1 Introduction

Testing gravity theories using observational data is an important task of relativistic astrophysics because gravity theories are a key concept in understanding the formation and evolution of the Universe, as well as the behavior of astronomical objects such as stars and galaxies, as well as the physics of gravitational compact relativistic objects: black holes and neutron stars.

One of the most important and well-tested gravity theories is general relativity, describing gravity as a curvature of spacetime, which was developed by Albert Einstein in the early 20th century. It has been incredibly successful in explaining a wide range of astronomical phenomena, such as the bending of light around massive objects [1, 2], the precession of Mercury’s orbit, and the existence of black holes. However, there are still a few open fundamental problems of current relativistic astrophysics related to the nature of dark matter and dark energy. On the other hand, astrophysical magnetic fields are also important for understanding the Universe. The magnetars, pulsars, white dwarfs, stars, and accretion disks of black holes may play the role of the source of a strong magnetic field. Thus, the effect of gravity is important in the study of powerful magnetic field sources.

In astrophysics, the magnetic fields play a vital role in explaining many astrophysical processes, such as the formation of stars and planets, the acceleration of cosmic rays, and the ejection of jets from black holes. Magnetic fields can also confine and heat plasma, the fourth state of matter that makes up most of the visible Universe.

The solutions of the Maxwell equations around both Schwarzschild and Kerr black holes immersed in an external asymptotically uniform magnetic field were first obtained by Wald in 1974 [3]. Petterson [4] proposed the dipole magnetic field, which may be formed using the circular current loops surrounding the black hole. The split-monopole magnetic field approximation, along with the parabolic field, has been explored in [5]. It should be noted that parabolic fields are typically the consequences of numerical modeling of toroidal configurations considered within the context of general relativistic magnetohydrodynamics [6].

In astrophysics, it is important to consider magnetic field effects on particle dynamics in the black holes accretion discs. However, astrophysical magnetic fields around black holes are difficult to observe directly. Thus, their effects can be observed in various ways, such as the emission of polarized light and the acceleration of charged and magnetized particles. Besides, the astrophysical compact entities’ magnetic fields are responsible for witnessing high-energy processes in their vicinity. This may be utilized to validate theories of gravity in an intense field environment by examining the dynamics of charged and magnetized particles. In this sense, gravity theories and astrophysical magnetic fields are important in understanding the Universe.

In past years, using the Walds approach, several studies have been performed through the dynamics and radiation of charged and magnetized particles [7,8,9,10,11,12,13,14,15]. Recently, Khan and Chen [16], by studying the dynamics of charged particles in the black hole split-monopole magnetosphere, discovered that the positive magnetic field boosts the stability of circular orbits. The motion of magnetized particles near black holes in the existence of an external magnetic field has been studied in [17,18,19,20,21,22,23].

Observations of electromagnetic fields around black holes in different bands have yielded valuable information about the dynamical processes around Sgr A*. The evaluated value of the magnetic field strength from the vicinity of Sgr A* to the innermost edge of its accretion is approximately 5–100 G [24, 25], and its estimated value around the orbit of PSR 1745–2900 is about few mG [26]. Another important issue is related to the magnetic field configuration. Although there exist numerous theoretical studies on astrophysical events in the vicinity of the supermassive black hole Sgr A*, the magnetic field configuration in the close environment of the black hole is still not defined [27].

Particle collisions around a revolving and charged black hole’s horizon is an essential and engaging process that can end up with high-energy emissions. To date, various physical mechanisms have been presented as black hole energy extraction techniques. Roger Penrose [28] was the pioneer in describing an easy mechanism for extracting energy from black holes. Piran et al. [29, 30] have investigated such mechanisms associated with the Penrose phenomenon. Banados, Silk, and West (BSW) [31] observed that collisions could yield high center-of-mass energy particles for an excessively spinning black hole. Such energy could be greater than Planckian energy in an idealistic set-up; therefore, black holes may be considered an ultra-high energy collider. By studying the charged Gauss–Bonnet-AdS black hole, Zahid et al. predicted that whenever a pair of neutral electrical test particles collide near the horizon of a black hole, it may act as a particle accelerator with arbitrarily enormous center-of-mass energy [32, 33]. Investigating this problem has attracted a great deal of attention in recent years [34,35,36,37,38,39].

Several mechanisms exist that minimize the possibility of large collision energy [40, 41]. In the case of fast-rotating black holes, the center-of-mass energy will be endlessly large. A little breach of the extremity criterion, which is virtually unavoidable in astrophysical applications, greatly reduces this impact. Particle trajectories require considerable fine-tuning, even in an idealized spinning situation. This mechanism is also heavily influenced by gravitational radiation [42].

In our previous works [43, 44], we have studied electrically charged particles with magnetic dipole moment in spacetime around Schwarzschild and Schwarzschild-MOG black holes located in external asymptotically uniform magnetic fields. In this work, we aim to extend the studies into magnetized rotating Kerr black hole spacetime.

Here, we plan to study the dynamics of magnetized and electrically charged particles around Kerr black holes in the presence of an external magnetic field. The paper is organized as follows: we briefly review magnetized Kerr black holes as a solution of the Maxwell equations in Sect. 2. We explore the equation of motion of charged/magnetized particles around Kerr black holes in the presence of magnetic fields in Sect. 3. Section 4 is devoted to the analysis of particle collisions near rotating magnetized black holes. We summarize the results obtained in Sect. 5. Throughout the paper, we use geometrized units \(c=G=1\) and run Latin indexes from 0 to 3 and Greek indexes from 1 to 3.

2 Magnetized Kerr black holes 

The spacetime geometry of the rotating Kerr black hole with the total mass M and rotating parameter a can be described by the following line element

$$\begin{aligned} ds^2=g_{tt}dt^2+2g_{t\phi }dtd\phi +g_{rr}dr^2+g_{\theta \theta }d\theta ^2+g_{\phi \phi }d\phi ^2, \end{aligned}$$
(1)

where the nonzero components of the metric tensor are defined as

$$\begin{aligned} g_{tt}= & {} -\left( 1-\frac{2 M r}{\Sigma }\right) ,~~g_{rr}=\frac{\Sigma }{\Delta },\ g_{\theta \theta }= \Sigma ,\\ g_{\phi \phi }= & {} \left( r^2+a^2 + \frac{2 M r}{\Sigma }a^2\sin ^2\theta \right) \sin ^2\theta ,\\ g_{t\phi }= & {} -\frac{2 M ra}{\Sigma }\sin ^2\theta \, \end{aligned}$$

with   \(\Sigma =r^2+a^2\cos ^2\theta \),   \(\Delta =r^2+a^2-2Mr\).

One may consider the test field around the rotating black hole. Magnetic fields around black holes satisfying the condition \(B\ll 10^{19}(M_\odot /M)\)G are considered to be weak, which cannot contribute to the space-time curvature of the black hole [45].

The structure of the external magnetic field near the black hole horizon could be highly complicated. Near the structure’s rotation axis, where jets might exist, a parabolic magnetic field, irrespective of the magnetic field’s original layout, is correlated. As a result, it may be appropriate to retain the asymptotically uniform magnetic field suggested by Wald [3] and used in numerous significant investigations of cosmic problems as the initial fundamental estimate. To maintain the space-time configuration’s symmetry, the lines of magnetic fields must be assumed to be oriented alongside the rotational axis.

With the assumption of an asymptotically uniform magnetic field of intensity B and lines directed alongside the z-axis in our model of the magnetosphere, the corresponding non-vanishing elements of the electromagnetic potential four-vector \(A_{mu}\) can be expressed as

$$\begin{aligned} A^\mu =\frac{B}{2} \left( \xi ^\mu _{(\phi }+2a \xi ^\mu _{(t)} \right) -\frac{Q}{2M}\xi ^\mu _{(t)}. \end{aligned}$$
(2)

In the expressions, (2) Q represents the induced electric charge of the black hole spacetime. In the case of chargeless black holes, the parameter \(Q = 0\) and the maximum produced black hole charge caused by the black hole spin and the magnetic field. This so-called Wald charge is defined as \(Q_W=2aB\), which results in a potential drop between the black hole center and infinity. Authors of Ref. [46] have estimated that the Wald charge is much lower than the critical charge, which may affect spacetime. Therefore, in our analyses, we consider the chargeless rotating black hole by setting \(Q=0\) as

$$\begin{aligned} A_t= & {} \frac{B}{2}\left( g_{t\phi } +2ag_{tt}\right) ,\quad A_\phi =\frac{B}{2}\left( g_{\phi \phi } +2a g_{t\phi }\right) . \end{aligned}$$
(3)

3 Equations of motion of magnetized particles with nonzero electric charge near magnetized Kerr black holes

The Hamilton-Jacobi method is a useful tool to determine the equations of charged and magnetized particles’ motion around a magnetized Kerr black hole in the following generalized form:

$$\begin{aligned} g^{\mu \nu }\left( \frac{\partial \mathcal{S}}{\partial x^{\mu }}+ eA_\mu \right) \left( \frac{\partial \mathcal{S}}{\partial x^{\nu }}+eA_{\nu }\right) = -\Big (m+U \Big )^2, \end{aligned}$$
(4)

where, the term \(2U=D^{\mu \nu }F_{\mu \nu }\) is responsible for the interaction between the magnetic dipole and the external magnetic field, \(D^{\mu \nu }\) and \(F_{\mu \nu }\) are the polarization tensors and the electromagnetic field, respectively. The expression for the tensor \(D^{\mu \nu }=\eta ^{\alpha \beta \sigma \nu }u_{\sigma }\mu _{\nu }\) satisfies the condition \(D^{\alpha \beta }u_{\beta }=0\), where \(\mu ^{\nu }\) refers to the vector of four dipole moment and \(u^{\nu }\) is the four-velocity of the particle.

The expression of the electromagnetic field tensor \(F_{\alpha \beta }=2u_{[\alpha }E_{\beta ]}+\eta _{\alpha \beta \sigma \gamma }u^{\sigma }B^{\gamma }\) helps to find non-zero components of the external magnetic field as

$$\begin{aligned} B^\alpha =\frac{1}{2}\eta ^{\alpha \beta \sigma \mu }F_{\beta \sigma }u_\mu \, \end{aligned}$$
(5)

where \( \eta ^{\alpha \beta \sigma \mu }=\sqrt{-g}\epsilon ^{\alpha \beta \sigma \mu }\) with the Levi–Civita symbol \(\epsilon ^{\alpha \beta \sigma \mu }\) and the metric determinant \(g=det(g_{\mu \nu })\).

In our further analysis, we assume the direction of the dipole moment is perpendicular to the equatorial plane and has the following components: \(\mu ^{\hat{i}}=(0,\mu ^{\hat{ \theta }},0)\), which is always parallel to the magnetic field lines and perpendicular to the equatorial plane. Thus, the interaction term U for the zero angular momentum observer (ZAMO) has the form \(U=2\mu ^{\hat{\theta }}B_{\hat{\theta }}\), where the magnetic field [46]

$$\begin{aligned} B^{\hat{\theta }}= & {} -\frac{B_0 r\sin ^2\theta \sqrt{\Delta }}{\Sigma ^2A} \nonumber \\{} & {} \times \Big \{ \Big [\Delta -\left( 1-\frac{M}{r}\right) \Sigma -a^2\sin ^2\theta \Big ]\nonumber \\ {}{} & {} \times a^2(1+\cos ^2\theta )-\Sigma ^2\Big \}. \end{aligned}$$
(6)

Finally, at the equatorial plane, the interaction term takes the form, \(U=2\mu B_0 F(r)\), where

$$\begin{aligned} F(r)=\frac{a^2M-r^3}{r(r^2+a^2)}\sqrt{1-\frac{2M}{r}+\frac{a^2}{r^2}}. \end{aligned}$$
(7)

The integrals of motion for electrically charged particles with magnetic dipole moment can be found using the time translation and the rotational symmetry of the geometry can be determined by the following form Lagrangian [47, see Eq.(7)]

$$\begin{aligned} \mathscr {L}=\frac{1}{2}\Big [(m+U)g_{\mu \nu }u^\mu u^\nu -kU\Big ]+eA_\mu u^\mu \, \end{aligned}$$
(8)

and the corresponding conserved quantities are the specific energy \(\mathcal{E}=E/m\) of the moving particle and its angular momentum \(\mathcal{L}=L/m\). Consequently, we have

$$\begin{aligned} -\mathcal{E}= & {} (1+\beta F(r))\left[ g_{tt}\dot{t}+g_{t\phi }\dot{\phi }\right] +qA_t, \end{aligned}$$
(9)
$$\begin{aligned} \mathcal{L}= & {} (1+\beta F(r))\left[ g_{\phi \phi }\dot{\phi }+g_{t\phi }\dot{t} \right] +qA_\phi , \end{aligned}$$
(10)

where \(\mathcal{L}=L/m\) is the specific angular momentum of the particles and \(\beta = \mu B/(2m)\) is the magnetic coupling parameter corresponding to the magnetic interaction between the magnetic dipole moment of the particles and \(q=e/m\) is the charge to mass relation.

The dynamics of electrically charged particles with magnetic dipole moment orbiting the magnetized Kerr black hole at the equatorial plane (i.e., \(\theta =\pi /2\) and \(u^{\theta }=\dot{\theta }=0\)) will be described by the following action:

$$\begin{aligned} \mathcal{S}=-Et+L\phi +\mathcal{S}_r. \end{aligned}$$
(11)

Then, we can derive the equation for the radial coordinate using the Hamilton–Jacobi equation (4), taking into account the Eqs. (9) and (10). With this in mind, we can write the variables in a separate form in the Hamilton-Jacobi equation. The radial motion of the particle can then be defined by

$$\begin{aligned} g_{rr}\dot{r}^2= & {} \alpha \mathcal{E}^2+2\delta \mathcal{E}+\gamma \nonumber \\= & {} \left[ \mathcal{{E}}-V^+_\textrm{eff}(r)\right] \left[ \mathcal{{E}}-V^-_\textrm{eff}(r)\right] \, \end{aligned}$$
(12)

where \(\alpha =-g^{tt}\), \(\delta =g^{t\phi } \left( \mathcal{L}+q A_{\phi }\right) +g^{tt}q A_t \) with

$$\begin{aligned} \gamma= & {} g^{tt}q^2 A_t^2+g^{\phi \phi }\left( \mathcal{L}+qA_\phi \right) ^2+2g^{t\phi }qA_t \left( \mathcal{L}+qA_\phi \right) \\{} & {} +\left( 1+\beta F(r)\right) ^2 \end{aligned}$$

and \(V_\textrm{eff}(r)\) for the circular motion (\(\dot{r}=0\)) of charged magnetized particles has the form

$$\begin{aligned} V_\textrm{eff}^\pm = \frac{-\delta \pm \sqrt{\delta ^2-4\alpha \gamma }}{2\alpha }. \end{aligned}$$
(13)

In this study, we consider the positive root of the effective potential, \(V^+_\textrm{eff}\), as it corresponds to the so-called positive root states with four-velocity oriented to future and positive energy relative to local observers (see, e.g.,   [48, 49]).

Fig. 1
figure 1

Radial profiles of \(V_\textrm{eff}\) for different values of \(\beta \), a and \(\omega _B=qB/2m\) parameters

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In Fig. 1, we present the radial dependency of the effective potential for different values of the black hole spin a, \(\omega _B\), and \(\beta \). It is observed that the presence of \(\omega _B>0\) (\(\omega _B<0\)) causes an increase (decrease) in the effective potential. While in the presence of the magnetic field and \(\beta \), the effective potential increases, particularly \(\beta \) greatly influenced it along the radial distance r. Our examination reflects that particles with magnetic dipole moment and electric charge have the highest unstable circular orbits in fast-rotating magnetized Kerr black holes with a compare to slow-rotating ones.

3.1 Circular orbits

This subsection aims to explore the circular motion of the magnetized and charged particles around magnetized Kerr black holes. In general, no radial forces/motion along the circular orbits or the present forces counterbalance each other at the appropriate angular momentum values of the particles.

The orbit circularity of magnetized test-charged particles orbiting around magnetized black holes may be investigated assuming the limitations \(V_\textrm{eff}=\mathcal{E}\) and \(V_\textrm{eff}'=0\), wherein the prime represents the partial derivative over r. As a result of these conditions, one can determine the angular momentum and energy of a particle in circular orbits.

Fig. 2
figure 2

The angular momentum of magnetized test-charged particles orbiting magnetized Kerr black holes at different values of the parameters \(\beta \), a and \(\omega _B\)

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Figure 2, depicts the graphical illustration of the angular momentum of the magnetized and charged particles for the different parametric values of \(\omega _B\), a, and \(\beta \). We find out that magnetic fields contribute to the minima of angular momentum in both positive and negative cases of the Larmor frequency parameter \(\omega _{B}\). One can also note that \(\omega _B<0\) has more impact on the minima of angular momentum in comparison with \(\omega _B>0\). Moreover, we found that black hole spin a significantly contributes to the minima of angular momentum. Surprisingly, the magnetized Kerr black holes have the smallest minima as compared to the Schwarzschild and Schwarzschild-MOG black holes (see Fig.2 in Refs. [43, 44]).

Fig. 3
figure 3

Graphical description of the Energy \(\mathcal{E}\) of the magnetized test-charged particles orbiting magnetized Kerr black holes at different values of the parameters \(\beta \), a and \(\omega _B\)

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In Fig. 3, we have graphically interpreted the behavior of magnetized test-charged particle’s energy orbiting magnetized Kerr black holes at different values of the parameters \(\beta \), a and \(\omega _B\). Our investigation revealed that the energy behaves quite similarly to the angular momentum under the impact of these parameters. In addition, one should note that here a also influences the energy minima. Our results showed that the magnetic field considerably decreases the energy and angular momentum of the magnetized test-charged particles orbiting magnetized Kerr black holes. Also, one noticed that the magnetized Kerr black holes have the least minima in comparison with the Schwarzschild and Schwarzschild-MOG black holes (see our previous papers [43, 44]). In addition, the \(\omega _B<0\) further enhances the minima in the orbital energy of the particles with magnetic dipole and electric charge.

3.2 Innermost stable circular orbits

The solution of \(V_\textrm{eff}'=0\) for r helps locate the position of the orbits, where the effective potential has its maximum values. The minima of effective potential corresponds to the stability of circular orbits. In other words, at \(V_\textrm{eff}''(r)<0\), the circular orbits become unstable, while all circular stable orbits fulfill the prerequisite of \(\partial _{rr}V_\textrm{eff}(r_\textrm{ISCO})>0\). On the contrary, ISCO complies with the requirements of \(\partial _{rr}V_\textrm{eff}(r_\textrm{ISCO})=0\).

Fig. 4
figure 4

Visual depiction of the ISCO radius vs \(\beta \) and \(\omega _B\), at various discrete values of black hole spin a and \(\omega _B\)

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Figure 4 demonstrates the graphical representation of ISCO radius against the Larmor frequency \(\beta \) (left panel) and \(\omega _B\) (right panel) for different values of the black hole spin. Our analysis reveals that the magnetic coupling parameter \(\beta \) contributes to the ISCO radius. From Fig. 4, it can also be observed that the black hole spin a and the Larmor frequency \(|\omega _B|\) result in decreasing the ISCO radius. Moreover, the Schwarzschild-MOG black holes (see Fig. 5 in [43]) have the highest ISCO as compared to the Schwarzschild black holes (see Fig. 3 in [44]) and magnetized Kerr black holes. Besides, we also observe that at a particular value of \(\omega _B\), the ISCO radius remains unaffected by the magnetic coupling parameter \(\beta \), providing the same values of ISCO radius (see red-dot dashed and black solid lines).

Figure 4 also shows the dependency of the ISCO radius along the magnetic coupling parameter \(\beta \). We observed that the ISCO radius significantly increases with the increase of \(\beta \) in the case of \(\omega _B=0\), while it decreases along the magnetic coupling \(\beta \) in the presence of Larmor frequency (\(\omega _B \ne 0\)). Also now, here we are interested in degeneracy values between Larmor frequency \(\omega _B\) and magnetic interaction parameters providing the same ISCO radius.

Fig. 5
figure 5

Visual depiction of \(\omega _B\) vs \(\beta \) at various values of ISCO radius for the case of \(r=6M\)

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In Fig. 5, we have plotted the relationships between \(\omega _B\) and \(\beta \) for the same values of ISCO radius. The graphical interpretation shows the relationships as contours: at larger magnetic field values, the contours became smaller. Also, the contour enlarges with increasing black hole spin.

3.3 The ISCO energy of particles

In the following subsection, we examine the impact of various values of the parameters a, \(\beta \), and \(\omega \) on the energy of the test particle in its ISCO.

Fig. 6
figure 6

Graphical illustration of the energy of particles at their ISCO at various values of \(\beta \) and \(\omega _B\)

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Figure 6 shows dependencies of the energy of charged magnetized particles orbiting magnetized Kerr black holes at the ISCO along the parameters \(\omega _B\) (upper row) and \(\beta \) (lower row). It can be seen that increases of \(\omega _B\) cause an increase in the energy at ISCO \(\mathcal{E}_{ISCO}\). Moreover, both the black hole spin and magnetic interaction parameters have a significant impact on the particle’s energy at ISCO and considerably decrease it. Also, one can observe from the top panel that at a certain negative value of \(\omega _B\), \(\mathcal{E}_{ISCO}\) take the same value.

3.4 The ISCO angular momentum

Here, we will study the effects of the magnetic interaction and spin parameters on the angular momentum of charged magnetized particles at their ISCOs.

Fig. 7
figure 7

Graphical illustration of the angular momentum of particles at ISCO at various \(\beta \), \(\omega _B\), and a parameter values

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In Fig. 7, we have plotted the behavior of the specific angular momentum of charged magnetized particles vs. \(\omega _B\) and \(\beta \) in the top and bottom rows, respectively. From where one can observe that the black hole spin a diminishes the ISCO’s angular momentum (\(\mathcal{L}_{ISCO}\)). While the Larmor frequency \(\omega _B\) increases the angular momentum, whereas \(\beta \) causes a decrease in it. Furthermore, the negative value of \(\omega _B\) decreases the angular momentum of the particles at ISCO compared to its positive value. Moreover, the Schwarzschild black hole has greater angular momentum at the ISCO as compared to the magnetized Kerr black holes, and it further decreases as the black hole rotates faster.

3.5 Energy efficiency

Another intriguing scenario emerges when examining a test particle within a Keplerian accretion disk as it descends towards the central black hole, releasing energy in the form of electromagnetic and/or gravitational radiation. The amount of radiated energy is dictated by the contrast between the particle’s rest energy, as observed locally, and its energy at the ISCO, which serves as a reflection of the spacetime characteristics. Consequently, one can compute the energy efficiency of the accretion disk using the expression provided by Novikov in 1973 [50].

Fig. 8
figure 8

Graphical behavior of the energy efficiency as a function of \(\omega _B\) (top panel) and \(\beta \) (bottom panel)

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The ISCO significance in the vicinity of black holes is linked to the inner boundary of an accretion disk. Surprisingly, as testing particles plunge into the center of the black hole within the Keplerian accretion disk, they extract a certain amount of energy that might be converted to electromagnetic and gravitational radiation upon limitations. The amount of released energy via radiations is often calculated as the difference between the particle’s rest energy (as determined by an appropriate viewer) and the particle’s ISCO energy (\(\mathcal{E}_\textrm{ISCO}\)) [50]. Indeed, the bolometric luminosity of the brightness emanating from the accretion disk is directly linked to the energy efficiency of the central black hole, as expressed by the equation \(\eta =L_\textrm{bol}/ (\dot{M}c^2)\), where \(\dot{M}\) signifies the accretion rate [51].

Here, we also study the efficiency of released energy at various parametric values of the magnetic coupling and magnetic interaction between the magnetic field and the particles’ electric charge, as well as the magnetic dipole moment. As a result, the efficiency of the accretion disk’s released energy can mathematically be represented as [50]

$$\begin{aligned} \eta =1-\mathcal{E}\vline _{r_\textrm{ISCO}}. \end{aligned}$$
(14)

In Fig. 8, we have plotted the behavior of the energy extraction efficiency along \(\omega _B\) and \(\beta \) in the upper and lower rows, respectively. Our graphical analysis demonstrated that black hole spin a contributes to the process of energy extraction. In addition, the parameters \(\omega _B\) and \(\beta \) considerably influence the process of energy extraction. One can extract much more energy in the case of \(\beta \ne 0\) compared to \(\beta =0\), whereas \(\omega _B<0\) contributes to the energy extraction while \(\omega _B > 0\) decreases it. Moreover, our analysis shows that more energy can be extracted from rotating black holes in comparison with static ones. Also, it is observed from the top panel that at a certain negative value of \(\omega _B\), the energy efficiency at different a takes the same value for \(\beta =0.1\).

4 Particle collisions near magnetized Kerr black holes

The collisional process near black holes remains a significant and intriguing general relativity topic. The extraction of energy may take place from a black hole throughout the collision process. A variety of methodologies may be employed to investigate the process of extracting energy from a spinning-charged black hole. Determining the entire amount of energy extracted via various methods taking place around black holes could clarify the reason why the intensity of AGN is on a scale of \( 10^{45}\mathrm{erg/s}\), which might be driven by supermassive black holes.

Among others, Penrose  [28] was the first who laid the foundation of an easy process according to which a particle entering the ergosphere of a spinning hole breaks down into a pair of particles: one of them crashes into the black hole while another travels to infinity with more energy compared to the original particle. The technique in question has recently been explored in various studies in the literature; for details, see [52,53,54]).

Banados–Silk–West (BSW) [55, 56] investigated particle collisions around the horizons of black holes as an approach to extract energy, which was further explored in  [56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71]. It is observed that the efficiency of energy extraction from the center black hole is proven to be greater through head-on collisions.

Here, we focus exploring the collisions of test electrically charged, magnetized, and neutral particles within the metric of magnetized rotating Kerr black holes. We use the generic equation presented in [55] for the center of mass energy \(E_\textrm{cm}\) of interacting particles.

$$\begin{aligned} \left( \frac{1}{\sqrt{-g_{tt}}} \ E_\textrm{cm}, 0, 0, 0\right) = m_{1}u_{(1)}^{\mu }+m_{2}u_{(2)}^{\nu }, \end{aligned}$$
(15)

where \(m_{i}\) and \(u_{(i)}^{\mu }\), respectively stands mass of the ith particle (\(i=1,2\)) and corresponding four-velocity. Using the normalization conditions \(g_{\mu \nu }u^{\mu }u^{\nu }=-1\), we can easily calculate the expression for \(E_\textrm{cm}\) as

$$\begin{aligned} \frac{E^2_{cm}}{m_1m_2}=\frac{m_1^2+m_2^2}{m_1m_2}-2g_{\mu \nu } u^{\mu }_1u^{\nu }_2 . \end{aligned}$$
(16)

We address straightforward situations by considering \(m_1=m_2=m\) (masses) in our subsequent analysis.

4.1 Critical angular momentum of colliding particles

In practice, in the case of particle collisions, the center of mass energy remains maximal at close orbits near the event horizon. At the same time, the particle collision’s axial velocity fulfills the requirements \(\dot{r}^2\ge 0\) as a function of the angular momentum \(\dot{r}^2(\mathcal {L})\) and the other parameters. The radial velocity and its first derivative with respect to r vanishes (\(\dot{r}=0\), and \(\partial _r(\dot{r}^2)=0\)) once the angular momentum reaches its threshold values. However, the radial velocity diminishes as the angular momentum increases. the challenging system of equations can be solved numerically.

Fig. 9
figure 9

Graphical behavior of the angular momentum’s critical value vs. black hole spin a

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In Fig. 9, we have plotted the behavior of the critical values of angular momentum at various parametric values of \(\beta \) in \(\omega _B=-0.01\) (top panel) and \(\omega _B=0.01\) (bottom panel). From the graphical behavior, we observed that the difference between the angular momentum’s critical values increases along the black hole’s spin and becomes larger in the fast-spinning black holes. On the other hand, both the black hole spin and the magnetic coupling parameter \(\beta \) contribute to increasing in \(\mathcal {L}_{cr}\).

4.2 Center of mass energy of particles in different scenarios.

This portion looks at the collisions between electrically neutral, magnetized, and electrically charged particles having a dipole magnetic moment. Taking into account particles of identical masses, the center of mass energy provided in Eq. (16) has the following form:

$$\begin{aligned} \frac{E^2_{cm}}{m^2}=2\left( 1-g_{\mu \nu } u^{\mu }_1u^{\nu }_2\right) . \end{aligned}$$
(17)

Which is in the dimensionless form simplifies to

$$\begin{aligned} \mathcal{E}^2_{cm}=E^2_{cm}/(2m^2)=1-g_{\mu \nu } u^{\mu }_1u^{\nu }_2. \end{aligned}$$
(18)

4.2.1 Neutral and electrically charged particles

We will initially look at the collisions between neutral and electrically charged particles. The aforementioned Eqs. (9)–(12) (equations of motion), simplifies the case of neutral particles with the substitution of \(\beta =\omega =0\).

Fig. 10
figure 10

Axial dependency of the center of mass energy in the case of neutral with electrically charged particle’s collision along the radial distance r

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Figure 10 depicts the axial dependency of the center of mass energy in the cases of neutral with neutral and electrically charged particles’ collision. Our graphical analysis shows that the center of mass energy decreases along the radial profile in both neutral and electrically charged particle collisions, but after some extent, in both cases, the energy increases with increasing values of \(| \omega _2 |\). Furthermore, \(\omega <0\) has a greater impact on the energy release rate compared to \(\omega >0\). In short, the electrically charged particles contribute to the center of mass energy.

4.2.2 Electrically neutral and magnetized particles

Here, in the second case, we consider collisions of electrically neural and magnetic particles.

Fig. 11
figure 11

Axial dependency of the center of mass energy in the case of electrically neutral with magnetized particle collisions

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In Fig. 11, we investigated the graphical behavior of the energy release rate in the case of electrically neutral particles with a collision of magnetized particles. Our results demonstrate that both \(\omega _2\) and the magnetic interaction parameter \(\beta _2\) greatly influence the energy release rate. In conclusion, \(\omega _2\) contributes, while \(\beta _2>0\) decreases the center of mass energy along the radial profile r.

4.2.3 Neutral, electrically charged and magnetized particles

In this subsection, we consider the collisions between neutral and electrically charged and magnetized particles.

Fig. 12
figure 12

Axial dependency of the center of mass energy in the case of neutral, electrically charged, and magnetized particle collisions

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In Fig. 12, we examine the center of mass energy’s graphical behavior in the collisions between neutral and electrically charged and magnetized particles. Here, we observed that the Larmor parameter \(\omega _2\) contributes to the energy release rate. On the other hand, \(\beta _2\) acts inversely as it diminishes the center of mass energy along the radial profile r.

4.2.4 Electrically charged and magnetized particles

This subsection of our article aims to explore the behavior of the center of mass energy in the collisions of electrically charged and magnetized particles.

Fig. 13
figure 13

The radial dependency of the center of mass energy in the collisions of electrically charged and magnetized particles

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The graphical behavior in Fig. 13 is shown for various combinations of \(\omega _1\) and \(\beta _2\). Our result demonstrates interesting behavior as \(\omega _1> 0\) contributes, while \(\omega _1< 0\) and \(\beta _2\) decreases the energy release rate along the radial profile r. In addition, we noted that the collision of neutral and electrically charged particles has more energy release rate as compared to the other cases of particle collisions.

4.2.5 Electrically charged and electrically charged magnetized particles

Fig. 14
figure 14

The radial dependency of the center of mass energy in the collisions of electrically charged and electrically charged magnetized particles

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The case of electrically charged and electrically charged magnetized particle collisions is plotted in Fig. 14. From our graphical analysis, one can see that \(\beta _2\) diminishes, the center of mass energy along the radial profile r.

4.2.6 Two magnetized particles

Here, we consider the collision between two magnetized particles only (Fig. 15).

Fig. 15
figure 15

The radial dependency of the center of mass energy in the collisions of two magnetized particles

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From the graphical descriptions, we can see that just like in the previous case the magnetic field influences the energy release rate in the same way, as it diminishes the energy release rate.

4.2.7 Two electrically charged and magnetized particles

In the present subsection, we assumed the collisions of two electrically charged and magnetized particles.

Fig. 16
figure 16

The radial dependency of the center of mass energy in the collisions of two electrically charged and magnetized particles

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In this case, we have plotted the behavior of the center of mass energy in Fig. 16 at various combinations of the electric and magnetic parameters values. Similarly to the previous case, here in the collisions of two electrically charged and magnetized particles, we observe a decrease in the center of mass energy for the magnetic field. In contrast, one can get more energy for the case of \(\omega _1 = \omega _2 > 0\) in comparison with \(\omega _1 = \omega _2 < 0\). Moreover, charged particle collision releases much more energy compared to the magnetized particle case near the horizon.

4.2.8 Magnetized, electrically charged and magnetized particles

This subsection of our article aims to explore the behavior of the center of mass energy in collisions of magnetized particles with electrically charged and magnetized particles (Fig. 17).

Fig. 17
figure 17

The radial dependency of the center of mass energy in the collisions of a magnetized particle with electrically charged and magnetized particles

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Our graphical results of the collisions of magnetized particles with electrically charged and magnetized particles show that just like in the previous case the magnetic field decreases the energy release rate. But here the impact of the magnetic field is much higher in comparison with previous cases.

5 Conclusion

Throughout this work, we have investigated the circular motion of test particles carrying electric charge and magnetic dipole moments around magnetized (uncharged) Kerr black holes. First, we have derived the effective potential for the circular motion of the particles, taking into account magnetic interaction with an electrically charged and magnetic dipole moment. We have analyzed the effective potential, angular momentum, and energy of the particles for various values of \(\omega _B\) and \(\beta \), which correspond to the interactions of external magnetic fields with the electric charge and magnetic moment of the particles. It is observed that Lorentz forces are proportional to \(\omega _B\) and are stronger than the magnetic dipole interaction forces. Our main findings are listed below:

  • In the case of fasting rotating magnetized Kerr black holes, the ISCO radius is smaller compared to the case of slowly rotating and Schwarzschild-MOG black holes. Moreover, \(\beta \) increases, whereas the black hole spin and \(\omega _B\) decrease the ISCO radius. Also, we have found that there are contour-like degeneracy values in \(\omega _B\) and \(\beta \) that provide the same ISCO radius.

  • At a particular value of \(\omega _B\), the ISCO radius remains unaffected by the magnetic coupling parameter \(\beta \), providing (almost) the same values of ISCO radius.

  • The ISCO energy of the particles increases with an increase of \(\omega _B\); however, \(\beta \) and \(\omega _B<0\) decrease it.

  • We observed that black hole spin a increases the process of energy extraction. The parameters \(\omega _B\) and \(\beta \) considerably affect the efficiency of the energy extraction process. As a result, one can extract much more energy in the case of \(\beta \ne 0\) compared to \(\beta =0\), whereas \(\omega _B<0\) contributes to the energy extraction while \(\omega _B > 0\) decreases it.

Further, we have focused on exploring the collisions of test electrically charged, magnetized, and neutral particles within the metric of magnetized rotating Kerr black holes. To happen the collisions in circular orbits, the radial velocity and its first derivative with respect to r must vanish with corresponding critical angular momentum.

Our numerical and graphical analyses have shown that \(\omega >0\) contributes, whereas \(\omega <0\) decreases the critical value of angular momentum. On the other hand, both spin a and the magnetic coupling parameter \(\beta \) contribute to \(\mathcal{L}_{cr}\). Finally, we have investigated the collisions between electrically neutral, magnetized, and electrically charged particles having a dipole magnetic moment by assuming the masses of the colliding particles are the same. We have studied various scenarios of the collisions and obtained the following results:

  • In the collisions between neutral and electrically charged and magnetized particles, we have obtained that \(\omega _B>0\) contributes to the energy release rate, while \(\beta \) acts inversely as it diminishes the center of mass energy.

  • In the collisions between neutral and electrically charged and magnetized particles, we observed that \(\omega _2\) contributes to the energy release rate; however, \(\beta _2\) acts inversely as it diminishes the center of mass energy.

  • For various combinations of \(\omega _1\) and \(\beta _2\) values: our results have shown interesting behavior as \(\omega _1>\)0 contributes, whereas \(\omega _1<\)0 and \(\beta _2>\)0 decreases the energy.

  • In the case of electrically charged and electrically charged magnetized particle collisions, we have obtained that the magnetic field diminishes the energy release rate.

  • In the absence of electric charge \((\omega _1=\omega _2=0)\), just like in the previous cases the positive magnetized particles diminish the center of mass energy.

  • In the collisions of electrically charged with electrically charged and magnetized particles, we have obtained a decrease in the center of mass energy for \(\beta _2 >0\).

  • The collisions between two electrically charged and magnetized particles: magnetic field diminish the energy release rate. In contrast, we get more energy in the case of \(\omega _1=\omega _2>0\) in comparison with \(\omega _1=\omega _2<0\). Moreover, charged particle collision releases much more energy compared to the magnetized particle case near the horizon.