# Magnetized black holes: the role of rotation, boost, and accretion in
twisting the field lines and accelerating particles111Based on an invited
contribution at 16th Marcel Grossmann Meeting, Session PT5 “Dragging is never
draggy: MAss and CHarge flows in GR” (id. #393), 5–10 July 2021.

Ondřej Kopáček and Vladimír Karas  Astronomical Institute, Czech Academy of
Sciences,
Boční II 1401, CZ-141 00 Prague, Czech Republic
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
http://astro.cas.cz

###### Abstract

Combined influence of rotation of a black hole and ambient magnetic fields
creates conditions for powerful astrophysical processes of accretion and
outflow of matter which are observed in many systems across the range of
masses; from stellar-mass black holes in binary systems to supermassive black
holes in active galactic nuclei. We study a simplified model of outflow of
electrically charged particles from the inner region of an accretion disk
around a spinning (Kerr) black hole immersed in a large-scale magnetic field.
In particular, we consider a non-axisymmetric magnetosphere where the field is
misaligned with the rotation axis. In this contribution we extend our previous
analysis of acceleration of jet-like trajectories of particles escaping from
bound circular orbits around a black hole. While we have previously assumed
the initial setup of prograde (co-rotating) orbits, here we relax this
assumption and we also consider retrograde (counter-rotating) motion. We show
that the effect of counter-rotation may considerably increase the probability
of escape from the system, and it allows more efficient acceleration of
escaping particles to slightly higher energies compared to the co-rotating
disk.

###### keywords:

black holes, magnetosphere, non-axisymmetry, dynamics of ionized matter,
acceleration of particles

##  1 Introduction

In this contribution we further study astrophysically relevant class of
escaping (jet-like accelerated) trajectories in a dynamical system of a
magnetized rotating black hole. Electrically charged particles following such
orbits may escape the attraction of the black hole in a collimated outflow and
they may be accelerated to very high energies. In Paper I[1] we adopted a
simple axisymmetric model consisting of a rotating black hole in an
asymptotically uniform magnetic field aligned with the rotation axis.
Initially neutral particles are supposed to follow circular orbits in the
equatorial plane (and freely fall below ISCO). At a particular ionization
radius, these particles obtain electric charge and their dynamics changes due
to the presence of electromagnetic field, and escape from the accretion disk
along high-velocity jet-like trajectories becomes possible. However, the
efficiency of the acceleration process remains only moderate in the
axisymmetric setup.[1]

In Paper II[2] we relax the assumption of the perfect spin-magnetic alignment
and consider a generalized non-axisymmetric model. The effect of non-
axisymmetry in electro-vacuum magnetospheres of compact objects is known to
introduce dynamically relevant features like magnetic null-points,[3] and
breaking the axial symmetry also appears to strongly affect the acceleration
process in the current model. In particular, the number of escaping orbits
generally increases and acceleration to ultrarelativistic energies becomes
possible in the oblique magnetosphere.[2, 4] Furthermore,[5] by employing a
boost transformation of the system into a frame in translation motion, we
explored the electromagnetic aspects of the linear (kick) velocity of the
black hole. Refer to enlightening discussions by various authors[6, 7] and see
also the relevant context of these solutions.[8]

In the above-mentioned previous papers, we numerically studied escaping orbits
and we assessed in detail the role of a broad variety of parameters of the
model. Nevertheless, the assumption that initially neutral matter co-rotates
with the central black hole, was adopted in all previous analyses. Here we
relax this assumption and we also consider counter-rotating (retrograde)
circular orbits as an initial setup of neutral particles. We compare the both
cases (of co-rotating vs. counter-rotating) in terms of the efficiency of the
formation and acceleration of the individual trajectories and the emerging
outflow.

The role of counter-rotation in processes of accretion and outflow near
compact objects has been only partially explored in the literature and various
relevant aspects of accretion of counter-rotating matter has been studied. In
particular, high-resolution hydrodynamic simulations suggest that counter-
rotating structure may largely increase the accretion rate.[9] Moreover,
general relativistic magnetohydrodynamic (GRMHD) simulations show that
accretion of counter-rotating matter may produce conditions for highly
efficient jet launching.[10] We also note that counter-rotating high-velocity
outflows are indeed reported in various systems, such as the spiral galaxy NGC
1068.[11]

The paper is organized as follows. In Section 2 we specify the employed model
of oblique black hole magnetosphere, we review the equations of motion and
describe the charging process of initially neutral matter. Jet-like
trajectories of charged particles escaping from counter-rotating orbits are
discussed in Section 3 with the focus on the direct comparison with the co-
rotating analogues. Results are summarized and briefly discussed in Section 4.
As the initial setup of the model investigated in the present paper is
analogous to that of Papers I and II, we refer to the more complete
introduction and references presented therein.

##  2 Model specification

We consider a spacetime around a rotating, axially-symmetric, uncharged black
hole of mass $M$ and spin $a$ described by the Kerr metric, which may be
expressed in Boyer–Lindquist coordinates $x^{\mu}=(t,\>r,\>\theta,\>\varphi)$
as follows:[12, 13]

$ds^{2}=-\frac{\Delta}{\Sigma}\>[dt-a\sin{\theta}\,d\varphi]^{2}+\frac{\sin^{2}{\theta}}{\Sigma}\>[(r^{2}+a^{2})d\varphi-a\,dt]^{2}+\frac{\Sigma}{\Delta}\;dr^{2}+\Sigma
d\theta^{2},$ (1)

where $\Delta(r)\equiv r^{2}-2Mr+a^{2}$ and $\Sigma(r,\theta)\equiv
r^{2}+a^{2}\cos^{2}\theta$. Roots of $\Delta(r)$ locate the outer (+) and the
inner (-) horizons as: $r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$. Geometrized units are
used throughout the paper; values of basic constants (gravitational constant
$G$, speed of light $c$, Boltzmann constant $k$, and Coulomb constant $k_{C}$)
therefore equal unity, $G=c=k=k_{C}=1$.

The black hole is supposed to be weakly magnetized by an external,
asymptotically uniform magnetic field with an asymptotic strength $B$ which
has an arbitrary inclination (angle $\alpha$) with respect to the rotation
axis. Vector potential $A_{\mu}=(A_{t},A_{r},A_{\theta},A_{\varphi})$ of a
corresponding test-field solution is given as:[14]

$\displaystyle A_{t}$ $\displaystyle=$
$\displaystyle\frac{B_{z}aMr}{\Sigma}\left(1+\cos^{2}\theta\right)-B_{z}a+\frac{B_{x}aM\sin\theta\cos\theta}{\Sigma}\left(r\cos\psi-a\sin\psi\right),$
(2) $\displaystyle A_{r}$ $\displaystyle=$ $\displaystyle-
B_{x}(r-M)\cos\theta\sin\theta\sin\psi,$ (3) $\displaystyle A_{\theta}$
$\displaystyle=$ $\displaystyle-
B_{x}a(r\sin^{2}\theta+M\cos^{2}\theta)\cos\psi-B_{x}(r^{2}\cos^{2}\theta$
$\displaystyle-Mr\cos 2\theta+a^{2}\cos 2\theta)\sin\psi,$ $\displaystyle
A_{\varphi}$ $\displaystyle=$ $\displaystyle
B_{z}\sin^{2}\theta\left[\frac{1}{2}(r^{2}+a^{2})-\frac{a^{2}Mr}{\Sigma}(1+\cos^{2}\theta)\right]$
$\displaystyle-
B_{x}\sin\theta\cos\theta\left[\Delta\cos\psi+\frac{(r^{2}+a^{2})M}{\Sigma}\left(r\cos\psi-a\sin\psi\right)\right],$

where $B_{z}$ denotes the component parallel to the rotation axis, while
$B_{x}$ corresponds to the perpendicular component, i.e.,
$B_{z}=B\cos{\alpha}$ and $B_{x}=B\sin{\alpha}$. Setting $B_{x}=0$ reduces the
above vector potential $A_{\mu}$ to the axisymmetric solution [15] employed in
Paper I. Azimuthal coordinate $\psi$ of Kerr ingoing coordinates is expressed
in Boyer–Lindquist coordinates as follows:

$\psi=\varphi+\frac{a}{r_{+}-r_{-}}\ln{\frac{r-r_{+}}{r-r_{-}}}.$ (6)

The Hamiltonian $\mathcal{H}$ of a particle of electric charge $q$ and rest
mass $m$ in the field $A_{\mu}$ and metric $g^{\mu\nu}$ is defined as:[12]

$\mathcal{H}=\textstyle{\frac{1}{2}}g^{\mu\nu}(\pi_{\mu}-qA_{\mu})(\pi_{\nu}-qA_{\nu}),$
(7)

where $\pi_{\mu}$ is the generalized (canonical) momentum. The equations of
motion are expressed as:

$\frac{{\rm d}x^{\mu}}{{\rm d}\lambda}\equiv
p^{\mu}=\frac{\partial\mathcal{H}}{\partial\pi_{\mu}},\quad\frac{d\pi_{\mu}}{d\lambda}=-\frac{\partial\mathcal{H}}{\partial
x^{\mu}},$ (8)

where $\lambda\equiv\tau/m$ is dimensionless affine parameter ($\tau$ denotes
the proper time). Employing the first equation we obtain the kinematical four-
momentum as: $p^{\mu}=\pi^{\mu}-qA^{\mu}$, and the conserved value of the
Hamiltonian is therefore given as: $\mathcal{H}=-m^{2}/2$. System is
stationary and the time component of canonical momentum $\pi_{t}$ is therefore
an integral of motion which equals (negatively taken) energy of the test
particle $\pi_{t}\equiv-E$. In the rest of paper we switch to specific
quantities $E/m\rightarrow E$, $q/m\rightarrow q$ which corresponds to setting
the rest mass of the particle $m=1$ in the formulas.

Figure 1: Position of the black hole’s outer horizon (black), marginally bound
circular orbit (cyan) and ISCO (green). Negative spin values correspond to the
counter-rotating orbits.

As an initial configuration of electrically neutral particles, we consider
circular Keplerian orbits specified by the values of constants of motion,
i.e., by specific energy $E_{\rm Kep}$ and specific angular momentum $L_{\rm
Kep}$ given as follows:[16]

$E_{\rm Kep}=\frac{r^{2}-2Mr\pm a\sqrt{Mr}}{r\sqrt{r^{2}-3Mr\pm
2a\sqrt{Mr}}},\;\;\;\;L_{\rm Kep}=\frac{\pm\sqrt{M}(r^{2}+a^{2}\mp
2a\sqrt{Mr})}{\sqrt{r(r^{2}-3Mr\pm 2a\sqrt{Mr})}},$ (9)

where the upper signs are valid for the prograde (co-rotating) orbits and the
lower ones for the retrograde (counter-rotating) orbits. Circular geodesics
remain stable only above the innermost stable circular orbit (ISCO):

$r_{\rm{ISCO}}=M\left(3+Z_{2}\mp\sqrt{(3-Z_{1})(3+Z_{1}+2Z_{2})}\right),$ (10)

where $Z_{1}\equiv
1+\left(1-\frac{a^{2}}{M^{2}}\right)^{1/3}\left[\left(1+\frac{a}{M}\right)^{1/3}+\left(1-\frac{a}{M}\right)^{1/3}\right]$
and $Z_{2}\equiv\sqrt{\frac{3a^{2}}{M^{2}}+Z_{1}^{2}}$. Positions of the ISCO
and marginally bound circular orbit[16] are plotted in Fig. 1.

Below ISCO the geodesics are supposed to turn into freely falling inspirals
maintaining the energy and angular momentum corresponding to the ISCO radius,
i.e., the particles keep $E=E_{\rm Kep}(r_{\rm ISCO})$ and $L=L_{\rm
Kep}(r_{\rm ISCO})$ during their infall while the radial velocity $u^{r}$ is
obtained from the normalization condition $u^{\mu}u_{\mu}=-1$.

In our model, we suppose that initially neutral elements undergo a sudden
charging process (e.g., due to photoionization) at a given radius $r_{0}$ and
obtain specific electric charge $q$. While the change of the rest mass $m$ is
considered as negligible, the particle dynamics changes due to the presence of
the electromagnetic field.[17] In particular, the conserved value of energy is
modified as:

$E=E_{\rm Kep}-qA_{t},$ (11)

while the values of spatial components of canonical momentum are changed as:

$\pi_{r}=\pi_{r}^{0}+qA_{r},\;\;\;\pi_{\theta}=qA_{\theta},\;\;\;\pi_{\varphi}=L_{\rm
Kep}+qA_{\varphi},$ (12)

where $\pi_{r}^{0}$ is zero for particles ionized above/at ISCO and for
infalling particles with ionization radius $r_{0}<r_{\rm{ISCO}}$ the value is
calculated from the normalization condition.

Figure 2: Comparison of initially co-rotating orbits (upper row) with counter-
rotating orbits (lower row). Color-coding: blue for plunging orbits, red for
stable ones, and yellow for escaping trajectories. The green circle denotes
the ISCO, the cyan circle marks the radius of the marginally bound orbit and
the white one shows the ergosphere. The inner black region marks the horizon
of the black hole. The parameters of the system are $\alpha=35\degree$ and
$qB=-5\,M^{-1}$. The magnetic field is inclined in the positive $x$-direction.

##  3 Escape zones

The dynamics of initially neutral particles on stable circular orbits (above
ISCO) or on a plunging trajectory (below ISCO) changes abruptly as they obtain
the electric charge and become affected by the electromagnetic field given by
Eqs. (2)-(2). As a result, some stable particles may become plunging, while
some plunging trajectories are stabilized by the field. Moreover, it appears
that particles may also become unbound and escape to infinity in jet-like
trajectories.

We have previously analyzed escaping particles in this model and, in
particular, in Paper I[1] we considered the axisymmetric system ($B_{x}=0$),
while in the Paper II[2] the general non-axisymmetric configuration was
studied. We have used the method of effective potential to find the necessary
conditions for the escape of particles. It appears that for the parallel
orientation of the spin axis $z$ and magnetic field component $B_{z}$, only
negatively charged particles may escape (while positively charged escape for
the anti-parallel orientation). Non-zero spin $a$ and $B_{z}$ are required for
the escape and final Lorentz factor $\gamma$ is found to be an increasing
function of the spin, specific charge $|q|$ and $|B_{z}|$ while it decreases
with the ionization radius $r_{0}$.

Figure 3: Comparison of initially co-rotating orbits (upper row) with counter-
rotating orbits (lower row). The color-coding and parameters as in Fig. 2.

However, as the above conditions are necessary but not sufficient, we need to
investigate the dynamic system numerically in order to determine which
particles actually follow the escaping trajectories. Initial locations of
escaping particles form escape zones whose emergence and evolution with
respect to the parameters of the system was discussed in our previous
papers.[4, 1, 2] Efficiency of the acceleration mechanism was quantified by
the final Lorentz factor of escaping particles leading to the conclusion that
also ultrarelativistic velocities with very high energies may be achieved
within the non-axisymmetric model.

In this contribution, we extend the previous discussion and consider a
modified setup. In particular, here we suppose that the initially neutral
matter may follow counter-rotating geodesics, i.e., unlike the previous
papers, here we also consider the lower signs in Eqs. (9) and (10). The ISCO
of counter-rotating orbits is located generally farther from the horizon
compared to the co-rotating ones (see Fig. 1). In particular, no stable
retrograde circular orbits around rotating black hole are allowed for $r\leq
6\,M$ and the ISCO location grows with the spin up to $r_{\rm ISCO}=9\,M$ for
the maximally rotating black hole with $a=M$.

We assume that below ISCO the initially neutral particle falls freely towards
the horizon keeping the energy and angular momentum corresponding to the ISCO
radius, while the radial velocity is computed from the normalization
condition. At the ionization radius $r_{0}$ the particle obtains the electric
charge and its dynamics changes due to magnetic field. Three main scenarios
may occur then; the particle plunges into the horizon, follows the stable
orbit around the black hole or escapes to infinity.

Analogously to our previous studies, we adopt the visual representation of
resulting orbits in escape boundary plots in which we assign blue dots to
plunging orbits, red dots to stable orbits and yellow dots to escaping ones.
In Figs. 2-4 we compare the co-rotating setup (upper rows) with the counter-
rotating case (bottom rows) in a series of escape boundary plots for a
particular set of parameters ($qB=-5\,M^{-1}$ and $\alpha=35\degree$) with
varying values of spin. In Fig. 2 we observe that for low spins the both cases
do not differ significantly. Corresponding escape zones have comparable shapes
and sizes regardless the sense of revolution. However, if we consider moderate
values of spin, the situation changes and significant differences between the
escape zones in co-rotating and counter-rotating disks arise as shown in Fig.
3. Most importantly, we observe that the number of escaping trajectories is
substantially higher for counter-rotating orbits forming considerably larger
escape zones. The tendency of more rapid rotation to favorize the escape from
counter-rotating orbits is confirmed in Fig. 4, where we notice that high spin
allows only narrow escape zones in the co-rotating disk while counter-rotating
zones become significantly wider. Also, these plots exhibit clear signatures
of a transition from the regular to chaotic system in agreement with previous
studies of similarly perturbed black hole systems.[18, 19, 20, 2]

Figure 4: Comparison of initially co-rotating orbits (upper row) with counter-
rotating orbits (lower row). The color-coding and parameters as in Fig. 2.

Figure 5: Comparison of escape zones in co-rotating (left panels) and counter-
rotating disks (right panels) for several values of magnetization parameter:
$qB=-M^{-1}$ (bottom panels), $qB=-5\,M^{-1}$ (middle panels) and
$qB=-10\,M^{-1}$ (upper panels). The inclination of the field is
$\alpha=35\degree$ and $\varphi_{0}=135\degree$. The color-coding as in Fig.
2.

Another convenient way to visualize the escape zones is to fix the initial
value of the azimuthal angle $\varphi_{0}$ and plot the trajectories in the
$r_{0}\times a$ plane (ionization radius vs. spin) using the same color-coding
of trajectories as in Figs. 2-4. The angle $\varphi$ is measured anticlockwise
from the positive direction of $x$-axis (which is the direction of the field
inclination). In particular, we choose the value $\varphi_{0}=135\degree$ for
which the structure of the escape zone seems especially complex. In Fig. 5 we
present the resulting plots for several values of magnetization parameter $qB$
and compare co-rotating escape zones (left column) with the counter-rotating
zones (right column). It confirms that counter-rotating orbits are generally
more prone to escape regardless the value of $qB$ while the both cases differ
most significantly for rapidly rotating black holes.

Figure 6: Final Lorentz factor $\gamma$ of escaping particles is encoded with
the colorscale comparing the co-rotating (left) and the counter-rotating disks
(right) while other parameters remain fixed as: $\alpha=35\degree$ and
$qB=-20\,M^{-1}$.

Previous analysis in Papers I and II shows that spin parameter is an important
factor for the acceleration of escaping particles. In particular, it appears
that final value of $\gamma$ for the fixed $qB$ grows with the spin and
decreases with the ionization radius $r_{0}$. Maximally accelerated particles
are thus expected to originate from the inner edge of the escape zone near
maximally rotating black hole with spin $a=M$.

In Fig. 5 we observe that counter-rotating escape zones for high spins are
considerably wider compared to co-rotating ones and, moreover, we notice that
counter-rotating zones are located closer to the horizon. We thus expect that
particles escaping from counter-rotating orbits could reach slightly higher
energies. This is indeed the case as shown in Fig. 6 where the final values of
the Lorentz factor are encoded with the color-scale. For this particular set
of parameters ($qB=-20\,M^{-1}$, $\varphi_{0}=135\degree$) we find that
maximally accelerated counter-rotating particle reaches $\gamma_{\rm max}=8.1$
being launched from $r_{0}=2.3\,M$ while for the co-rotating disk we find
$\gamma_{\rm max}=7.3$ for the particle escaping from $r_{0}=2.5\,M$.

##  4 Conclusions

We have numerically analyzed a simplified model of the outflow of electrically
charged particles from the inner region of the accretion disk near a weakly
magnetized rotating black hole. Stable circular orbits (freely falling below
ISCO) of initially neutral particles were perturbed by the ionization process
occuring in the non-axisymmetric magnetosphere. As a result, the escaping jet-
like trajectories may be realized for some range of parameters and
acceleration to high energies becomes possible. While we have previously
studied various aspects of given model and investigated the formation of
escape zones in detail,[2, 1] we have assumed the initial setup of co-rotating
accretion disk. In particular, the neutral particles above ISCO were supposed
to follow prograde circular orbits and freely fall below ISCO.

In this contribution we have generalized the previous analysis by considering
also counter-rotating (retrograde) Keplerian orbits as an initial setup of
neutral matter. We have studied the escape zones emerging in the counter-
rotating disks which were directly compared with the corresponding co-rotating
versions. Based on the analyzed set of escape zones we may conclude that
counter-rotation tends to increase the efficiency of the outflow. While its
effect becomes negligible for low spins, for moderate to high spin values we
observe that counter-rotating disks allow the formation of significantly wider
escape zones compared to their co-rotating analogues. The position of the zone
is also affected as counter-rotating zones are shifted towards the horizon
which makes the acceleration process more efficient. Although the difference
is rather small, we observe that counter-rotating particles may be accelerated
to higher energies.

## Acknowledgements

Authors acknowledge the support from the Inter-Excellence mobility program in
science of the Czech Ministry of Education, Youth and Sports (projects ref.
8JCH 1080 and LTC 18058). OK has been supported by the project “Lumina
Quaeruntur” (ref. LQ100032102) of the Czech Academy of Sciences. VK thanks the
Czech Science Foundation grant “Mass and charge currents in general relativity
and astrophysics” (ref. 21-11268S).

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