Charged string loops in Reissner–Nordström black hole background
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Abstract
We study the motion of current carrying charged string loops in the Reissner–Nordström black hole background combining the gravitational and electromagnetic field. Introducing new electromagnetic interaction between central charge and charged string loop makes the string loop equations of motion to be nonintegrable even in the flat spacetime limit, but it can be governed by an effective potential even in the black hole background. We classify different types of the string loop trajectories using effective potential approach, and we compare the innermost stable string loop positions with loci of the charged particle innermost stable orbits. We examine string loop small oscillations around minima of the string loop effective potential, and we plot radial profiles of the string loop oscillation frequencies for both the radial and vertical modes. We construct charged string loop quasiperiodic oscillations model and we compare it with observed data from microquasars GRO 165540, XTE 1550564, and GRS 1915+105. We also study the acceleration of current carrying string loops along the vertical axis and the string loop ejection from RN black hole neighbourhood, taking also into account the electromagnetic interaction.
1 Introduction
Detailed studies of relativistic currentcarrying string loops moving axisymmetrically along the symmetry axis of Kerr or Schwarzschild–de Sitter black holes appeared currently [8, 9, 10]. Tension of such string loops prevents their expansion beyond some radius, while their worldsheet current introduces an angular momentum barrier preventing collapse into the black hole. Such a configuration was also studied in [7, 13, 21]. There is an important possible astrophysical relevance of the currentcarrying string loops [8] as they could in a simplified way represent plasma that exhibits associated stringlike behavior via dynamics of the field lines in the plasma [4, 19] or due to thin isolated flux tubes of plasma that could be described by an onedimensional string [5, 19, 20].
In the previously mentioned articles the string loop was electromagnetically neutral and there was no external electromagnetic field. Motion of electromagnetically charged string loops in combined external gravitational and electromagnetic fields has been recently studied [30, 31]. Now we would like to extend such research and we examine dynamic properties of electromagnetically charged and current carrying string loop also in combined electromagnetic and gravitational fields of Reissner–Nordström background representing a pointlike electric charge Q source. Our work demonstrates the effect of the black hole charge Q on the string loop dynamic in general; the discussion of the black hole charge relevance is given in Appendix A. We discuss two astrophysically crucial limiting cases of the dynamics of the charged string loops related to phenomena observed in microquasars: small oscillations around equilibrium radii that can be relevant for the observed quasiperiodic highfrequency oscillations, and strong acceleration of the string loops along the symmetry axis of the black hole–string loop system that can be relevant for creation of jets.
The general dynamics of motion for relativistic current and charge carrying string loop with tension \(\mu \) and scalar field \(\varphi \) was introduced by [13] for the spherically symmetric Schwarzschild BH spacetimes, for the Kerr spacetimes it is discussed in [8, 10, 11, 14]. General Hamiltonian form for all axially symmetric spacetimes also with electromagnetic field is introduced by [13]. To show properly how the string loops interact electromagnetically, we will compare charged particle motion with the charged string loop in the same Reissner–Nordström black hole background, using results already obtained in [1, 2, 17, 18]. We show that there are similarities in the dynamics of the charged string loops and charged test particles, as the dynamics can be described in both cases by the Hamiltonian formalism with a relatively simple effective potential. There is a fundamental difference in the RN backgrounds: while the test particle motion is regular, the string loop motion has in general chaotic character [10, 13], where “islands” of regularity occur only for small oscillations near the string loop stable equilibrium points.
Throughout the present paper we use the spacelike signature \((,+,+,+)\), and the system of geometric units in which \(G = 1 = c\). However, for expressions having an astrophysical relevance we use the constants explicitly. Greek indices are taken to run from 0 to 3.
2 Dynamics in spherically symmetric spacetimes
 Flat
There is no black hole and hence no gravitational interaction. The electric field of the charge Q is so weak, that it will not contribute to the metric. We will use the flat metric (1), with \(M=0,Q=0\), while the electromagnetic interaction will be given by (5). Discussed in Sect. 3.1.
 RN
There will be black hole, with the gravitational field influenced by the strong electromagnetic field. We will use full RN metric (1) with electromagnetic interaction given by (5). Discussed in Sect. 3.2.
2.1 Hamiltonian formalism for charged particle motion
The dynamics of axially symmetric charged current carrying string loops can be enlightened by comparison with charged test particle motion, as both these dynamics can be formulated in the framework of Hamiltonian formalism. Recall that evolution of axisymmetric string loops adjusted to axisymmetric backgrounds can be represented by evolution of a single point of the string [10, 21].
We can also compare electromagnetic forces acting on charge test particles or string loops. Since the motion of a charged test particle in the RN black hole background has been intensively studied in literature [1, 2, 17, 18], we will give just short summary.
2.2 Hamiltonian formalism for relativistic string loop
2.3 Conserved quantities and effective potential for string loop dynamics
 \(\omega =1\)

There is no electric current on the string loop, \(n=0\), only negative electric charge \(\Omega <0\) uniformly distributed along the loop. Since we consider the central object charge \(Q>0\) to be positive, there acts an electromagnetic attractive force between the central object and the string loop.
 \(\omega =0\)

There is no charge on the string loop, \(\Omega =0\), only current n, and there is no electromagnetic interaction between the string loop and the central object electric charge Q. The black hole charge Q can affect the string loop dynamic only through changes in (1) metric. This case was already studied in [22] for the so called “tidal charge” black hole scenario.
 \(\omega =1\)

There is no electric current on the string loop, \(n=0\), only positive electric charge \(\Omega >0\) uniformly distributed along the loop. there will be electromagnetic repulsive force between the central object and the string loop.
The electric force between the central object with charge Q is attractive for \(1\le \omega <0\), while it is repulsive for \(0<\omega \le ~1\). We will focus on string loop dynamics for \(\omega \in \{1,0,1\}\) limiting values, and we will assume the string loop behaviour for another value of \(\omega \), will be combination of the limiting values. It is interesting that for all three limiting cases \(\omega \in \{1,0,1\}\), the string loop angular momentum L is zero (23).
3 String loop in combined electric and gravitational field
3.1 Charged string loop in flat spacetime
We discuss the flat spacetime case separately, as establishing the flat space limit requires \(M=0\) and \(Q=0\) simultaneously, but this means vanishing of the electromagnetic field.
In Fig. 1 the left graph represents the string loop effective potential function \(V_\mathrm{eff}(x,z=0)\) as section at the equatorial plane. In the xdirection, we have one minimum of the effective potential, depending on values of J an \(\omega \). In middle graph is plotted the string loop effective potential function \(V_\mathrm{eff}(x=x_\mathrm{min},z)\) as section at its equatorial minimum. The stationary points of the \(V_\mathrm{eff}(x,z)\) function are located in the equatorial plane, \(z=0\), only; in the zdirection, for \(\omega =1\) we have minima, for \(\omega =0\) we have constant behaviour in z direction, and for \(\omega =1\) we have saddle point. The behaviour of the effective potential along the vertical z axis \(V_\mathrm{eff}(x=x_{0},z)\), as section at \(x_0=2\) (right Fig.), is also plotted for all three limiting values of \(\omega \) parameter.
For visualizing the regions where the string loop motion is possible, we demonstrate in Fig. 2. the \(E=\mathrm{const.}\) sections of the effective potential full 2D function \(V_\mathrm{eff}(x,z)\) for both x and z coordinates. In the left, picture we give the \(V_\mathrm{eff}\) profile for \(\omega =1\), \(E=3.5\) case. The electrostatic interaction between the black hole and string loop charges is attractive and the string loop is trapped in closed area (light grey) – string loop is located in effective potential “lake”. In the middle figure, the case \(\omega =0\), \(E=4.5\) is presented. In this case, trapped motion in x axis is observed, and there is no motion in vertical z axis, if we consider zero initial velocity in z direction, the effective potential (32) does not depend on z in absence of the electric interaction between the black hole and string loop. In the right picture, we shown the case \(\omega =1\), \(E=5\). The string loop is allowed to oscillate in a limited x interval, while, due to the electrostatic repulsion between the black hole and string loop charges, the string loop is escaping along the vertical z axis. Depending on the initial position, the string loop can move in the upper or lower half spaces and it can never cross the equatorial plane.
Coming from Fig. 2, we draw in Fig. 3 the trajectories of the string loop within their energy boundaries for the same values of the parameters J, Q and \(\omega \). We can conclude that in the case of opposite (attractive) charges (\(\omega =1\)), electric attraction resists the string loop to escape to infinity. In the absence of electric interaction (\(\omega =0\)), there is no force in vertical direction and the trajectory of the string loop is always on the plane parallel to the z plane. In the repulsively charged case (\(\omega =1\)), the electric repulsive potential barrier pushes the string loop away from the center.
3.2 Charged string loops in Reissner–Nordström background
Depending on the black hole charge Q and string loop charge parameter \(\omega \), there can exist one, two or three stationary points of the effective potential in the equatorial plane. The sign of \(\frac{dJ_\mathrm{ext}}{dx}\) defines type of the extrema, the positive derivation term, \(\frac{dJ_\mathrm{ext}}{dx}>0\), determines the effective potential equatorial minima while negative derivative, \(\frac{dJ_\mathrm{ext}}{dx}<0\), determines the maxima. The extremal point of the \(J_\mathrm{ext}\) function, given by \(\frac{dJ_\mathrm{ext}}{dx}=0\), defines the innermost stable string loop position.
In Fig. 5 we use the diagonal pictures of Fig. 4 and construct the corresponding effective potential along xaxis and zaxis at \(x_\mathrm{min}\) given for the chosen values of J, where \(x_\mathrm{min}\) is position of the effective potential minima. In Fig. 6 we give some typical trajectories of the string loop motion. In the top row of the Fig. 5, we consider \(Q=0.3\), \(\omega =1\) case. First we take \(J=10\) as it is crossing the \(J_\mathrm{ext}\) profile at two points (Fig 5). In this case, there is one stable and one unstable equilibrium position.
In Fig. 6a we show trajectories of string loop’s motion crosssection along with the boundary energy profiles – we observe that the motion is trapped in the closed region. Then we consider \(J=7.44\), as this value of J is touching the extremal point of the function \(J_\mathrm{ext}\) (Fig 5), corresponding to the innermost stable equilibrium position (ISEP) – any small deviation from this position causes the string to collapse to the black hole. String loop’s trajectory for this case is given in Fig. 6b; we can conclude that the motion is finite in the zdirection and the energy boundary profile is open to the black hole, and the string finally falls down to the black hole. And last case of \(Q=0.3\), \(\omega =1\) configuration, we take \(J=4\) value as it is not crossing the \(J_\mathrm{ext}\) profile at all. In this case, there is no possible trapped motion and the string loop has to escape to infinity in the vertical direction (Fig. 6c). Another possible string loop trajectory around the black hole is given for \(Q=0.3\), \(\omega =1\), \(J=10\) case in Fig. 6d, with string escaping to infinity.
4 Quasiperiodic oscillations of string loops
The quasiperiodic oscillatory motion of the string loops trapped in a toroidal space (or in “lake”) around the minima of the effective potential \(V_\mathrm{eff}(x,z)\) function could be used to interpret interesting astrophysical phenomenon – highfrequency quasiperiodic oscillations (HF QPOs). Most of compact Xray binaries that contain a black hole or a neutron star demonstrate quasiperiodic variability of the Xray flux in the kHz frequency range. Some of these HF QPOs appear in pairs as upper and lower frequencies (\(\nu _\mathrm{U}, \nu _\mathrm{L}\)) and in Fourier spectra are observed twin peaks. Since the peaks of high frequencies are close to the orbital frequency of the marginally stable circular orbit representing the inner edge of Keplerian discs orbiting black holes (or neutron stars), the strong gravity effects must be relevant to interpret HF QPOs [23]. So far, many models have been proposed to explain HF QPOs in black hole binaries: the relativistic precession model, the warped disc model, resonance model [16, 24, 25, 26, 28]. Usually, Keplerian orbital and epicyclic (radial and latitudinal) frequencies of geodetical circular motion are assumed in models explaining the HF QPOs in both black hole and neutron star systems [27]. However, neither of these models is able to explain the HF QPOs in all microquasars [29]. On the other hand, there is possibility of the relevance of string loop’s oscillations, characterized by their radial and vertical (latitudinal) frequencies that are comparable to the epicyclic geodetical frequencies, but slightly different, enabling thus some corrections to the predictions of the models based on the geodetical epicyclic frequencies. Of course, the frequencies of string loops oscillations in physical units have to be related to distant observers.
Observed twin HF QPOs data for three microquasars, and the restrictions on mass of black holes located in them, based on independent measurements on the HF QPO measurements
Source  GRO 165540  XTE 1550564  GRS 1915+105 

\(\nu _U\) (Hz)  447–453  273–279  165–171 
\(\nu _L\) (Hz)  295–305  179–189  108–118 
\(M/M_{\odot }\)  6.03–6.57  8.5–9.7  9.6–18.4 
For fixed black hole charge Q and fixed string loop charge parameter \(\omega \), upper frequency of the twin HF QPOs can be given as a function of black hole’s mass M. If the black hole mass is restricted by separated observations, as is commonly the case, we obtain some restrictions on the string loop resonant oscillations model, as illustrated in Fig. 10. Here, the situation is demonstrated for several values of black hole’s charge Q and limits on the black hole mass as given in Table 1. We can see that for the Schwarzschild black hole (\(Q=0\)), the string loop model can explain only the HF QPOs in GRS 1915+105. Introducing black hole charge Q and parameter \(\omega \), the string loop resonant oscillations model widens the area of its applicability. For \(\omega =1\) and \(Q=0.5, 0.8\) case, the model fully describes observed values from GRO 165540 source. It contains the whole range of expected mass range from Table 1. Nevertheless, the string loop resonant oscillation model in Reissner–Nordström background can not explain the observed values from XTE 1550564 source. For any value of \(\omega \) parameter and for any low values of black hole charge Q, the string loop model can not fit observed mass range for the XTE 1550564 source and an additional influence of the black hole rotation has to be expected.
Moreover, in Fig. 10 we can clearly see that the predicted value of the black hole mass is increasing with the black hole charge Q increase. It will become harder and harder to fit the observed HF QPOs as the Q parameter increases, hence we can conclude that introducing new parameter Q into the string loop HF QPOs model is not successfully efficient in explaining the observed HF QPOs in microquasars, and inclusion of the black hole spin that can be sufficiently efficient as demonstrated in [23] is necessary.
5 String loop acceleration and asymptotical ejection speed
Clearly, \(E_\mathrm{x}=E_0\) and \(E_\mathrm{z}\) are constants of string motion in the flat spacetime and transmutation between energy modes are not possible there. However, in the vicinity of black holes, the kinetic energy of oscillating string can be transformed into the kinetic energy of the translational linear motion.
In Fig. 12 we give escaping trajectories of the transmitted string loops in the RN background and flat spacetime for attracting, \(\omega =1\), not interacting, \(\omega =0\), and repulsing, \(\omega =1\), types of the string loop interaction with the charged black hole and central point charge in flat spacetime. It is expected to observe bigger ejection speed in repulsing case(\(\omega =1\)) than attracting one(\(\omega =1\)). However, surprisingly the string loop acceleration is higher in \(\omega =1\) case than \(\omega =1\) case. This can be explained by studying their trajectories within the energy boundaries. In the \(\omega =1\) case, due to attraction by the black hole, the string loops enters deeper in a black hole’s potential well and the transition effect of oscillating energy to escaping translational energy in the zdirection becomes more effective. Due to the chaotic nature of string loop dynamics, we can expect completely different set of velocities for different set of initial conditions.
6 Conclusion
The astrophysically relevant problems of current carrying string loops in spherically symmetric spacetimes have been studied recently [9, 21, 22]. In the present paper we investigate the relevant issues for Reissner–Nordström background, giving the attention on the influence of black hole charge Q and its electromagnetic interaction with string loop charge \(\omega \) created by scalar field \(\varphi \) living on the string loop.
Scalar field \(\varphi \), living on the string loops and represented by the angular momentum parameter J, is essential for creating the centrifugal forces, and therefore for existence of stable string loop positions. In RN background is the charged string loop innermost stable equilibrium position (ISEP) located between the photon circular orbit at \(r_\mathrm{ph}\) and innermost stable charged particle orbit (ISCO). The condition \(r_\mathrm{ph}<r_\mathrm{ISSP}<r_\mathrm{ISCO}\), already proven for rotating Kerr black hole background [10], is supporting consideration of string loop model as a composition of charged particles and their electromagnetic fields [5].
We have shown different types of string loop energy boundaries and different string loop trajectories in RN backgrounds. There are not any new types of the string loop motion for RN black hole background [8], but in the field of RN naked singularities two closed toroidal regions for the string loop motion are possible (\(Q=1.0677\), \(\omega =1\), \(J=10\)) .
String loop harmonic oscillations around stable equilibria, defined by Eq. (47), could be one of the perspective explanations of the HF QPOs observed in binary systems containing black holes or neutron starts. In the present paper, we applied the string loop resonant oscillations model to fit observed data from GRO 165540, XTE 1550564, GRS 1915+105 microquasar sources. Our fittings are substantially compatible with the observed data from the GRO 165540 source and partially coincide with the GRS 1915+105 data. For the latter source the values of \(\omega \) parameter are significant. Observed data from XTE 1550564 can be explained only for \(Q\sim 0.9\) values. We can conclude that the twin HF QPOs could be efficiently explained by the string loop oscillatory model, if we consider interaction of an electrically charged current carrying axisymmetric string loop with the combined gravitational and electromagnetic fields of Kerr–Newman black hole where due to the combination of the black hole spin even small electric charge of the black hole can cause relevant modifications of the frequencies of the string loop oscillations.
String loop acceleration to the relativistic escaping velocities in the black hole neighbourhood, is one of the possible explanations of relativistic jets coming from AGN. We have studied the effect of black hole and string loop charges interaction on to the acceleration process. Due to the chaotic character of equations of motion, the positively charged string loop \(\omega >0\) can be ejected from equatorial plane even in flat background. The RN black hole charge Q does not contribute to the string loop acceleration speeds due to electrostatic repulsion, it only modifies the effective potential \(V_\mathrm{eff}\) and allows the string loop to came closer to the black hole, where the transmutation is more effective. This implies a surprising phenomena: the transmutation effect is more efficient, and the string loop is more significantly accelerated for the electric attraction of the string loop and the black hole, as the transmission process can occur in deeper regions of the gravitational potential well than in the case of electric attraction. Note that contrary to the standard Blandford–Znajek mechanism of jet acceleration to high velocities [3], where fast rotating black hole must be assumed, in the stringloop acceleration model rotation of the black hole is nor required.
The RN solution is simple and elegant solution of combined Einstein and Maxwell equations and by studying charged string loop dynamics in this solution, we would like to just complete our previous string loop studies in order to map potential role of the Coulombic electric interaction. We explore theoretical properties of charged string loop motion in RN background and we show that unrealistically high values of RN charge are needed to explain real astrophysical data. In some dynamic situations, as those corresponding to unstable states of accretion disks wider ionization processes, the electric charge could be momentarily larger as indicated above for stationary situations.
Notes
Acknowledgements
T.O. acknowledges the Silesian University in Opava Grant no. SGS/14/2016, M.K. acknowledges the Czech Science Foundation Grant no. 1603564Y and Z.S. acknowledges Albert Einstein Centre for Gravitation and Astrophysics supported by the Czech Science Foundation Grant no. 1437086G.
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