Figure 1

Energy boundary function $E_{\mathrm{b}}(x,y;J,\omega ,a)$ for string loops with $J=0$ (a)–(c) and $J\ne 0$ (d)–(i), given for characteristic values of spin $a$. We chose parameter $\omega =0$ in all the presented cases, but the function $E_{\mathrm{b}}(x,y;J,\omega )$ depends only slightly on $\omega$ for small values of $J$, and its behavior is of the same character for the other values of $\omega$. The maxima (“horns”) are located at the singularity points $[\pm a,0]$. Empty space represents the dynamical region (41), where $E_{\mathrm{b}}$ is not defined. For angular momenta small enough ($J=0.1$), the repulsive centrifugal barrier is located at the ring $x\le a$, while for angular momentum large enough ($J=1.1$), it is located above the ring singularity.Reuse & Permissions

Figure 2

Equatorial ($y=0$) sections of the energy boundary function $E_{\mathrm{b}}(x,y;J,\omega )$ for Kerr BH ($a=0.99$) and NS ($a=1.1$) in the neighborhood of the Kerr ring singularity (dotted line). Particular values of the parameter $\omega$ are distinguished as $-1$ (dashed), 0 (plain) and 1 (dash-dotted) curve. Dynamical region of the Kerr BH is dashed. Notice that in the case $\omega =-1$, corresponding to axial angular momentum $L<0$, negative energies $E<0$ are allowed.Reuse & Permissions

Figure 3

String loops with negative energy ($E<0$) can exist only inside the ergosphere and for string loops with $\omega =-1$. Contours $E_{\mathrm{b}}(x,y)=0$ with typical values of the parameter $J$ (solid curves) are plotted; for $J=0$ they coincide with the outer horizon, and for $J\to \infty$ they coincide with the boundary of the ergosphere, the so-called surface of the static limit (thick dotted line).Reuse & Permissions

Figure 4

Function $J_{(\mathrm{e})\mathrm{eq}}^2(r;a,\omega )$ (black thick curve) denoting the local extrema of the energy boundary function $E_{\mathrm{b}}(r,\theta ;J,\omega ,a)$ located in the equatorial plane. We plotted the function $J_{(\mathrm{e})\mathrm{eq}}^2(r;a,\omega )$ for characteristic combinations of the parameters $a$ and $\omega$. The dotted curves represent the functions $J_{\mathrm{L}1}(r;a,\omega )$ and $J_{\mathrm{L}2}(r;a,\omega )$ determining the regions from which the string loops can escape to the infinity along the $y$ axis. In the shaded regions the string loops are trapped by closed energy boundaries, given by the sections $E=E_{\mathrm{b}}(x,y;J,\omega ,a)$. Note that string loops can be captured by Kerr BH, and therefore, we have to exclude such cases from the gray region; for Kerr NS only the trapped states (gray) can exist for all values of the parameter $J$ satisfying condition $J_{\mathrm{L}1}<J<J_{\mathrm{L}2}$, since no capture of string loops by Kerr NS is possible. Hatched are the dynamical regions of the Kerr spacetime. (a) $a=0$ represents string loops in the Schwarzschild spacetime and corresponds to Fig. 8a presented in 2.Reuse & Permissions

Figure 5

Loci of the local extrema of the energy boundary function $E_{\mathrm{b}}(r,\theta ;J,\omega ,a)$ in the equatorial plane in dependence on the rotation parameter $a$, allowing us to classify different regimes of behavior of the energy boundary function $E_{\mathrm{b}}$. Thin solid curves give the position of the horizons and restrict the dynamical region (hatched). Gray color denotes the region where the function $J_{(\mathrm{e})\mathrm{eq}}^2(r;\omega ,a)$ is positive—extrema of energy boundary function $E_{\mathrm{b}}(r,\theta ;J,\omega ,a)$ can exist in this area, while in the white areas extrema of $E_{\mathrm{b}}(r,\theta ;J,\omega ,a)$ cannot exist. The regions with maxima of $E_{\mathrm{b}}$ and regions with minima of $E_{\mathrm{b}}$ are separated by a thick black curve, which is given by the extrema of the $J_{(\mathrm{e})\mathrm{eq}}^2(r,\omega ,a)$ function. It is useful to compare this figure with Fig. 4, where exact functions $J_{(\mathrm{e})\mathrm{eq}}^2(r;\omega ,a)$ are plotted. It can be seen that for $\omega \in \{-1,1\}$ only two types of behavior of $E_{\mathrm{b}}$ exist: Kerr BH and Kerr NS, while three types of this behavior occur for $\omega =0$: Kerr BH, Kerr NSa, and Kerr NSb.Reuse & Permissions

Figure 6

Extrema of the $J_{(\mathrm{e})\mathrm{eq}}^2(r;\omega ,a)$ function in dependence on the Kerr rotational parameter $a$. Particular values of the parameter $\omega$ are distinguished as $-1$ (dashed), 0 (plain), and 1 (dash-dotted) curves.Reuse & Permissions

Figure 7

Characteristic radii of the string loop dynamics in the Kerr spacetimes introduced in this paper are compared to characteristic radii of the circular equatorial orbits of test particles and photons. We plotted loci of the outer BH horizon $r_{\mathrm{h}}$, photon circular orbits $r_{\mathrm{ph}}$, innermost bound circular orbits $r_{\mathrm{mb}}$, and innermost stable circular orbits $r_{\mathrm{ms}}$; we use dashed curves for the direct (corotating) orbits $(+)$ and dotted curves for the retrograde orbits $(-)$—compare with Fig. 1 from 39. The innermost stable positions of string loops $s_{\mathrm{ms}}$ are determined by the local minima of the $J_{(\mathrm{e})\mathrm{eq}}(r;\omega ,a)$ function. The innermost bound positions of string loops $s_{\mathrm{mb}}$ are given by the minimal value of the radial coordinate $r$ for which the function $J_{(\mathrm{e})\mathrm{eq}}(r;\omega ,a)$ exists; see Figs. 4, 5. The innermost stable and innermost bound positions are defined separately for string loops having $\omega =-1$, 0, $+1$.Reuse & Permissions

Figure 8

Positions of the off-equatorial local extrema (minima) of the energy boundary function $E_{\mathrm{b}}(x,y;J,\omega ,a)$, numerically calculated from Eq. (68), are presented for representative values of the rotation parameter $a>1$ (Kerr NS). For $a\le 1$ (Kerr BH) no of-equatorial local extrema can be found.Reuse & Permissions

Figure 9

Evolution of the energy boundary function $E_{\mathrm{b}}(x,y;J,\omega ,a)$ (black thick curves) around the off-equatorial minima (denoted by black crosses) demonstrated for fixed parameters $a=2$, $\omega =0$. Contours of the $E_{\mathrm{b}}(x,y;J,\omega ,a)$ function are represented for various energy levels by gray curves; the ring singularity is located at the point $[a,0]$.Reuse & Permissions

Figure 10

Extrema of the boundary energy function $E_{\mathrm{b}}(x,y;J,\omega ,a)$ illustrated for Schwarzschild and Kerr BH spacetimes as functions of the angular momentum $J$. Thick solid curves correspond to the maxima and minima of the energy boundary function $E_{\mathrm{b}}(x,y;J,\omega ,a)$ in the equatorial plane. The minimum of the $E_{\mathrm{b}}$ function for the flat spacetime $E=2J$ (escape to infinity) is represented by the dashed line. Regions where the “lakes” corresponding to the trapped states can exist are shaded. The numbers in (a) denote the examples of different types of the characteristic $E=\mathrm{const}$ sections of the boundary energy function $E_{\mathrm{b}}(x,y;J,\omega ,a)$, illustrated in Fig. 11.Reuse & Permissions

Figure 11

Constant energy sections of the energy boundary function $E=E_{\mathrm{b}}(x,y;J,\omega ,a)$ governing the string loop dynamics in the Kerr or Schwarzschild BH spacetimes. Four different types of the behavior of the string loop dynamics are possible: collapse or escape to infinity (type 1), collapse to the BH (type 2), trapping in some lakelike region (type 3), escape to infinity without possibility to collapse to the BH (type 4); see Fig. 10.Reuse & Permissions

Figure 12

Extrema of the boundary energy function $E_{\mathrm{b}}(x,y;J,\omega ,a)$ illustrated for the Kerr NS spacetimes as functions of the angular momentum $J$. The upper row of figures represents the large scale behavior, while the lower row figures, constructed for small values of $E$ and $J$, demonstrate details representing the off-equatorial minima. Thick solid curves correspond to the maxima and minima of the energy boundary function $E_{\mathrm{b}}$ in the equatorial plane. The boundary for the string loop escape to infinity along the $y$ axis, given by the relation $E=2J$, is illustrated by the dashed line. Shaded are the regions where the constant energy sections of the energy boundary function $E=E_{\mathrm{b}}(x,y;J,\omega ,a)$ form closed curves (lakes) around the minima of the function in the equatorial plane; hatched are the regions representing the off-equatorial lakes that exist around the off-equatorial minima of the energy boundary function. The off-equatorial minima can exist only for string loops with sufficiently small values of $J$ and $E$. The thin solid curve represents the limiting values of the energy boundary function just below the ring singularity, i.e., $E_{\mathrm{b}}(r\to 0,\pi /2;J,\omega ,a)$; such values could correspond in the Kerr NS spacetimes to the lowest allowed values of the energy $E$ of the string loops with $\omega \in \{-1,0\}$. There is no maximum of the energy boundary function $E_{\mathrm{b}}$ in the case of $\omega =1$; negative energies are allowed only for $\omega =-1$. Regions of the forbidden combinations of the $E$ and $J$ parameters are represented by the symbol $\nexists$. The numbers correspond to the characteristic examples of the constant energy sections of the $E_{\mathrm{b}}(x,y;J;\omega ,a)$ function as illustrated in Fig. 13.Reuse & Permissions

Figure 13

Constant energy sections of the energy boundary function governing the string loop dynamics in the Kerr NS spacetimes. The string loops escape to infinity (type 1 and type 3a), or are trapped in closed boundaries (lakes) formed around minima of the $E_{\mathrm{b}}(x,y;J,\omega ,a)$ function in the equatorial plane (types 1 and 3a–3c), or are trapped out of the equatorial plane (3d)—see Fig. 12 for detailed classification. The lakes are approaching the ring singularity in the cases 1, 3a, and partially in the 3b case, while lakes completely separated from the ring singularity occur in the cases 3b, 3c, and 3d.Reuse & Permissions

Figure 14

Comparison of the energy boundary function $E_{\mathrm{b}}$ for characteristic values of the spin parameter $a=0$ (thick solid curvee), $a=0.99$ (black solid curve), and $a=1.1$ (dashed curve). The functions $E_b$ differ significantly near the origin of the coordinates for given cases; we also depict the area of the horizon for black holes (gray). The string loop motion is represented for appropriately chosen characteristic cases. Because of the sensitivity of the string loop motion to the initial conditions, we have to compare the whole set of trajectories to see the effect of the rotation parameter $a$. We “shoot” the string loops from some position $x_s\sim 25$, $y_s\in (0,13)$, with current $J=2$ and energy $E=25$ toward the BH (NS). We continuously change the coordinate $y_s$ that serves as an impact parameter and plot the resulting gamma factor; results for different $a$ and $\omega$ are given in Fig. 15.Reuse & Permissions

Figure 15

The asymptotic speed of transmitted string loops given for all three values of the string loop parameter $\omega$ and characteristic values of the spacetime spin parameter $a$. The Lorentz factor $\gamma$ (vertical axis) is calculated for string loops with energy $E=25$ and current $J=2$, starting from the rest with different initial position $y_0\in (0,13)$ (horizontal axis) while $x_0$ is calculated from (38). Gray points correspond to the string loops collapsed to the black hole. Maximal acceleration (75) for this case gives us the limiting gamma factor ${\gamma }_{\max }=6.25$. We show the topical gamma factor that is numerically found in the sample, ${\gamma }_{\mathrm{top}}$, and also the efficiency of the transmutation effect, $R_{\gamma (\max )/3}$, given by the relative number of accelerations when the final Lorentz factor is larger than ${\gamma }_{\max }/3$.Reuse & Permissions

Figure 16

Maximal and extremal (topical) Lorentz gamma factor $\gamma$ in dependence on the angular momentum parameter $J$ for fixed string loop energy $E=25$, parameter $\omega =-1$, 0, $+1$, and the spin parameter $a$. The solid curve shows the predicted maximal acceleration (given by the theoretical limit) represented by the Lorentz factor ${\gamma }_{\max }$ in dependence on $J$, Eq. (75), while the markers are numerically calculated extremal (topical) Lorentz factors ${\gamma }_{\mathrm{top}}$ from the sample under consideration.Reuse & Permissions

Figure 17

Topical Lorentz factors ${\gamma }_{\mathrm{top}}$ given for all three possible values of $\omega =-1$, 0, $+1$ of string loops in dependence on the spin parameter $a$. Notice the sharp change at the BH/NS transition. String loops are assumed with energy $E=25$ and angular momentum $J=2$; maximal acceleration is limited by ${\gamma }_{\max }=6.25$.Reuse & Permissions

Figure 18

String loop acceleration in the Schwarzschild BH spacetime, plotted for various starting points of the string loop and its energy. The string loop is starting from rest at the points $x_{\mathrm{s}}\in (0.1,30.1)$, $y_{\mathrm{s}}\in (0.1,30.1)$ with the angular momentum fixed to values of $J=0.1$ or $J=11$, but with different energy $E$ that is determined by the starting point. The energy can be estimated from the first subfigure, where different levels of the $E_{\mathrm{b}}(x,y;J,\omega ,a)$ function are plotted. The presented figures correspond to Fig. 22 from 2, but here we colored every point according to the asymptotic Lorentz factor of the translational string loop motion in the $y$ direction. Black denotes the region below the horizon; gray regions correspond to the string loops collapsed to the black hole. The regions of white correspond to string loops that do not reach “infinity” located at $r=1000$ for the given time $\zeta =200$ and remain oscillating around the BH.Reuse & Permissions

Figure 19

String loop Lorentz factors of accelerated motion in the Kerr BH ($a=0.99$), upper row, and Kerr NS ($a=1.1$), lower row, spacetimes for $J=0.1$ and $J=11$ angular momenta; different values of the parameter $\omega$ are considered. The starting positions of the string loop agree with those in Fig. 18 plotted for $a=0$; see the description there. In the Kerr BH case with tje string loop parameter $\omega =-1$, we observe a significantly extended region of the gray color, leading to the conclusion that the string loop configurations with $L<0$ are less stable and tend to collapse to the Kerr BH more rapidly. In the Kerr NS spacetimes, the region of collapsed trajectories (gray) is substituted by the region of accelerated trajectories.Reuse & Permissions

Figure 20

Transition from the regular to the chaotic regime of the string loop motion as a solution to the “focusing” of the trajectories problem [1]. All four calculated cases (a)–(d) demonstrate an increase of chaoticity as the string loop energy $E$ grows. We use the Poincaré sections of the $r-p_r$ ($\theta -p_{\theta }$) spaces, and the Fourier spectrum of the coordinates $r(\zeta )$ [$\theta (\zeta )$]. The string loop is starting from the point $P_1$ ($r_0=r_{\mathrm{h}}+1.25$, ${\theta }_0=\pi /2+0.02$) in the Kerr BH with $a=0.99$ with various values of energy $E$ (and current $J$). The parameter $\omega$ is fixed to be $\omega =1$. The vertical lines in the Fourier spectra are the frequencies ${\omega }_r/(2\pi ),{\omega }_{\theta }/(2\pi )$. The boxed column of figures “regular” (b) and “chaotic” (d) are corresponding to the cases 9(a) and 8(a) from 1.Reuse & Permissions

Figure 21

Transition from the regular to the chaotic regime of the string loop motion. We use Poincaré sections of the $r-p_r$ ($\theta -p_{\theta }$) space and the Fourier spectrum of the coordinates $r(\zeta )$ [$\theta (\zeta )$]. The string loop is starting from the point $P_2$ ($r_0=r_{\mathrm{ISCO}}$, ${\theta }_0=\pi /2+0.02$) in the Kerr BH with $a=0.99$ with various values of energy $E$ (and current $J$). The parameter $\omega$ is fixed to be $\omega =1$. The boxed column of figures regular (b) and chaotic (d) correspond to the cases 9(c) and 8(c) from 1. There is no focusing of string trajectories in this case, but only an increase of chaoticity of the motion.Reuse & Permissions

Figure 22

Growing of the maximal Lyapunov exponent in dependence on the increasing string loop energy $E$. For small energies the motion is regular, but above some critical energy $E_{\mathrm{c}(\mathrm{P}1)}\sim 8.2$ ($E_{\mathrm{c}(\mathrm{P}2)}\sim 5.7$) the chaotic regime starts. Arrows with letters mark individual cases taken from Figs. 20, 21.Reuse & Permissions

Figure 23

Radial profiles of fundamental radial and vertical frequencies ${\nu }_r={\Omega }_r/(2\pi )$, ${\nu }_{\theta }={\Omega }_{\theta }/(2\pi )$ measured by distant observers, determined for harmonic string loop oscillations around stable equatorial equilibrium states. The string loops are considered with the parameter $\omega =0$, oscillating around a compact object with mass $2M_{\mathrm{sun}}$. The loci where resonances with the ratios $3:2$ and $2:3$ of the frequencies ${\nu }_{\theta }:{\nu }_r$ occur are denoted. These could be relevant for the explanation of the HF QPOs observed in microquasars [53, 55]. Both fundamental frequencies ${\nu }_r$, ${\nu }_{\theta }$ and loci of their resonance ratios are only slightly depending on the Kerr spacetime spin parameter $a$.Reuse & Permissions