaystyle \frac{w^2}{w^2-w_e^2}},$$

(36)

Using the equation above, we can illustrate the relation between the deflection angle and impact parameter b for different values of parameters $Q_m$, Q ,and $w_e^2/w^2$, as illustrated in Figure 8. We also investigated the relationship between deflection angle and plasma parameters By varying BH parameters Q and $Q_m$, as shown in Figure 8, Figure 9, Figure 10 and Figure 11.

4.2. Case 2: In the Presence of Non-Uniform Plasma

Based on our study, a model depicting a non-singular isothermal sphere (SIS) with a singularity at it is center appears to be the most suitable for comprehending the unique behaviors exhibited by photons that experience weak gravitational lensing in the surroundings of BHs. An SIS is cloud of gas in a spherical shape, featuring a singularity located at the center of the spherical shape where density becomes extremely high. The density distribution of the SIS plasma field is stated in [40], as given below:

$$\rho \left(r\right)={\displaystyle \frac{{\sigma }_v^2}{2\pi r^2}},$$

(37)

where ${\sigma }_v^2$ represents uni-dimensional dispersion of velocity. Concentrations of plasma allows us to have the following analytical expression:

$$N\left(r\right)={\displaystyle \frac{\rho \left(r\right)}{k{m}_p}},$$

(38)

where k is a constant coefficient, which is dimensionless and $m_p$ is proton mass. So, we have an expression for the frequency of plasma as:

$$w_e^2=K_{e}N\left(r\right)={\displaystyle \frac{K_e{\sigma }_v^2}{2\pi k{m}_p{r}^2}}.$$

(39)

Now, we investigate the influence of $SIS$ plasma on deflection angle. One can have an expression for deflection angle in $SIS$ plasma field as follows [41]:

$${\widehat{\alpha }}_{SIS}={\widehat{\alpha }}_{SIS1}+{\widehat{\alpha }}_{SIS2}+{\widehat{\alpha }}_{SIS3}.$$

(40)

By combining Equations (30), (34), and (40), we can obtain deflection angle in the following expression:

$$\begin{array}{c}\hfill {\widehat{\alpha }}_{SIS}={\displaystyle \frac{3G{Q}^2}{16b^2}}+{\displaystyle \frac{3G{Q}_m^2}{16b^2}}-{\displaystyle \frac{2G{R}_s}{b}}-{\displaystyle \frac{R_s^2{w}_c^2}{2b^2{w}^2}}+{\displaystyle \frac{3G{Q}^2{\sigma }^2{K}_e}{64b^{4}km\pi w^2}}+{\displaystyle \frac{3G{Q}_m^2{\sigma }^2{K}_e}{64b^{4}km\pi w^2}}-{\displaystyle \frac{G{\sigma }^2{K}_e{R}_s^2}{3b^{3}km\pi w^2}}.\end{array}$$

(41)

From [43], we have:

$$w_e^2={\displaystyle \frac{K_e{\sigma }_v^2}{2\pi k{m}_p{R}_s^2}}.$$

(42)

Utilizing Equation (41), we have illustrated the relation of deflection angle and impact parameter b for different choices for values of Q, $Q_m$, and $w_e^2/w^2$. Figure 12 depicts deflection angle in non-uniform plasma medium using EEH BH solution. In Figure 12 and Figure 13, we show how a non-uniform plasma affects the angle of deflection of the beam of light near BH. Furthermore, we investigated impacts of plasma parameter on deflection angle as shown in Figure 14 and Figure 15.

5. Comparison

Comparison of deflection angles in both cases is illustrated by Figure 16. Our work shows that for the given fixed BH parameters in the case of EEH BH, uniform plasma has a higher impact on deflection angle than non-uniform plasma.

6. Summary

This article explores the role of the effective Lagrangian of Euler–Heisenberg in developing Einstein–Euler–Heisenberg BH solution, alongside the Maxwell Lagrangian. Within this study, we have explored the optical characteristics of BH in EEH theory for massless and massive particles, and also, we discussed weak gravitational lensing. If we replace $Q^2+Q_m^2\to Q^2$ in Equation (5), we will obtain the BH solution that coincides with the four-dimensional Reissner–Nordström (RN) one with the charge Q. Some similar works have been obtained regarding RN-like BH solutions in the literature [43,44,45,46,47]. However, our BH solution is different one, as it contains two charges, the electric charge Q and magnetic charge $Q_m$, instead of single charge Q, which is not addressed in RN solutions. In this study, we analyzed the the individual effect these charges instead of combined effect. Additionally, we integrate quantum electrodynamics corrections that introduce screening effects for electric charges and paramagnetic effects for magnetic charges, resulting in novel physical insights. In contrast to prior investigations of RN black holes, our analysis delineates the individual and synergistic effects of Q and $Q_m$ on effective potential, photon orbits, inner circular orbits, and weak gravitational lensing, revealing substantial differences and new outcomes attributable to the EEH framework. These results enhance our comprehension of BH physics and underscore the distinctive contributions of EEH theory. Our results summary is as follows:

• We worked on BH’s horizon structure using the effective Lagrangian of EEH BH solution. The findings show that increasing the parameter Q and $Q_m$ decreases the event horizon radius, as shown in Figure 1.
• We discussed the particle motion effective potential near BH. We investigated the relation between massive and massless particle effective potential along radial coordinate r for various values of EEH BH parameters Q and $Q_m$. We noticed that effective potential decreases by increasing these parameters.
• We explored the ISCO radius for massive particles surrounding BH. It has been shown that with an increases in the values of EEH BH parameters Q and $Q_m$, the ISCO radius decreases (see Figure 5).
• The photon’s sphere radius is also studied near BH using the effective Lagrangian of EEH BH. Photon orbits decrease with increase in BH parameters, as shown in Figure 7.
• We have also studied deflection angle near EEH BH with plasma (homogeneous, and non-homogeneous fields). Our work shows that for fixed BH parameters, uniform plasma has the greater impact on deflection angle than non-uniform plasma.

This study’s findings reveal new opportunities for investigating the interaction between electric and magnetic charges in BH physics, especially considering quantum electrodynamics corrections. Subsequent research may expand upon this study by examining the thermodynamic features, performing stability analyses, and exploring potential observational signatures of BHs within the Einstein–Euler–Heisenberg framework, so enhancing our understanding of the quantum characteristics of gravity and electromagnetism.