# Topological study of equatorial timelike circular orbit for spherically
symmetric (hairy) black holes

Xu Ye, Shao-Wen Wei 111Corresponding author. E-mail<EMAIL_ADDRESS>1Lanzhou 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,
2Institute of Theoretical Physics $\&$ Research Center of Gravitation, Lanzhou
University, Lanzhou 730000, People’s Republic of China

###### Abstract

Topological approach has been shown a promising study toward to the light ring
in a generic black hole background. We in this paper generalize it to the
study of timelike circular orbits in static, spherically symmetric, and
asymptotic flat black holes. By making use of the equation of motion of the
massive particles, we construct a vector with its zero points exactly relating
with the timelike circular orbits. Since each zero point of the vector can be
endowed with a winding number, the topology of the timelike circular orbits is
established. Stable and unstable timelike circular orbits respectively have
winding number +1 and -1. In particular, different from that of the light
ring, the timelike circular orbits possess vanishing topological number and
appear in pairs for given angular momentum. Moreover, we apply the study to
the Schwarzschild, scalarized Einstein-Maxwell, and dyonic black holes, which
have distinct topological configurations. It is shown that although the
existence of scalar hair and quasi-topological term leads to richer
topological configurations of the timelike circular orbits, whereas they have
no influence on the topological number. These results indicate that the
topological approach indeed provides us a novel way to understand the timelike
circular orbits. More information is expected to be disclosed when other
different topological configurations are present.

Classical black hole, timelike circular orbit, topology

###### pacs:

04.20.-q, 04.25.dg, 04.70.Bw

##  I Introduction

The gravitational wave detections by LIGO and Virgo Collaborations ligo16a ;
ligo16b ; ligo17  provide strong evidence that astrophysical black holes exist
and merge. Via such binary black hole mergers, the nature of the black hole
can be well studied with the inspiral, merger, and ringdown waveforms. On the
other hand, through the observations of shadow imaging EHTL1 ; EHTL5 ; EHTL6 ,
the information near the black hole horizon geometry can be tested.

Extensive studies have shown that the ringdown and shadow observables are both
intimately connected to a special set of null circular orbits, known as the
light rings (LRs) Cardoso2016 ; Cunha2018 . Apart from the null geodesics, the
timelike geodesics of massive particles can also form the circular orbit
around the black holes. Such timelike circular orbits (TCOs) are also one kind
fundamental characteristic orbits. These massive particles dropped from far
away from the black hole accumulate on these stable TCOs and form an accretion
disk with its inner edge measuring by the innermost stable circular orbit
(ISCO) Abramowicz2011 .

On account of that these characteristic orbits are directly related to the
motion of particles that hide valuable information on spacetime background,
several different methods are developed to deal with these circular orbits.
The most common one is to solve the geodesic equations via Lagrangian, and
obtain the circular orbit by formulating the effective potential. This
treatment has a wide range of applications such as studying the photon sphere
in static, stationary, or dynamical spacetime Hasse2001 ; Koga2022 ;
Johannsen2013 . The other one is called the quasi-local approach, through
which the first quasi-local definition of the photon surface was given by
Claudel, Virbhadra, and Ellis Claudel2000 , and then it is extended to the
trapped surface Shiromizu2017 .

Quite differently, the topological method can also be applied in the analysis
of circular orbit whether its center is a black hole or a horizonless
ultracompact object. In Ref. Cunha2017 , Cunha, Berti, and Herdeiro first
proposed a topological approach and proved a theorem that if an axisymmetric
and stationary solution of the Einstein field equation obeys the null energy
condition, the ultracompact objects formed from the classical gravitational
collapse of matter must have at least two LRs, one of which is stable and the
other is unstable. Such study showed the great success of topological approach
without knowing the specific locations of LRs. Subsequently, such treatment
was generalized to a stationary axisymmetric, asymptotically flat black hole
Cunha2020 . The result stated that there is at least one standard unstable LR
outside the black hole horizon for each rotation sense. For the static,
spherically symmetric black holes in asymptotically flat, dS, and AdS
spacetime, such property still held Wei2020 . Even when more LRs or photon
spheres are presented, there is always one more unstable LR or photon sphere.
Other relevant studies can also be found in Refs. Guo2020 ; Junior2021 ;
Ghosh2021 ; Junior2021b .

On the other hand, the equatorial circular orbits for the photons and massive
particles are closely related to each other. In Refs. Delgado2021 ;
Lehebel2022 , it was found that an unstable (stable) LR delimits a region of
unstable (stable) TCOs radially above (below) it. Moreover, the corresponding
corollary was discussed for both horizonless ultracompact objects and black
holes. However one significant difference of TCO from LR is that it not only
depends on the black hole parameters, but also on the angular momentum and
energy of the particles. It seems that such feature makes it impossible to
establish the topology for the TCOs.

However, very recently, it was first noted in our previous work Ref. Wei2022
that by constructing an appropriate vector, the topology can be well-behaved
for the TCOs. Although the angular momentum and energy of the particles modify
the location of the TCO or the zero points of the vector, they do not alter
the asymptotic behaviors at the boundaries of the parameter space, and thus
the corresponding topological argument was meaningful. Considering a
stationary black hole background, we found that the topological number of TCOs
$W=0$, quite different from $W$=-1 of LRs. This suggests that if there exist
TCOs, they must appear in pairs for given angular momentum. In particular,
stable and unstable TCOs, respectively, have positive or negative winding
numbers. These results were exactly confirmed by further applying the topology
to the Kerr black holes, where there will be no TCO, a pair TCOs with the
increase of the angular momentum.

As pointed out in Ref. Wei2020 , the static black holes have richer LR
structures than the stationary black holes. Here it is interesting to examine
whether these topological results of TCOs still hold for the static,
spherically symmetric black holes when different topological configurations
are given.

An outline of the present paper is as follows. In. Sec. II, we start with a
general spherically symmetric, asymptotic flat black hole and obtain the
effective potential for the motion of massive particles via its Lagrangian.
Then, we construct a vector $\phi$. Its zero points are exactly related to the
TCOs, which allows us to establish a corresponding topology. After examining
the asymptotic behavior of the vector, the topological number is found to
vanish, implying that the TCOs always come in pairs for given angular
momentum. Furthermore, the relation between the stability and the winding
number is analyzed. In Sec. III, this generic study is applied to the
Schwarzschild black holes. As expected, the result is consistent with the
general analysis. In Sec. IV, the topological study is carried out for the
scalarized Einstein-Maxwell black holes. There exhibit five different
topological situations of TCOs. For small angular momentum, the TCO does not
exist. However for large angular momentum, two pairs TCOs can be observed.
Nevertheless, the topological number always keeps zero. In Sec. V, the dyonic
black hole with a quasi-topological term is investigated. Although the quasi-
topological term leads to the existence of TCOs for arbitrarily small angular
momentum, the total topological charge still vanishes. Finally, we summarize
and discuss our results in Sec. VI. In this paper, the geometrized unit system
is chosen $c=\hbar=G=1$.

##  II General set up

In four-dimensional asymptotic flat spacetime, the static and spherically
symmetric black holes can be described by the following line element

$ds^{2}=g_{tt}dt^{2}+g_{rr}dr^{2}+r^{2}d\Omega_{2}^{2},$ (1)

where $d\Omega_{2}^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ describes unit
2-sphere. In this black hole background, the Lagrangian of a free test
particle reads

$\mathcal{L}=\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-\frac{1}{2}\mu^{2},$
(2)

where the dot denotes the derivative with respect to an affine parameter, and
$\mu^{2}=1$, $0$, $-1$ are for the timelike, null, and spacelike geodesics,
respectively. Via the Legendre transformation, the Hamiltonian of the test
particle can be obtained

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\pi_{\mu}\dot{x}^{\mu}-\mathcal{L}$ (3) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(g_{tt}\dot{t}^{2}+g_{rr}\dot{r}^{2}+g_{\theta\theta}\dot{\theta}^{2}+g_{\phi\phi}\dot{\phi}^{2}\right),$

where
$\pi_{\mu}\equiv\frac{\partial\mathcal{L}}{\partial\dot{x}^{\mu}}=g_{\mu\nu}\dot{x}^{\nu}$
is the corresponding conjugate momentum of the canonical coordinate $x^{\mu}$.

After a simple rearrangement, the Lagrangian (2) is reexpressed as

$g_{rr}\dot{r}^{2}+g_{\theta\theta}\dot{\theta}^{2}=-g_{tt}\dot{t}^{2}-g_{\phi\phi}\dot{\phi}^{2}-\mu^{2}.$
(4)

Since $\dot{r}$ and $\dot{\theta}$ in (4) are related to the radial motion and
polar angle motion, we can regard the left-hand side of the equation as the
kinetic energy term of a test particle. Outside the horizon, both $g_{rr}$ and
$g_{\theta\theta}$ are positive, leading to a non-negative kinetic energy term
as expected. At the same time, the right-hand side can be defined as the
effective potential

$\mathcal{V}=-g_{tt}\dot{t}^{2}-g_{\phi\phi}\dot{\phi}^{2}-\mu^{2}.$ (5)

It should be emphasized that the motion of particles is completely governed by
the effective potential. On the other hand, there are two Killing vectors
$\xi^{\mu}=(\partial_{t})^{\mu}$ and $\psi^{\mu}=(\partial_{\phi})^{\mu}$
associating with two conservation quantities, the energy $E$ and angular
momentum $l$ of the test particle,

$\displaystyle-E$ $\displaystyle=g_{\mu\nu}u^{\mu}\xi^{\nu}=g_{tt}\dot{t},$
(6) $\displaystyle l$
$\displaystyle=g_{\mu\nu}u^{\mu}\psi^{\nu}=g_{\phi\phi}\dot{\phi},$ (7)

where $u^{\mu}$ is the four-velocity of a particle with respect to the affine
parameter.

In terms of energy (6) and angular momentum (7), the effective potential
becomes

$\mathcal{V}=-\frac{E^{2}}{g_{tt}}-\frac{l^{2}}{g_{\phi\phi}}-\mu^{2}.$ (8)

If $\mu^{2}=0$, this expression would precisely reduce to the corresponding
effective potential of photon. Another feature is that the effective potential
is symmetric under $l\to-l$. Without loss of generality, we only focus on the
positive angular momentum $l$. On the other hand, the formula (8) is a
quadratic form of energy $E$, and can be factorized as Wei2022

$\mathcal{V}=-\frac{1}{g_{tt}}(E-e_{1})(E-e_{2}),$ (9)

with

$e_{1}=\sqrt{\frac{-g_{tt}\left(l^{2}+g_{\phi\phi}\mu^{2}\right)}{g_{\phi\phi}}},\quad
e_{2}=-\sqrt{\frac{-g_{tt}\left(l^{2}+g_{\phi\phi}\mu^{2}\right)}{g_{\phi\phi}}}.$
(10)

The timelike circular orbit of a massive particle ($\mu^{2}=1$) requires

$\mathcal{V}(r)=0,\qquad\frac{\partial\mathcal{V}(r)}{\partial r}=0.$ (11)

For given $l$, one can obtain the radius of the TCOs through the second
condition. Then the energy of the particle relating with the TCOs will be
calculated via the first condition. Considering that $e_{2}$ is negative, we
abandon it here. As a result, the conditions (11) turn to

$E=e_{1},\quad\partial_{r}\,e_{1}=0.$ (12)

###  II.1 Vector and asymptotic behaviors

As previously stated, the TCOs are completely controlled by $e_{1}$. In this
subsection, by making use of $e_{1}$, we would like to construct a vector
relating with the topology of TCOs. Then the asymptotic behaviors of the
vector will be examined at the boundaries of ($r$, $\theta$) plane.

Following Ref. Cunha2020 , it is convenient to introduce the following vector

$\phi^{r}=\frac{\partial_{r}e_{1}}{\sqrt{g_{rr}}},\qquad\phi^{\theta}=\frac{\partial_{\theta}e_{1}}{\sqrt{g_{\theta\theta}}}.$
(13)

In a spacetime with $\mathcal{Z}_{2}$ symmetry of $\theta$, the zero points of
vector $\phi$ locate at $\theta=\pi/2$ and $\partial_{r}e_{1}=0$, which
exactly correspond to the equatorial TCOs.

Here, we consider one closed loop $\mathcal{I}$ displayed in Fig. 1 in
($r,\theta$) plane which consists of four line segments, $I_{1}$, $I_{2}$,
$I_{3}$, and $I_{4}$. Under the infinity limit ($r\to\infty$), axis limit
($\theta\to$ 0, $\pi$), and horizon limit ($r\to r_{h}$), the loop
$\mathcal{I}$ tends to the boundary $\partial\Sigma$, where $\Sigma$ denotes
the complete parameter space in ($r$, $\theta$) plane.

Case 1: axis-limit $(\theta\to 0$ and $\pi$, or $\delta\to 0$)

In spherical polar coordinates ($\rho$, $\phi$, $z$), $z=r\cos\theta$ is an
axis perpendicular to the equatorial plane and $\rho=r\sin\theta$ is an
orthogonal projection distance from the fixed origin. Performing the expansion
of the metric function near $\rho=0$, one has Cunha2020

$g_{tt}\sim g_{tt}^{0}+\mathcal{O}(\rho),\quad g_{rr}\sim
g_{rr}^{0}+\mathcal{O}(\rho),\quad g_{\theta\theta}=r^{2},\quad
g_{\phi\phi}=r^{2}\sin^{2}\theta,$ (14)

where $g^{0}_{tt}$ and $g^{0}_{rr}$ are the zeroth terms and are constants.
Using (10), we obtain

$e_{1}\sim\frac{|l|\sqrt{-g_{tt}^{0}}}{r\sin\theta}.$ (15)

The ratio $\phi^{\theta}/\phi^{r}$ is

$\frac{\phi^{\theta}}{\phi^{r}}=\frac{\sqrt{g_{rr}}}{\sqrt{g_{\theta\theta}}}\frac{\partial_{\theta}e_{1}}{\partial_{r}e_{1}}=\frac{\sqrt{g_{rr}}}{\sqrt{g_{\theta\theta}}}\frac{\partial_{\rho}e_{1}}{\partial_{\rho}e_{1}}\frac{\partial_{\theta}\rho}{\partial_{r}\rho}=\frac{\sqrt{g_{rr}}}{\sqrt{g_{\theta\theta}}}\;r\cot\theta\propto\sqrt{g_{rr}^{0}}\cot\theta.$
(16)

Since $|\cot\theta|=\infty$ for $\theta=0$ or $\pi$, it is straightforward to
confirm $\phi^{\theta}\gg\phi^{r}$. The argument of $\phi$ will rely on the
sign of $\phi^{\theta}$

$\arg\phi\propto\arctan\phi^{\theta}\propto\arctan(-\frac{|l|\sqrt{-g_{tt}^{0}}\cot\theta\csc\theta}{r^{2}})=\begin{cases}-\frac{\pi}{2},&\theta\to
0,\\\ +\frac{\pi}{2},&\theta\to\pi.\end{cases}$ (17)

Therefore, the orientation of the vector $\phi$ is always downward along the
path $I_{4}$ and upward along path $I_{2}$ for axis limit $\delta\to 0$.

Case 2: horizon-limit ($r\to r_{h}$)

Figure 1: (a) Closed loop $\mathcal{I}$ and its asymptotic boundary
$\partial\Sigma$ in spherical symmetric and asymptotic flat spacetime. (b)
Path $I_{3}$ for horizon limit in spherical coordinate. Vector $\vec{R}$ is
used to describe path $I_{3}$.

Now, let us move to the horizon limit. In this case, path $I_{3}$ approaches
horizon when $r_{0}\to r_{h}$ shown in Fig. 1. The parametric equation
$\vec{R}(\theta)$ corresponding to path $I_{3}$ is given by

$\vec{R}(\theta)=\frac{\vec{x}}{\sin(\theta)}.$ (18)

Taking $\vec{x}\to 0$, the horizon limit will be realized. The argument of
$\phi$ is

$\arg\phi=\arctan\frac{\phi^{\theta}}{\phi^{r}}=\arctan\left(\frac{\sqrt{g_{rr}}}{\sqrt{g_{\theta\theta}}}\frac{d\vec{r}}{d\theta}\right)\propto\arctan\left(\frac{\sqrt{g^{0}_{rr}}}{r}\frac{d\vec{R}(\theta)}{d\theta}\right),$
(19)

where we have used $\vec{r}=\vec{r}_{h}+\vec{R}$. Performing the derivative of
$\vec{R}(\theta)$, one gets

$\arg\phi=\arctan\left(-\frac{\sqrt{g_{rr}^{0}}}{r}x\cot\theta\csc\theta\right)\bigg{|}_{\vec{x}\to
0}=0,$ (20)

which obviously implies that $\phi^{\theta}\ll\phi^{r}$. In order to determine
the direction of $\phi$, we clearly show the component $\phi^{r}$ by using
(13)

$\phi^{r}=\frac{2l^{2}g_{tt}\csc^{2}\theta-
rg_{tt}^{\prime}(r)\left(l^{2}\csc^{2}\theta+\mu^{2}r^{2}\right)}{2r^{2}\sqrt{-g_{tt}\;g_{rr}\left(l^{2}\csc^{2}\theta+\mu^{2}r^{2}\right)}}.$
(21)

Notice that $g_{tt}>0$ and $<0$ inside and outside horizon, it is reasonable
to infer that $g_{tt}(r_{h})=0$ and $g_{tt}^{\prime}(rh)<0$ at the
neighborhood of the horizon. So $\phi^{r}$ becomes

$\phi^{r}\propto\frac{-rg_{tt}^{\prime}(r)}{\sqrt{-g_{tt}\;g_{rr}}}\bigg{|}_{r\to
r_{h}}>0.$ (22)

Combined (20) and (22), it is easy to know that the direction of vector $\phi$
is to the right on the path $I_{3}$.

Case 3: infinity limit ($r\to\infty$)

Finally, let us examine the infinity limit corresponding to the path $I_{1}$.
Given that the spacetime is asymptotic flat, the metric can be expanded as

$g_{tt}\sim-1+\frac{2m}{r}+\mathcal{O}(\frac{1}{r^{2}}),\quad g_{rr}\sim
1+\frac{2m}{r}+\mathcal{O}(\frac{1}{r^{2}}),\quad g_{\theta\theta}=r^{2},\quad
g_{\phi\phi}=r^{2}\sin^{2}\theta,$ (23)

when $r\to\infty$. Positive constant $m$ is relevant to the mass of a
spherically symmetric black hole. Employing (10) and (13), we obtain

$\displaystyle
e_{1}\sim\sqrt{\frac{(r-2m)(r^{2}+l^{2}\csc^{2}\theta)}{r^{3}}},$ (24)
$\displaystyle\phi^{\theta}\sim\frac{-l^{2}\cot\theta\csc^{2}\theta}{r^{3}}+\mathcal{O}(\frac{1}{r^{4}}),\qquad\phi^{r}\sim\frac{m}{r^{2}}+\mathcal{O}(\frac{1}{r^{3}}).$
(25)

The argument of the vector $\phi$ is evaluated as

$\arg\phi\propto\arctan\left(\frac{-l^{2}\cot\theta\csc^{2}\theta}{m\,r}\right)\bigg{|}_{r\to\infty}\sim
0.$ (26)

Considering that $\phi^{r}>0$ and $\arg\phi=0$, the direction of the vector
$\phi$ on the path $I_{1}$ is to the right as the same as the horizon limit.

In summary, we obtain the direction of the vector $\phi$ at the boundaries
$I_{1-4}$ of ($r$, $\theta$) plane sketched in Fig. 2 in spherically symmetric
spacetime. As we shall show, this result has an important application on the
topological approach of the TCOs.

Figure 2: (a) The direction (purple arrows) of the vector $\phi$ at the
asymptotic boundaries. (b) Function $\phi^{r}(r)$. The blue dots stand for the
locations of TCOs. One is unstable with the winding number $-1$ and the other
is stable with winding number $+1$.

###  II.2 Topology, stability, and bifurcation point

In this subsection, we will study the topological properties of TCOs by
utilizing Duan’s $\phi$-mapping topological current theory, and then examine
the relationship between the stability and the topological charge of TCOs.
Furthermore, the marginally stable circular orbits (MSCOs) acting as
bifurcation points will be explored.

In Duan’s $\phi$-mapping topological current theory Duan1984 , a moving point-
like particle corresponds to the zero point of the vector $\phi$. Moreover,
the conservation of particle number is guaranteed by the total topological
charge. As shown above, the TCOs exactly locate at the zero points of vector
$\phi$. Thus, we can endow each TCO with a topological charge. This allows us
to establish the topology for the TCOs. Then, the topological properties will
be uncovered as expected. Following Ref. Duan1984 , the topological current
relating with the topological charge reads

$j^{\mu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}\frac{\partial
n^{a}}{\partial x^{\nu}}\frac{\partial n^{b}}{\partial x^{\rho}},$ (27)

where $x^{\mu}=(t,r,\theta)$ and
$n^{a}=(\frac{\phi^{r}}{|\phi|},\frac{\phi^{\theta}}{|\phi|})$ is the unit
vector of $\phi$. It is easy to check that this current is conserved, i.e.,
$\partial_{\mu}j^{\mu}$=0. After a simple algebra, one reaches

$j^{\mu}=\delta^{2}(\phi)J^{\mu}\left(\frac{\phi}{x}\right),$ (28)

with Jacobi tensor
$\epsilon^{ab}J^{\mu}\left(\frac{\phi}{x}\right)=\epsilon^{\mu\nu\rho}\partial_{\nu}\phi^{a}\partial_{\rho}\phi^{b}$.
Significantly, $j^{\mu}$ is nonzero only at the zero points of the vector
$\phi$. Denoting the $i$-th zero point as $\vec{x}=\vec{z}_{i}$, we have the
density $j^{0}$ of the topological current Duan1984

$j^{0}=\sum_{i}^{N}=\beta_{i}\eta_{i}\delta^{2}(\vec{x}-\vec{z}_{i}),$ (29)

where $\beta_{i}$ and $\eta_{i}$ are the Hopf index and Brouwer degree of the
$i$-th zero point. By integrating the density $j^{0}$ of the topological
current over the giving region $\Sigma$, one obtains the topological number

$W=\int_{\Sigma}j^{0}d^{2}x=\sum_{i}^{N}\beta_{i}\eta_{i}=\sum_{i}^{N}w_{i}.$
(30)

Here $w_{i}$ denotes the winding number of the $i$-th zero point of the vector
$\phi$ enclosed in $\Sigma$. For each zero point, there is a winding number
acting as a topological charge. Employing it, these zero points can be
classified by the values of their winding numbers. This gives the local
topological property for the zero point. While if $\Sigma$ covers all the
possible regions, the global topological property will be disclosed by the
number $W$.

Considering the change of the vector direction, the topological number can
also be calculated by

$\displaystyle W=\frac{1}{2\pi}\oint_{\mathcal{I}}\mathrm{d}\Omega,$ (31)

where $\Omega$ is a deflection angle of vector $\phi$ along the
counterclockwise closed path $\mathcal{I}=\partial\Sigma$. If taking $\Sigma$
covering all the region of ($r$, $\theta$) plane, we easily have

$\displaystyle W=0,$ (32)

by making use of the asymptotic behaviors of vector $\phi$ sketched in Fig. 2.
This result strongly implies that, for each angular momentum $l$, the TCOs
always come in pairs for the static, spherically symmetric, and asymptotic
flat black holes. This is quite different from the topology of LRs, which have
topological number $W=-1$ indicating that there exists at least one standard
LR.

Now we turn to investigate the local stability of the TCOs. Compared with LR,
the TCO depends not only on the black hole parameter, but also on the energy
and angular momentum of the particle. Solving
$\mathcal{V}(r)=\partial_{r}\mathcal{V}(r)=0$, one obtains the energy and
angular momentum of TCOs on the equatorial plane Witek

$l_{t}=\sqrt{\frac{r^{3}g_{tt}^{\prime}(r)}{rg_{tt}^{\prime}(r)-2g_{tt}(r)}},\qquad
E_{t}=\sqrt{\frac{2g_{tt}(r)^{2}}{rg_{tt}^{\prime}(r)-2g_{tt}(r)}}.$ (33)

More importantly, the stability of TCOs can be determined by the second
derivative of the effective potential

$\mathcal{V}^{\prime\prime}(r)=\frac{(e_{1}(r)-e_{2}(r))}{g_{tt}(r)}e_{1}^{\prime\prime}(r)=\frac{(e_{1}(r)-e_{2}(r))\sqrt{g_{rr}}}{g_{tt}(r)}\partial_{r}\phi^{r}(r).$
(34)

Since $e_{1}>0$, $e_{2}<0$, and $g_{tt}<0$ outside the horizon, we have

$\begin{cases}\mathcal{V}^{\prime\prime}(r)>0\Leftrightarrow\text{unstable},&\text{if
}\partial_{r}\phi^{r}(r)<0,\\\
\mathcal{V}^{\prime\prime}(r)<0\Leftrightarrow\text{stable},&\text{if
}\partial_{r}\phi^{r}(r)>0.\end{cases}$ (35)

This states that the stable and unstable TCOs, respectively, correspond to
$\partial_{r}\phi^{r}>0$ and $\partial_{r}\phi^{r}<0$. So, one can gain the
local stability of TCOs based on this result.

Next, let us examine the relationship between the stability and the
topological charge for the TCOs. As we shown, for each angular momentum, the
TCOs always come in pairs. For convenience, we consider that there are two
TCOs denoted by TCO1 and TCO2 with radii $r_{1}$ and $r_{2}$ in Fig. 2.
Without loss of generality, we assume $r_{1}<r_{2}$. Further considering that
$\phi^{r}(r)$ is positive at the black hole horizon and infinity, one obtains
the result that $\phi^{r}$ turns from positive to negative at $r_{1}$ and from
negative to positive at $r_{2}$. As a result, the following result holds

$\partial_{r}\phi^{r}(r_{1})<0\quad\text{and}\quad\partial_{r}\phi^{r}(r_{2})>0.$
(36)

Combining with (35), it is easy to know that TCO1 is unstable while TCO2 is
stable. In particular, following the study of Ref. Wei2022 , one can find the
winding number of TCO1 and TCO2 are $-1$ or $+1$, respectively. Therefore, the
TCO with positive or negative winding number is locally stable or unstable.

If more than one pair TCOs is present, we can further obtain an interesting
generalization that the TCOs nearest and farthest to the horizon have negative
and positive winding numbers, and thus they are locally unstable and stable,
respectively. More interestingly, any two arbitrary nearby TCOs have the
opposite winding numbers.

Among the TCOs, there is a special kind, the MSCO, which satisfies Delgado2021

$\mathcal{V}(r)=0,\quad\mathcal{V}^{\prime}(r)=0,\quad\mathcal{V}^{\prime\prime}(r)=0,$
(37)

or equivalently,

$E=e_{1},\quad\partial_{r}e_{1}(r)=0,\quad\partial_{r,r}e_{1}(r)=0.$ (38)

Solving the second condition, we get $l=l_{t}(r)$ corresponding to the zero
points of $\phi$, and its explicit form is given in (33). Formulating the
third condition, one has $\partial_{r}l_{t}(r)=0$, which exactly gives the
MSCO with ($l_{t}^{*},r^{*}$). As shown in Ref. Wei2022 , the MSCOs can be
treated as bifurcation points Fu2000 . Near it, we can carry out the Taylor’s
expansion for $l_{t}(r)$

$l_{t}=l_{t}^{*}+\frac{1}{2}\frac{d^{2}l_{t}}{d\,r^{2}}\bigg{|}_{r^{*}}(r-r^{*})^{2}+\mathcal{O}\left((r-r^{*})^{3}\right).$
(39)

Supposing that expansion coefficient $l_{t}^{\prime\prime}(r^{*})>0$, one has

$l>l^{*}_{t}\Leftrightarrow\text{Exist TCOs},\quad
l<l^{*}_{t}\Leftrightarrow\text{No TCOs},$ (40)

which implies that the TCOs can exist when $l>l^{*}$, while absent when
$l<l^{*}$. So such MSCO corresponds to the generated point. Conversely,
negative $l_{t}^{\prime\prime}(r^{*})<0$ leads to the annihilated point.

Here, we would like to address some comments on the MSCO and ISCO. Both them
are characteristic kinds of TCOs. ISCO is the stable TCO with the smallest
radius among all the stable TCOs. While MSCO denotes the stable TCO, which has
the smallest radius and can be continuously connected to spatial infinity by a
set of stable TCOs. In general, the MSCO and ISCO coincide with each other,
for example in Schwarzschild or Kerr black holes. However, for the hairy black
holes or generic ultracompact objects, they do not coincide Delgado2021 .

To summarize, the study of the topological charge of TCOs reveals that, for
each angular momentum, the TCOs come in pairs and possess opposite values of
the winding numbers. Positive or negative winding number corresponds to that
the TCO is local stable or unstable. Moreover, whether the TCOs present or
not, the topological number $W$ always vanishes. Of particular interest, MSCOs
act as the generated points with positive $l_{t}^{\prime\prime}(r^{*})$, or
annihilated points with negative $l_{t}^{\prime\prime}(r^{*})$.

In the following sections, we would like to examine the topology for three
characteristic black hole examples, and to see whether our general results
hold.

##  III Schwarzschild black holes

In this section, we shall carry out the topological study of the TCOs for the
Schwarzschild black hole.

###  III.1 Effective potential and asymptotic behaviors

The Schwarzschild black hole is a static spherically symmetric vacuum solution
of the Einstein field equation, and it can be described by the element (1)
with

$g_{tt}=-\left(1-\frac{2M}{r}\right),\quad
g_{rr}=\left(1-\frac{2M}{r}\right)^{-1},$ (41)

where the black hole mass $M$ corresponds to the event horizon as $r_{h}=2M$.

From (8), the effective potential $\mathcal{V}(r)$ reads

$\mathcal{V}(r)=-\frac{l^{2}\csc^{2}\theta}{r^{2}}-\frac{rE^{2}}{2M-r}-\mu^{2}.$
(42)

Reformulating it, one gets $e_{1}$ and $e_{2}$ via the equation (10)

$e_{1,2}=\pm\sqrt{\frac{(r-2M)(r^{2}\mu^{2}+l^{2}\csc^{2}\theta)}{r^{3}}}.$
(43)

Following the definition (13), the vector $\phi$ is

$\phi^{r}=\frac{Mr^{2}+(3M-r)l^{2}\csc^{2}\theta}{r^{3}+\sqrt{r^{2}+l^{2}\csc^{2}\theta}},\quad\phi^{\theta}=-\frac{l^{2}\cot\theta\csc^{2}\theta\sqrt{r-2M}}{r^{5/2}\sqrt{r^{2}+l^{2}\csc\theta}},$
(44)

where we have taken $\mu^{2}=1$ for the timelike geodesics.

Here we would like to examine the asymptotic behavior of the vector $\phi$ in
Schwarzschild black hole background and confirm whether it is consistent with
our analysis in Sec. II.

Case 1: axis-limit $(\theta\to 0$ and $\pi$)

First, let us consider the limit $\theta\to 0$. Expanding the vector near
$\theta=0$, we have

$\displaystyle\phi^{r}$ $\displaystyle=\frac{l(3M-r)}{\theta
r^{3}}+\frac{\theta\left(-3Mr^{2}+3Ml^{2}+3r^{3}-rl^{2}\right)}{6r^{3}l}+\mathcal{O}\left(\theta^{3}\right),$
(45) $\displaystyle\phi^{\theta}$
$\displaystyle=-\frac{l\sqrt{r-2M}}{\theta^{2}r^{5/2}}+\frac{\sqrt{r-2M}\left(3r^{2}+l^{2}\right)}{6r^{5/2}l}-\frac{\theta^{2}\sqrt{r-2M}\left(45r^{4}+30r^{2}l^{2}-7l^{4}\right)}{120r^{5/2}l^{3}}+\mathcal{O}\left(\theta^{3}\right).$
(46)

Abandoning the higher $\theta$ term, the ratio $\phi^{\theta}/\phi^{r}$ can be
calculated as

$\frac{\phi^{\theta}}{\phi^{r}}=-\frac{\sqrt{r(r-2M)}}{\theta(3M-r)}\bigg{|}_{\theta\to
0}\to\infty.$ (47)

Combining with $\phi^{\theta}<0$, one can see that the direction of vector
$\phi$ when $\theta\to 0$ is downward which exactly agrees with the above
result (17).

On the other hand, expanding the vector $\phi$ near $\theta=\pi$, one gets

$\displaystyle\phi^{r}$
$\displaystyle=-\frac{l(3M-r)}{(\theta-\pi)r^{3}}+\frac{(\theta-\pi)\left(3Mr^{2}-3Ml^{2}-3r^{3}+rl^{2}\right)}{6r^{3}l}+\mathcal{O}\left((\theta-\pi)^{3}\right),$
(48) $\displaystyle\phi^{\theta}$
$\displaystyle=\frac{l\sqrt{r-2M}}{(\theta-\pi)^{2}r^{5/2}}-\frac{\sqrt{r-2M}\left(3r^{2}+l^{2}\right)}{6r^{5/2}l}+\frac{(\theta-\pi)^{2}\sqrt{r-2M}\left(45r^{4}+30r^{2}l^{2}-7l^{4}\right)}{120r^{5/2}l^{3}}+\mathcal{O}\left((\theta-\pi)^{3}\right).$
(49)

The ratio reads

$\frac{\phi^{\theta}}{\phi^{r}}=\frac{\sqrt{r(r-2M)}}{(\pi-\theta)(3M-r)}\bigg{|}_{\theta\to\pi}\to\infty.$
(50)

As a result, the direction of vector $\phi$ is upward when $\theta=\pi$ by
considering $\phi^{\theta}>0$.

Case 2: horizon-limit ($r\to r_{h}$)

Taking $r=r_{h}+\epsilon$, the horizon limit will turn to $\epsilon\to 0$.
Under this limit, the vector reads

$\displaystyle\phi^{r}$
$\displaystyle=\frac{\sqrt{4M^{2}+l^{2}\csc^{2}\theta}}{8M^{2}}+\frac{\epsilon\left(-8M^{2}-5l^{2}\csc^{2}\theta\right)}{16M^{3}\sqrt{4M^{2}+l^{2}\csc^{2}\theta}}+\mathcal{O}(\epsilon^{2}),$
(51) $\displaystyle\phi^{\theta}$
$\displaystyle=\frac{-\epsilon^{1/2}\left(l^{2}\cot\theta\csc^{2}\theta\right)}{4M^{5/2}\sqrt{8M^{2}+2l^{2}\csc^{2}\theta}}+\mathcal{O}(\epsilon^{3/2}).$
(52)

The corresponding outcome of $\phi$ is

$\frac{\phi^{\theta}}{\phi^{r}}=-\frac{\sqrt{2}l^{2}\sqrt{\epsilon}\cot(\theta)\csc^{2}(\theta)}{\sqrt{M}\left(l^{2}\csc^{2}(\theta)+4M^{2}\right)}\bigg{|}_{\epsilon\to
0}=0,\quad\text{and}\quad\phi^{r}>0,$ (53)

leading to the result that the direction of vector $\phi$ at the event horizon
is to the right.

Case 3: infinity limit ($r\to\infty$)

When $r\to\infty$, one has

$\displaystyle\phi^{r}$
$\displaystyle=\frac{M}{r^{2}}-\frac{l^{2}\csc\theta^{2}}{r^{3}}+\mathcal{O}\left(\frac{1}{r^{4}}\right),$
(54) $\displaystyle\phi^{\theta}$
$\displaystyle=\frac{-l^{2}\cot\theta\csc\theta^{2}}{r^{3}}+\mathcal{O}\left(\frac{1}{r^{4}}\right).$
(55)

After a simple algebra calculation, we have

$\frac{\phi^{\theta}}{\phi^{r}}=\frac{-l^{2}\cot\theta\csc^{2}\theta}{Mr}\bigg{|}_{r\to\infty}=0,\quad\text{and}\quad\phi^{r}>0,$
(56)

indicating that the direction of vector $\phi$ at infinity is to the right.

In summary, we see that the asymptotic behaviors of the vector is the same as
that we obtained above for general black hole backgrounds. Therefore, we have
the topological number of TCOs for the Schwarzschild black hole

$W=0.$ (57)

So, for a certain angular momentum, if there are TCOs, they must appear in
pairs.

###  III.2 Topology of TCOs and evolution of control parameter

Solving the zero points of the vector, we obtain the angular momentum of the
TCOs

$l_{t}=\sqrt{\frac{r^{2}M}{r-3M}},$ (58)

for the Schwarzschild black holes. Further solving $\partial_{r}l_{t}=0$, one
obtains the radius and angular momentum of the ISCO or MSCO

$r_{ISCO}=6M\quad\text{and}\quad l_{ISCO}=2\sqrt{3}M.$ (59)

Note that for the Schwarzschild black hole, the ISCO and MSCO coincide, and
thus we will not distinguish them here. In what follows, we would like to
figure out three characteristic cases according to the angular momentum:

  * •

$0\leq l<l_{ISCO}$,

  * •

$l=l_{ISCO}$,

  * •

$l_{ISCO}<l<\infty$.

For simplicity, we shall take $M=1$ for our following study.

First, let us consider $0<l=3.2<l_{ISCO}$. The effective potential
$\mathcal{V}(r)$ is plotted in Fig. 3 when $E=0.9$, $0.92$, $0.94$, and $0.96$
from the bottom to top, respectively. For each curve, we find that,
$\mathcal{V}(r)$ decreases monotonically with $r$. So for this case, there is
no TCO. Also, we show the unit vector field $n$ on a portion of the
$\theta$-$r$ plane in Fig. 3. Obviously, the vector is outwards at $\theta=0$
and $\pi$. On the equatorial plane $\theta=\pi/2$, the direction of the vector
is always toward to the right, and no zero point can be found. So the
topological number must vanish, i.e., $W=0$.

Figure 3: The effective potential $\mathcal{V}(r)$ and unit vector field of
$\phi$ for the Schwarzschild black holes for the case one with $l=3.2$. (a)
$\mathcal{V}(r)$ as a function of $r$ with $E=0.9$, $0.92$, $0.94$, and $0.96$
from the bottom to top. (b) The unit vector field $n$ on a portion of the
$\theta$-$r$ plane.

Figure 4: The effective potential $\mathcal{V}$, unit vector field $n$ and
deflection angle $\Omega(\psi)$ for the Schwarzschild black holes. (a)
Effective potential $\mathcal{V}$ for case two with $l=l_{ISCO}$. The energy
$E=0.91$, $0.925$, $0.9428$ and $0.96$ from bottom to top. (b) Effective
potential $\mathcal{V}$ for case three with $l=3.7$. The energy $E=0.942$,
$0.9535$, $0.9649$ and $0.978$ from bottom to top. (c) The unit vector field
$n$ on a portion of the $\theta$-$r$ plane with $l=l_{ISCO}$. “$IP_{1}$”
denotes the ISCO at $r_{ISCO}$=6. The closed loop $C_{1}$ has parametric
coefficients ($c_{0}$, $c_{1}$, $c_{2}$)=($r_{ISCO}$, 0.4, 0.5). (d) The unit
vector field $n$ on a portion of the $\theta$-$r$ plane with $l=3.7$.
“$TP_{1}$” and “$TP_{2}$” are two TCOs located at $r_{t}$=4.44 and 9.25. The
closed loops $C_{2}$ and $C_{3}$ have parametric coefficients ($c_{0}$,
$c_{1}$, $c_{2}$)=(4.44, 0.4, 0.5) and (9.25, 1.2, 0.2). (e) Deflection angle
$\Omega(\psi)$ along $C_{1}$. (f) Deflection angle $\Omega(\psi)$ along
$C_{2}$ and $C_{3}$.

For the second characteristic case, we take $l=l_{ISCO}=3.4641$. The effective
potential and the unit vector field $n$ on a portion of the $\theta$-$r$ plane
are shown in Figs. 4 and 4. The ISCO exactly locates at $r=6$ with
$E=2\sqrt{2}/3$. From Fig. 4, it can be found that for $E>2\sqrt{2}/3$, no
extremal point is present for the effective potential $\mathcal{V}(r)$, while
two extremal points are present for $E<2\sqrt{2}/3$. In Fig. 4, it is easy to
find that the unit vector field $n$ has a similar behavior as that given in
Fig. 3. Although the direction of $n$ seems to be always towards to the right
on the equatorial plane, it indeed vanishes at the point marked with the dot.
Nevertheless, $\partial_{r,r}\mathcal{V}=0$ is satisfied. Here we wonder
whether the winding number still vanishes as that case of $l=3.2$. In order to
answer this question, we turn to evaluate its winding number by constructing a
closed loop $C_{1}$ with following parameterized form Wei2021

$\begin{cases}r=c_{1}\cos\psi+c_{0},\\\
\theta=c_{2}\sin\psi+\frac{\pi}{2},\end{cases}$ (60)

where ($c_{0}$, $c_{1}$, $c_{2}$)=(3.4641, 0.4, 0.5). Note that all the closed
loops will be parameterized with this form while with different values of
($c_{0}$, $c_{1}$, $c_{2}$). Then along the closed loop, the deflection angle
$\Omega(\psi)$ can be calculated by

$\Omega(\psi)=\int_{C}\epsilon_{ab}\,n^{a}\,\mathrm{d}n^{b}.$ (61)

The winding number shall be $w=\Omega(2\pi)/2\pi$ as expected. We list the
deflection angle $\Omega(\psi)$ in Fig. 4. With the increase of $\psi$ from 0
to $2\pi$, we see that $\Omega(\psi)$ first increases, then decreases, and
finally increases. Nevertheless, $\Omega(2\pi)$ vanishes, strongly implying
that the winding number $w=0$ for the ISCO.

Next we turn our attention to $l=3.7>l_{ISCO}$. The effective potential
$\mathcal{V}$ is plotted in Fig. 4. For different values of energy, two
extremal points can be observed on each curve. However, they do not denote the
TCOs unless they have vanished potential. According to it, we find that there
are two TCOs marked with dots at $r$=4.44 and 9.25, which correspond to
$E=0.9535$ and $0.9649$, respectively. Also, from the behaviors of the
effective potential, one easily reaches that the TCO at $r=4.44$ is local
unstable, while the other one is stable. As we shall see, this result will
also be confirmed by their winding numbers.

The unit vector field $n$ is also described on a portion of the $\theta$-$r$
plane in Fig. 4. Obviously, there are two zero points, which are exactly
consistent with these shown in Fig. 4. In order to calculate their winding
numbers, we construct two closed loops $C_{2}$ and $C_{3}$ parametrized by the
form (60) with ($c_{0}$, $c_{1}$, $c_{2}$)=(4.44, 0.4, 0.5) and (9.25, 1.2,
0.2). Then we show $\Omega(\psi)$ for them in Fig. 4. With the increase of
$\psi$, $\Omega(\psi)$ increases along $C_{3}$, while decreases $C_{2}$. The
winding number is easily got, for example $w$=1 for $TP_{2}$ and -1 for
$TP_{1}$, which implies that a positive or negative winding number corresponds
to a local stable or unstable TCO as expected. As a result, the total
topological number $W$=+1-1=0, keeping the same with the first two cases.

After studying the topological charge of ISCO and TCO, we concentrate on the
evolution of TCO radius $r_{t}(l)$, from the idea that the angular momentum
can be treated as a time control parameter Wei2022 . Expanding the angular
momentum $l_{t}$ of the zero points of the vector $\phi$ at $r_{ISCO}$, one
has

$\displaystyle l_{t}$
$\displaystyle=2\sqrt{3}+\frac{1}{12\sqrt{3}}(r-6)^{2}+\mathcal{O}\left((r-6)^{3}\right).$
(62)

Since $l_{t}^{\prime\prime}(r_{ISCO})=1/(12\sqrt{3})>0$, the bifurcation point
must be a generated point. In order to make it more clear, we display the
radius $r_{t}$ of the TCOs as a function of $l$ in Fig. 5. For small $l$, no
branch of the TCO can be found. While after the ISCO point, two TCO branches
emerge from the ISCO with the opposite winding numbers. Such behavior
obviously addresses that the ISCO is a generated point with the angular
momentum. More interestingly, whether there are TCOs or not, the total
topological number always vanishes for arbitrary values of the angular
momentum. Furthermore, we sketch the behavior of the winding number and
topological number in Fig. 5.

Figure 5: (a) The evolution of the TCO radius $r_{t}$ as a function of angular
momentum $l$ for the Schwarzschild black holes. Obviously, the point ${\rm
IP}_{1}$ is a generated point. The signs $\pm$ denote the winding number are
$\pm 1$ for these TCO branches. (b) The topological number (black line) and
winding number of TCO branches (green lines). For different values of $l$, we
see the topological number $W$ always vanishes.

In summary, we in this section show that the topological number $W$=0 of the
TCOs in the Schwarzschild black hole background. Positive or negative winding
number indeed indicates that the TCO is local stable or unstable. These
results exactly support our general conclusion in Sec. II.

##  IV Scalarized Einstein-Maxwell black holes

In this section, we would like to consider another characteristic example, the
scalarized Einstein-Maxwell black holes. Even though their total topological
number is the same as the Schwarzschild black holes, the topological
configurations of the TCOs are quite different.

###  IV.1 Scalarized Einstein-Maxwell black holes

The Einstein-Maxwell-scalar model describes a real scalar field $\phi$
coupling to Einstein’s gravity and Maxwell’s electromagnetism. The scalarized
Einstein-Maxwell black hole can be described by the following action Silva2017
; Konoplya2019

$\mathcal{S}=\int
d^{4}x\sqrt{-g}(R-2g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-f(\phi)F_{\mu\nu}F^{\mu\nu}),$
(63)

where $R$ and $\phi$ are the Ricci scalar and scalar field. Maxwell tensor
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. The last term
represents a non-minimal coupling term between scalar field and the Maxwell
electric field.

The line element of the spherically symmetric scalarized Einstein-Maxwell
black holes is assumed to be

$ds^{2}=-N(r)e^{-2\delta(r)}dt^{2}+\frac{dr^{2}}{N(r)}+r^{2}d\Omega^{2},$ (64)

where the metric function $N(r)$ and $\delta(r)$ are only radially dependent.
The four-potential $A_{\mu}$ of the electromagnetic field is

$A_{\mu}(x)dx^{\mu}=V(r)dr.$ (65)

The effective Lagrangian in the Einstein-Maxwell-scalar model are
Fernandes2019

$\mathcal{L}_{eff}=-\frac{1}{2}e^{-\delta}\left(rN^{\prime}+N-1\right)-\frac{1}{2}e^{-\delta}r^{2}N\phi^{\prime}(r)^{2}+\frac{1}{2}e^{\delta}f(\phi)r^{2}V^{\prime}(r)^{2}.$
(66)

Here the radially dependent is omitted for notation simplicity. By making use
of the Lagrangian, the equations of motion are Fernandes2019

$\displaystyle
N^{\prime}-\frac{1-N}{r}=-\frac{Q^{2}}{r^{3}f(\phi)}-r(\phi^{\prime})^{2}N,\;\qquad\delta^{\prime}=-r(\phi^{\prime})^{2},$
(67)
$\displaystyle(r^{2}N\phi^{\prime})^{\prime}=-\frac{f^{\prime}(\phi)Q^{2}}{2f^{2}(\phi)r^{2}}-r^{3}(\phi^{\prime})^{3}N,\qquad
V^{\prime}=\frac{Q}{f(\phi)r^{2}}e^{-\delta}.$ (68)

Further, we assume this black hole spacetime is asymptotic flat

$\lim_{r\to\infty}N(r)=1,\qquad\lim_{r\to\infty}\delta(r)=0.$ (69)

To numerically solve the differential equations (67) and (68), the exponential
coupling are chosen as Guo2021b

$f(\phi)=e^{\alpha\phi^{2}},\quad\text{where $\alpha=0.9$},$ (70)

and the value of scalar field at the event horizon

$\phi({r_{h}})=2.25859.$ (71)

The numerical results of the metric functions $N(r)$, $\delta(r)$, and
$\phi(r)$ are exhibited in Fig. 6 by taking $r_{h}=1$. We observe that
$\delta(r)$ and $\phi(r)$ decrease with $r$, while $N(r)$ shows a nonmonotonic
behavior. After obtaining these functions, the explicit form of
$\mathcal{V}(r)$ and energy $e_{1}$, as well as the vector $\phi$ will be
given. However, we cannot express them analytically, so we will not show them
here. Other studies concerning the black hole solutions and potential
observations of multi-sphoton sphere can also be found in Refs. Guo2021b ;
Fernandes ; Pappas ; Gan2021 ; Guo2022ba

Figure 6: The numerical results of functions $N(r)$, $\phi(r)$, and
$\delta(r)$ for scalarized Einstein-Maxwell black holes.

###  IV.2 Topology of TCOs and winding number

Now, let us turn to examine the topology of the TCO for the scalarized
Einstein-Maxwell black holes.

As we have shown in the last section, ISCOs or MSCOs acting as bifurcation
points, have an important impact on the topological configurations of TCOs. So
it is key to determine them. By solving the conditions
$\mathcal{V}(r)=\mathcal{V}^{\prime}(r)=0$ and
$\mathcal{V}^{\prime\prime}(r)=0$, or alternatively (38),

$\phi^{r}(r)=0\quad\text{and}\quad\phi^{r\prime}(r)=0,$ (72)

we obtain the locations of ISCO and MSCO, which are given by

$\displaystyle r_{ISCO}$ $\displaystyle=2.3294\quad\text{and}\quad
l_{ISCO}=6.4043,$ $\displaystyle r_{MSCO}$
$\displaystyle=15.753\quad\text{and}\quad l_{MSCO}=12.314.$ (73)

Further, they must satisfy

$\mathcal{V}^{\prime\prime}(r)=0\quad\text{and}\quad\mathcal{V}^{\prime\prime\prime}(r)<0,$
(74)

or,

$\phi^{r\prime}(r)=0\quad\text{and}\quad\phi^{r\prime\prime}(r)>0.$ (75)

A simple algebra gives

$\mathcal{V}^{\prime\prime\prime}(r)=\frac{\left(e_{1}(r)-e_{2}(r)\right)\sqrt{g_{rr}(r)}\phi_{r}^{\prime\prime}(r)}{g_{tt}},$
(76)

which implies that
$\mathcal{V}^{\prime\prime\prime}(r)\sim-\phi_{r}^{\prime\prime}(r)$. Note
that the ISCO also satisfies condition (74), but it has the smallest radius
among all stable TCOs. Besides, the significant discrepancy between the ISCO
and MSCO is that the MSCO can be continuously connected to spatial infinity by
a set of stable TCOs, whereas the ISCO fails. If only one ISCO occurs such as
for a Schwarzschild black hole, the ISCO will coincide with MSCO.

To check whether the ISCO and MSCO hold the condition (75), we calculate the
derivatives of $\phi^{r}(r)$ at the $r_{ISCO}$ and $r_{MSCO}$

$\begin{split}\phi^{r\prime}(r_{ISCO})&=0,\quad\phi^{r\prime\prime}(r_{ISCO})=0.0455786,\\\
\phi^{r\prime}(r_{MSCO})&=0,\quad\phi^{r\prime\prime}(r_{MSCO})=0.00007001.\end{split}$
(77)

To show this result more clearly, we show $\phi^{r}(r)$ and its first and
second derivatives in Fig. 7.

Figure 7: The function $\phi^{r}(r)$ and its derivatives $\phi^{r\prime}(r)$
and $\phi^{r\prime\prime}(r)$ for scalarized Einstein-Maxwell black holes. (a)
$l=l_{ISCO}$. (b) $l=l_{MSCO}$. The black dots ${\rm IP}_{2}$ and ${\rm
MP}_{1}$ stand for the locations of the ISCO or MSCO, respectively.

According to the values of $l_{ISCO}$ and $l_{MSCO}$, we divide the parameter
region of the angular momentum into the following five types:

  * •

$0\leq l<l_{ISCO}$,

  * •

$l=l_{ISCO}$,

  * •

$l_{ISCO}<l<l_{MSCO}$,

  * •

$l=l_{MSCO}$,

  * •

$l_{MSCO}<l$.

Figure 8: Case two with $l=l_{ISCO}$ for the scalarized Einstein-Maxwell black
holes. (a) The unit vector field $n$ on a portion of the $\theta$-$r$ plane.
“$IP_{2}$” denotes the ISCO of the black hole with $r$=2.329. The closed loops
$C_{4}$ has parametric coefficients ($c_{0}$, $c_{1}$, $c_{2}$)=(2.329, 0.6,
0.3). (b) Deflection angle $\Omega(\psi)$ along $C_{4}$.

Figure 9: Case three with $l_{ISCO}<l=10<l_{MSCO}$ for the scalarized
Einstein-Maxwell black holes. (a) The unit vector field $n$ on a portion of
the $\theta$-$r$ plane. “$TP_{3}$” and “$TP_{4}$” denote the TCOs at $r$=1.944
and 3.189. The closed loops $C_{5}$ and $C_{6}$ have parametric coefficients
($c_{0}$, $c_{1}$, $c_{2}$)=(1.944, 0.5, 0.2) and (3.189, 0.3, 0.4). (b)
Deflection angle $\Omega(\psi)$ along $C_{5}$ and $C_{6}$.

For the first case $0\leq l<l_{ISCO}$, it is easy to find that there is no
TCO, quite similar to that of the Schwarzschild black hole with small angular
momentum, see Fig. 3. Therefore the topological number $W=0$. For the second
case, we set the angular momentum $l=l_{ISCO}$. Then the unit vector field $n$
and deflection angle $\Omega(\psi)$ are shown in Figs. 8 and 8. From Fig. 8,
one can find that the direction of the vector does not change when it crosses
the ISCO point. Further by constructing the closed loop $C_{4}$, the winding
number $w=\Omega(2\pi)$=0. When increasing the angular momentum such that
$l_{ISCO}<l<l_{MSCO}$, we shall see that two TCOs emerge from the ISCO. As an
example, we take $l=10$. The unit vector $n$ is displayed in Fig. 9. Two zero
points $TP_{3}$ and $TP_{4}$ corresponding to the TCOs are easily observed at
$r$= 1.944 and 3.189. By, respectively, constructing two closed loops $C_{5}$
and $C_{6}$, we get that the small radius TCO has $w$=-1 and the large radius
TCO has $w$=+1, see Fig. 9. As expected, such pattern gives the vanishing
topological number $W=-1+1=0$, the same with that of the Schwarzschild black
hole.

Figure 10: Case four $l=l_{MSCO}$ for the scalarized Einstein-Maxwell black
holes. (a) and (b) are for the unit vector field $n$ on a portion of the
$\theta$-$r$ plane near the TCOs marked with $TP_{5}$ and $TP_{6}$, and MSCO
with $MP_{1}$. The closed loops $C_{7}$, $C_{8}$, and $C_{9}$ are
respectively, surround them. Their parametric coefficients ($c_{0}$, $c_{1}$,
$c_{2}$)=(1.911, 0.4, 0.25), (3.394, 0.3, 0.4), and (15.753, 0.5, 0.2). (c)
and (d) are for the deflection angle $\Omega(\psi)$ along these closed loops.

Now we consider the fourth case with $l=l_{MSCO}$. The unit vector $n$ is
plotted near the TCOs and MSCO in Figs. 10 and 10. Obviously, these certain
orbits are the zero points of $n$ marked with black dots in the figures.
Further constructing these closed loops $C_{7}$, $C_{8}$, and $C_{9}$, we show
the deflection angle $\Omega(\psi)$ along them in Figs. 10 and 10, which
implies that the small and large radii TCOs denoted with $TP_{5}$ and
$TP_{6}$, respectively, have $w$=-1 and 1 while MSCO’s winding number
vanishes. Summing up, we have the topological number $W=-1+1+0=0$, which is
still the same with the previous cases.

Finally, we take $l=12.42>l_{MSCO}$ for the fifth case. Similarly, we exhibit
the unit vector $n$ and the deflection angle $\Omega(\psi)$ in Fig. 11. More
clearly, there are four zero points located at $r$=1.911, 3.405, 13.813, and
18.253. Taking advantage of $\Omega(\psi)$, we see the winding number $w$=-1,
+1, -1, and +1 for these zero points with values of $r$ from small to large.
Summing these winding numbers, we have the topological number $W=-1+1-1+1=0$.
This result is the same as that of the Schwarzschild black hole and our above
analysis in Sec. II.

Figure 11: Case five with $l=12.42$ for the scalarized Einstein-Maxwell black
holes. (a) and (b) are for the unit vector field $n$ on a portion of the
$\theta$-$r$ plane. (c) and (d) are for the deflection angle $\Omega(\psi)$.
For this case, there are four TCOs at $TP_{7}$, $TP_{8}$, $TP_{9}$, and
$TP_{10}$. These closed loops $C_{10-13}$ have parametric coefficients
($c_{0}$, $c_{1}$, $c_{2}$)=(1.910, 0.3, 0.4), (3.404, 0.5, 0.25), (13.813,
0.6, 0.25), and (18.283, 0.3, 0.4).

###  IV.3 Evolution of TCOs

Here by considering the angular momentum as a control parameter, we depict the
evolution of the zero point of the vector corresponding to the TCOs in Fig.
12. The ISCO and MSCO are marked with the dots. Near them, we expand the
angular momenta as follows

$\begin{split}l_{t}&=l_{ISCO}+7.26668(r-r_{ISCO})^{2}+\mathcal{O}\left((r-r_{ISCO})^{3}\right),\\\
l_{t}&=l_{MSCO}+0.021906(r-r_{MSCO})^{2}+\mathcal{O}\left((r-r_{MSCO})^{3}\right).\end{split}$
(78)

Obviously, both $l_{t}^{\prime\prime}(r_{ISCO})$ and
$l_{t}^{\prime\prime}(r_{MSCO})$ are positive, which suggests that both the
bifurcation points are generated points. It is also can be seen that, for each
of them, two TCO branches emerge, one of which has positive winding number
$w=1$ while another has negative winding number $w=-1$ marked with “+” and “-”
in the figure. Nevertheless, the topological number $W$ always vanishes.

Meanwhile, the winding number with respect to the angular momentum is
illustrated in Fig. 12. For $l_{ISCO}<l<l_{MSCO}$, two TCOs with opposite
winding numbers are described as a green line, resulting in $W=0$. When
$l>l_{MSCO}$, Four TCOs appear, two stable of them have $W=+2$, whereas the
other two unstable ones have $W=-2$, thus their total winding number is still
zero.

Figure 12: (a) The evolution of TCO radius $r_{t}$ vs. the angular momentum
$l$ for the scalarized Einstein-Maxwell black holes. ${\rm IP}_{2}$ and ${\rm
MP}_{1}$ are two generated points. The “$\pm$” denotes that the winding number
is $\pm 1$ for these TCO branches. (b) The total topological number (black
line) and winding number of TCO branches (green and purple lines).

In total, despite the existence of the MSCO and ISCO, the topological number
$W$=0 always holds for the scalarized Einstein-Maxwell black holes.

##  V Dyonic Black Holes

As shown above, for the scalarized Einstein-Maxwell black holes, the MSCO and
ISCO are present, which gives a topology configuration of TCOs different from
that of the Schwarzschild black holes, see Figs. 12 and 5. In this section, we
would like to exhibit another characteristic topological configuration of
TCOs, where the ISCO needs not to satisfy the condition
$\partial_{r,r}e_{1}(r)=0$. For this characteristic case, we also expect to
check whether the topological number $W$ still vanishes.

###  V.1 Dyonic black holes

Here we focus on the Dyonic black holes with the quasi-topological
electromagnetism, while it has no contributions to the Maxwell equation and
energy-momentum tensor.

The Lagrangian with the quasi-topological electromagnetism is written as
Liu2019

$\mathcal{L}=\sqrt{-g}(R+\alpha_{1}U^{(1)}-\alpha_{2}U^{(2)}),$ (79)

where $\alpha_{1}$ and $\alpha_{2}$ are coupling constants. $U^{(1)}=-F^{2}$
is the conventional Maxwell Lagrangian and $U^{(2)}=-2F^{4}+(F^{2})^{2}$ is a
quasi-topological electromagnetism term. Here $F^{2}=F^{\mu\nu}F_{\mu\nu}$ and
$F^{4}=F^{\mu}_{\;\nu}F^{\nu}_{\;\rho}F^{\rho}_{\;\sigma}F^{\sigma}_{\;\mu}$.
From the Lagrangian (79), the Bianchi identity and Maxwell equation of motion
read

$\begin{split}&\text{BI: }\nabla_{[\mu}F_{\nu\rho]}=0,\qquad\text{EOM:
}\nabla_{\mu}\tilde{F}^{\mu\nu}=0,\\\
&\tilde{F}^{\mu\nu}=4\alpha_{1}F^{\mu\nu}+8\alpha_{2}(F^{2}F^{\mu\nu}-2F^{\mu\rho}F^{\sigma}_{\;\rho}F_{\sigma}^{\;\nu}).\end{split}$
(80)

Simultaneously, the Einstein field equations are

$\begin{split}&R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}=T_{\mu\nu},\\\
&T_{\mu\nu}=\alpha_{1}(2F_{\mu\rho}F_{\nu}^{\;\rho}-\frac{1}{2}F^{2}g_{\mu\nu})+\alpha_{2}(4F^{2}F_{\mu\rho}F_{\nu}^{\;\rho}-8F_{\mu\rho}F^{\rho}_{\;\sigma}F^{\sigma}_{\;\lambda}F^{\lambda}_{\;\nu}-\frac{1}{2}((F^{2})^{2}-2F^{4})g_{\mu\nu}).\end{split}$
(81)

The quasi-topological electromagnetism admits spherically symmetric dyonic
black hole solution described by the following line element Liu2019

$\displaystyle ds^{2}$
$\displaystyle=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\Omega^{2},$ (82)
$\displaystyle f(r)$
$\displaystyle=1-\frac{2M}{r}+\frac{\alpha_{1}p^{2}}{r^{2}}+\frac{q^{2}}{\alpha_{1}r^{2}}{\;}_{2}F_{1}\left(\frac{1}{4},1;\frac{5}{4};-\frac{4p^{2}\,\alpha_{2}}{r^{4}\,\alpha_{1}}\right),$
(83)

where $p$ and $q$ correspond to the magnetic charge and electric charge of the
black holes. ${\;}_{2}F_{1}$ is the hypergeometric function. Coupling
constants $\alpha_{1}$ and $\alpha_{2}$ are associated with Maxwell theory and
quasi-topological electromagnetism term, respectively. For a characteristic
case, we set $\alpha_{1}=1$, $q=6.85$, $p=\sqrt{\frac{396}{443}}$,
$\alpha_{2}=\frac{196249}{1584}$, and $M=6.7$. Accordingly, the effective
potential reads

$\mathcal{V}(r)=\frac{\alpha_{1}E^{2}r^{2}}{q^{2}\,_{2}F_{1}\left(\frac{1}{4},1;\frac{5}{4};-\frac{4p^{2}\alpha_{2}}{r^{4}\alpha_{1}}\right)+\alpha_{1}\left(-2Mr+\alpha_{1}p^{2}+r^{2}\right)}-\frac{l^{2}\csc^{2}\theta}{r^{2}}-\mu^{2},$
(84)

by making use of Eq. (8).

###  V.2 Topology of TCOs and winding number

In order to investigate the topology of the TCO for the dyonic black hole, we
evaluate $e_{1}$ and $e_{2}$ via Eq. (10)

$e_{1,2}=\pm\sqrt{\frac{\left(l^{2}\csc^{2}\theta+\mu^{2}r^{2}\right)\left(q^{2}\,_{2}F_{1}\left(\frac{1}{4},1;\frac{5}{4};-\frac{4p^{2}\alpha_{2}}{r^{4}\alpha_{1}}\right)+\alpha_{1}\left(-2Mr+\alpha_{1}p^{2}+r^{2}\right)\right)}{\alpha_{1}r^{4}}}.$
(85)

Based on them, the components $\phi^{r}$ and $\phi^{\theta}$ of the vector can
be easily calculated. However, they are in complicated forms and we will not
show them here. Adopting Eq. (72), the radius and angular momentum of MSCO for
dyonic black hole is

$r_{MSCO}=25.3799,\quad\text{and}\quad l_{MSCO}=18.52.$ (86)

Note that this MSCO is not the innermost TCO, so the ISCO does not coincide
with this MSCO. The actual ISCO locates at $r_{ISCO}=6.0928$ with $l_{ISCO}=0$
and $E_{ISCO}=0.1393$.

According to the angular momentum of the MSCO and ISCO, we can divide them
into following three characteristic cases,

  * •

$0\leq l<l_{MSCO}$,

  * •

$l=l_{MSCO}$,

  * •

$l_{MSCO}<l<\infty$.

For the first case with small angular momentum, the study shows that there are
two TCOs denoted by “$TP_{11}$” and “$TP_{12}$”. This is quite different from
that of the Schwarzschild black holes and scalarized Einstein-Maxwell black
holes, where no TCOs can be found. So the existence of the pair TCOs is mainly
caused by the quasi-topological electromagnetism term. More interestingly,
they are not generated from an ISCO or MSCO. Taking $0<l=2<l_{MSCO}$ as an
example, the radius of TCOs locate at

$r_{TP_{11}}=2.63492,\qquad r_{TP_{12}}=6.11503.$ (87)

Their stability is evaluated via the first derivative

$\partial_{r}\phi^{r}(r_{TP_{11}})=-0.07678,\qquad\partial_{r}\phi^{r}(r_{TP_{12}})=0.01462,$
(88)

which significantly implies that $TP_{11}$ is unstable while $TP_{12}$ is
stable.

For the second case $l=l_{MSCO}$, we find the vector satisfies

$\phi^{r\prime}(r_{MSCO})=0,\quad\text{and}\quad\phi^{r\prime\prime}(r_{MSCO})=0.00001835.$
(89)

at $r_{MSCO}$, which indicates that MSCO obeys the condition (75). For the
third case with $l=18.58$ for an example, four TCOs will be observed with two
being stable and other two unstable.

Next, we turn to the topology for the TCOs. For the first case with $l=2$, we
observe that there are two zero points of the unit vector field
$n=(n^{r},n^{\theta})$ marked with black dots in Fig. 13. By constructing two
closed loops $C_{14}$ and $C_{15}$, we calculate the deflection angle
$\Omega(\psi)$, and plot them in Fig. 13 for these two loops. Along $C_{15}$
and $C_{14}$, $\Omega(\psi)$ increases from 0 to 2$\pi$, or decreases from 0
to $-2\pi$. This result suggests that the winding number of $TP_{11}$ and
$TP_{12}$ are -1 and +1, respectively. Summing them, one obtains the
topological number $W=-1+1=0$. This result indicates that although the quasi-
topological electromagnetism term produces two new TCOs, the topological
number stays unchanged.

Figure 13: Case one with $l=2.0$ for the dyonic black holes. (a) The unit
vector field $n$ on a portion of the $\theta$-$r$ plane. “$TP_{11}$” and
“$TP_{12}$” are two TCOs at $r$=2.635 and 6.115. The closed loops $C_{14}$ and
$C_{15}$ have parametric coefficients ($c_{0}$, $c_{1}$, $c_{2}$)=(2.635, 0.9,
0.15) and (6.115, 0.45, 0.4). (b) Deflection angle $\Omega(\psi)$ along
$C_{14}$ and $C_{15}$.

Concentrating on the other two cases, we plot the unit vector $n$ and the
deflection angle $\Omega(\psi)$ in Figs. 14 and 15. When $l=l_{MSCO}$, the
vector $n$ admits three zero points, two of them are the TCOs and the large
one is MSCO, see Figs. 14 and 14. By constructing three loops enclosing these
three zero points, we find these two small TCOs have winding numbers -1 and
+1, respectively, via $\Omega(\psi)$ shown in Fig. 14. While the MSCO does not
attribute to the winding number, see Fig. 14. As a result, the topological
number $W=-1+1+0=0$ as expected. When taking $l=18.38>l_{MSCO}$, we observe
four zero points of the vector $n$, which, respectively, locate at $r$=
2.3125, 6.3071, 23.4766 and 27.6136, displayed in Figs. 15 and 15. After
constructing the closed loops, we find their winding numbers are -1, +1, -1,
+1 through the deflection angle $\Omega(\psi)$ in Figs. 15 and 15. Summing
them, we obtain the topological number $W=-1+1-1+1=0$, which remarkably
indicates that $W$ does not change for large angular momentum.

Figure 14: Case two with $l=l_{ISCO}$ for the dyonic black holes. (a) The unit
vector field $n$ on a portion of the $\theta$-$r$ plane. (b) Deflection angle
$\Omega(\psi)$. “$TP_{13}$” and “$TP_{14}$” are two TCOs, and “$MP_{2}$” is
the MSCO. The closed loops $C_{16}$, $C_{17}$, and $C_{18}$ have parametric
coefficients ($c_{0}$, $c_{1}$, $c_{2}$)=(2.313, 0.32, 0.51), (6.307, 0.3,
0.5), and (25.380, 0.9, 0.4).

Figure 15:  Case three with $l=18.58$ for the the dyonic black holes. (a) The
unit vector field $n$ on a portion of the $\theta$-$r$ plane. (b) Deflection
angle $\Omega(\psi)$. “$TP_{15-18}$” are four TCOs. The closed loops
$C_{19-22}$ have parametric coefficients ($c_{0}$, $c_{1}$, $c_{2}$)=(2.313,
0.22, 0.5), (6.307, 0.22, 0.48), (23.477, 0.6, 0.35), and (27.614, 0.6, 0.35).

###  V.3 Evolution of TCOs

In the previous section, we have examined the topology of the TCOs for the
dyonic black holes. With different angular momentum $l_{t}$, the number of the
TCOs changes. So here we turn to study the evolution of the TCOs as a function
of $l_{t}$.

First, we perform the Taylor’s expansion near $r_{MSCO}$ denoting a
bifurcation point,

$l_{t}=l_{MSCO}+0.0140657(r-r_{MSCO})^{2}+\mathcal{O}\left((r-r_{MSCO})^{3}\right).$
(90)

Since $l_{t}^{\prime\prime}(r_{ISCO})=0.0140657>0$, the MSCO acts as a
generated point. This behavior can also be clearly observed in Fig. 16. After
$l_{MSCO}$, two TCO branches originate. A simple calculation shows that the
upper and lower branches have positive and negative winding numbers. Their sum
of the winding numbers keeps zero, the same with that of the MSCO.

On the other hand, there are two extra TCOs at small $r$ caused by the quasi-
topological electromagnetism term. Both them start at $l=0$ and extend to
large $l$ forming two new TCO branches described by green curves in Fig. 16.
Interestingly, they have different values of the winding number.

Such characteristic behavior of the TCOs is significantly different from that
of the Schwarzschild black holes and scalarized Einstein-Maxwell black holes.
For convenience, we sum the winding numbers for the TCOs in Fig. 16. For
$l<l_{MSCO}$, there is a pair TCOs, respectively having $w$=1 and -1. When the
angular momentum is beyond $l_{MSCO}$, a new pair TCOs emerge. However the
topological number $W$ always vanishes.

In summary, for a spherically symmetric dyonic black hole with the quasi-
topological term, there is a differential topological configuration from other
black holes. However, the topological number still vanishes. This result also
obviously supports our general result given in Sec. II.

Figure 16: (a) The evolution of TCO radius $r_{t}$ vs. the angular momentum
$l$ for the dyonic black holes. The “$\pm$” denote the positive or negative
winding numbers for the TCO branches. (b) The topological number (black line)
and winding numbers of the TCO branches (green and purple lines).

##  VI Conclusions

In this work, we studied the topology of the TCOs for generic spherically
symmetric and asymptotic flat black holes. The results suggest that the total
topological charge of TCOs is zero for each given angular momentum. Then we
extended the study to the Schwarzschild black holes, scalarized Einstein-
Maxwell black holes, and dyonic black holes, which exhibit three
characteristic topological configurations.

At first, we considered a static, spherically symmetric, and asymptotic flat
black hole. Starting with the Lagrangian of a massive test particle, we
obtained the corresponding effective potential, through which the TCOs can be
well determined. By making use of the effective potential, we constructed a
vector $\phi$ in the $r-\theta$ plane with its zero points exactly denoting
the TCOs. Employing this property, we established the topology for the TCOs.
Each TCO is endowed with a winding number and the stability of the TCOs can be
reflected via it. Globally, the topological number $W$ defined as the sum of
all the winding numbers can give us information on the number of the TCOs. By
examining the asymptotic behaviors of the vector, we observed that, for each
angular momentum, the vector $\phi$ is outwards at $\theta=0$ and $\pi$ and to
the right at the horizon and infinity. Such a pattern leads to a vanishing
total topological number. Consequently, the TCOs always come in pairs for the
fixed angular momentum. Locally, we also found that the stable and unstable
TCOs have positive or negative winding numbers, respectively. Meanwhile, the
MSCO can be treated as the bifurcation point with vanishing winding number.

Then we generalized the results to three kinds of black holes, which have
three different topological configurations.

For the Schwarzschild black holes, we divided them into three cases: $0\leq
l<l_{ISCO}$, $l=l_{ISCO}$, and $l_{ISCO}<l<\infty$ according to the angular
momentum of the ISCO. For the first case, no TCO can be observed. For the
second case, the ISCO exists. And when the angular momentum beyond this
values, a pair TCOs appears, one of which is stable and the other one is
unstable. For each case, we obtained the topological number $W$. The results
indicate that $W=0$ and is independent of the angular momentum. It confirms
our generic result given in Sec. II.

Further, we took the scalarized Einstein-Maxwell black hole as an example.
Different from the Schwarzschild black holes, the MSCO and ISCO do not
coincide due to the presence of the scalar hair, and which allows an
interesting topological configuration of the TCOs. With the increase of the
angular momentum, we observed that there may be no TCO, one pair TCOs, and two
pairs TCOs. Nevertheless, the topological number $W$ still vanishes, keeping
the same with the Schwarzschild black holes.

As a third example, we considered the dyonic black holes with a quasi-
topological term. Due to the quasi-topological term, the pattern of the TCOs
modifies. For the previous two cases, the TCO is absent for small angular
momentum. However, there will be at least one pair TCOs for the dyonic black
holes, providing a novel topological configuration. As a result, there will be
one pair, two pairs TCOs with the increase of the angular momentum. More
interestingly, the innermost and outermost TCOs have $w=-1$ and 1,
respectively. Further combining with the winding number of each TCO, we
obtained that the topological number $W$ still vanishes, and which is
independent of the angular momentum. This further confirms our general
results.

In conclusion, we in this paper established the topology of the TCOs for a
spherically symmetric and asymptotic flat black hole. For each angular
momentum, the TCOs always come in pairs, independent of the certain
topological configurations. Moreover, stable and unstable TCOs, respectively,
have winding number $w$=1 and -1. We expect to generalize this study to other
black hole backgrounds and to disclose more information of the TCOs from the
viewpoint of topology.

## Acknowledgements

We thank Prof. Peng Wang for sharing their Mathematica file of hairy black
holes. This work was supported by the National Natural Science Foundation of
China (Grants No. 12075103 and No. 12247101).

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