# Novel topological phenomena of timelike circular orbits for charged test
particles

Xu Ye, Shao-Wen Wei111Corresponding author. E-mail<EMAIL_ADDRESS>1Key
Laboratory of Quantum Theory and Applications of MoE, Lanzhou University,
Lanzhou 730000, China
2Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, School of Physical Science and Technology, Lanzhou
University, Lanzhou 730000, People’s Republic of China,
3Institute of Theoretical Physics $\&$ Research Center of Gravitation, Lanzhou
University, Lanzhou 730000, People’s Republic of China

###### Abstract

The topological approach has recently been successfully employed to
investigate timelike circular orbits for massive neutral test particles. The
observed vanishing topological number implies that these timelike circular
orbits occur in pairs. However, the behavior of charged test particles in this
regard remains unexplored. To address this issue, our study focuses on
examining the influence of particle charge on the topology of timelike
circular orbits within a spherically symmetrical black hole spacetime holding
a nonvanishing radial electric field. We consider four distinct cases based on
the charges of the particle and the black hole: unlike strong charge, unlike
weak charge, like weak charge, and like strong charge. For each case, we
calculate the corresponding topological number. Our results reveal that when
the charge is large enough, the topological number takes a value of -1 instead
of 0, which differs from the neutral particle scenario. Consequently, in cases
of small charges, the timelike circular orbits appear in pairs, whereas in
cases of larger charges, an additional unstable timelike circular orbit
emerges. These findings shed light on the influence of the particle charge on
the topological properties and number of timelike circular orbits.

Classical black hole, timelike circular orbit, topological charge

##  I Introduction

Topology has emerged as a promising approach in the exploration of the
underlying information of black hole physics. By disregarding the local
properties of the black hole system, topology offers insights into its global
properties. Notably, this approach has successfully been employed to study the
light ring, timelike circular orbits, and thermodynamics of black holes,
leading to the discovery of even more intriguing properties Cunha2017 ;
Cunha2020azh ; Wei2022mzv ; Wei2022dzw .

In the recent study Cunha2017 , Cunha, Berti, and Herdeiro constructed a
characteristic vector field that relates to the topology of light ring (LR).
Through this construction, they discovered that the zero points of the vector
field correspond exactly to the LRs on the equatorial plane in stationary
axisymmetric spacetimes. Moreover, they established a universal relationship
between the stability of the LRs and the winding number of its zero points for
these ultracompact objects. The LRs always come in pairs with one of them
being stable. Subsequently, this topological method was extended to the
rotating black holes Cunha2020azh . In contrast to ultracompact objects,
stationary, axisymmetric, asymptotically flat black holes must have at least
one unstable LR outside the event horizon.

In addition to the LRs, the circular orbit of the massive particles is another
crucial aspect to consider. In Ref. Delgado2021jxd , the authors highlighted
the significance of the timelike circular orbits (TCOs). The results show that
there exists a close relation between the LRs and TCOs. For example, the
energy and angular momentum of the TCOs diverge if they coincide with the LRs.
Furthermore, the radial stability of TCOs can be effectively tested near the
LRs. Specifically, in the vicinity of a stable LR, TCOs with a slightly larger
radius exhibit radial instability. Conversely, in the vicinity of an unstable
LR, TCOs with a slightly smaller radius are stable. Importantly, no TCOs can
survive in the radial region between a stable LR and an unstable LR.

Given the relationship between LRs and TCOs, one might wonder whether the
topological approach can be extended to TCOs. However, unlike LRs, the
location of TCOs is closely dependent on the energy and angular momentum of
the massive particles within a given spacetime. This particular characteristic
makes the LR approach unsuitable for analyzing TCOs. Fortunately, we have
discovered a solution to this issue in Ref. Wei2022mzv . It is revealed that
if the angular momentum of the TCO is known, a vector can be defined in such a
way that its zero points precisely correspond to the locations of TCOs. This
discovery allows for the establishment of a well-defined and effective
topology to characterize TCOs. Our study reveals that the total topological
number of the TCOs for a generic stationary axisymmetric black hole is zero.
This observation implies that the stable and unstable TCOs either do not exist
or occur in pairs since they possess opposite topological numbers.
Significantly, when the angular momentum of the particles is selected as a
control parameter, the ISCO acts as a bifurcation point Fu2000pb . Even in
black hole solutions with the scalar hair and quasitopological terms, more
diverse topological configurations for TCOs are exhibited. Nevertheless, the
total topological number remains unchanged Ye2023gmk .

Topology also finds important applications in black hole thermodynamics,
particularly in the exploration of critical points and phase transitions.
Recent studies have revealed the existence of two distinct critical points,
indicating different phase structures of black holes. To differentiate between
these critical points, a topology is constructed in Ref. Wei2021vdx  by using
Duan’s topological current $\phi$-mapping theory Duan1984ws . By calculating
the winding number, the critical points can be classified as either novel or
conventional. Additionally, black hole systems can be categorized into
different topological classes based on the topological number. Furthermore,
these study has been extended to other black hole backgrounds, leading to the
discovery of even more intriguing topological properties Sadeghi2023dsg ;
Hazarika2023iwp ; Hazarika2024cpg . Soon afterwards, a fascinating new idea
was proposed. By utilizing the generalized off-shell free energy Wei2022dzw ,
black hole solutions can be treated as topological defects in the
thermodynamic parameter space. Each black hole is then associated with a
topological charge known as the winding number. Positive or negative winding
number indicates that the black hole is thermodynamical stable or unstable.
Moreover, by summing these winding numbers at a given temperature, one can
obtain a global topological number. From a thermodynamic perspective, black
hole solutions can be classified into different classes. For instance, the
four-dimensional Schwarzschild, charged, and charged AdS black holes have
total topological numbers of -1, 0, and 1, respectively. This suggests that
these black holes belong to different topological classes. Subsequently, this
topological approach has been extended to explore massive gravity, Lovelock
gravity, and other modified gravities Chen2023ddv ; Bai2022klw ; Hung2023ggz .

Recently, the dynamics of massive charged particles surrounding compact
objects has gained significant attention, particularly in the field of
relativistic astrophysics. The motion of these charged particles is directly
influenced by the gravitational properties of the sources, allowing us to
extract valuable information about the compact objects through the study of
the geometric and physical characteristics of their motion trajectories. In
Ref. Pugliese2011py , Pugliese, Quevedo, and Ruffini examined the circular
motion of charged test particles in Reissner-Nordström (RN) black holes. Their
results demonstrated that by analyzing the geometric structure of stable
accretion disks composed solely of charged particles moving along the circular
orbits, it is possible to clearly distinguish the black holes and naked
singularities. Furthermore, they provided a comprehensive classification of
circular orbits by varying the charge-to-mass ratios Pugliese2017xfa .
Notably, other studies have also explored the behavior of charged particles in
the black hole background with magnetic fields Wald1974np ; Grunau2010gd ;
Shaymatov2021qvt ; Garcia2013zud .

For the circular orbits, previous topological studies have mainly focused on
the photons and massive neutral particles, while leaving the case of charged
particles unexplored. However, if black holes possess electric charge, the
presence of Coulomb interaction presents the potential case for a balance
between gravity and electromagnetism. This leads to a distinct structure of
TCO. Another notable example is the violation of the chaos bound by the TCOs
of charged particles in RN black holes Zhao2018wkl ; Lei2021koj .
Consequently, one can expect the emergence of novel topological
configurations. Whether topological phase transitions occur is also a valuable
question to explore. In this paper, we focus on these issues and demonstrate
that when the Coulomb interaction dominates over gravity, the topological
number of the TCOs changes from 0 to -1, indicating an underlying topological
phase transition.

This paper is organized as follows. In Sec. II, we solve the radial effective
potential for charged particles in a generic spherically symmetric spacetime
with a radial electric field, and then examine the asymptotic behavior of the
vector field at the boundaries of the spacetime. The total topological number
is obtained. In Sec. III, we focus specifically on the RN black holes. Sec. IV
is devoted to the study of the topology of the TCOs, where four cases
characterized by distinct charge-to-mass ratios are considered. Finally, we
summarize and discuss our results in Sec. V. Throughout this paper, we adopt
the geometrized unit system where $c=\hbar=G=1$.

##  II Generic spherically symmetrical charged black hole

In this section, we would like to explore the geodesics of charged particles
and derive the effective potential for the radial motion, then construct the
corresponding vector of the timelike circular orbits. By examining the
behavior of the vector field at the boundaries, the topological number shall
be obtained for a spherically symmetrical spacetime.

###  II.1 Effective potential and timelike circular orbit

The Lagrangian of a charged test particle under the curved spacetime is
Chandrasekhar1985kt

$\mathcal{L}=\frac{1}{2}g_{\alpha\beta}\dot{x}^{\alpha}\dot{x}^{\beta}+qA_{\alpha}\dot{x}^{\alpha}.$
(1)

The parameter $q$ represents the charge per unit mass of the particle, which
can have either a positive or negative value. The dot denotes the derivative
with respect to the affine parameter.

Assuming that the electromagnetic potential has only the $t$ component and
takes the following form

$A_{\mu}=\left(\frac{Q}{r},\,0,\,0,\,0\right),$ (2)

under which, one can derive the radial electric field $E_{r}=-\nabla
A_{t}=Q/r^{2}$, where $Q$ represents the electric charge of the black hole
solution.

In general, the line element of a spherically symmetrical black hole spacetime
reads

$ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2},$ (3)

where metric function $f(r)$ only depends on the radial coordinate $r$.

By employing Eqs. (2) and (3), we can reformulate the Lagrangian (1) as

$\mathcal{L}=\frac{1}{2}\left(-f(r)\dot{t}^{2}+\frac{1}{f(r)}\dot{r}^{2}+r^{2}\dot{\theta}^{2}+r^{2}\sin^{2}\theta\dot{\phi^{2}}\right)+\frac{qQ}{r}\dot{t}.$
(4)

Due to the spherical symmetry of spacetime, two Killing vectors, namely
$\partial/\partial_{t}$ and $\partial/\partial_{\phi}$, correspond to the
conservation laws of energy $E$ and angular momentum $L$ for the charged
particles, serving as the conjugate momenta of the coordinates $t$ and $\phi$,
respectively

$\displaystyle
P_{t}:=\frac{\partial\mathcal{L}}{\partial\dot{t}}=\frac{qQ}{r}-f(r)\dot{t}=E,$
(5) $\displaystyle
P_{\phi}:=\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=r^{2}\sin^{2}\theta\dot{\phi}=L.$
(6)

Therefore, we can easily obtain $\dot{t}$ and $\dot{\phi}$

$\dot{t}=\frac{qQ-Er}{f(r)r},\qquad\dot{\phi}=\frac{L\csc^{2}\theta}{r^{2}}.$
(7)

For the timelike geodesics, we have $g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-1$
leading to

$f(r)\dot{t}^{2}-\frac{1}{f(r)}\dot{r}^{2}-r^{2}\dot{\theta}^{2}-r^{2}\sin^{2}\theta^{2}\dot{\phi}^{2}=1.$
(8)

Substituting Eq. (7) into (8), it is easy to obtain

$\frac{\dot{r}^{2}}{f(r)}+r^{2}\dot{\theta}^{2}+V_{eff}=0,$ (9)

where the effective potential reads

$V_{eff}:=1-\frac{(qQ-Er)^{2}}{f(r)r^{2}}+\frac{L^{2}\csc^{2}\theta}{r^{2}}.$
(10)

In particular, it can be further factored into

$V_{eff}=\frac{-1}{f(r)}(E-E_{1})(E-E_{2}),$ (11)

where

$\displaystyle
E_{1}=\frac{qQ}{r}+\frac{\sqrt{f(r)\csc^{2}(\theta)\left(2L^{2}-r^{2}\cos(2\theta)+r^{2}\right)}}{\sqrt{2}r},$
(12) $\displaystyle
E_{2}=\frac{qQ}{r}-\frac{\sqrt{f(r)\csc^{2}(\theta)\left(2L^{2}-r^{2}\cos(2\theta)+r^{2}\right)}}{\sqrt{2}r}.$
(13)

It should be noted that the conditions for a TCO determined by
$V_{\text{eff}}(r)=0$ and $V_{\text{eff}}^{\prime}(r)=0$, can be alternatively
expressed in terms of $E_{1}$ and $E_{2}$

$E=E_{1}\;\wedge\;E_{1}^{\prime}=0,\quad\text{or}\quad
E=E_{2}\;\wedge\;E_{2}^{\prime}=0.$ (14)

The prime denotes the derivative with respect to the radial coordinate $r$.

###  II.2 Energy and angular momentum

Solving $E_{1}^{\prime}=0$, we can obtain the angular momentum

$\displaystyle L_{1A}$
$\displaystyle=\pm\sqrt{\frac{2f(r)\left(r^{3}f^{\prime}(r)+q^{2}Q^{2}\right)-2\sqrt{q^{2}Q^{2}f(r)^{2}\left(-2r^{3}f^{\prime}(r)+4r^{2}f(r)+q^{2}Q^{2}\right)}-r^{4}f^{\prime}(r)^{2}}{\left(rf^{\prime}(r)-2f(r)\right)^{2}}},$
(15) $\displaystyle L_{1B}$
$\displaystyle=\pm\sqrt{\frac{2f(r)\left(r^{3}f^{\prime}(r)+q^{2}Q^{2}\right)+2\sqrt{q^{2}Q^{2}f(r)^{2}\left(-2r^{3}f^{\prime}(r)+4r^{2}f(r)+q^{2}Q^{2}\right)}-r^{4}f^{\prime}(r)^{2}}{\left(rf^{\prime}(r)-2f(r)\right)^{2}}}.$
(16)

The presence of the upper and lower signs $\pm$ in $L_{1A}$ and $L_{1B}$ is a
consequence of the spherical symmetry of the background spacetime. This
symmetry implies that the TCO remains unchanged under the reflection
transformation $L\to-L$.

Putting (15) and (16) back into (12), the corresponding energy are

$\displaystyle E_{1A}$
$\displaystyle=\frac{qQ}{r}+\frac{f(r)}{r}\sqrt{\frac{4r^{2}f(r)-2\sqrt{q^{2}Q^{2}\left(-2r^{3}f^{\prime}(r)+4r^{2}f(r)+q^{2}Q^{2}\right)}-2r^{3}f^{\prime}(r)+2q^{2}Q^{2}}{\left(rf^{\prime}(r)-2f(r)\right)^{2}}},$
(17) $\displaystyle E_{1B}$
$\displaystyle=\frac{qQ}{r}+\frac{f(r)}{r}\sqrt{\frac{4r^{2}f(r)+2\sqrt{q^{2}Q^{2}\left(-2r^{3}f^{\prime}(r)+4r^{2}f(r)+q^{2}Q^{2}\right)}-2r^{3}f^{\prime}(r)+2q^{2}Q^{2}}{\left(rf^{\prime}(r)-2f(r)\right)^{2}}}.$
(18)

Adopting the similar calculation, we obtain the angular momentum from
$E_{2}^{\prime}=0$. However, the results are identical with Eqs. (15) and (16)

$L_{2A}=L_{1A},\qquad L_{2B}=L_{1B}.$ (19)

Substituting $L_{2A}$ and $L_{2B}$ back into (13), the distinct TCO energy
takes the form

$\displaystyle E_{2A}$
$\displaystyle=\frac{qQ}{r}-\frac{f(r)}{r}\sqrt{\frac{4r^{2}f(r)-2\sqrt{q^{2}Q^{2}\left(-2r^{3}f^{\prime}(r)+4r^{2}f(r)+q^{2}Q^{2}\right)}-2r^{3}f^{\prime}(r)+2q^{2}Q^{2}}{\left(rf^{\prime}(r)-2f(r)\right)^{2}}},$
(20) $\displaystyle E_{2B}$
$\displaystyle=\frac{qQ}{r}-\frac{f(r)}{r}\sqrt{\frac{4r^{2}f(r)+2\sqrt{q^{2}Q^{2}\left(-2r^{3}f^{\prime}(r)+4r^{2}f(r)+q^{2}Q^{2}\right)}-2r^{3}f^{\prime}(r)+2q^{2}Q^{2}}{\left(rf^{\prime}(r)-2f(r)\right)^{2}}}.$
(21)

As a result, within this static spherically symmetric spacetime characterized
by a radial electric field, there exist at most four distinct combinations of
energy and angular momentum for the massive charged test particles: ($E_{1A}$,
$L_{1A}$), ($E_{1B}$, $L_{1B}$), ($E_{2A}$, $L_{2A}$), and ($E_{2B}$,
$L_{2B}$). This situation is more complicated compared to the scenario for the
neutral particles, where only a single solution is present, as shown in Ref.
Ye2023gmk . However, it is important to note that the four solution sets are
not uniformly well behaved for a specific black hole background. In certain
regions of the parameter space, they are valid solutions, while in other
regions, they are not. This gives significant challenges to our investigation.
To address this issue, it becomes necessary to reformulate the solution.
Fortunately, the solutions corresponding to the four sets can be expressed in
the following form:

$\displaystyle
E_{t1}=-\frac{\sqrt{f(r)^{2}\left(-2r^{3}f^{\prime}(r)+4r^{2}f(r)+q^{2}Q^{2}\right)}+qQ\left(f(r)-rf^{\prime}(r)\right)}{r^{2}f^{\prime}(r)-2rf(r)},$
(22) $\displaystyle
L_{t1}=\sqrt{\frac{2f(r)\left(r^{3}f^{\prime}(r)+q^{2}Q^{2}\right)-r^{4}f^{\prime}(r)^{2}-2qQf(r)\sqrt{-2r^{3}f^{\prime}(r)+4r^{2}f(r)+q^{2}Q^{2}}}{\left(2f(r)-rf^{\prime}(r)\right)^{2}}};$
(23) $\displaystyle
E_{t2}=\frac{\sqrt{f(r)^{2}\left(-2r^{3}f^{\prime}(r)+4r^{2}f(r)+q^{2}Q^{2}\right)}+qQ\left(rf^{\prime}(r)-f(r)\right)}{r^{2}f^{\prime}(r)-2rf(r)},$
(24) $\displaystyle
L_{t2}=\sqrt{\frac{2f(r)\left(r^{3}f^{\prime}(r)+q^{2}Q^{2}\right)-r^{4}f^{\prime}(r)^{2}+2qQf(r)\sqrt{-2r^{3}f^{\prime}(r)+4r^{2}f(r)+q^{2}Q^{2}}}{\left(2f(r)-rf^{\prime}(r)\right)^{2}}};$
(25) $\displaystyle
E_{t3}=-\frac{\sqrt{f(r)^{2}\left(-2r^{3}f^{\prime}(r)+4r^{2}f(r)+q^{2}Q^{2}\right)}+qQ\left(f(r)-rf^{\prime}(r)\right)}{r^{2}f^{\prime}(r)-2rf(r)},$
(26) $\displaystyle
L_{t3}=-\sqrt{\frac{2f(r)\left(r^{3}f^{\prime}(r)+q^{2}Q^{2}\right)-r^{4}f^{\prime}(r)^{2}-2qQf(r)\sqrt{-2r^{3}f^{\prime}(r)+4r^{2}f(r)+q^{2}Q^{2}}}{\left(2f(r)-rf^{\prime}(r)\right)^{2}}};$
(27) $\displaystyle
E_{t4}=\frac{\sqrt{f(r)^{2}\left(-2r^{3}f^{\prime}(r)+4r^{2}f(r)+q^{2}Q^{2}\right)}+qQ\left(rf^{\prime}(r)-f(r)\right)}{r^{2}f^{\prime}(r)-2rf(r)},$
(28) $\displaystyle
L_{t4}=-\sqrt{\frac{2f(r)\left(r^{3}f^{\prime}(r)+q^{2}Q^{2}\right)-r^{4}f^{\prime}(r)^{2}+2qQf(r)\sqrt{-2r^{3}f^{\prime}(r)+4r^{2}f(r)+q^{2}Q^{2}}}{\left(2f(r)-rf^{\prime}(r)\right)^{2}}}.$
(29)

Obviously, we find the following relation

$\displaystyle E_{t1}=E_{t3}\quad\text{and}\quad L_{t1}=-L_{t3},$ (30)
$\displaystyle E_{t2}=E_{t4}\quad\text{and}\quad L_{t2}=-L_{t4}.$ (31)

Considering the spherical symmetry of the spacetime background, it suffices to
analyze the solutions for the TCOs with positive angular momentum.
Consequently, we obtain two distinct sets of solutions: $E_{t1}$ and $L_{t1}$,
as well as $E_{t2}$ and $L_{t2}$.

Subsequently, we turn our attention to the ISCO as it is closely related to
the stability of the TCOs. In general, if the radius of the TCO is greater
than $r_{\text{ISCO}}$, the TCO is radially stable, otherwise, it is unstable.
For a generalization, we would like to refer to Ref. Delgado2021jxd  if MSCO
appears. The ISCO is usually determined by

$V_{eff}(r)\big{|}_{r=r_{ISCO}}=0\;\;\wedge\;\;V_{eff}^{\prime}(r)\big{|}_{r=r_{ISCO}}=0\;\;\wedge\;\;V_{eff}^{\prime\prime}(r)\big{|}_{r=r_{ISCO}}=0.$
(32)

Solving the third condition (32), we have

$\begin{split}V_{eff}^{\prime\prime}(r)=&\frac{-2rf^{\prime}(r)^{2}(qQ-
Er)^{2}+f(r)(qQ-Er)\left(rf^{\prime\prime}(r)(qQ-
Er)-4qQf^{\prime}(r)\right)}{r^{3}f(r)^{3}}\\\
&+\frac{6L^{2}f(r)+2qQ(2Er-3qQ)}{r^{4}f(r)}=0.\end{split}$ (33)

By substituting $E_{t1}$ and $L_{t1}$, as well as $E_{t2}$ and $L_{t2}$, into
(33), we derive two equations associated with the ISCO

$\displaystyle
f(r)^{2}\left(2r^{4}f^{\prime\prime}(r)+6r^{3}f^{\prime}(r)-2q^{2}Q^{2}\right)-qQf(r)\left(r^{2}f^{\prime\prime}(r)-2rf^{\prime}(r)+2f(r)\right)\sqrt{-2r^{3}f^{\prime}(r)+4r^{2}f(r)+q^{2}Q^{2}}$
$\displaystyle\quad+r^{2}f^{\prime}(r)^{2}\left(2r^{3}f^{\prime}(r)-q^{2}Q^{2}\right)+rf(r)\left(q^{2}Q^{2}rf^{\prime\prime}(r)-7r^{3}f^{\prime}(r)^{2}+f^{\prime}(r)\left(2q^{2}Q^{2}-r^{4}f^{\prime\prime}(r)\right)\right)=0,$
(34) $\displaystyle
f(r)^{2}\left(2r^{4}f^{\prime\prime}(r)+6r^{3}f^{\prime}(r)-2q^{2}Q^{2}\right)+qQf(r)\left(r^{2}f^{\prime\prime}(r)-2rf^{\prime}(r)+2f(r)\right)\sqrt{-2r^{3}f^{\prime}(r)+4r^{2}f(r)+q^{2}Q^{2}}$
$\displaystyle\quad+r^{2}f^{\prime}(r)^{2}\left(2r^{3}f^{\prime}(r)-q^{2}Q^{2}\right)+rf(r)\left(q^{2}Q^{2}rf^{\prime\prime}(r)-7r^{3}f^{\prime}(r)^{2}+f^{\prime}(r)\left(2q^{2}Q^{2}-r^{4}f^{\prime\prime}(r)\right)\right)=0.$
(35)

The above equations indicate that the ISCO relies on the black hole charge
$Q$, the charge-to-mass ratio $q$ of the timelike particle, and the metric
function $f(r)$. If a specific $f(r)$ is given, we shall obtain the radius of
the ISCO in terms of $Q$, $q$, and other black hole parameters. Henceforth, in
the subsequent discussion, we will refer the solutions to Eqs. (II.2) and
(II.2) as $r_{\text{ISCO}}(E_{t1},L_{t1})$ and
$r_{\text{ISCO}}(E_{t2},L_{t2})$, respectively.

###  II.3 Vector and topological number

In this subsection, our aim is to construct two vectors
$\vec{\phi}_{1}=(\phi^{r}_{1},\phi^{\theta}_{1})$ and
$\vec{\phi}_{2}=(\phi^{r}_{2},\phi^{\theta}_{2})$ separately using $E_{1}$ and
$E_{2}$, such that their zero points exactly correspond to the TCOs. Treating
these zero points as topological defects, we can assign a local topological
charge to each of them, known as the winding number. By summing up all these
topological charges, we obtain a topological number that characterizes the
TCOs. This topological number reveals the global characteristic of the
spacetime under consideration.

At first, we examine the scenario where the vector field is constructed using
$E_{1}$. Then, the radial and angular components of vector $\vec{\phi}_{1}$
are defined as Wei2022mzv

$\vec{\phi}_{1}=\left(\phi^{r}_{1},\;\phi^{\theta}_{1}\right):=\left(\frac{\partial_{r}E_{1}}{\sqrt{g_{rr}}},\;\frac{\partial_{\theta}E_{1}}{\sqrt{g_{\theta\theta}}}\right),$
(36)

where $g_{rr}=\frac{1}{f(r)}$ and $g_{\theta\theta}=r^{2}$.

Employing with Eq. (12), the explicit expressions of $\phi^{r}_{1}$ and
$\phi^{\theta}_{1}$ are given by

$\displaystyle\phi^{r}_{1}=$
$\displaystyle\;\frac{\sqrt{2}\csc(\theta)f^{\prime}(r)\sqrt{2L^{2}-r^{2}\cos(2\theta)+r^{2}}}{4r}-\frac{qQ\sqrt{f(r)}}{r^{2}}-\frac{\sqrt{2}L^{2}f(r)\csc(\theta)}{r^{2}\sqrt{2L^{2}-r^{2}\cos(2\theta)+r^{2}}},$
(37) $\displaystyle\phi^{\theta}_{1}=$
$\displaystyle\;-\frac{\sqrt{2f(r)}L^{2}\cot(\theta)\csc(\theta)}{r^{2}\sqrt{2L^{2}-r^{2}\cos(2\theta)+r^{2}}}.$
(38)

Furthermore, when combined with the conditions (14), it becomes evident that
the zero points of the vector, i.e., $\phi^{r}_{1}=\phi^{\theta}_{1}=0$, are
located at $r=r_{\text{tco}}$ and $\theta=\pi/2$.

Next, we aim to investigate the boundary behaviors of the vector
$\vec{\phi}_{1}=(\phi^{r}_{1},\phi^{\theta}_{1})$ and analyze the topological
charge in the context of this generic spherically symmetric spacetime. The
considered spacetime boundary is denoted as $\partial\Sigma$, which
encompasses the region $\Sigma$

$\Sigma=\left\\{I_{r}\times
I_{\theta}\subset\mathbb{R}^{2}\bigg{|}I_{r}=\left[r_{h},\;\infty\right)\subset\mathbb{R},I_{\theta}=\left[0,\;\pi\right]\subset\mathbb{R}\right\\}.$
(39)

Therefore, the boundary $\partial\Sigma$ consists of four segments, the event
horizon $r=r_{h}$, spatial infinity $r=\infty$, the polar axis at $\theta=0$
and $\pi$.

i) Asymptotic limit ($r\to\infty$)

We assume that spacetime is asymptotical Schwarzschild case, which gives

$f(r)=1-\frac{2M}{r}+\mathcal{O}\left(r^{-2}\right).$ (40)

as $r\to\infty$. Here $M$ measures the black hole mass and is always positive.

By performing Taylor expansion of $\vec{\phi}_{1}$ near $r=\infty$, we obtain
the following results

$\displaystyle\phi^{r}_{1}=\frac{M-qQ}{r^{2}}+\mathcal{O}\left(r^{-3}\right),$
(41)
$\displaystyle\phi^{\theta}_{1}=-\frac{L^{2}\cot(\theta)\csc^{2}(\theta)}{r^{3}}+\mathcal{O}\left(r^{-4}\right).$
(42)

The argument of the vector reads

$\arctan\left(\frac{\phi^{\theta}_{1}}{\phi^{r}_{1}}\right)=\arctan\left(-\frac{L^{2}\cot(\theta)\csc^{2}(\theta)}{r(M-qQ)}\right)\bigg{|}_{r\to\infty}=0.$
(43)

This observation suggests that $\phi^{\theta}_{1}$ is significantly smaller
than $\phi^{r}_{1}$, indicating that the vector field $\vec{\phi}_{1}$
predominantly aligns horizontally along the $r$-direction. Additionally, we
observe that the sign of $\phi^{r}_{1}$ depends on the difference between
$M-qQ$. Consequently, two distinct scenarios arise concerning the direction of
the vector field

$\begin{cases}qQ<0\;\;\text{or}\;\;0<qQ<M,&\vec{\phi}_{1}\;\;\text{points to
the right};\\\ 0<M<qQ,&\vec{\phi}_{1}\;\;\text{points to the
left}.\end{cases}$ (44)

ii) Horizon limit ($r\to r_{h}$)

Near the event horizon $r=r_{h}$, the metric function $f(r)$ can be expanded
as

$f(r)=f\left(r_{h}\right)+\left(r-r_{h}\right)f^{\prime}\left(r_{h}\right)+\frac{1}{2}\left(r-r_{h}\right){}^{2}f^{\prime\prime}\left(r_{h}\right)+\mathcal{O}\left(\left(r-r_{h}\right){}^{3}\right).$
(45)

Note that $f(r_{h})=0$ and $f^{\prime}(r_{h})>0$. By making use of it, we have

$\displaystyle\phi^{r}_{1}=\frac{\csc(\theta)f^{\prime}(r_{h})\sqrt{r_{h}^{2}+2L^{2}-r_{h}^{2}\cos(2\theta)}}{2\sqrt{2}r_{h}}+\mathcal{O}\left(\left(r-r_{h}\right)\right),$
(46)
$\displaystyle\phi^{\theta}_{1}=-\frac{L^{2}\sqrt{2f(r_{h})}\cot(\theta)\csc(\theta)}{r_{h}^{2}\sqrt{r_{h}^{2}+2L^{2}-r_{h}^{2}\cos(2\theta)}}+\mathcal{O}\left(\left(r-r_{h}\right)\right).$
(47)

Since the angle between $\phi^{\theta}_{1}$ and $\phi^{r}_{1}$ is determined
by the ratio $\phi^{\theta}_{1}/\phi^{r}_{1}$, we just focus on calculating
this ratio as follows

$\frac{\phi^{\theta}_{1}}{\phi^{r}_{1}}=\frac{4L^{2}\cot(\theta)\sqrt{f(r_{h})}}{r_{h}f^{\prime}\left(r_{h}\right)\left(r_{h}^{2}\cos(2\theta)-r_{h}^{2}-2L^{2}\right)}\bigg{|}_{f(r_{h})\to
0}=0.$ (48)

Hence, considering that $\phi^{\theta}_{1}$ is significantly smaller than
$\phi^{r}_{1}$ and
$\phi^{r}_{1}\propto\sqrt{r_{h}^{2}\left(1-\cos(2\theta)\right)+2L^{2}}>0$, we
can conclude that the vector field $\vec{\phi}_{1}$ on the horizon $r_{h}$
points to the right.

iii) Axial limit ($\theta\to 0$ and $\pi$)

Near $\theta\to 0$, $\phi^{r}_{1}$ and $\phi^{\theta}_{1}$ can be expanded as

$\displaystyle\phi^{r}_{1}=\frac{|L|\left(rf^{\prime}(r)-2f(r)\right)}{2\theta
r^{2}}+\mathcal{O}\left(\theta^{0}\right),$ (49)
$\displaystyle\phi^{\theta}_{1}=-\frac{|L|\sqrt{f(r)}}{\theta^{2}r^{2}}+\mathcal{O}\left(\theta^{0}\right).$
(50)

As a result, we have

$\frac{\phi^{\theta}_{1}}{\phi^{r}_{1}}=-\frac{2\sqrt{f(r)}}{\theta\left(rf^{\prime}(r)-2f(r)\right)}\bigg{|}_{\theta\to
0}=\infty.$ (51)

Combining with $\phi^{\theta}_{1}\gg\phi^{r}_{1}$ and $\phi^{\theta}_{1}<0$,
we conclude that the direction of vector field is downward at $\theta=0$.

Performing the similar calculation, we obtain

$\displaystyle\phi^{r}_{1}=-\frac{|L|\left(rf^{\prime}(r)-2f(r)\right)}{2(\theta-\pi)r^{2}}+\mathcal{O}\left((\theta-\pi)^{0}\right),$
(52)
$\displaystyle\phi^{\theta}_{1}=\frac{|L|\sqrt{f(r)}}{(\theta-\pi)^{2}r^{2}}+\mathcal{O}\left((\theta-\pi)^{0}\right).$
(53)

near $\theta=\pi$. The corresponding ratio reads

$\frac{\phi^{\theta}_{1}}{\phi^{r}_{1}}=-\frac{2\sqrt{f(r)}}{(\theta-\pi)\left(rf^{\prime}(r)-2f(r)\right)}\bigg{|}_{\theta\to\pi}=\infty.$
(54)

Further considering $\phi^{\theta}_{1}>0$, the direction of vector field is
upward at $\theta=\pi$.

In summary, at $r=r_{h}$, $\theta=0$, and $\theta=\pi$, the vector
$\vec{\phi}_{1}=(\phi^{r}_{1},\phi^{\theta}_{1})$ points rightward, downward,
and upward, respectively. However, the direction of the vector at $r=\infty$
is not unique and depends on the sign of $M-qQ$ as indicated in Eq. (44).
Specifically, if $M-qQ$ is positive, the vector points to the right, resulting
in a global topological number (winding number) of $W=0$. Instead, if
$M-qQ<0$, the vector points to the left, yielding a topological number $W=-1$.
A schematic diagram illustrating the vector constructed from $E_{1}$ at the
boundary $\partial\Sigma$ is depicted in Fig. 1. It is worth pointing out that
the topological number $W$ counts the number of the loops that the vector
makes in the vector space when the one moves along the closed boundary
$\partial\Sigma$ in a counterclockwise direction.

Figure 1: The boundary behaviors of vector field
$\vec{\phi}_{1}=(\phi^{r}_{1},\phi^{\theta}_{1})$ constructed from $E_{1}$ on
($r,\,\theta$) plane. The purple arrows represent the direction of vector
field, which are uniquely determined at $r\to r_{h}$ and $\theta\to 0$ or
$\pi$. However the situation with $r\to\infty$ depends on the sign of $M-qQ$.
If $M-qQ>0$, the arrow direction is to the right and the topological number
$W=0$. Otherwise, the arrow points to left and $W=-1$. The full boundaries
include four parts $I_{1}$, $I_{2}$, $I_{3}$, and $I_{4}$ corresponding to
$r=\infty$, $\theta=\pi$, $r=r_{h}$, and $\theta=0$, respectively.

It is important to note that our previous discussions focus only on the vector
derived from $E_{1}$. However, considering the expression (13), the energy
$E_{2}$ may also take positive values under some suitable parameter
conditions. Consequently, it becomes necessary to investigate the vector field
$\vec{\phi}_{2}=(\phi^{r}_{2},\phi^{\theta}_{2})$ constructed by $E_{2}$.
Similarly, we have the following $r$\- and $\theta$-components of
$\vec{\phi}_{2}$

$\displaystyle\phi^{r}_{2}=$
$\displaystyle\;\frac{\partial_{r}E_{2}}{\sqrt{g_{rr}}}=\frac{-\sqrt{2}\csc(\theta)f^{\prime}(r)\sqrt{2L^{2}-r^{2}\cos(2\theta)+r^{2}}}{4r}-\frac{qQ\sqrt{f(r)}}{r^{2}}+\frac{\sqrt{2}L^{2}f(r)\csc(\theta)}{r^{2}\sqrt{2L^{2}-r^{2}\cos(2\theta)+r^{2}}},$
(55) $\displaystyle\phi^{\theta}_{2}=$
$\displaystyle\;\frac{\partial_{\theta}E_{2}}{\sqrt{g_{\theta\theta}}}=\frac{\sqrt{2f(r)}L^{2}\cot(\theta)\csc(\theta)}{r^{2}\sqrt{2L^{2}-r^{2}\cos(2\theta)+r^{2}}}.$
(56)

Performing the similar treatment, we can obtain the boundary behavior of the
vector $\vec{\phi}_{2}=(\phi^{r}_{2},\phi^{\theta}_{2})$. The detailed results
show that the direction of the vector $\vec{\phi}_{2}$ exhibits an opposite
direction at $r=r_{h}$, $\theta=0$, and $\theta=\pi$ when one compares to the
$E_{1}$ case. This is mainly caused by the additional minus sign of
$\phi^{\theta}_{2}$, see Eq. (56). On the other hand, the direction of the
vector at $r=\infty$ depends on the sign of $-M-qQ$ instead, which is closely
related to the asymptotic behavior of $E_{2}$ at infinity. The corresponding
schematic diagram of the vector
$\vec{\phi}_{2}=(\phi^{r}_{2},\phi^{\theta}_{2})$ is illustrated in Fig. 2.

Figure 2: The boundary behaviors of vector field
$\vec{\phi}_{2}=(\phi^{r}_{2},\;\phi^{\theta}_{2})$ constructed from $E_{2}$
on ($r,\,\theta$) plane. The orange arrows represent the direction of vector
field, which are uniquely determined at $r\to r_{h}$ and $\theta\to 0$ or
$\pi$. However, the situation with $r\to\infty$ depends on the sign of
$-M-qQ$. If $-M-qQ>0$, the arrow direction is to the right and topological
number $W=-1$. Otherwise, the arrow points to left and $W=0$. The full
boundary includes four parts $I_{1}$, $I_{2}$, $I_{3}$, and $I_{4}$
corresponding to $r=\infty$, $\theta=\pi$, $r=r_{h}$, and $\theta=0$,
respectively.

To provide a clearer understanding of the relationship between the sign of
$M-qQ$ ($-M-qQ$) for $E_{1}$ ($E_{2}$) and the global topological number $W$
of the vector $\vec{\phi}_{1}$ ($\vec{\phi}_{2}$), we summarize the key
findings in Table 1. In this table, we categorize the charge-to-mass ratio and
the black hole charge, $qQ$, into four distinct cases:

$\displaystyle\text{(1) unlike strong charge:}\;qQ<-M<0;$
$\displaystyle\text{(2) unlike weak charge:}\;-M<qQ<0;$
$\displaystyle\text{(3) like weak charge:}\;0<qQ<M;$ $\displaystyle\text{(4)
like strong charge:}\;0<M<qQ.$ (57)

It is worth emphasizing that these four regimes correspond to distinct
configurations of the TCO related to energy or angular momentum, as will be
further discussed in Section IV.

Table 1: The sign of the radial component of $\vec{\phi}_{1}$ and $\vec{\phi}_{2}$ at the spatial infinity and the corresponding topological number $W$ within the distinct $qQ$ regimes. Here $\pm$ denote $\phi^{r}>0$ or $\phi^{r}<0$, and the number 0 and -1 are the topological number. $E_{1}:\mathrm{Sign}(M-qQ)$ signifies that $\phi^{r}_{1}$ constructed from $E_{1}$ and its sign relies on $M-qQ$. $E_{2}:\mathrm{Sign}(-M-qQ)$ has analogous meaning. $W_{tot}$ is defined as the sum of $W_{1}$ and $W_{2}$ for $E_{1}$ and $E_{2}$. | $qQ<-M<0$ | $-M<qQ<0$ | $0<qQ<M$ | $0<M<qQ$
---|---|---|---|---
$E_{1}:\mathrm{Sign}(M-qQ)$ | + | + | + | -
$W_{1}$ | 0 | 0 | 0 | -1
$E_{2}:\mathrm{Sign}(-M-qQ)$ | + | - | - | -
$W_{2}$ | -1 | 0 | 0 | 0
$W_{tot}:=W_{1}+W_{2}$ | -1 | 0 | 0 | -1

In Table 1, the influence of strong and weak charges on the topological number
is clearly illustrated by combining the results obtained from $E_{1}$ and
$E_{2}$. We denote $W_{tot}$ as total topological number, and $W_{1}$ or
$W_{2}$ are responsible for vector field $\vec{\phi}_{1}$ and $\vec{\phi}_{2}$
constructed from $E_{1}$ or $E_{2}$, respectively. It is evident that in the
case of weak charges ($0<|qQ|<M$), the topological number vanishes regardless
of whether the charges of the particle and black hole are like or unlike. This
result is consistent with the findings for the uncharged particles discussed
in previous reference Ye2023gmk , strongly indicating that the TCOs always
come in pairs. However, in the case of strong charges ($M<|qQ|$), the
topological number changes to -1, suggesting the existence of one more
unstable TCO to the stable ones. This novel pattern in the topology of the TCO
is noteworthy.

Now, we will provide a brief introduction to the concept of topological charge
and present some essential conclusions. For a detailed, we would like to refer
readers to Ref. Duan1984ws . In principle, to determine the topological charge
$W$ associated with a particular zero point of the vector, we need to perform
the integral over a considered parameter region $\Sigma$ that only contains
that zero point

$\displaystyle
W=\int_{\Sigma}j^{0}\;d^{2}x=\frac{1}{2\pi}\oint_{\partial\Sigma}\epsilon_{ab}\;n^{a}\;dn^{b},$
(58)

where $j^{0}$ is the zero-component of the topological current defined as
Duan1984ws

$j^{\mu}:=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}\frac{\partial
n^{a}}{\partial x^{\nu}}\frac{\partial n^{b}}{\partial x^{\rho}}.$ (59)

Here $x^{\mu}=(t,\,r,\,\theta)$ and the Latin indexes $a$, $b$ take values $r$
and $\theta$. The corresponding unit vector is
$\vec{n}=(\phi^{r},\phi^{\theta})/\sqrt{(\phi^{r})^{2}+(\phi^{\theta})^{2}}$.
In particular, the topological current is conserved.

In this study, we encountered three distinct types of topological
configurations near the zero points. We summarize several key conclusions
below:

  1. $\clubsuit$

$W=0$. The vector flows outward from its zero point in the $\theta$-direction
and maintains a consistent orientation (either pointing to the left or right)
along the $r$-direction. This scenario corresponds to either the absence of
the TCOs or the presence of the ISCOs acting as the bifurcation points.

  2. $\spadesuit$

$W=+1$. The vector flows outward from the zero point in both the
$\theta$-direction and the $r$-direction. This configuration of the zero point
signifies the presence of the stable TCOs.

  3. $\blacklozenge$

$W=-1$. The vector flows outward from the zero point in $\theta$-direction,
and inward at the $r$-direction. This configuration signifies the unstable
TCOs.

##  III Reissner-Nordström black hole

In the previous section, we have studied the topology of the TCOs for the
charged particles within a spherically symmetrical black hole spacetime
characterized by the electromagnetic potential (2). In the following sections,
we will apply the general method outlined in Section II to a specific
background, namely the four-dimensional Reissner-Nordström (RN) black hole,
which represents a static spherically symmetrical solution of Einstein field
equation coupling with the Maxwell field minimally.

The line element of the RN black hole can also be described by (3), and the
metric function is given by

$f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}.$ (60)

The electromagnetic potential $A_{\mu}$ of Maxwell field takes the same form
with Eq. (2).

A brief calculation shows that the effective potential governing the motion of
the charged test particles reads

$V_{eff}=\frac{-1}{1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}}(E-E_{1})(E-E_{2}),$
(61)

where

$\displaystyle
E_{1}=\frac{2qQr+\sqrt{2}\csc(\theta)\sqrt{\left(r^{2}-2rM+Q^{2}\right)\left(2L^{2}-r^{2}\cos(2\theta)+r^{2}\right)}}{2r^{2}},$
(62) $\displaystyle
E_{2}=\frac{2qQr-\sqrt{2}\csc(\theta)\sqrt{\left(r^{2}-2rM+Q^{2}\right)\left(2L^{2}-r^{2}\cos(2\theta)+r^{2}\right)}}{2r^{2}}.$
(63)

Via Eqs. (37) and (38), the radial and angular components of the vector
$\vec{\phi}_{1}$ associated with $E_{1}$ are in the following form

$\displaystyle\phi^{r}_{1}=$
$\displaystyle\frac{\csc(\theta)\left(2L^{2}\left(3Mr-2Q^{2}-r^{2}\right)+r^{2}\left(1-\cos(2\theta)\right)\left(Mr-Q^{2}\right)\right)}{\sqrt{2}r^{4}\sqrt{2L^{2}-r^{2}\cos(2\theta)+r^{2}}}-\frac{qQ\sqrt{r(r-2M)+Q^{2}}}{r^{3}},$
(64) $\displaystyle\phi^{\theta}_{1}=$
$\displaystyle-\frac{\sqrt{2}L^{2}\cot(\theta)\csc(\theta)\sqrt{r^{2}-2Mr+Q^{2}}}{r^{3}\sqrt{2L^{2}-r^{2}\cos(2\theta)+r^{2}}}.$
(65)

Using the similar manner, the $\vec{\phi}_{2}$ relating to $E_{2}$ can also be
obtained from the (55) and (56)

$\displaystyle\phi^{r}_{2}=$
$\displaystyle\frac{-\csc(\theta)\left(2L^{2}\left(3Mr-2Q^{2}-r^{2}\right)+r^{2}(1-\cos(2\theta))(Mr-Q^{2})\right)}{\sqrt{2}r^{4}\sqrt{2L^{2}-r^{2}\cos(2\theta)+r^{2}}}-\frac{qQ\sqrt{r(r-2M)+Q^{2}}}{r^{3}},$
(66) $\displaystyle\phi^{\theta}_{2}=$
$\displaystyle\frac{\sqrt{2}L^{2}\cot(\theta)\csc(\theta)\sqrt{r^{2}-2rM+Q^{2}}}{r^{3}\sqrt{2L^{2}-r^{2}\cos(2\theta)+r^{2}}}.$
(67)

The zero points of the vector $\vec{\phi}_{1}$ and $\vec{\phi}_{2}$ correspond
to the locations of the TCOs. By solving these zero points, we can easily
determine the energy and angular momentum of the TCOs

$\displaystyle
E_{t1}=\frac{qQr(r-4M)+3qQ^{3}+\left(r^{2}-2rM+Q^{2}\right)\sqrt{4r(r-3M)+\left(q^{2}+8\right)Q^{2}}}{2r\left(r^{2}-3rM+2Q^{2}\right)},$
(68) $\displaystyle
L_{t1}=r\sqrt{\tfrac{Q^{2}r\left(\left(q^{2}-2\right)r-2M\left(q^{2}-5\right)\right)-qQ\left(r^{2}-2rM+Q^{2}\right)\sqrt{4r(r-3M)+\left(q^{2}+8\right)Q^{2}}+2Mr^{2}(r-3M)+\left(q^{2}-4\right)Q^{4}}{2\left(r^{2}-3rM+2Q^{2}\right)^{2}}};$
(69) $\displaystyle
E_{t2}=\frac{qQr(r-4M)+3qQ^{3}-\left(r^{2}-2rM+Q^{2}\right)\sqrt{4r(r-3M)+\left(q^{2}+8\right)Q^{2}}}{2r\left(r^{2}-3rM+2Q^{2}\right)},$
(70) $\displaystyle
L_{t2}=r\sqrt{\tfrac{Q^{2}r\left(\left(q^{2}-2\right)r-2M\left(q^{2}-5\right)\right)+qQ\left(r^{2}-2rM+Q^{2}\right)\sqrt{4r(r-3M)+\left(q^{2}+8\right)Q^{2}}+2Mr^{2}(r-3M)+\left(q^{2}-4\right)Q^{4}}{2\left(r^{2}-3rM+2Q^{2}\right)^{2}}}.$
(71)

From above obtained expressions for the energy and angular momentum, we have
three specific radii as follows

$\displaystyle r_{L}=\frac{3M+\sqrt{9M^{2}-8Q^{2}-q^{2}Q^{2}}}{2},$ (72)
$\displaystyle r_{P}=\frac{3M+\sqrt{9M^{2}-8Q^{2}}}{2},$ (73) $\displaystyle
r_{S}=\frac{MQ^{2}(q^{2}-1)+Q^{2}\sqrt{q^{2}(q^{2}-1)(M^{2}-Q^{2})}}{q^{2}Q^{2}-M^{2}}.$
(74)

For $r<r_{L}$, both energies $E_{t1}$ and $E_{t2}$ become imaginary.
Therefore, we only restrict our attention to $r>r_{L}$. The radius $r_{P}$
corresponds to the case where $r^{2}-3rM+2Q^{2}=0$, resulting in divergences
of $E_{t1}$, $L_{t1}$, $E_{t2}$, and $L_{t2}$. On the other hand, the radius
$r_{S}$ indicates situations where either $L_{t1}=0$ or $L_{t2}=0$, signifying
the absence of angular momentum.

Using (II.2) and (II.2), the conditions determined the radius of the ISCO,
i.e., $r_{ISCO}(E_{t1},L_{t1})$ and $r_{ISCO}(E_{t2},L_{t2})$, reduce to

$\displaystyle
Q^{2}r^{2}\left(6M^{2}\left(q^{2}+13\right)-2M\left(2q^{2}+11\right)r+q^{2}r^{2}\right)+Q^{4}r\left(\left(q^{2}+8\right)r-6M\left(q^{2}+10\right)\right)-2Mr^{3}\left(18M^{2}-9Mr+r^{2}\right)$
$\displaystyle+qQ\left(r^{2}\left(12M^{2}-8Mr+r^{2}\right)+Q^{2}r(7r-18M)+6Q^{4}\right)\sqrt{4r(r-3M)+\left(q^{2}+8\right)Q^{2}}+2\left(q^{2}+8\right)Q^{6}=0,$
(75) $\displaystyle
Q^{2}r^{2}\left(6M^{2}\left(q^{2}+13\right)-2M\left(2q^{2}+11\right)r+q^{2}r^{2}\right)+Q^{4}r\left(\left(q^{2}+8\right)r-6M\left(q^{2}+10\right)\right)-2Mr^{3}\left(18M^{2}-9Mr+r^{2}\right)$
$\displaystyle-
qQ\left(r^{2}\left(12M^{2}-8Mr+r^{2}\right)+Q^{2}r(7r-18M)+6Q^{4}\right)\sqrt{4r(r-3M)+\left(q^{2}+8\right)Q^{2}}+2\left(q^{2}+8\right)Q^{6}=0.$
(76)

These two equations demonstrate that the ISCO is determined by the black hole
mass $M$ and charge $Q$, as well as the charge-to-mass ratio $q$ of the
charged particle. In the case of neutral particles, where we take the limit
$q\to 0$, these two equations become dependent on each other. However, in the
more general scenario where $q\neq 0$, one can solve $r_{ISCO}(E_{t1},L_{t1})$
and $r_{ISCO}(E_{t2},L_{t2})$. Nevertheless, it is important to note that one
of these solutions yields a negative energy, as discussed in Section IV.

##  IV Topological configurations of timelike circular orbits

In this section, we investigate the topological configurations of the TCOs in
four distinct regimes based on the $qQ$ classification (II.3). These regimes
are as follows: the unlike strong charge regime ($qQ<-M<0$), the unlike weak
charge regime ($-M<qQ<0$), the like weak charge regime ($0<qQ<M$), and the
like strong charge regime ($0<M<qQ$). For each regime, we will determine the
topological configuration of the zero point of the vector $\vec{\phi}_{1}$ and
$\vec{\phi}_{2}$ against the $E_{1}$ and $E_{2}$ on the ($r,\theta$) plane,
respectively. We will then analyze the energy and angular momentum of the
TCOs. Finally, we will discuss the representation of TCO radius and the
topological charge as a function of angular momentum.

###  IV.1 Unlike Strong Charge Regime: $qQ<-M<0$

We begin by examining the regime $qQ<-M<0$, where the electric charge type of
the charged particle is opposite to that of the charged black hole. Throughout
our analysis, we focus on non-extremal charged black holes, specifically
$0<|Q|<M$, while fixing the parameters $Q=0.6$ and $M=1$ without loss of
generality.

For a strong charge $q=-3$, the asymptotic analysis of the vector
$\vec{\phi}_{1}=(\phi^{r}_{1},\phi^{\theta}_{1})$ for $E_{1}$ at the boundary
$\partial\Sigma$ reveals a vanishing topological charge $W_{1}=0$, as shown in
Table 1. This corresponds to two distinct topological configurations of the
unit vector $\vec{n}_{1}$ on the $(r,\theta)$ plane, which are depicted in
Figs. 3 and 3. In the first case, considering an angular momentum value of
$L=5.5<L_{ISCO}=5.8976$, there are no zero points presented in the region
$\Sigma$. In Fig. 3, the arrows of $\vec{n}_{1}$ consistently point towards
the right along the $r$-direction at $\theta=\pi/2$. This pattern confirms the
previous conclusion $\clubsuit$, indicating the absence of the TCOs and
resulting in a vanishing topological charge. On the other hand, by increasing
the angular momentum to $L=6.5>L_{ISCO}$, we can clearly observe two zero
points located at $r=4.2902$ and $10.6808$, and enclosed by the red closed
path in Fig. 3. Examining the left zero point, one can see that the vector
arrows flow into it along the $r$-direction and out of it along the
$\theta$-direction. Referring to the conclusion $\blacklozenge$, this zero
point corresponds to an unstable TCO with topological charge $W=-1$. As for
the right zero point, the vector arrows all flow out of it in both the $r$ and
$\theta$ directions, which indicates that it corresponds to a stable TCO with
$W=+1$. Summing these topological charges yields the topological number
$W_{1}=-1+1=0$ as expected.

Figure 3: Unit vector
$\vec{n}_{1}=(\phi^{r}_{1},\phi^{\theta}_{1})/||\vec{\phi}_{1}||$ constructed
from $E_{1}$ on ($r,\theta$) plane in the unlike strong charge regime with
$q=-3$. (a) Angular momentum $L=5.5<L_{ISCO}$. (b) $L=6.5>L_{ISCO}$. Two green
zero points of $\vec{n}$ locate at $r=4.2902$ and $10.6808$. Here
$L_{ISCO}=5.8976$.

From the Table 1, we already know that the total topological number of TCO in
the unlike strong charge regime is $W_{tot}=-1$. Nevertheless, the above
result on vector field $\vec{\phi}_{1}$ built from the $E_{1}$ is just
$W_{1}=0$. Thereby, we should continue to consider the situation of vector
field $\vec{\phi}_{2}$ coming from $E_{2}$. We show the topological
configuration of the unit vector
$\vec{n}_{2}=\vec{\phi}_{2}/||\vec{\phi}_{2}||$ in Fig. 4. Contrary to the
case of $E_{1}$, here only one green zero point can be found at $r=2.3545$ for
$\vec{n}_{2}$. Based on the conclusion $\blacklozenge$, this zero point
represents an unstable TCO with topological number $W_{2}=-1$. Even thought
the angular momentum here we chose is $L=2.2$, the outcome of $W_{2}=-1$ holds
for arbitrary $L>0$. Combining $W_{1}$ and $W_{2}$ for the vector field
$\vec{\phi}_{1}$ and $\vec{\phi}_{2}$, we end up with the total topological
number $W_{tot}=W_{1}+W_{2}=-1$ in the unlike strong charge regime. This
strongly verifies our analysis in Section II.

Figure 4: Unit vector
$\vec{n}_{2}=(\phi^{r}_{2},\phi^{\theta}_{2})/||\vec{\phi}_{2}||$ constructed
from $E_{2}$ on ($r,\theta$) plane in the unlike strong charge regime with
$q=-3$. Here $L=2.2$ and the green zero point locates at $r=2.3545$. The
vector configuration is similar for arbitrary positive angular momentum.

Now, let us proceed with the analysis of the energy and angular momentum of
the TCOs. As expected, we have two sets of solutions, namely $(E_{t1},L_{t1})$
and $(E_{t2},L_{t2})$ given in Eqs. (68)-(71). It is worth noting that we
split the radial range of TCO into two parts, i.e., $[r_{L},r_{P}]$ and
$[r_{P},\infty)$ using the (72) and (73). The solutions $(E_{t1},L_{t1})$ and
$(E_{t2},L_{t2})$ in these two intervals are separately presented in Figs. 5
and 5. In Fig. 5, the red and blue solid curves denoting $L_{t1}$ and $L_{t2}$
are connected at $r_{L}$, while $L_{t1}$ is always increasing from $r_{L}$ to
$r_{P}$ and the value of $L_{t2}$ eventually decays to zero at $r_{S}$. Hence
$L_{t1}$ and $L_{t2}$ can form a sole TCO branch in $L\in[0,\infty)$ within
the interval $[r_{L},r_{P})$. If we focus on $E_{t1}$ and $E_{t2}$ described
by red and blue dashed curves, we observe that they are both negative and
exist in all $[r_{L},r_{P}]$. Different from the decreasing $E_{t1}$, we
notice that $E_{t2}$ can extend $r_{P}$ to $r=\infty$, so we label the
position of $E_{t2}$ at $r_{P}$ with a blue point, indicating this curve can
exceed it. In Fig. 5, the new branch of $L_{t1}$ and $E_{t1}$ appears in the
range $[r_{P},\infty)$. Both of them are positive, and the curves firstly
descend with $r$ and then slowly grow. The minimal point of the curves
corresponds to ISCO, and its location, energy and angular momentum can be
solved from the Eq. (75)

$\displaystyle r_{ISCO}=6.1593,\quad E_{t1}=0.853,\quad L_{t1}=5.8976.$ (77)

It is noteworthy that $L_{t1}=5.8976$ represents a critical value
distinguishing two distinct topological configurations of the unit vector
$\vec{n}_{1}$ depicted in Figs. 3 and 3. Despite the negative $E_{t2}$ can
emerge in Fig. 5, it not provides us extra information on TCO because of the
absence of $L_{t2}$.

Figure 5: The solutions $(E_{t1},L_{t1})$ and $(E_{t2},L_{t2})$ of TCOs in the
unlike strong charge regime with $q=-3$. (a) Interval [$r_{L},r_{P}$]. (b)
Interval [$r_{P},\infty$). The blue point indicates that this curve can be
extended and surpass $r_{P}$. The location of ISCO is denoted as the black
point. Here $r_{L}=2.3485$, $r_{P}=2.7369$, and $r_{S}=2.3766$.

One important fact is that the combined branch $L_{t1}\cup L_{t2}$ and the
branch $L_{t1}$ with ISCO in Figs. 5 and 5 can separately interpret the
topological number $W_{1}=0$ and $W_{2}=-1$ of TCO given by Figs. 3 and 4.

For RN spacetime, if we focus exclusively on the positive energy values of
TCO, then by disregarding the negative energy portion, we obtain the valid
energy $E_{tco}$ and angular momentum $L_{tco}$, which can be expressed as

$\displaystyle E_{tco}(r)=\left\\{E_{t1}(r_{a})\bigcup
E_{t2}(r_{b})\,\bigg{|}\,r_{a}\in(r_{P},\infty),r_{b}=\varnothing,qQ<-M<0\right\\},$
(78) $\displaystyle L_{tco}(r)=\left\\{L_{t1}(r_{a})\bigcup
L_{t2}(r_{b})\,\bigg{|}\,r_{a}\in(r_{P},\infty),r_{b}=\varnothing,qQ<-M<0\right\\}.$
(79)

We present them in Fig. 6. It is easy to see that $E_{t2}$ and $L_{t2}$ do not
contribute to $E_{tco}$ and $L_{tco}$. The contribution only arises from the
segment of $L_{t1}$ and $E_{t1}$ with $r\in(r_{P},\infty)$.

Figure 6: Valid energy $E_{tco}$ and angular momentum $L_{tco}$ of TCO in the
unlike strong charge regime with $q=-3$. The location of ISCO is denoted as
the black point. Here $r_{P}=2.7369$.

With regard to the valid energy and angular momentum, we further present the
TCO radius $r_{t}$ as a function of the angular momentum $L$ in Fig. 7. It
becomes evident that when $L<L_{ISCO}$, no TCO can emerge. However, if
$L>L_{ISCO}$, both stable and unstable TCO branches originate from the ISCO.
This observation leads us to consider the ISCO as a bifurcation point Fu2000pb
. Moreover, in Fig. 7, we depict the topological charge of the TCO branches
with orange lines and the total topological number with black line. The stable
TCOs possess a topological charge of $W=+1$, while the unstable TCOs have a
topological charge of $W=-1$. Obviously, when $L>L_{ISCO}$, both the stable
and unstable TCOs branches are generated simultaneously, thus the total
topological number is given by $W_{RN}=+1-1=0$ when we neglect the negative
energy sector. Regardless of the value of the angular momentum, the total
topological number always vanishes, as illustrated by the black line in
figure. This suggests that the valid topological number $W_{RN}=0$ for RN
black hole is significant distinguished from $W_{tot}=-1$ in Table 1.

Figure 7: (a) TCO radius $r_{t}$ as a function of angular momentum $L$. (b)
Topological number $W_{RN}$ as a function of $L$. Here $W=\pm 1$ represent
stable and unstable TCOs. The black point denotes ISCO that is a generated
bifurcation point. The orange line stands for TCO branch.

###  IV.2 Unlike Weak Charge Regime: $-M<qQ<0$

For this case, we would like to take the charge-to-mass ratio $q=-0.5$ as an
example. Due to the weak charge $q$, the Coulomb attractive force is expected
to be weaker than the gravitational interaction of the RN black holes.

According to the Table 1, the boundary behavior of the vector field
$\vec{\phi}_{1}$ for $E_{1}$ indicates that the topological number in the
region $\Sigma$ is $W_{1}=0$, which is consistent with the presence of the two
distinct topological configurations of the unit vector $\vec{n}_{1}$ depicted
in Figs. 8 and 8. The first configuration, characterized by an angular
momentum of $L=3.73$, does not possess a zero point. In this case, the
direction of the vector always points towards the right at $\theta=\pi/2$ with
$r$. Utilizing the conclusion $\clubsuit$, we can easily obtain the result
that the topological charge $W=0$ for this particular configuration.

By increasing the angular momentum, such as $L=3.78$, two zero points of the
unit vector $\vec{n}_{1}$, marked in green color, emerge at $r=4.9549$ and
$6.1876$. The vector arrows near the left zero point flow into it along the
$r$-direction and flow out of it along the $\theta$-direction. This behavior
is consistent with the conclusion $\blacklozenge$, and gives a topological
charge $W=-1$ for this zero point. As for the right zero point, all arrows
flow out from it in both $r$ and $\theta$ directions, resulting in a
topological charge $W=+1$. Combining with these results, we have the
topological number $W=-1+1=0$ for the second configuration. Consequently, the
results obtained for these two configurations are consistent with $W_{1}=0$
for $E_{1}$ presented in Table 1.

Figure 8: Unit vector
$\vec{n}_{1}=(\phi^{r}_{1},\phi^{\theta}_{1})/||\vec{\phi}_{1}||$ constructed
from $E_{1}$ on ($r,\,\theta$) plane in the unlike weak charge regime with
$q=-0.5$. (a) Angular momentum $L=3.73<L_{ISCO}$. (b) $L=3.78>L_{ISCO}$. Two
green zero points locate at $r=4.9549$ and $6.1876$. Here $L_{ISCO}=3.7561$.

After investigating the vector field $\vec{\phi}_{1}$ built from $E_{1}$, we
continue to explore the scenario of vector $\vec{\phi}_{2}$ for $E_{2}$. In
this case, the topological number based on the Table 1 is $W_{2}=0$ for
$E_{2}$ as same as $W_{1}$. This outcome corresponds to two kinds of
topological configuration of $\vec{n}_{2}$, and we exhibit them in Figs. 9 and
9, respectively. It is quite obvious that when $L<L_{ISCO}$ there is no zero
points, whereas for $L>L_{ISCO}$ two green zero points separately appear at
$r=4.969$ and 6.083 with topological number $W=-1$ and +1 by applying the
previous conclusions $\blacklozenge$ and $\spadesuit$. Therefore, the
topological number $W_{2}=0$ always holds for these two distinct
configurations. Regardless of vector field constructed from $E_{1}$ and
$E_{2}$, the result of vanishing topological number leads to total topological
number $W_{tot}=W_{1}+W_{2}=0$ in the unlike weak charge regime as predicted
in Table 1.

Figure 9: Unit vector
$\vec{n}_{2}=(\phi^{r}_{2},\phi^{\theta}_{2})/||\vec{\phi}_{2}||$ constructed
from $E_{2}$ on ($r,\,\theta$) plane in the unlike weak charge regime with
$q=-0.5$. (a) Angular momentum $L=2.75<L_{ISCO}$. (b) $L=2.78>L_{ISCO}$. Two
green zero points locate at $r=4.969$ and 6.083. Here $L_{ISCO}=2.7631$

Furthermore, we investigate the energy and angular momentum of the TCOs within
the regime of weak unlike charge. Two sets of the solutions $(E_{t1},L_{t1})$
and $(E_{t2},L_{t2})$ situating at the intervals $[r_{L},r_{P}]$ and
$[r_{P},\infty)$ are clearly presented in Figs. 10 and 10, respectively. In
Fig. 10, we see that $E_{t1}$ and $L_{t1}$ both start at $r_{L}$ and terminate
at $r_{P}$, while they show diverse trends, namely $L_{t1}>0$ keeps increase
and $E_{t1}<0$ decays within this interval. Notably, positive $L_{t2}$ and
negative $E_{t2}$ at starting point $r_{L}$ connect with $L_{t1}$ and
$E_{t1}$, respectively. Furthermore, these two blue solid and dashed curves
can surpass the restriction of $r_{P}$ and extend to $[r_{P},\infty)$ showed
in Fig. 10. The blue points at $r_{P}$ mean that the corresponding curves can
continue to be extended. Let us turn our attention to the Fig. 10. There are
new $L_{t1}$ and $E_{t1}$ branches presented in $[r_{P},\infty)$. Both of them
have positive values and originally reduce and then gradually rise with the
TCO radius $r$. Additionally, the extremal points of $L_{t1}$ and $E_{t1}$
stand for an ISCO. On the other hand, the segments of $E_{t2}$ and $L_{t2}$
locating in [$r_{P},\infty$) also possess another ISCO. Through solving the
constraint relations (75) and (76), the radial radius, energy, and angular
momentum of these two ISCOs can be given by

$\displaystyle r_{ISCO1}=5.5021,\quad E_{t1}=0.9204,\quad L_{t1}=3.7561;$ (80)
$\displaystyle r_{ISCO2}=5.4094,\quad E_{t2}=-0.9555,\quad L_{t2}=2.7631.$
(81)

It is important to note that the branch $L_{t1}$ with $r_{ISCO1}$ and the
combined branch $L_{t1}\cup L_{t2}$ with $r_{ISCO2}$ in Figs. 10 and 10 can
separately interpret the topological number $W_{1}=0$ and $W_{2}=0$ of TCO
given by Figs. 8 and 9.

Figure 10: The solutions $(E_{t1},L_{t1})$ and $(E_{t2},L_{t2})$ of TCOs in
the unlike weak charge regime with $q=-0.5$ (a) Interval [$r_{L},r_{P}$]. (b)
Interval [$r_{P},\infty$). The blue point indicates that this curve can be
extended and surpass $r_{P}$. The location of ISCO is denoted as the black
point. Here $r_{L}=2.7278$ and $r_{P}=2.7369$.

Since we only focus on the TCOs with positive energy, the valid energy
$E_{tco}$ and angular momentum $L_{tco}$ should be

$\displaystyle E_{tco}(r)=\left\\{E_{t1}(r_{a})\bigcup
E_{t2}(r_{b})\,\bigg{|}\,r_{a}\in(r_{P},\infty),r_{b}=\varnothing,-M<qQ<0\right\\},$
(82) $\displaystyle L_{tco}(r)=\left\\{L_{t1}(r_{a})\bigcup
L_{t2}(r_{b})\,\bigg{|}\,r_{a}\in(r_{P},\infty),r_{b}=\varnothing,-M<qQ<0\right\\}.$
(83)

These two expressions are shown in Fig. 11 and indicates that only $E_{t1}$
and $L_{t1}$ in interval $(r_{P},\infty)$ make a valid contribution to TCOs.

Figure 11: Valid energy $E_{tco}$ and angular momentum $L_{tco}$ of TCO in the
unlike weak charge regime with $q=-3$. The location of ISCO is denoted as the
black point. Here $r_{P}=2.7369$.

Considered the valid energy and angular momentum, we now examine the
relationship between the TCO radius $r_{t}$ and the angular momentum $L$, as
illustrated in Fig. 12. The connection between the ISCO and TCO branches
becomes evident. If $L<L_{ISCO}$, no TCOs can emerge. However, by increasing
the angular momentum to $L>L_{ISCO}$, stable and unstable TCOs appear in
pairs. These observations are consistent with these two distinct topological
configurations of the vector depicted in Figs. 8 and 8. Note that
$r_{ISCO}(E_{t1},L_{t1})$, represented by the black point in the $r_{t}-L$
relation, acts as a generated bifurcation point. The topological charge of the
TCO branches (depicted in orange) is shown in Fig. 12. When TCO branches are
absent, the topological number vanishes. In the presence of TCO branches
($L>L_{ISCO}$), since the stable and unstable TCOs possess opposite
topological charges, the total topological number still vanishes. Therefore,
the total topological number always remains zero, i.e., $W_{RN}=0$ within the
regime of weak unlike charge, as indicated by the black line in Fig. 12. This
result is in accordance with $W_{tot}=0$ in the Table 1 which do not take into
account the negative energy.

Figure 12: (a) TCO radius $r_{t}$ as a function of angular momentum $L$. (b)
Topological number $W_{RN}$ as a function of $L$. Here $W=\pm 1$ represent
stable and unstable TCOs. The black point denotes ISCO that is a generated
bifurcation point. The orange line stands for TCO branch.

###  IV.3 Like Weak Charge Regime: $0<qQ<M$

As demonstrated earlier, we have studied the case of unlike charge with
$qQ<0$. Now, let us turn our attention to the like charge case with $qQ>0$. To
illustrate it, we consider a specific example where $q=1.2$ within the regime
of like weak charge, i.e., $0<qQ<M$.

In this scenario, two distinct types of topological configurations of the unit
vector $\vec{n}_{1}$ constructed from $E_{1}$ with the angular momentum
$L=1.85$ and $1.95$ are depicted in Figs. 13 and 13. In the former case, there
is an absence of the TCOs, whereas in the latter case, unstable and stable
TCOs appear in pairs at $r=4.8549$ and $7.3882$. It is noteworthy that these
two configurations are similar to the previous case depicted in Figs. 3 and 3,
as well as Figs. 8 and 8. Regardless of the occurrence of the TCOs, the
topological number remains zero, i.e., $W_{1}=0$ for the vector of $E_{1}$.
Furthermore, the angular momentum $L_{ISCO}$ of the ISCO can be used to split
these two topological configurations.

Figure 13: Unit vector
$\vec{n}_{1}=(\phi^{r}_{1},\phi^{\theta}_{1})/||\vec{\phi}_{1}||$ constructed
from $E_{1}$ on ($r,\,\theta$) plane in the like weak charge regime with
$q=1.2$. (a) Angular momentum $L=1.85<L_{ISCO}$. (b) $L=1.95>L_{ISCO}$. Two
green zero points locate at $r=4.8549$ and 7.3882. Here $L_{ISCO}=1.9160$.

Apart from the situation of unit vector $\vec{n}_{1}$, we also need to focus
on $\vec{n}_{2}=\vec{\phi}_{2}/||\vec{\phi}_{2}||$ generated from $E_{2}$. In
this case, based on the angular momentum of another ISCO, two dissimilar
topological configurations of $\vec{n}_{2}$ can be depicted in Figs. 14 and
14, respectively. Analogous to the case of $E_{1}$ in the regime $0<qQ<M$,
such a pattern of vector configurations using the previous conclusions
$\clubsuit$, $\spadesuit$, and $\blacklozenge$ exhibits the vanishing
topological number $W_{2}=0$ before and after $L_{ISCO}$. Combining the result
from $E_{1}$ and $E_{2}$, the total topological number is
$W_{tot}=W_{1}+W_{2}=0$ in the like weak charge regime, which is consistent
with the result in Table 1.

Figure 14: Unit vector
$\vec{n}_{2}=(\phi^{r}_{2},\phi^{\theta}_{2})/||\vec{\phi}_{2}||$ constructed
from $E_{2}$ on ($r,\,\theta$) plane in the like weak charge regime with
$q=1.2$. (a) Angular momentum $L=4.2<L_{ISCO}$. (b) $L=4.5>L_{ISCO}$. Two
green zero points locate at $r=4.6481$ and 7.2591. Here $L_{ISCO}=4.3881$.

The energy and angular momentum of the TCOs, denoted as $(E_{t1},L_{t1})$ and
$(E_{t2},L_{t2})$, are plotted in Figs. 15 and 15 corresponding to intervals
$[r_{L},r_{P}]$ and $[r_{P},\infty)$, respectively. Significantly, it is
observed that $E_{t2}$ and $L_{t2}$ are divided into two distinct branches,
the one in Fig. 15 and the other in Fig. 15, quite different from $E_{t1}$ and
$L_{t1}$, which can be extended from $r_{P}$ to $r=\infty$. The red points at
$r_{P}$ mean that the corresponding curves are jointed at this point. Besides,
we noticed that the solution $(E_{t2},L_{t2})$ can link with $(E_{t1},L_{t1})$
at $r_{L}$ in Fig. 15. Considering the scenario in Fig. 15, it is clearly
observed that $E_{t1}$, $L_{t1}$, $E_{t2}$, and $L_{t2}$ both exist the
minimal value in the range $[r_{P},\infty)$ corresponding to two diverse
ISCOs. Calculating the constraint relations (75) and (76), the radial radius,
energy and angular momentum of these two ISCO can be achieved as

$\displaystyle r_{ISCO1}=5.8871,\quad E_{t1}=0.9835,\quad L_{t1}=1.9160;$ (84)
$\displaystyle r_{ISCO2}=5.6653,\quad E_{t2}=-0.8991,\quad L_{t2}=4.3881.$
(85)

It is worth emphasizing that the second ISCO has negative energy $E_{t2}<0$.
At the same time, for the solution $(E_{t2},L_{t2})$, we also find that
$E_{t2}<0$ within the interval $[r_{P},\infty)$.

Figure 15: The solutions $(E_{t1},L_{t1})$ and $(E_{t2},L_{t2})$ of TCOs in
the like weak charge regime with $q=1.2$. (a) Interval [$r_{L},r_{P}$]. (b)
Interval [$r_{P},\infty$). The red point indicates that this curve can be
extended and surpass $r_{P}$. The location of ISCO is denoted as the black
point. Here $r_{L}=2.6833$ and $r_{P}=2.7369$.

One important fact is that the combined branch $L_{t1}\cup L_{t2}$ with
$r_{ISCO1}$ and the branch $L_{t2}$ with $r_{ISCO2}$ in Figs. 15 and 15 can
separately interpret the topological number $W_{1}=0$ and $W_{2}=0$ of TCO
given by Figs. 13 and 14.

After abandoning the negative energy portion of TCOs and combining two sets of
solution, the valid energy $E_{tco}$ and angular momentum $L_{tco}$ are shown
as

$\displaystyle E_{tco}(r)=\left\\{E_{t1}(r_{a})\bigcup
E_{t2}(r_{b})\,\bigg{|}\,r_{a}\in[r_{L},\infty),r_{b}\in[r_{L},r_{P}),0<qQ<M\right\\},$
(86) $\displaystyle L_{tco}(r)=\left\\{L_{t1}(r_{a})\bigcup
L_{t2}(r_{b})\,\bigg{|}\,r_{a}\in[r_{L},\infty),r_{b}\in[r_{L},r_{P}),0<qQ<M\right\\},$
(87)

which are depicted in Fig. 16.

Figure 16: Valid energy $E_{tco}$ and angular momentum $L_{tco}$ of TCO in the
like weak charge regime with $q=1.2$. The $E_{t1}$ and $E_{t2}$ together with
$L_{t1}$ and $L_{t2}$ are connected at $r_{L}$. Here $r_{L}=2.6833$ and
$r_{P}=2.7369$.

Remarkably, the red and blue curves representing $L_{t1}(r)$ and $L_{t2}(r)$
shown in Fig. 16 exhibit a connection point at $r=r_{L}$. This unique
characteristic is a notable novelty observed within the regime of like weak
charge, which distinguishes it from the cases depicted in Figs. 6 and 11,
where $E_{t2}$ and $L_{t2}$ do not contribute to $E_{tco}$ and $L_{tco}$.
Furthermore, in Fig. 16, the curves of $E_{tco}(r)$ and $L_{tco}(r)$ appear as
double-valued functions within the range of $[r_{L},r_{P})$. However, it is
essential to emphasize that $E_{tco}$ and $L_{tco}$ are not single, smooth,
and continuous functions resulting from the union of (86) and (87). Instead,
they consist of two distinct parts originating from TCOs with $E_{t1}$ and
$L_{t1}$ or TCOs with $E_{t2}$ and $L_{t2}$.

Although the topological configurations of the unit vector $\vec{n}_{1}$
appear quite simple, see Figs. 13 and 13, the behaviors of $E_{tco}$ and
$L_{tco}$ to TCOs suggest that there are additional hidden details within the
topological configurations of $\vec{n}_{1}$ that need to be unveiled. In
contrast to the previous cases depicted in Figs. 7 and 12, where only a single
$r_{t}(L)$ curve was presented, two $r_{t}(L)$ curves emerge here. We now
consider three distinct situations: $r_{t1}(L_{t1})$, $r_{t2}(L_{t2})$, and
their combination $r_{t}(L_{t})$. We, respectively, illustrate these
situations in Figs. 17, 17, and 17.

Considering the curve $r_{t1}(L_{t1})$ depicted in Fig. 17, it is evident that
the ISCO, denoted by the black point, divides $r_{t1}(L_{t1})$ of the TCOs
into two distinct branches. The unstable branch, characterized by radii
$r<r_{ISCO}$, originates at $L_{ISCO}$ and terminates at a finite angular
momentum value of $L=10.709$. On the other hand, the stable branch,
corresponding to larger radii $r>r_{ISCO}$, also starts at $L_{ISCO}$ but
extends indefinitely to $L=\infty$. Consequently, the total topological number
$W_{t1}$ is equal to zero when $L<L_{ISCO}$ and $L_{ISCO}<L<10.709$. However,
when $L>10.709$, $W_{t1}$ becomes +1 due to the presence of a single stable
TCO, as indicated by the black line in Fig. 17. This observation strongly
suggests the occurrence of a topological phase transition, where $W_{t1}$
changes from $0$ to $+1$ precisely at $L=10.709$. This phenomenon is
remarkably intriguing and has not been previously observed.

Regarding the curve $r_{t2}(L_{t2})$ depicted in Fig. 17, no ISCO is present,
and only one unstable TCO branch appears within the range of
$L\in[10.709,\infty)$. Consequently, when $L<10.709$, no TCOs exist, resulting
in a vanishing topological number $W_{t2}=0$. However, when $L>10.709$, the
topological number becomes $W_{t2}=-1$, as clearly illustrated in Fig. 17. The
transition of the topological number from $W_{t2}=0$ to -1 signifies the
occurrence of a topological phase transition precisely at $L=10.709$. It is
noteworthy that the topological phase transitions in $W_{t1}(L_{t1})$ and
$W_{t1}(L_{t2})$ both occur at the same angular momentum value $L=10.709$.

Next, we examine the combination of $r_{t1}(L_{t1})$ and $r_{t2}(L_{t2})$ in
$r_{t}(L_{t})$ as illustrated in Fig. 17. In this case, when $L<L_{ISCO}$, no
TCOs are present. However, as the angular momentum increases within the range
$L_{ISCO}<L<10.709$, $r_{t1}(L_{t1})$ exhibits a pair TCO branches.
Subsequently, in the interval $L\in[10.709,\infty)$, $r_{t1}(L_{t1})$ and
$r_{t2}(L_{t2})$ contribute a stable and an unstable TCO, respectively. As a
result, the total topological number $W_{RN}=0$ remains unchanged both before
and after $L=10.709$, as depicted in Fig. 17. This result is in agreement with
$W_{tot}=0$ in the Table 1 which do not take into account the negative energy.

Figure 17: (a) TCO radius $r_{t1}$ as a function of angular momentum $L_{t1}$.
(b) Topological number $W_{t1}$ (black line) as a function of $L_{t1}$. (c)
$r_{t2}-L_{t2}$. (d) $W_{t2}-L_{t2}$. (e) $r_{t}-L_{t}$. (f) $W_{RN}-L_{t}$.
Here $W=\pm 1$ represent stable and unstable TCOs. The black point denotes
ISCO that is a generated bifurcation point. The orange and purple lines stand
for TCO branch.

###  IV.4 Like Strong Charge Regime: $0<M<qQ$

In this regime of like strong charge case $0<M<qQ$, the Coulomb repulsive
force may surpass the attraction of the RN black hole. Additionally, as
indicated in Table 1, the total topological number is $W_{tot}=-1$ within this
specific regime. Notably, this differs from the weak charge regimes of
$0<|qQ|<M$ that we have previously discussed.

It is noteworthy that we further subdivide this regime into two distinct
subcases: (1) $M<qQ<q_{t}Q$ and (2) $q_{t}Q<qQ<\infty$, based on the differing
energy and angular momentum of the TCOs depicted in Figs. 20 and 25. Here,
$q_{t}$ represents a special charge-to-mass ratio, and its analytical
expression is given by Pugliese2011py

$q_{t}=\frac{1}{\sqrt{2}Q}\sqrt{5M^{2}-4Q^{2}+\sqrt{25M^{2}-24Q^{2}}}.$ (88)

####  IV.4.1 Subcase: $M<qQ<q_{t}Q$

We now investigate the first subcase, where $M<qQ<q_{t}Q$, and consider a
specific charge-to-mass ratio $q=2.1$. In Fig. 18, we present the topological
configuration of the unit vector $\vec{n}_{1}$ for $E_{1}$ with $L=10$ in the
$(r,\theta)$ plane. Notably, one can see a zero point of $\vec{n}_{1}$, which
is indicated by the green marker and surrounded by a closed red path.

It holds true that within this subcase of the like strong charge regime, only
one type of topological configuration of the unit $\vec{n}_{1}$ can be found,
regardless of the specific value of the angular momentum. In this
configuration, the vector arrows flow towards the zero point along the
$r$-direction and outwards from it along the $\theta$-direction. From the
conclusion $\blacklozenge$, we can conclude that the zero point possesses a
topological charge $W_{1}=-1$ for $E_{1}$, corresponding to an unstable TCO.
Obviously, this configuration is exactly consistent with our asymptotic
analysis of the $\vec{\phi}_{1}$ constructed from $E_{1}$ at $\partial\Sigma$
depicted in Fig. 1.

Figure 18: Unit vector
$\vec{n}_{1}=(\phi^{r}_{1},\phi^{\theta}_{1})/||\vec{\phi}_{1}||$ constructed
from $E_{1}$ with angular momentum $L=10$ on $(r,\theta)$ plane in the first
subcase $M<qQ<q_{t}Q$ with $q=2.1$. The topological configuration is similar
for arbitrary $L>0$. One green zero point represents an unstable TCO and
locates at $r=2.6009$.

We further examine the scenario of vector $\vec{n}_{2}$ that is calculated
from $E_{2}$ and the corresponding topological configuration of unit vector
$\vec{n}_{2}$ can be illustrated in Figs. 19 and 19. As we have seen earlier
in other charge regimes, these two configurations are split by angular
momentum of ISCO. In the former case with $L=5.15<L_{ISCO}$, no any zero
points exist and this results in the topological number $W=0$. While for the
latter case with $L>L_{ISCO}$, we can observe that two green zero points
emerge concurrently, and they separately possess topological number $W=-1$ and
+1 according to previous conclusions $\blacklozenge$ and $\spadesuit$. Summing
over these two number, the topological number for the second configuration of
unit vector also remain $W=0$. As a result, $W_{2}=0$ is expected for the
vector $\vec{n}_{2}$ built from $E_{2}$. Combining the outcome from $E_{1}$
and $E_{2}$, we end up with total topological number $W_{tot}=W_{1}+W_{2}=-1$,
which keeps same with the result in Table 1.

Figure 19: Unit vector
$\vec{n}_{2}=(\phi^{r}_{2},\phi^{\theta}_{2})/||\vec{\phi}_{2}||$ constructed
from $E_{2}$ on ($r,\,\theta$) plane in the first subcase $M<qQ<q_{t}Q$ with
$q=2.1$. (a) Angular momentum $L=5.15<L_{ISCO}$. (b) $L=5.2>L_{ISCO}$. Two
green zero points locate at $r=5.2455$ and 6.7564. Here $L_{ISCO}=5.1587$.

Let us now turn our attention to the energy and angular momentum of the TCOs.
We plot $(E_{t1},L_{t1})$ and $(E_{t2},L_{t2})$ in Figs. 20 and 20
corresponding to intervals $[r_{L},r_{P}]$ and $[r_{P},\infty)$, respectively.
In Fig. 20, We see that $L_{t1}$ and $E_{t1}$ both are positive and initiate
at $r_{L}$, whereas they have dissimilar ending points. Specifically, $L_{t1}$
and $E_{t1}$ extends from $r_{P}$ to $r_{s}$ and $\infty$, respectively. Note
that $L_{t1}=0$ at the position $r_{S}$. Furthermore, the TCOs characterized
by positive $E_{t1}$ and vanishing $L_{t1}$ at $r_{s}$ do not exhibit any
relative rotation with respect to distant observers, which is exactly the
static point orbit observed in Ref. Wei2023bgp  for a uncharged particle. Here
the red points on the red solid and dashed curves indicate that the curves can
continue to stretch. As for $E_{t2}$ and $L_{t2}$, they are divided into two
sectors against radius $r_{P}$ displayed in Figs. 20 and 20, respectively. In
the left figure, both $L_{t2}$ and $E_{t2}$ are positive and they link with
$L_{t1}$ and $E_{t1}$ at the starting point $r_{L}$. These two curves increase
with the TCO radius $r$. However, in the right figure, $E_{t2}$ becomes
negative and they separately show an extreme point corresponding to ISCO. The
radial radius, energy, and angular momentum of ISCO can be gained from the Eq.
(76)

$r_{ISCO}=5.9067,\quad E_{t2}=-0.9561,\quad L_{t2}=2.7728.$ (89)

Notice that the energy $E_{t2}$ of this ISCO is negative.

Figure 20: The solutions $(E_{t1},L_{t1})$ and $(E_{t2},L_{t2})$ of TCOs in
the first subcase of the like strong charge regime with $q=2.1$. (a) Interval
[$r_{L},r_{P}$]. (b) Interval [$r_{P},\infty$). The red point indicates that
this curve can be extended and surpass $r_{P}$. The location of ISCO is
denoted as the black point. Here $r_{L}=2.5644$, $r_{S}=3.9898$, and
$r_{P}=2.7369$.

One important fact is that the combined branch $L_{t1}\cup L_{t2}$ and the
branch $L_{t2}$ with $r_{ISCO}$ in Figs. 20 and 20 can separately interpret
the topological number $W_{1}=-1$ and $W_{2}=0$ of TCO given by Figs. 18 and
19.

After excluding the negative energy portion, the physically meaningful energy
$E_{tco}$ and angular momentum $L_{tco}$ of the TCOs can be defined as
follows:

$\displaystyle E_{tco}(r)=\left\\{E_{t1}(r_{a})\bigcup
E_{t2}(r_{b})\,\bigg{|}\,r_{a}\in[r_{L},r_{S}],r_{b}\in[r_{L},r_{P}),M<qQ<q_{t}Q\right\\},$
(90) $\displaystyle L_{tco}(r)=\left\\{L_{t1}(r_{a})\bigcup
L_{t2}(r_{b})\,\bigg{|}\,r_{a}\in[r_{L},r_{S}],r_{b}\in[r_{L},r_{P}),M<qQ<q_{t}Q\right\\}.$
(91)

The $E_{tco}$ and $L_{tco}$ are illustrated in Fig. 21. In this figure,
$E_{t1}$ and $E_{t2}$, as well as $L_{t1}$ and $L_{t2}$, are connected at
$r=r_{L}$. When comparing with them depicted in Fig. 16, we observe that in
the present case, $E_{tco}$ and $L_{tco}$ must terminate at $r_{S}$ due to the
vanishing angular momentum rather than extend to radial infinity at
$r=\infty$.

Figure 21: Valid energy $E_{tco}$ and angular momentum $L_{tco}$ of TCO in the
first subcase of the like strong charge regime with $q=2.1$. The $E_{t1}$ and
$E_{t2}$ together with $L_{t1}$ and $L_{t2}$ are connected at $r_{L}$ and
terminate at $r_{S}$. Here $r_{P}=2.7369$ and $r_{S}=3.9898$.

Here, we aim to examine the relationship between the TCO radius and the
angular momentum based on $E_{tco}$ and $L_{tco}$. As $L_{tco}$ is an union of
both $L_{t1}$ and $L_{t2}$, we need to consider three scenarios:
$r_{t1}(L_{t1})$, $r_{t2}(L_{t2})$, and their union $r_{t}(L_{t})$,
respectively. In Fig. 22, $r_{t1}(L_{t1})$ initiates at $L=0$ and terminates
at $L=4.834$. It represents an unstable TCO branch that exhibits a topological
number $W_{t1}=-1$ for $L<4.834$. However, in the absence of TCOs for
$L>4.834$, the topological number becomes $W_{t1}=0$, as depicted in Fig. 22.
Consequently, a topological phase transition occurs at the angular momentum
value of $L=4.834$, where the topological number turns from $W_{t1}=-1$ to
$0$.

In the previous case, we also have the topological phase transition as
depicted in Fig. 17. However, it should be noted that the existence of the
ISCO differs from the current case. In contrast, the $r_{t2}(L_{t2})$ curve in
Fig. 22 originates at a non-zero angular momentum value of $L=4.834$ and
extends indefinitely to $L=\infty$. This TCO branch is also unstable, and the
corresponding topological number $W_{t2}=0$ and $-1$ before and after
$L=4.834$ as illustrated in Fig. 22. The occurrence of a topological phase
transition in the $W_{t2}$-$L_{t2}$ diagram leads to a variation in the
topological number from $W_{t2}=0$ to $-1$. Remarkably, this phenomenon
exhibits a striking analogy to the situation observed in $W_{t1}$ while with a
different sign.

The final result for the $r_{t}$-$L_{t}$ relation is simply the combination of
$r_{t1}(L_{t1})$ and $r_{t2}(L_{t2})$, as shown in Fig. 22. It is important to
note that both of these TCO branches represent the unstable TCOs.
Consequently, the total topological number remains constant $W_{RN}=-1$, as
indicated by the black line in Fig. 22. This result is consistent with the
result $W_{tot}=-1$ obtained from Table 1 where we do not take into account
the negative energy.

Figure 22: (a) TCO radius $r_{t1}$ as a function of angular momentum $L_{t1}$.
(b) Topological number $W_{t1}$ (black line) as a function of $L_{t1}$. (c)
$r_{t2}-L_{t2}$. (d) $W_{t2}-L_{t2}$. (e) $r_{t}-L_{t}$. (f) $W_{RN}-L_{t}$.
Here $W=-1$ represents unstable TCOs. The orange and purple lines stand for
TCO branch.

####  IV.4.2 Subcase: $q_{t}Q<qQ<\infty$

In the second subcase $q_{t}Q<qQ<\infty$, we consider a specific charge-to-
mass ratio example of $q=3.8$ to investigate the topological configuration of
TCOs. Referring to the results in Table 1, we have already established that
the total topological number remains constant $W_{tot}=-1$ in the like strong
charge regime. Therefore, this subcase also exhibits an identical topological
number.

In Fig. 23, we present the unit vector $\vec{n}_{1}$ for $E_{1}$ in the
$(r,\theta)$ plane for $L=2.9$. A zero point marked in green at $r=2.2389$ is
observed. Through further adjustments of $L$, we find that only a single zero
point exists in the $(r,\theta)$ plane. This observation suggests that the
topological configuration with a single TCO is the unique possibility for any
positive angular momentum. The vector arrows in the vicinity of the zero point
flow into it along the $r$-direction and out of it along the
$\theta$-direction. Based on the conclusion $\blacklozenge$, it is evident
that this zero point represents an unstable TCO, and its topological number is
$W_{1}=-1$.

Figure 23: Unit vector
$\vec{n}_{1}=(\phi^{r}_{1},\phi^{\theta}_{1})/||\vec{\phi}_{1}||$ constructed
from $E_{1}$ with angular momentum $L=2.9$ on $(r,\,\theta)$ plane in the
second subcase $q_{t}Q<qQ<\infty$ with $q=3.8$. The topological configuration
is similar for arbitrary positive angular momentum. The green zero point
represents an unstable TCO and locates at $r=2.2389$.

Let us move to the situation of the unit vector $\vec{n}_{2}$ built from
$E_{2}$. The corresponding topological configurations are depicted in Figs. 24
and 24. In Fig. 24, we can see that no zero point can be found in $(r,\theta)$
plane when $L=6.52<L_{ISCO}$, and the vector arrows always point to the left
at $\theta=\pi/2$. It implies that the topological number in this case is
$W_{2}=0$. For the second configuration with $L=6.55>L_{ISCO}$, unit vector
$\vec{n}_{2}$ exists two zero points marked in green and their radial
coordinates are $r=5.9755$ and 6.8511, respectively. In addition, applying the
conclusions $\blacklozenge$ and $\spadesuit$ earlier, these two zero points
separately have topological number $W=-1$ and +1. Thereby, the eventually
topological number remain the constant $W_{2}=0$ for the second configuration.
As a result, $W_{tot}=W_{1}+W_{2}=-1$ in the regime of $q_{t}Q<M<\infty$,
which coincide with the result given by Table 1.

Figure 24: Unit vector
$\vec{n}_{2}=(\phi^{r}_{2},\phi^{\theta}_{2})/||\vec{\phi}_{2}||$ constructed
from $E_{2}$ on ($r,\,\theta$) plane in the second subcase $q_{t}Q<qQ<\infty$
with $q=3.8$. (a) Angular momentum $L=6.52<L_{ISCO}$. (b) $L=6.55>L_{ISCO}$.
Two green zero points locate at $r=5.9755$ and 6.8511. Here $L_{ISCO}=6.5356$.

We now turn our attention to the energy and angular momentum of TCOs, namely
$(E_{t1},L_{t1})$ and $(E_{t2},L_{t2})$, as depicted in Figs. 25 and 25. These
two figures correspond to intervals $[r_{L},r_{P}]$ and $[r_{P},\infty)$,
respectively. A particularly unusual situation arises where $L_{t1}$ becomes
an imaginary number and is absent in Fig. 25 and 25, while $E_{t1}$ remains a
positive real number that starts at $r_{L}$ and ends at $r=\infty$. The
presence of an imaginary $L_{t1}$ indicates that TCOs characterized by
$L_{t1}$ and $E_{t1}$ do not contribute to the topological configuration of
the vector. On the other hand, positive $L_{t2}$ can be divided into two
distinct sectors. One part of $L_{t2}$ monotonically increases from $r_{S}$ to
$r_{P}$ shown in Fig. 25, while the other part has a minimal value and extends
from $r_{P}$ to radial infinity $r=\infty$, as shown in Fig. 25. The energy
$E_{t2}$ exhibits increasing behavior both in the intervals $[r_{L},r_{P})$
and $(r_{P},\infty)$, and takes positive and negative values in the former and
latter intervals, respectively. The radial radius, energy, and angular
momentum of ISCO displayed in Fig. 25 can be achieved from the constraint
relation (76)

$r_{ISCO}=6.3849,\quad E_{t2}=-0.9574,\quad L_{t2}=2.7954.$ (92)

Note that this particular ISCO possesses a negative energy $E_{t2}<0$.

Figure 25: The solutions $(E_{t1},L_{t1})$ and $(E_{t2},L_{t2})$ of TCOs in
the second subcase of the like strong charge regime with $q=3.8$. (a) Interval
[$r_{L},r_{P}$]. (b) Interval [$r_{P},\infty$). The $L_{t1}$ is an imaginary
number and disappears in the figure. The red point indicates that this curve
can be extended and surpass $r_{P}$. The location of ISCO is denoted as the
black point. Here $r_{L}=1.98$, $r_{S}=2.1087$, and $r_{P}=2.7369$.

One important fact is that the monotonously branch $L_{t2}$ in Fig. 25 and the
branch $L_{t2}$ with $r_{ISCO}$ in Fig. 20 can separately interpret the
topological number $W_{1}=-1$ and $W_{2}=0$ of TCO given by Figs. 23 and 24.

After excluding the imaginary $L_{t1}$ and the negative energy portion of
$E_{t2}$, the valid energy $E_{tco}$ and angular momentum $L_{tco}$ can be
defined as follows

$\displaystyle E_{tco}(r)=\left\\{E_{t1}(r_{a})\bigcup
E_{t2}(r_{b})\,\bigg{|}\,r_{a}\in\varnothing,\,r_{b}\in[r_{S},r_{P}),\,q_{t}Q<qQ<\infty\right\\},$
(93) $\displaystyle L_{tco}(r)=\left\\{L_{t1}(r_{a})\bigcup
L_{t2}(r_{b})\,\bigg{|}\,r_{a}\in\varnothing,\,r_{b}\in[r_{S},r_{P}),\,q_{t}Q<qQ<\infty\right\\}.$
(94)

They are depicted in Fig. 26 and exhibit a monotonic increase from $r_{S}$ to
$r_{P}$. Additionally, $L_{t2}$ equals to zero at the radius $r_{S}$.

Figure 26: Valid energy $E_{tco}$ and angular momentum $L_{tco}$ of TCOs in
the second subcase of the like strong charge regime with $q=3.8$. Here
$r_{S}=2.1087$ and $r_{P}=2.7369$.

Now, we turn our attention to the relationship between the TCO radius $r_{t}$
and the angular momentum $L$ according to $E_{tco}$ and $L_{tco}$, and then
investigate the topological number $W_{RN}$ as a function of $L$. The
representation of $r_{t}$-$L$ in Fig. 27 consistently increases from $L=0$ to
$L=\infty$, indicating the absence of any ISCO in this subcase. Moreover, this
TCO branch corresponds to an unstable TCO. Consequently, the total topological
charge remains constant, i.e., $W_{RN}=-1$, in all the parameter space. While
in the previous subcase $M<qQ<q_{t}Q$, a topological phase transition can
occur in $W_{t1}(L_{t1})$ and $W_{t2}(L_{t2})$ shown in Figs. 22 and 22, we do
not observe such a transition in the $q_{t}Q<qQ<\infty$ subcase. The result of
$W_{RN}=-1$ is consistent with $W_{tot}=-1$ presented from Table 1 where we do
not take into account the negative energy.

Figure 27: (a) TCO radius $r_{t}$ as a function of angular momentum $L$. (b)
Topological number $W_{RN}$ as a function of $L$. Here $W=-1$ represents
unstable TCO. The orange line stands for TCO branch.

Before closing this section, we would like to give a summary on topological
number $W_{RN}$ within distinct $qQ$ regimes via considering valid energy
$E_{tco}$ and angular momentum $L_{tco}$. The completely results are collected
into Table 2. From this table, we notice that the unique discrepancy between
$W_{RN}$ and $W_{tot}$ shown in Tables 2 and 1 is that $W_{RN}=0\neq
W_{tot}=-1$ in the regime of unlike strong charge. The reason for it is that
if we only focus on the positive energy, the branch combining with $L_{t1}$
and $L_{t2}$ presented in Fig. 5 must be discarded, which corresponds to an
unstable TCO with topological number $W=-1$. On the other hand, we find that
$E_{tco}$ and $L_{tco}$ can consist of three possible situations: (1)
$(E_{t1},L_{t1})$; (2) $(E_{t2},L_{t2})$; (3) the combination of
$(E_{t1},L_{t1})$ and $(E_{t2},L_{t2})$. According to our previous
investigation, only the last situation (3) can exhibit topological phase
transition, as displayed in Figs. 16 and 21, which correspond to the charge
regimes: $0<qQ<M$ and $M<qQ<q_{t}Q$, respectively. The specific details of the
topological phase transitions that occur in these two charge regimes can be
found in the Figs. 28 and 29.

Table 2: The elements of valid energy $E_{tco}$ and angular momentum $L_{tco}$
of TCO. within the distinct $qQ$ regimes and the corresponding topological
numbers $W_{RN}$ for RN black holes. Here “PT” stands for whether the TCO
occurs phase transition.

| $qQ<-M<0$ | $-M<qQ<0$ | $0<qQ<M$ | $M<qQ<\infty$
---|---|---|---|---
| $M<qQ<q_{t}Q$ | $q_{t}Q<qQ<\infty$
$E_{tco}$ | $E_{t1}$ | $E_{t1}$ | $E_{t1}\cup E_{t2}$ | $E_{t1}\cup E_{t2}$ | $E_{t2}$
$L_{tco}$ | $L_{t1}$ | $E_{t1}$ | $L_{t1}\cup L_{t2}$ | $L_{t1}\cup L_{t2}$ | $L_{t2}$
PT | No | No | Yes | Yes | No
$W_{RN}$ | 0 | 0 | 0 | -1 | -1
-1

Figure 28: Topological phase transition of TCO within the charge regime
$0<qQ<M$. The topological numbers of the TCO branches $r_{t1}$ and $r_{t2}$
change from $W_{t1}=0$ to +1 and from $W_{t2}=0$ to -1 at $L=10.709$,
respectively.

Figure 29: Topological phase transition of TCO within the charge regime
$M<qQ<q_{t}Q$. The topological numbers of the TCO branches $r_{t1}$ and
$r_{t2}$ change from $W_{t1}=-1$ to 0 and from $W_{t2}=0$ to -1 at $L=4.834$,
respectively.

##  V Conclusions

In this paper, we mainly focused on studying the influence of the charge-to-
mass ratio $q$ of massive charged test particles on the topological number of
TCOs within a static spherically symmetric spacetime with the electromagnetic
potential $A_{\mu}=(Q/r,0,0,0)$. It is important to emphasize that in the case
of neutral particles ($q=0$) within the same spacetime, the topological number
is $W=0$, implying that either no TCOs occur or TCOs appear in pairs with
opposite topological charges. However, we here mainly aimed at the case $q\neq
0$, which shows a different behavior from the neutral case.

Starting with a generic Lagrangian for a massive charged test particle in this
spacetime, we solved the equation of motion and obtained the effective
potential $V_{\text{eff}}$, which is a quadratic function of energy. Using the
decomposed energy $E_{1}$, we constructed a characteristic vector field
$\vec{\phi}_{1}=(\phi^{r}_{1},\phi^{\theta}_{1})$ that can describe the
topological properties of the TCOs. Subsequently, in order to obtain the
topological number, we examined the boundary behavior of $\vec{\phi}_{1}$ in
the region $\Sigma=[r_{h},\infty)\times[0,\pi]\subset\mathbb{R}^{2}$. The
results reveal that the vector arrows of $\vec{\phi}_{1}$ point upwards,
downwards at $\theta=\pi$ and $0$, and to the right at $r=r_{h}$. These
directions are independent of the charge, and are uniquely determined depicted
in Fig. 1. However, the direction of the vector at $r\to\infty$ closely
depends on the sign of $M-qQ$. When it is positive, the topological number is
$W_{1}=0$, otherwise it is $W_{1}=-1$ for $E_{1}$. Besides that, we also
investigated the boundary behaviors of the vector $\vec{\phi}_{2}$ constructed
from $E_{2}$. In this case, the vector arrows of $\vec{\phi}_{2}$ at
$r=\infty$ rely on the sign of $-M-qQ$, as displayed in Fig. 2. When
$-M-qQ>0$, the topological number is $W_{2}=-1$, otherwise it is $W_{2}=0$ for
$E_{2}$. Consequently, combining the analysis of vector $\vec{\phi}_{1}$ and
$\vec{\phi}_{2}$, we arrived at our main conclusion: in a static spherically
symmetric spacetime with a radial electric field, the presence of electric
charge in massive particles can change the topology of TCOs from $W=0$ to
$W=-1$ when $qQ>M$ or $qQ<-M$ for the decomposed energy $E_{1}$ or $E_{2}$,
respectively.

According to the sign and value of $qQ$, we classified the parameter space
into four distinct regimes: (a) unlike strong charge regime; (b) unlike weak
charge regime; (c) like weak charge regime; (d) like strong charge regime.
Furthermore, to illustrate our findings, we listed the total topological
number $W_{tot}$ of TCOs within these $qQ$ regimes in Table 1, and considered
RN black holes as an example. For each regime, we carefully examined the
topological configuration of the unit vector $\vec{n}_{1}$ and $\vec{n}_{2}$
with respect to $E_{1}$ and $E_{2}$ in the ($r,\theta$) plane, respectively.
The energy and angular momentum of TCOs are also examined. Importantly, we
carried out the detailed study of the TCO radius $r_{t}$ and the topological
number $W_{RN}$ as function of the the angular momentum $L$, via which one can
obtain whether there exists topological phase transition.

In regimes (a) and (b), we identified two distinct topological configurations
of the unit vector $\vec{n}_{1}$: $L<L_{\text{ISCO}}$ and $L>L_{\text{ISCO}}$
for $E_{1}$. In both cases, the topological number is $W_{1}=0$. When we
further considered the situation of $\vec{n}_{2}$, the topological number
$W_{2}=-1$ and 0 in regimes (a) and (b), respectively. That is because the
configuration of $\vec{n}_{2}$ in regime (a) just has one possibility for any
positive angular momentum. Therefore, the total topological number is
$W_{tot}=W_{1}+W_{2}=-1$ and 0 for regimes (a) and (b), as presented in Table
1. Interestingly, we observed the existence of negative energy sectors
represented by $E_{t1}$ and $E_{t2}$. By excluding them, we obtained the valid
energy $E_{\text{TCO}}$ and angular momentum $L_{\text{TCO}}$ by combining
both $(E_{t1},L_{t1})$ and $(E_{t2},L_{t2})$. We found that the contribution
to the topological number $W_{RN}=0$ without involving negative energy in
regimes (a) and (b) only comes from $E_{t1}$ and $L_{t1}$. It is worth
emphasizing that the topological number $W_{RN}=0\neq W_{tot}=-1$ in the
regime (a) due to the fact that an unstable TCO branch with negative energy
has to be discarded.

On the other hand, in regime (c), the topological configuration of
$\vec{n}_{1}$ and $\vec{n}_{2}$ built from $E_{1}$ and $E_{2}$ is identical to
that in regimes (b). Therefore, the topological number remains zero, i.e.,
$W_{1}=W_{2}=0$. As a result, $W_{tot}=W_{1}+W_{2}=0$ is same as that in Table
1. However, the difference is that both $(E_{t1},E_{t2})$ and
$(L_{t1},L_{t2})$ have contribution to $E_{\text{TCO}}$ and $L_{\text{TCO}}$.
This observation reveals underlying topological phase transitions in
$W_{t1}(L_{t1})$ and $W_{t2}(L_{t2})$, where the topological numbers change
from $W_{t1}=0$ to +1 and from $W_{t2}=0$ to -1 at $L=10.709$, respectively.
This nontrivial feature constitutes a significant finding in this study. In
the regime (c), $W_{RN}=0$ is consistent with the result of $W_{tot}$.

In the last regime (d), the topological number is $W_{tot}=-1$. We observed a
single type of topological configuration for $\vec{n}_{1}$ against $E_{1}$,
which represents an unstable TCO branch with $W_{1}=-1$ for arbitrary $L>0$.
However, regard to $E_{2}$ the configuration of $\vec{n}_{2}$ has two kinds
with $W_{2}=0$ according to $L_{ISCO}$. It is important to note that regime
(d) needs to be further divided into two subcases: $M<qQ<q_{t}Q$ and
$q_{t}Q<qQ<\infty$. In the former subcase, $E_{\text{TCO}}$ and
$L_{\text{TCO}}$ are composed of $(E_{t1},L_{t1})$ and $(E_{t2},L_{t2})$.
Similarly to regime (c), a topological phase transition occurs in
$W_{t1}(L_{t1})$ and $W_{t2}(L_{t2})$, resulting in a change of the
topological numbers from $W_{t1}=-1$ to 0 and from $W_{t2}=0$ to -1 at
$L=4.834$. However, the topological number $W_{RN}=-1$ remains unchanged. In
the latter subcase, $L_{t1}$ becomes an imaginary number. Thus, $E_{t1}$ and
$L_{t1}$ have no contribution to $E_{\text{TCO}}$ and $L_{\text{TCO}}$.
Furthermore, $W_{RN}=-1$ is in accordance with the first subcase. Our study
shows that no topological phase transition occurs in this case, and the
outcome of $W_{RN}=-1$ equals to $W_{tot}$ exhibited in Table 1. Meanwhile,
all the topological number $W_{RN}$ focusing on positive energy within
distinct $qQ$ regimes are organized in Table 2.

In conclusion, we have presented a comprehensive analysis of the topological
structures of TCOs in the presence of test particles with nonvanishing charge.
Our study has revealed a significant influence of the charge on the
topological number. Furthermore, there are other issues for further
exploration, such as extending the analysis to spinning black hole backgrounds
and considering different forms of the electromagnetic potential. These
directions offer promising opportunities to further understand the topology of
TCOs.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China
(Grants No. 12075103, No. 12247101).

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