Regular Bardeen Black Holes in Anti-de Sitter Spacetime versus Kerr Black Holes through Particle Dynamics

Abstract

In this work, test particle dynamics around a static regular Bardeen black hole (BH) in Anti-de Sitter spacetime has been studied. It has been shown for neutral test particles that parameters of a regular Bardeen black hole in Anti-de Sitter spacetime can mimic the rotation parameter of the Kerr metric up to the value $a\approx 0.9$ providing the same innermost stable circular orbit (ISCO) radius. We have also explored the dynamics of magnetized particles with a magnetic dipole moment around a magnetically charged regular Bardeen black hole in Anti-de Sitter spacetime. As a realistic astrophysical scenario of the study, we have treated neutron stars orbiting a supermassive black hole (SMBH), in particular, the magnetar PSR J1745-2900 orbiting Sgr A* with the parameter $\beta =10.2$, as magnetized test particles. The magnetized particles dynamics shows that the parameter $\beta$, negative values of cosmological constant and magnetic charge parameter of the central BH cause a decrease in the ISCO radius. We have compared the effects of the magnetic charge of the Bardeen BH with the spin of rotating Kerr BH and shown that magnetic charge parameter can mimic the spin in the range $a/M\simeq (0,0.7896)$ when $\Lambda =0$ at the range of its values $g/M\simeq (0,0.648)$.

1. Introduction

The curvature singularity at the origin of compact objects is formed as a result of a gravitational collapse of the end state of the evolution of massive stars. It is the one of the main fundamental problems of the general relativity being the classical theory. One of the first attempts to solve this issue has been proposed by Bardeen [1]. In their pioneering work, it was suggested that the regular black hole solution describing the nonsingular compact object surrounded with the event horizon satisfies the weak energy conditions. The solution obtained by Bardeen is not a vacuum solution of the Einstein equations, and its physical interpretation has been delayed for decades. The physical interpretation of this type of solutions has been proposed in the re-obtained solution of regular black hole by Ayon–Beato and Garcia [2,3,4,5]. They have obtained the regular black hole solution introducing the nonlinear magnetic monopole charge of the black hole, which affects the gravitational field of the latter. This is a new, charged black hole solution of the Einstein field equations coupled with nonlinear electrodynamics. The thermodynamic properties of Bardeen’s solution have been investigated in [6]. The properties of the regular black hole solutions with nonlinear electric and magnetic charges have been studied in [7,8,9,10,11,12,13,14,15,16,17,18]. In the framework of tensor-vector-scalar (TeVeS) gravity proposed by Moffat and called Modified Gravity (MOG), the regular black hole solutions have been also discussed [19]. Hayward [20] has also proposed an alternative regular black hole solution that is free of an electric charge. The test particle motion around a regular black hole has been studied in [21,22,23,24,25,26]. The extension of the study of particle motion to the case of no-horizon highly curved spacetimes, being complementary to the regular black hole spacetimes, has been described in [27]. The motion of test particles around a no-horizon regular black hole is similar to one around the naked singularity Reissner–Nordström [28,29,30,31] or Kehagias–Sfetsos spacetimes [32,33,34]. However, there are fundamental differences observed in geodesic motion around the Kerr naked singularity [35,36,37]. Deformed spacetime properties have been studied in Refs. [38,39,40,41,42,43,44,45,46,47]. The prospects of cosmological dark energy for a dynamical theory have been discussed in [48], and the cosmological transitions with a change in metric signature have been investigated in [49]. Bardeen black hole spacetime is not a solution of Einstein field equations. However, regular black hole solutions being the solutions of Einstein field equations also exist in the literature, see, e.g., the non-commutative black holes considered in [50,51].

Einstein’s theory of gravity has been tested in both weak and strong field regimes using Solar system tests and observations of the shadow of a supermassive black hole [52,53] at the center of elliptical galaxy M87. It has also been concluded that the analysis of detected gravitational waves by LIGO-Virgo collaboration [54,55] also justifies the general relativity within the current accuracy of the gravity wave experiments. The GRAVITY experiment has justified the gravitational redshift of S2 star infrared (IR) radiation orbiting Sgr A* in the strong field regime. On the other hand, these experiments allow testing and obtain constraints on modified and/or alternative theories of gravity. One of the possible ways of testing the gravity theory is to analyze the X-ray observations from Active galactic nuclei (AGN), and numerous works have been completed, e.g., in Refs. [56,57,58]. The Kerr black hole having two parameters, namely the total mass and angular momentum, is the most favorable in relativistic astrophysics. However, a recent study performed has indicated that the test electric charge of the supermassive black hole can mimic the spin of the black hole through the test particle’s dynamics. It has been shown that the unscreened charge of supermassive black hole Sgr A* at the center of our galaxy, estimated as $3\times {10}^8\mathrm{C}$, can effectively mimic the spin parameter of a black hole up to $0.6$ of its maximal value, which is reflected in a considerable shift of the innermost stable circular orbits for charged particles. Consequently, the models estimating the spin of a black hole may require reconsideration due to arising uncertainty in the measurements of the black hole spin (see, e.g., [59]). These uncertainties motivated us to explore black holes with possible extra parameters beyond the standard ones, and a regular Bardeen black hole with a magnetic charge is the case studied in this work.

The effects of the electromagnetic field surrounding the compact object on electrically and magnetically charged particles are also essential. Astrophysical black hole solutions in general relativity according to the no-hair theorem cannot have their own intrinsic magnetic field [60]. However, a black hole can be considered as immersed in an external asymptotically uniform magnetic field [61], and due to spacetime curvature, the structure of the electromagnetic field will be changed in the black hole close environment. These modifications, at the same time, affect different properties of a black hole [62,63,64,65,66,67].

We refer the readers to Refs. [67,68,69,70,71,72,73,74,75,76,77,78,79,80] for detailed information about the structure of the spacetime and test particles motion around black holes in the presence of a magnetic field. For more studies on the electromagnetic field structure and particle motion around black holes in different gravity models, see, e.g., [22,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114].

The magnetized particles motion around non-rotating and rotating black holes in the presence of an external magnetic field have been studied in [115,116]. Further developments of the magnetized particles motion around a black hole in different gravity models can be found in [117,118,119,120,121,122,123,124,125,126,127,128,129,130].

In this paper, we study the neutral and magnetized particles motion around a Bardeen regular black hole in Anti-de Sitter spacetime. The paper is organized as follows: Test particle motion around Bardeen BH in Anti-de Sitter spacetime is studied in Section 2. Section 3 is devoted to studying the magnetically charged particle motion around the Bardeen regular black hole in Anti-de Sitter spacetime. The dynamics of the dipolar magnetized particles around the regular black hole in the presence of a cosmological constant is studied in Section 4. We summarize our results in Section 5. We use the system where $G=1=c$, and Greek (Latin) indices run values $0,1,2,3(1,2,3)$.

2. Test Particle Motion around Bardeen BH in Anti-de Sitter Spacetime

There are several ways of obtaining the regular black hole solutions using the nonlinear electrodynamics, including the Born–Infield extension of the Maxwell theory [131,132,133,134]. On the other hand, the explored Bardeen regular black hole solution in GR coupled with nonlinear electrodynamics has the right limiting cases and better describes astrophysical scenarios compared to the regular black solutions in the Born–Infield gravity theory. For this reason, in this work, we consider the special extension to the nonlinear Electrodynamics and the corresponding regular black hole solution obtained in [135]

$$\begin{array}{ccc}\hfill d{s}^2& =& -f\left(r\right)d{t}^2+f{\left(r\right)}^{-1}d{r}^2+r^{2}d{\Omega }^2,\hfill \\ \hfill f\left(r\right)& =& 1-\frac{2M{r}^2}{{(r^2+g^2)}^{\frac{3}{2}}}-\frac{\Lambda r^2}{3},\hfill \end{array}$$

(1)

where M, g and $\Lambda$ are the mass of a central object, magnetic charge parameter and cosmological constant, respectively. One can check the regularity of the spacetime of the central object by calculating the Kretschman scalar that is the square of the Riemann tensor $\mathcal{K}=R_{\alpha \beta \mu \nu }{R}^{\alpha \beta \mu \nu }$, which has the following expression for the spacetime metric above

$$\begin{array}{ccc}& & \mathcal{K}=\frac{8{\Lambda }^2}{3}+\frac{8\Lambda g^{2}M}{{\left(g^2+r^2\right)}^{7/2}}\left(4g^2-r^2\right)\hfill \\ \hfill & & +\frac{12M^2}{{\left(g^2+r^2\right)}^7}\left(8g^8-4g^6{r}^2+47g^4{r}^4-12g^2{r}^6+4r^8\right).\hfill \end{array}$$

(2)

It immediately comes out that the metric is indeed regular everywhere, including $r=0$.

The event horizon of a black hole is characterized by the condition $g^{rr}=0$ or equivalently $f\left(r\right)=0$ for this spacetime. From this condition, one can plot the dependence between the event horizon radius and the parameters of the spacetime metric g and $\Lambda$, as shown in Figure 1. From the figure, it is apparent that the increase of the magnetic charge parameter decreases the radius of the event horizon, while the increase of the cosmological constant makes it bigger. Since both parameters can change the event horizon radius, one can assume that these parameters can replace the effect of the spin parameter a of the well-known Kerr spacetime. However, we will come to this later when we start to deal with the unique observable quantity called the innermost stable circular orbits (ISCOs) of the test particles.

The equation of motion of a test particle can be obtained by solving the Hamilton–Jacobi equation that reads

$$\begin{array}{c}\hfill g^{\alpha \beta }\frac{\partial S}{d{x}^{\alpha }}\frac{\partial S}{d{x}^{\beta }}=-m^2,\end{array}$$

(3)

where S defines the action for the test particle, and m is its mass.

The action of the test particle moving around the central object can be easily constructed from the symmetrical properties of the spacetime that reads

$$\begin{array}{c}\hfill S=-Et+L\varphi +S_{\theta }+S_r,\end{array}$$

(4)

with E and L being the energy and angular momentum of the test particle, respectively. Then the equation of motion (3) takes the following form

$$\begin{array}{ccc}\hfill & & -\frac{E^2}{1-\frac{2M{r}^2}{{\left(g^2+r^2\right)}^{3/2}}-\frac{\Lambda r^2}{3}}+\left(1-\frac{2M{r}^2}{{\left(g^2+r^2\right)}^{3/2}}-\frac{\Lambda r^2}{3}\right){\left(\frac{\partial S}{dr}\right)}^2\hfill \\ & & +\frac{L^2}{{sin}^2\theta r^2}+\frac{1}{r^2}{\left(\frac{\partial S}{d\theta }\right)}^2=-m^2.\hfill \end{array}$$

(5)

The trajectory of the test particle can be plotted from the equation above, as shown in Figure 2, for fixed values of the magnetic charge parameter and cosmological constant. One can see that the increase of the magnetic charge and the decrease of the cosmological constant makes the average radius bigger. We can play with these quantities such that the rotation parameter of the spinning Kerr black hole can be replaced by these quantities, which supplies the idea that these parameters can totally or partially mimic the rotation parameter a of the Kerr one.

The effective potential of the test particle can be obtained from the equation of motion and has the following form for the motion at the equatorial plane ($\theta =\pi /2$):

$$\begin{array}{c}\hfill V_{\mathrm{eff}}=\left(1-\frac{2M{r}^2}{{\left(g^2+r^2\right)}^{3/2}}-\frac{\Lambda r^2}{3}\right)\left(1+\frac{L^2}{r^2}\right).\end{array}$$

(6)

For fixed values of magnetic charge parameter g and cosmological constant $\Lambda$, one can plot the radial dependence for the obtained effective potential as in Figure 3.

For circular orbits of the test particle at the equatorial plane, one can set conditions that should be satisfied by the effective potential

$$\begin{array}{c}\hfill V_{\mathrm{eff}}\left(r\right)=\mathcal{E},V_{\mathrm{eff}}^{\prime }\left(r\right)=0,\end{array}$$

(7)

with specific energy, $\mathcal{E}=E/m$. From these conditions, the dependence of the energy from the circular orbit radius of a test particle can be immediately constructed, as presented in Figure 4. One can see that the increase of the magnetic charge parameter decreases the energy near the central object, while the increase of the absolute value of negative cosmological constant makes the energy to increase infinitely since it has an infinite contribution to the spacetime metric as $-\frac{\Lambda r^2}{3}$ that increases infinitely for $r\to \infty$.

Conditions for effective potential can also be used to obtain the radial dependence for angular momentum of the test particle as in Figure 5. It is well known that the minimum value of the angular momentum of the test particle appears at the innermost circular orbits of the test particles and here one can see that the increase of the magnetic charge parameter g reduces this radius, which is similar to the behaviour of the rotation parameter of the Kerr metric. It is also worth noting that for the fixed negative values of cosmological constant, one can see the infinite increase of the angular momentum of the test particle, as in the case for the energy of the test particle also.

We have said that the combination of the magnetic charge parameter and cosmological constant can somehow mimic the rotation parameter of the Kerr metric. Now it is time to investigate the possible degeneracy between the rotation parameter and the parameters of regular Bardeen metric in Anti-de Sitter spacetime. The importance of this idea is that it is deeply believed that the black holes in the universe are described by the Kerr metric that defines the rotating gravitational compact object. However, theoretically, it might be possible to have alternate spacetime metric of the black hole that can provide similar behavior of the central massive object with different parameters that could mimic the rotation parameter of the Kerr one. In this work, we focus on the idea that if the black holes in the universe can be described by the regular Bardeen metric in Anti-de Sitter spacetime, then the parameters included in such metric can produce the same ISCO location as if it would be the Kerr black hole. Now our task is to determine how well the rotation parameter can be replaced by parameters included in the regular Bardeen metric in Anti-de Sitter spacetime. First, we see the dependence of the ISCO radius from such parameters. To do so, we add an auxiliary condition

$$\begin{array}{ccc}\hfill V_{\mathrm{eff}}^{″}\left(r\right)& =& 0\hfill \end{array}$$

to Equation (7). All together, these conditions provide the sought dependence between the ISCO radius of the test particle orbiting around the regular Bardeen metric in Anti-de Sitter spacetime, as shown in the left panel of Figure 6. It is clearly seen from the left panel that the increase of the magnetic charge parameter in the absence of the cosmological constant can reduce the ISCO radius up to around $3M$ corresponding to the value $a/M\approx 0.8$ of the rotation parameter of the Kerr spacetime, while the presence of the cosmological constant can “help” to reduce it even better, up to $2M$, which matches with the value of the rotation parameter $a/M\approx 0.9$.

However, one needs to take into account the fact that the ISCO radius we are using is coordinate-dependent and has no real physical meaning. In order to compare the effects of two different spacetime metrics, one needs to use an invariant quantity, which is coordinate-independent. For this purpose, we introduce the location of ISCO defined as

$$\begin{array}{c}\hfill R_{ISCO}=\frac{l_{\varphi }}{2\pi }.\end{array}$$

(8)

Here,

$$l_{\varphi }={\int }_0^{2\pi }d{s}_{\varphi },$$

gives us the circumference of the ISCO, and it is an invariant quantity formed from the line element $d{s}_{\varphi }=\sqrt{g_{\varphi \varphi }}{|}_{r=r_{ISCO}}d\varphi$ for the constant r, $\theta =\pi /2$, and t. Now, for the matching values of the location of ISCO, $R_{ISCO}$, of test particles, one can get the desired degeneracy between the spacetime parameters of two different metrics.

The estimated maximum value of the rotation parameter that can be replaced by the combinations of the magnetic charge parameter of the black hole and cosmological constant leads to the exact dependence between these parameters, which provides the same location of ISCO. To do so, one can obtain the degeneracy plot between the parameters of interest, as illustrated in the right panel of Figure 6 for the matching $R_{ISCO}$. From the figure, it is clearly seen that the magnetic charge itself (i.e., with zero cosmological constant) can mimic the rotation parameter up to $a/M\approx 0.8$, which matches with $g/M\approx 1$. One can also see that in the combination, these parameters can mimic the rotation parameter up to $a/M\approx 0.9$. Based on the idea that (i) a black hole in the universe can be explained not only as the rotating Kerr one but also as the static regular Bardeen black hole in Anti-de Sitter spacetime, and (ii) based on the same ISCO location, one can conclude that a black hole in the universe with a rotation parameter up to $a/M\approx 0.9$ can be either a Kerr or Bardeen one. It should be mentioned that, when $a=0.9$, the ISCO equation reduces to a regular Bardeen BH, which is found to be in [136].

Figure 7 presents the combination of two parameters g and $\Lambda$ could give the same ISCO location (panel a) or equivalently the same rotation parameter a (panel b).

3. Magnetically Charged Particle Motion

Having investigated the neutral particle motion around the regular Bardeen black hole in Anti-de Sitter spacetime, now we turn to the magnetically charged particle motion characterized with the non-zero magnetic monopole. We follow the same idea and corresponding steps as it has been performed in the previous section. Namely, we start from the construction of the equation of motion. For the magnetically charged and electrically neutral particle, the Hamilton–Jacobi equation of motion reads

$$\begin{array}{c}\hfill g^{\alpha \beta }\left(\frac{\partial \mathcal{S}}{\partial x^{\alpha }}+i{q}_m{A}_{\alpha }^{★}\right)\left(\frac{\partial \mathcal{S}}{\partial x^{\beta }}+i{q}_m{A}_{\beta }^{★}\right)=-m^2,\end{array}$$

(9)

where $q_m$ is the magnetic charge of the test particle, and four vector $A_{\alpha }^{★}$ defines the dual vector potential that has the following non-vanishing component (see [80])

$$\begin{array}{c}\hfill A_t^{★}=-\frac{ig}{r}.\end{array}$$

(10)

The equation of motion then becomes

$$\begin{array}{ccc}\hfill & & -\frac{{\left(E-\frac{g{q}_m}{r}\right)}^2}{1-\frac{2M{r}^2}{{\left(g^2+r^2\right)}^{3/2}}-\frac{\Lambda r^2}{3}}+\frac{L^{2}csc\theta }{r^2}+\frac{1}{r^2}{\left(\frac{\partial S}{d\theta }\right)}^2\hfill \\ & & +\left(1-\frac{2M{r}^2}{{\left(g^2+r^2\right)}^{3/2}}-\frac{\Lambda r^2}{3}\right){\left(\frac{\partial S}{dr}\right)}^2=-m^2.\hfill \end{array}$$

(11)

The effective potential of the magnetically charged particle moving at the equatorial plane ($\theta =\pi /2$) can be written in the following form

$$\begin{array}{c}\hfill V_{\mathrm{eff}}=\frac{q_{m}g}{r}+\sqrt{\left[1+\frac{{\mathcal{L}}^2}{r^2}\right]\left[1-\frac{2M{r}^2}{{\left(r^2+g^2\right)}^{\frac{3}{2}}}-\frac{\Lambda r^2}{3}\right]},\end{array}$$

(12)

(with specific angular momentum, $\mathcal{L}=L/m$), which recovers the standard Schwarzschild(-AdS) effective potential in the case where $g=0$. The radial dependence of this effective potential is shown in Figure 8. It is clearly seen from the figures that the additional term in the effective potential due to the electromagnetic interaction between the magnetically charged particle and the field produced by the magnetic charge of the black hole shifts the lines accordingly with the value and sign of the magnetic charge of the test particle.

Equation (7) provides the circular orbit conditions that lead the angular momentum of the magnetically charged test particle to have the radial dependence as plotted in Figure 9. It is seen from the plots that the increase of the magnetic charge of the test particle shifts the minimum of the lines towards the bigger circular orbits. It demonstrates that the increase of such parameter makes the ISCO radius of the test particle smaller. We also see that the shape of the lines does not experience considerable changes compared with the case of a neutral test particle.

The idea of this work is based on the investigation of the ISCO radius of the magnetically charged particle moving around the regular Bardeen black hole in Anti-de Sitter spacetime. From the additional condition in Equation (8), one can obtain the aimed relation between the ISCO radius and the parameters of the spacetime metric, together with the magnetic charge of the test particle, as plotted in Figure 10. In the given range of the magnetic charge parameter of the test particle, one can see the almost radial dependence between this parameter and the ISCO radius. One can also find that the lines in the figures do not have considerable differences in shape compared to the neutral particle motion that comes from the fact that the interaction between the magnetic field and the magnetically charged test particle is much weaker than the gravitational force of the gravitating object.

Now we turn to the final part of the section, investigating how the extra interaction due to the influence of the additional parameters embedded in the spacetime metric together with the interaction between magnetically charged test particle and magnetic field around the central black hole can mimic the role of the rotation parameter of the Kerr black hole. Based on the same idea as in the previous section, one can get the degeneracy plots between the rotation parameter a of the Kerr black hole and the parameters g, $\Lambda$, and $q_m$ for the orbiting magnetically charged test particle moving around the regular Bardeen black hole in Anti-de Sitter spacetime as plotted in Figure 11. We see that the additional force due to the electromagnetic interaction between the black hole and the magnetically charged test particle does not “help” the metric parameters mimic the rotation parameter of the Kerr black hole.

4. Dynamics of Dipolar Magnetized Particles in Regular Black Hole Vicinity

By solving the vacuum Maxwell equations for the electromagnetic field around the central object, one could find the potential of the electromagnetic field as

$$\begin{array}{c}\hfill A_{\alpha }=-{\delta }_{\alpha }^{\varphi }gcos\theta ,\end{array}$$

(13)

where ${\delta }_{\alpha }^{\varphi }$ is Kronecker delta. The non-zero component of the electromagnetic field tensor $F_{\mu \nu }$ can be written through the electromagnetic potential (13), in the following form

$$\begin{array}{c}\hfill F_{\theta \varphi }=-gsin\theta .\end{array}$$

(14)

One may express the magnetic field of a magnetically charged Bardeen black hole using by the well-known relation

$$\begin{array}{c}\hfill B^{\alpha }=\frac{1}{2}{\eta }^{\alpha \beta \sigma \mu }{F}_{\beta \sigma }{w}_{\mu },\end{array}$$

(15)

where $w_{\mu }$ is the four velocity of the observer, and ${\eta }_{\alpha \beta \sigma \gamma }$ is the pseudo-tensorial form of the Levi-Civita symbol ${ϵ}_{\alpha \beta \sigma \gamma }$ defined as

$$\begin{array}{c}\hfill {\eta }_{\alpha \beta \sigma \gamma }=\sqrt{-g}{ϵ}_{\alpha \beta \sigma \gamma },{\eta }^{\alpha \beta \sigma \gamma }=-\frac{1}{\sqrt{-g}}{ϵ}^{\alpha \beta \sigma \gamma },\end{array}$$

(16)

with $g=\det |g_{\mu \nu }|=-r^4{sin}^2\theta$ for the spacetime metric (1) and

$$\begin{array}{c}\hfill {ϵ}_{\alpha \beta \sigma \gamma }=\left\{\begin{array}{c}+1,\mathrm{for}\mathrm{even}\mathrm{replacements},\hfill \\ -1,\mathrm{for}\mathrm{odd}\mathrm{replacements},\hfill \\ 0,\mathrm{for}\mathrm{the}\mathrm{other}\mathrm{combinations}.\hfill \end{array}\right.\end{array}$$

(17)

Finally, the orthonormal radial component of the magnetic field generated by the magnetic charge of regular Bardeen BH can be expressed using four-potential (Equation (13)) as

$$B^{\widehat{r}}=\frac{g}{r^2}.$$

(18)

The radial component of the magnetic field around magnetically charged black holes formally coincides with the standard Newtonian expression.

4.1. Equation of Motion

One may investigate the motion of a magnetized particle with the magnetic dipole moment around a magnetically charged black hole using the following Hamilton–Jacobi (H-J) Equation (see, for example, [115,128])

$$\begin{array}{c}\hfill g^{\mu \nu }\frac{\partial \mathcal{S}}{\partial x^{\mu }}\frac{\partial \mathcal{S}}{\partial x^{\nu }}=-{\left(m-\frac{1}{2}{\mathcal{D}}^{\mu \nu }{F}_{\mu \nu }\right)}^2,\end{array}$$

(19)

where the term ${\mathcal{D}}^{\mu \nu }{F}_{\mu \nu }$ is responsible for the interaction between the magnetic field generated by the magnetic charge of the regular Bardeen black hole and magnetic dipole moment ${\mu }^{\nu }$ of the magnetized particles.

Let the magnetized particles have a polarization tensor ${\mathcal{D}}^{\alpha \beta }$, which satisfies the following condition,

$$\begin{array}{c}\hfill {\mathcal{D}}^{\alpha \beta }={\eta }^{\alpha \beta \sigma \nu }{u}_{\sigma }{\mu }_{\nu },{\mathcal{D}}^{\alpha \beta }{u}_{\beta }=0.\end{array}$$

(20)

One can determine the interaction term ${\mathcal{D}}^{\mu \nu }{\mathcal{F}}_{\mu \nu }$ using the following relation between the electromagnetic field tensor $F_{\alpha \beta }$ and components of electric $E_{\alpha }$ and magnetic $B^{\alpha }$ fields

$$\begin{array}{c}\hfill F_{\alpha \beta }=w_{\alpha }{E}_{\beta }-w_{\beta }{E}_{\alpha }-{\eta }_{\alpha \beta \sigma \gamma }{w}^{\sigma }{B}^{\gamma }.\end{array}$$

(21)

One may immediately calculate the interaction part of the H-J Equation (19) using the condition given in Equation (20) and the non-zero components of the electromagnetic field tensor as

$$\begin{array}{c}\hfill {\mathcal{D}}^{\alpha \beta }{F}_{\alpha \beta }=2{\mu }_{\alpha }{B}^{\alpha }=2{\mu }^{\widehat{\alpha }}{B}_{\widehat{\alpha }}.\end{array}$$

(22)

Moreover, one has to assume that the direction of the magnetic dipole moment of the magnetized particle lies at the equatorial plane and parallel to the magnetic field generated by the magnetic charge of the magnetically charged Bardeen BH and the orthonormal components of the magnetic dipole of the magnetized particle is ${\mu }^{\widehat{i}}=({\mu }^{\widehat{r}},0,0)$, and it can provide a stable equilibrium to the magnetic interaction between the magnetized particle and the magnetic field. The interaction term can be expressed using Equations (22) and (14) in the following form,

$$\begin{array}{c}\hfill {\mathcal{D}}^{\alpha \beta }{F}_{\alpha \beta }=\frac{2\mu g}{r^2},\end{array}$$

(23)

where ${\mu }^2={\mu }_{\widehat{i}}{\mu }^{\widehat{i}}$ is the norm of the magnetic dipole moment of the magnetized particle.

Now it is possible to study dynamics of the magnetized particles around the magnetically charged Bardeen BH at the equatorial plane, where $\theta =\pi /2$, with $p_{\theta }=0$, using conserved quantities of motion of the magnetized particles, such as energy $p_t=-E$ and angular momentum $p_{\varphi }=l$, and the energy of the particle can be obtained using Equations (22), (19) and the action (4) in the following form

$$\begin{array}{c}\hfill {\dot{r}}^2={\mathcal{E}}^2-V_{\mathrm{eff}}(r;l,\Lambda ,\beta ,g),\end{array}$$

(24)

where the effective potential for radial motion of the magnetized particles around the regular Bardeen black hole in Anti-de Sitter spacetime has the following form

$$\begin{array}{c}\hfill V_{\mathrm{eff}}(r;l,\Lambda ,\beta ,g)=\left(1-\frac{2M{r}^2}{{\left(r^2+g^2\right)}^{\frac{3}{2}}}-\frac{\Lambda r^2}{3}\right)\left[{\left(1-\beta \frac{Mg}{r^2}\right)}^2+\frac{{\mathcal{L}}^2}{r^2}\right],\end{array}$$

(25)

where the parameter $\beta =\mu /\left(mM\right)$ indicates the parameters of the magnetized particle and central object. Neutron stars with their nonzero magnetic moment ∼${10}^{32}\mathrm{G}·{\mathrm{cm}}^3$ and typical mass of $1.4M_{\odot }$, together with supermassive black holes with the mass in the range $({10}^6-{10}^{10})M_{\odot }$, can be treated as the system of a black hole and a test particle with a magnetic dipole. This makes the investigation of motion of magnetic dipoles around a black hole astrophysically relevant. One may estimate the coupling parameter $\beta$, assuming the dipole magnetic moment of the neutron star $\mu =(1/2)B{R}^3$ with the surface magnetic field B and radius R as

$$\begin{array}{c}\hfill \beta \simeq 0.128\left(\frac{B}{{10}^{12}\mathrm{G}}\right)\left(\frac{R}{{10}^6\mathrm{cm}}\right){\left(\frac{m}{1.4M_{\odot }}\right)}^{-1}{\left(\frac{M}{{10}^6{M}_{\odot }}\right)}^{-1},\end{array}$$

(26)

where m and M are the masses of the neutron star and the central SMBH, respectively. The interaction parameter $\beta$ for the magnetar SGR (PSR) J1745–2900 with the magnetic dipole moment $\mu \simeq 1.6\times {10}^{32}\mathrm{G}·{\mathrm{cm}}^3$ and mass $m\simeq 1.4M_{\odot }$ orbiting the SMBH Sgr A* ($M\simeq 3.8\times {10}^6{M}_{\odot }$) has the following value [137]

$$\begin{array}{c}\hfill \beta =\frac{{\mu }_{\mathrm{PSR}\mathrm{J}1745-2900}}{m_{\mathrm{PSR}\mathrm{J}1745-2900}{M}_{\mathrm{SgrA}*}}\approx 10.2.\end{array}$$

(27)

Since the magnetar PSR J1745-2900 is a single-neutron star close to SMBH Sgr A* observed in reality, we will use the parameter $\beta =10.2$ in our further calculations.

Now, one may calculate the characteristics of circular stable orbits for the magnetized particle around the central magnetically charged Bardeen BH using the standard conditions as

$$\begin{array}{c}\hfill V_{\mathrm{eff}}^{\prime }=0,V_{\mathrm{eff}}^{″}\ge 0.\end{array}$$

(28)

The specific angular momentum and energy of the magnetized particle being responsible for circular orbits can be expressed as

$$\begin{array}{ccc}\hfill {\mathcal{L}}^2& =& \frac{1-\beta \frac{gM}{r^2}}{\frac{2}{r^3}-\frac{6Mr}{{\left(g^2+r^2\right)}^{5/2}}}\{\left(1-\beta \frac{gM}{r^2}\right)\left(\frac{2Mr\left(r^2-2g^2\right)}{{\left(g^2+r^2\right)}^{5/2}}-\frac{2\Lambda r}{3}\right)\hfill \\ & -& \beta \frac{4gM}{3r^3}\left[r^2\left(\frac{6M}{{\left(g^2+r^2\right)}^{3/2}}+\Lambda \right)-3\right]\},\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill {\mathcal{E}}^2& =& \frac{r^4-{\beta }^2{g}^2{M}^2}{9r^4\left(g^6+3g^4{r}^2-3M{r}^4\sqrt{g^2+r^2}+3g^2{r}^4+r^6\right)}\hfill \\ & \times & {\left[{\left(g^2+r^2\right)}^{3/2}\left(\Lambda r^2-3\right)+6M{r}^2\right]}^2.\hfill \end{array}$$

(30)

Figure 12 illustrates the radial dependence of the specific angular momentum of magnetized particles for circular motion around a magnetically charged Bardeen black hole in Anti-de Sitter spacetime for the different values of cosmological constant and the black hole charge. Since the magnetar PSR J1745-2900 is treated as a magnetized particle, the value of the parameter $\beta =10.2$ is fixed. One may see from the figure the increase of the absolute value of the cosmological constant and the specific angular momentum with the distance from the central object. The increase of the BH charge decreases the minimal radius for circular motion. The distance where the angular momentum has a minimum comes close to the central object with the increase of both the cosmological constant and the magnetic charge parameter of the central object.

Figure 13 demonstrates the radial dependence of the specific energy of the magnetized particles moving along circular orbits with the parameter $\beta =10.2$ for the different values of the cosmological constant and magnetic charge of the regular Bardeen black hole. One may see from the figure that the minimum of the energy increases with the increase of the absolute value of the cosmological constant, while with the increase of the magnetic charge, the minimum of the energy decreases. However, the distance where the energy is minimum decreases with the increase (decrease) of the magnetic charge of the BH (cosmological constant).

4.2. Stable Circular Orbits

One can find the equation that allows finding the ISCO radius using the second condition of Equation (28) in the following complicated form.

$$\begin{array}{ccc}& & 18{\beta }^2{g}^2{M}^4{r}^6\left(4g^2+5r^2\right)+3{\beta }^2{g}^2{M}^3{r}^4{\left(g^2+r^2\right)}^{3/2}\hfill \\ \hfill & & \times \left[4g^2\left(\Lambda r^2-9\right)-r^2\left(\Lambda r^2+21\right)\right]+6M^2\left[2{\beta }^2{g}^2{\left(g^2+r^2\right)}^5-3r^{12}\right]\hfill \\ \hfill & & -4\Lambda r^6{\left(g^2+r^2\right)}^5+3M{r}^6\sqrt{g^2+r^2}\left[g^2{r}^4\left(5\Lambda r^2+9\right)+5\Lambda r^8+r^6-8g^6\right]\ge 0.\hfill \end{array}$$

(31)

Here we provide analyses, presenting them in plot form.

In order to see the effects of the cosmological parameter and magnetic charge of the regular Bardeen BH, we plot the ISCO radius as a function of the magnetic charge parameter of the BH in Figure 14 for the different values of the cosmological constant and the parameter $\beta$ for the magnetized particle.

One can see from the figure that the increase of the magnetic charge causes a decrease in the ISCO radius, and the ISCO radius increases (decreases) with the increase in the parameter $\beta$ (cosmological constant).

Another interesting and actual problem in relativistic astrophysics is testing different gravity theories in the study of test (neutral, charged and magnetized) particles. The spacetime exterior to an astrophysical black hole is described by the Kerr metric, having a non-zero spin parameter a. However, the effects of spin of Kerr black holes on the ISCO location of the test particles can be reflected by the parameters of alternative or/and modified theories of gravity, and this may cause the results of analysis to become indistinguishable.

4.3. Regular Bardeen BH in dS Spacetime versus Kerr BH

Here, we perform detail analysis for degeneracy values of the spin of Kerr black holes and the magnetic charge of regular Bardeen black holes in Anti-de Sitter spacetime, which may provide the same values of ISCO for magnetized particles by treating the magnetar PSR 1745-2900 orbiting Sgr A* as a magnetized particle with the parameter $\beta =10.2$ for the different values of the cosmological constant.

The ISCO radius of the magnetized particle around a rotating Kerr black hole reads by following the expression in the absence of the external magnetic field (magnetized particle is considered as a neutral test particle) for the corresponding prograde and retrograde orbits as [138]

$$\begin{array}{c}\hfill r_{\mathrm{isco}}=3+Z_2\pm \sqrt{(3-Z_1)(3+Z_1+2Z_2)},\end{array}$$

(32)

where

$$\begin{array}{ccc}\hfill Z_1& =& 1+\left(\sqrt[3]{1+a}+\sqrt[3]{1-a}\right)\sqrt[3]{1-a^2},\hfill \\ \hfill Z_2^2& =& 3a^2+Z_1^2.\hfill \end{array}$$

Now, we look for possible ways to distinguish the effects of the magnetic charge of regular Bardeen black hole in Anti-de Sitter spacetime and the rotating Kerr black hole on the dynamics of the magnetized particles with the parameter $\beta =10.2$ for the different values of the cosmological constant.

Figure 15 demonstrates the correspondence between the spin and magnetic charge parameters for the same values of the ISCO location of a magnetized particle with the parameter $\beta =10.2$ for the different values of the cosmological constant. One can see that the magnetic charge of the regular Bardeen BH can mimic the spin of Kerr BH up to $a/M\simeq 0.7896$, with its value in the range of $g/M\in (0,0.648)$ when the cosmological constant is zero. However, as the absolute value of cosmological constant, the mimic range of the rotation parameter decreases. For example, when the cosmological constant $\Lambda =-0.01$ ($\Lambda =-0.05$), the magnetic charge parameter of the Bardeen BH can mimic the spin parameter of the Kerr BH in the range of $a/M\in (0.3012,0.7896)$ ($a/M\in (0.4236,0.7896)$).

4.4. Regular Bardeen BH in AdS Spacetime versus Kerr BH for the Matching Radiative Efficiency

Particles falling into a BH from the accretion disk around the BH defines radiation energy released from the accretion disk [139]. The efficiency of the energy release in the accretion disk can be calculated using the expression in the single-particle model,

$$\eta =1-{\mathcal{E}}_{\mathrm{ISCO}},$$

(33)

where ${\mathcal{E}}_{\mathrm{ISCO}}$ corresponds to the specific energy of a particle orbiting at the ISCO.

One can analyze the effect of the magnetic charge of the Bardeen BH in Anti-de Sitter spacetime on the radiative efficiency given by Equation (33).

In Figure 16, we have presented the dependence of the radiative efficiency of the Bardeen BH in de-Sitter spacetime from the BH’s magnetic charge for the different values of the cosmological constant. One can see from the figure that the efficiency grows with the increase in the charge parameter, up to 40%. It has been also shown in the figure that for smaller values of the cosmological constant, the radiative efficiency also becomes smaller.

One of the ways to test/compare gravity theories using astrophysical observations is connected with the study of the efficiency of the energy release from the accretion disk around a BH. Since the dependence of the radiative efficiency of the Kerr BH from the spin is well known [138], one can now obtain the degeneracy between the spin parameter of the Kerr BH and the magnetic charge of the Bardeen BH in AdS spacetime for the matching values of the radiative efficiency, as shown in Figure 17.

One can clearly see from Figure 17 that the range of the mimicking values of the spin parameter increases as the absolute values of the cosmological constant increase and completely mimics the spin when the cosmological constant is $\Lambda =-0.05$. We see that the degeneracy plots presented for the magnetic dipole in Figure 15 and Figure 17 do not recover each other for the given ranges of the parameters. This, in turn, indicates that theoretically, it is possible to distinguish the Bardeen BH in AdS spacetime from that of the Kerr BH based on the two measurements, namely the measurement of the radius of the inner edge of the accretion disk that matches with the ISCO and the measurement of the radiative efficiency of the accretion disk around a BH.

5. Conclusions

This work is devoted to the investigation of the test particle motion around a static regular Bardeen black hole in Anti-de Sitter spacetime. Based on the idea that if a black hole in the universe is believed to be a Bardeen one with a cosmological constant rather than the Kerr one, then the same ISCO location for matching values of the spin parameter of the Kerr metric can be produced by the parameters included in the spacetime metric of interest.

It has been shown for neutral test particles that parameters of a regular Bardeen black hole in Anti-de Sitter spacetime can mimic the rotation parameter of the Kerr metric up to the value $a/M\approx 0.9$ based at the same ISCO location in both cases. It leads to the assumption that black hole candidates assumed to be rotating Kerr ones with a rotation parameter up to this value can also be static ones with a magnetic charge parameter in the presence of a negative cosmological constant.

The similar calculations for the magnetically charged test particles have shown that the additional force due to the electromagnetic interaction influences the results when compared to the neutral ones. Based on the obtained results, it has been justified that this kind of additional interaction does not cause drastic changes in the situation. The black hole characterized by the Bardeen metric in Anti-de Sitter spacetime can mimic the Kerr one with a considerably high rotation parameter $a>0.9$. It shows that the force acting on the test particle is strongly dominated by the spacetime parameters compared to the electromagnetic interaction.

Moreover, we have explored the dynamics of magnetized particles with a non-zero magnetic dipole moment, around a magnetically charged regular Bardeen black hole in Anti-de Sitter spacetime. As a realistic astrophysical scenario of the study, we have treated neutron stars orbiting SMBH, in particular, the magnetar PSR J1745-2900 orbiting Sgr A* with the parameter $\beta =10.2$, as the test magnetized particles. The performed study of the ISCO radius of the magnetized particles has shown that the parameter $\beta$, negative values of cosmological constant and magnetic charge of the central BH causes the ISCO radius to decrease. Finally, as an important issue of relativistic astrophysics, we have compared the effect of the magnetic charge of the Bardeen BH with one of the spin parameter of rotating Kerr BHs and shown that the magnetic charge parameter can mimic the spin parameter of Kerr BHs in the range $a/M\simeq (0,0.7896)$ when $\Lambda =0$ is at the range of its values $g/M\simeq (0,0.648)$, while with the increase of the absolute value of cosmological constant, the mimic range of the spin parameter decreases.

The performed study has indicated the possible degeneracy between the parameters of black holes in the dynamics of test particles in a close BH environment. It is concluded that the various BH parameter extraction methods from observational data have to be applied for correctly probing the nature of the astrophysical black holes.