# Topology of light rings for extremal and non-extremal Kerr-Newman Taub-NUT
black holes without $\mathbb{Z}_{2}$ symmetry

Shan-Ping Wu, Shao-Wen Wei 111Corresponding author. E-mail<EMAIL_ADDRESS>1Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, and Key Laboratory of Quantum Theory and Applications of
MoE, Lanzhou University, Lanzhou, Gansu 730000, China,
2Institute of Theoretical Physics $\&$ Research Center of Gravitation, Lanzhou
University, Lanzhou 730000, People’s Republic of China

###### Abstract

Understanding the light ring, one kind fundamental orbit, shall provide us
with novel insight into the astronomical phenomena, such as the ringdown of
binary merger and shadow of black holes. Recently, topological approach has
preliminarily demonstrated its potential advantages on the properties of the
light rings. However, for the black holes without $\mathbb{Z}_{2}$ symmetry
and extremal spinning black holes are remained to be tested. In this paper, we
aim at these two issues. Due to the NUT charge, the Kerr-Newman Taub-NUT
solution has no $\mathbb{Z}_{2}$ symmetry. By constructing the corresponding
topology for the non-extremal spinning black holes, we find the topological
number keeps unchanged. This indicates that $\mathbb{Z}_{2}$ symmetry has no
influence on the topological number, while it indeed affects the locations of
the light rings and deviates them off the equatorial plane. For the extremal
spinning black holes, we find its topology is critically dependent of the
leading term of the vector’s radial component at the zero point of its angular
component on the black hole horizon. The findings state that there exists a
topological phase transition, where the topological number changes, for the
prograde light rings. While no phase transition occurs for the retrograde
light rings. Our study uncovers some universal topological properties for the
extremal and non-extremal spinning black holes with or without
$\mathbb{Z}_{2}$ symmetry. It also has enlightening significance on
understanding the light rings in a more general black hole background.

Classical black hole, topology, light ring, Taub-NUT black hole

###### pacs:

04.20.-q, 04.20.-g, 04.70.Bw

###### Contents

  1. I Introduction
  2. II  Kerr-Newman Taub-NUT spacetime
  3. III Topology for non-extremal black holes
    1. III.1 Light rings
    2. III.2 Topological charge for light rings
  4. IV Solutions of light rings
  5. V Topology for extremal black holes
    1. V.1 Example 1: Extremal Kerr-Newman black holes
    2. V.2 Example 2: Extremal Kerr-Newman Taub-NUT black holes
  6. VI Conclusions and discussions
  7. A The light rings with $g_{\phi\phi}=0$
  8. B Vector and topological number for $n\neq 0$
  9. C Proof for inequality $\partial_{\theta}g$($r$, $\theta$)$<0$ at $g$($r$, $\theta$)=0
    1. C.1 Case: $a=0$
    2. C.2 Case: $a>0$
  10. D Topological phase transition with varying $n$

##  I Introduction

The characteristic orbits around the ultra-compact objects and black holes are
very important on understanding the nature of strong gravity and quantum
gravity. Accompanied by the recent observations of the gravitational waves
abbott2016observation ; abbott2017gw170817 ; abbott2020gw190521 ;
PhysRevX.9.031040  and black hole images akiyama2019firstshadow ;
akiyama2019firstIm , such issue is further highlighted. In particular, the
light rings of axi-symmetric black hole (or photon sphere in spherically
symmetric spacetime) are closely related to the ring down stage of the black
hole mergers cardoso2009geodesic ; berti2014black  and shadows Chandrasekhar .

Similar to the event horizon, the light rings also seem to be intrinsic
circular orbits for the photons around the ultra-compact objects or black
holes, which is because that the light rings are only dependent of the black
hole parameters, while independent of the energy and angular momentum of the
photons. For the Schwarzschild black hole, the radius of the photon sphere is
3$M$ with $M$ being the black hole mass, which leads to a shadow with size
$3\sqrt{3}M$. The photon sphere shall be broken into two light rings by the
black hole spin. If the direction of the orbit angular momentum and black hole
spin are the same, we refer them to the prograde light rings which has a
smaller radius than that of the radius of the Schwarzschild black hole. While
if they are opposite, we call them retrograde light rings with a larger
radius. In particular, for different black holes, the sizes of the light rings
or photon spheres behave quite differently, which shall imprint potential
observed phenomena in black hole mergers or shadows.

Recently, a novel study, topology approach, was first introduced on exploring
the ultra-compact object light rings by Cunha, Berti, and Herdeiro
Cunha:2017qtt . Soon, this study was generalized to the black hole backgrounds
cunha2020stationary . Instead focusing on the size of the light rings, they
proposed a topological charge corresponding them. The results imply that, for
a four dimensional non-extremal axi-symmetric, asymptotically flat black hole,
there exists, at least, one unstable light ring outside the horizon and ergo-
surface for each rotation sense. Or, we always have one more unstable light
ring than the stable one. Other work on the light ring number can be found in
Refs. Hod2018plb ; Hod2018epjc ; Guo ; Ghosh ; Junior ; Sanchis ;
Hod:2022EBH_LR . For the Schwarzschild or Kerr black holes, this result is
straightforward because there exist only one photon sphere or light ring (for
each sense). Nevertheless, we in Ref. Wei:TC_BH_PhotonSpheres  showed that
when more than one photon spheres are presented, the topological number for
the dyonic black hole is -1, indicating that the unstable light ring must
exist and its number is one more than the stable one. Taking specific example,
we exactly confirmed this result. Although, in the radial direction, the light
rings could be stable or unstable, they are all stable in angular direction
Guo . For the Schwarzschild-Melvin solution, it was found that there exists a
topological phase transition at certain value of the dilatonic coupling due to
the modification of the asymptotic behavior at radial infinity Junior . Other
related work can also be found in Refs. Ghosh ; Bargueno:2022EBL_S_LR . Such
topological approach was also generalized to the study of the time-like
circular orbits Wei:2022mzv ; Ye:2023gmk , black hole thermodynamics
Wei:2021vdx ; Yerra:2022alz ; Wu:2022whe ; Fan:2022Topologicalphasetransition
; Bai:2022Topolthermodynamics ; Yerra:2022TopologHPtransition  and so on.

Among the study of the light rings, these black hole solutions considered all
possess a $\mathbb{Z}_{2}$ symmetry, i.e., $\theta\rightarrow\pi-\theta$. This
leads to that all the light rings locate on the equatorial plane. One may
wonder whether these topological results hold if the $\mathbb{Z}_{2}$ symmetry
of the concerned solutions is broken. Another issue worths to explore is for
the extremal black holes. Although it is extensively known that the
topological number for the retrograde light rings always holds, the results
for the prograde light rings are remained to be tested. These two issues are
the main purpose of this paper.

Interestingly, the Taub-NUT black hole solutions without $\mathbb{Z}_{2}$
symmetry provide a good test to our above concerns. The spinning Kerr-Newman
Taub-NUT black holes are characterized by four quantities, the mass $m$,
electric charge $q$, spin $a$, and NUT charge $n$ taub1951empty ;
newman1963empty . For non-zero NUT charge, it was found that the spacetime is
not globally asymptotically flat for its string singularity Misner:1963fr ;
manko2005physical , meanwhile, the $\mathbb{Z}_{2}$ symmetry is also broken.
In Refs. Wei:Strongfield_GL_KTNspacetime ; Pradhan:2014zia ; cebeci2016motion
; Vandeev:2022Geodesicdeviation , the influences of the NUT charge on the
motion of particles were investigated. The size and distortion of the shadows
were also closely related with the NUT charge. Hence, the light rings as one
kind characteristic orbits must be influenced. As a result, in this paper, we
wonder whether the topology of light rings heavily rely on the NUT charge.
Moreover, for the extremal Kerr-Newman Taub-NUT black holes, the topological
properties shall also be examined in details.

The organization of this paper is as followings. In Sec. II, we briefly
introduce the solution of the Kerr-Newman Taub-NUT black hole. In Sec. III, we
study the equation of motion for a photon in a non-extremal black hole
background, and then the total topological number defining as the sum of
winding numbers for zero points are obtained. It was shown that the location
of the light rings is shifted off the equatorial plane by the NUT charge. The
topological number still remains unchanged. The location of light ring in the
$r$-$\theta$ plane is solved in Sec. IV for both the prograde and retrograde
cases. Moreover, for the extremal Kerr-Newman Taub-NUT black holes, we explore
their light rings in Sec. V. By expanding the constructed vector near the
degenerate horizon, we find that the topological number closely depends on the
dominate term of the vector’s radial component at the zero point of the
angular component. For the retrograde case, there is at least one unstable
light rings. While for the prograde case, there may be one or no light ring,
indicating a topological phase transition. Finally, we summarize and discuss
our results in Sec. VI.

##  II  Kerr-Newman Taub-NUT spacetime

Let us start with the black hole solution. In Boyer-Lindquist coordinate, the
line element of the Kerr-Newman Taub-NUT black hole can be written as
Griffiths:2005familyofSolutions ; Pradhan:2014zia ; cebeci2016motion

$ds^{2}=-\frac{\Delta}{\Sigma}(dt-{\chi}d{\phi})^{2}+\frac{\sin^{2}{\theta}}{\Sigma}\left((r^{2}+n^{2}+a^{2})d{\phi}-adt\right)^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma
d{\theta}^{2},$ (1)

where $\Sigma$, $\Delta$, and $\chi$ are, respectively, defined by

$\displaystyle\Sigma$ $\displaystyle=r^{2}+(n+a\cos\theta)^{2},$ (2)
$\displaystyle\Delta$ $\displaystyle=r^{2}-2mr+a^{2}+q^{2}-n^{2},$ (3)
$\displaystyle\chi$ $\displaystyle=a\sin^{2}\theta-2n\cos\theta.$ (4)

Parameters $m$, $a$, $q$, and $n$ are the black hole mass, spin, electric
charge, and NUT charge, respectively. The electromagnetic 2-form field is
given by

$\displaystyle F_{ab}$
$\displaystyle=\frac{q}{\Sigma^{2}}(r^{2}-(n+a\cos\theta)^{2})(dr)_{a}\wedge\left((dt)_{b}-\chi(d\phi)_{b}\right)$
$\displaystyle+\frac{2aqr\sin\theta\cos\theta}{\Sigma^{2}}(d\theta)_{a}\wedge((r^{2}+a^{2}+n^{2})(d\phi)_{b}-a(dt)_{b}).$
(5)

Importantly, metric (1) and electromagnetic field (5) are solutions to the
Einstein field equation, but the spacetime is not globally asymptotically flat
Misner:1963fr ; manko2005physical . From Eqs. (2) and (4), the NUT charge
breaks $\mathbb{Z}_{2}$ symmetry defined by the transformation
$\theta\rightarrow\pi-\theta$. This symmetry breaking induces that some
physical quantities may be no longer mirror symmetric about the equatorial
plane. Besides, when $r$ approaches infinity, the component $g_{t\phi}$ of
metric will be asymptotic to $-2n\cos\theta$, which is also a little
difference from the case of the Kerr-Newman black hole. But, on the other
hand, similar to Kerr-Newman black hole, Kerr-Newman Taub-NUT black hole can
also have two horizons at

$r_{\pm}=m\pm\sqrt{m^{2}+n^{2}-a^{2}-q^{2}},$ (6)

for $m^{2}+n^{2}>a^{2}+q^{2}$. Besides, $m^{2}+n^{2}=a^{2}+q^{2}$ corresponds
to the extreme black holes. In the following sections, we will discuss both
the non-extremal and extremal black holes. We write $r_{+}$ as $r_{h}$ to
avoid symbol confusion with other contents.

##  III Topology for non-extremal black holes

Topological number are discrete, which can be used to distinguish different
system structures. For a vector, its zero points, often relating with physical
sources, can be treated as defects, and thus we can endow each of them with a
topological charge, with which these zero points can be classified. In Refs.
Cunha:2017qtt ; cunha2020stationary , it was shown that the light rings can be
cast to the zero points of the constructing vector. Following this idea, the
winding number acting as topological charge of the light rings can uncover
some underlying properties of the black hole solutions.

In this section, we will briefly review the equations of the light rings and
study the corresponding topological charges for the black hole without
$\mathbb{Z}_{2}$ symmetry.

###  III.1 Light rings

For a photon, its motion can be determined by the Hamiltonian

$\mathcal{H}=\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu}=0,$ (7)

where $p_{\mu}$ is 4-momentum of the photon. Considering the black hole
solution (1), there are two Killing vectors, $\partial_{t}$ and
$\partial_{\phi}$. The corresponding conserved quantities are $p_{t}=-E$ and
$p_{\phi}=\Phi$. Combining with these results, the equation (7) can be
rewritten as

$\displaystyle T+V=0,$ (8)

where

$\displaystyle T$ $\displaystyle=$ $\displaystyle
g^{rr}p_{r}^{2}+g^{\theta\theta}p_{\theta}^{2},$ (9) $\displaystyle V$
$\displaystyle=$
$\displaystyle-\frac{1}{D}(g_{\phi\phi}E^{2}+2g_{t\phi}E{\Phi}+g_{tt}{\Phi}^{2}),$
(10)

with $D=g_{t\phi}^{2}-g_{tt}g_{\phi\phi}$. For a light ring, $p_{r}$,
$p_{\theta}$ and $\dot{p}_{\mu}$ should vanish. Therefore, photon will move
around the black hole and keep $p_{a}=(-E,0,0,\Phi)$ uncharged. By using Eqs.
(8), (9), (10) and Hamiltonian canonical equation, the light rings satisfy the
following conditions

$V=\partial_{r}V=\partial_{\theta}V=0.$ (11)

Moreover, the light ring is stable or unstable in $x^{\mu}$ direction
($\mu=r$, $\theta$) if $\partial_{\mu}^{2}V>0$ or $\partial_{\mu}^{2}V<0$.
Here, we only focus on the light rings with $\Phi\neq 0$. Therefore, it is
reasonable to convert formula (10) into the following form

$V=-\frac{{\Phi}^{2}g_{\phi\phi}}{D}(E/\Phi-H_{+})(E/\Phi-H_{-}),$ (12)

where $H_{+}$ and $H_{-}$ correspond to the prograde and retrograde orbits,
and are given by

$H_{\pm}=\frac{-g_{t\phi}\pm\sqrt{D}}{g_{\phi\phi}}.$ (13)

The direction of angular momentum for the prograde and retrograde photons are
the same and opposite with the black hole spin. So if $a\geq 0$, we shall have
$\Phi>0$ for prograde case and $\Phi<0$ for the retrograde case. Without loss
of generality, we only concern $a\geq 0$. Employing with formula (12), the
original three conditions of light rings in Eq. (11) can be transformed into
two following conditions (for each rotating sense)

$\partial_{r}H_{\pm}=\partial_{\theta}H_{\pm}=0.$ (14)

Note that for $H_{+}$, the stable and unstable light rings correspond to
$\partial_{\mu}^{2}H_{+}>0$ and $\partial_{\mu}^{2}H_{+}<0$, respectively.
While it reverses for $H_{-}$. It seems that Eq. (14) can not be obtained from
Eqs. (11) and (12) if $g_{\phi\phi}=0$. However, this is not the case, and we
show it in Appendix A. Besides, in the above discussion, we assume $D>0$, so
the conclusion here may not apply to the case that light ring coincides with
or hides behind the event horizon.

###  III.2 Topological charge for light rings

For each rotating sense given in Eq. (14), these light rings can be well
described by constructing the corresponding topological charge, or winding
number. For the first step, we need to introduce a normalized field $v$

$v_{r}^{\pm}(r,\theta)=\frac{1}{\sqrt{g_{rr}(r,\theta)}}\partial_{r}H_{\pm}(r,\theta),\quad
v_{\theta}^{\pm}(r,\theta)=\frac{1}{\sqrt{g_{\theta\theta}(r,\theta)}}\partial_{\theta}H_{\pm}(r,\theta),$
(15)

which can be treated as a map from the physical space $X$

$X=\left\\{\left(r,\theta\right)\middle|r_{h}<r<\infty,0<\theta<\pi\right\\},$
(16)

to vector space $V$

$V=\left\\{\left(v_{r},v_{\theta}\right)\middle|-\infty<v_{r}<\infty,-\infty<v_{\theta}<\infty\right\\}.$
(17)

Space $X$ represents the physical space outside the event horizon, ensuring
$D>0$ and validity of our discussion. The corresponding parameters
$(r,\theta)$ of light rings in space $X$ will be mapped to the points at
$(v_{r},v_{\theta})=0$. Thus, we can use the winding number

$w=\frac{1}{2\pi}\oint_{\tilde{C}}d\Omega$ (18)

to characterize each zero point. The angle $\Omega$ measures the direction of
the vector and satisfies

$\displaystyle\cos\Omega(r,\theta)={v_{r}(r,\theta)}/v(r,\theta),\
\sin\Omega(r,\theta)={v_{\theta}(r,\theta)}/v(r,\theta),$ (19) $\displaystyle
v(r,\theta)^{2}=v_{r}(r,\theta)^{2}+v_{\theta}(r,\theta)^{2}.$ (20)

The contour $\tilde{C}$ is in space $V$, and it is given by the image of
mapping from contour $C$ in space $X$. The winding number of different points
can be calculated by constructing contour $C$ enclosing them:

  * •

If curve $C$ encloses a normal space point (not a light ring), the winding
number is trivial.

  * •

If curve $C$ encloses a zero points of vector (a light ring), the winding
number is $\pm 1$. The “+1” and “-1” represent local extremal point and saddle
point, respectively.

When we consider a large contour that encloses $N$ zero points of the vector,
the integral (18) shall give the total topological number

$\displaystyle W=\sum_{i=1}^{N}w_{i}.$ (21)

If this large contour covers the whole space of $X$, $W$ will contain the
contributions of all zero points of the vector $v$. As a result, it shall give
the global topological property of the concerned space. Therefore, we can
study the topological properties of the light rings in Taub-NUT black hole by
both the winding number and topological number from local and global
perspectives.

Choosing the contour $C$ as the boundary of region $X$, the integral (18) can
be decomposed as

$W=\frac{1}{2\pi}\left(\int_{C_{0}}d\Omega+\int_{C_{\infty}}d\Omega+\int_{C_{\pi}}d\Omega+\int_{C_{h}}d\Omega\right),$
(22)

if the boundary is smooth. For clarity, we sketch the contour in Fig. 1(a)
with $C=C_{0}\cup C_{\infty}\cup C_{\pi}\cup C_{h}$.

(a) (b)

Figure 1:  Representation of the contour $C$ which encloses
$X=\left\\{\left(r,\theta\right)\middle|r_{h}<r<\infty,0<\theta<\pi\right\\}$.
The complete $C$ contains four segments, i.e., $C=C_{0}\cup C_{\infty}\cup
C_{\pi}\cup C_{h}$. (a) The black arrows indicate the direction of the contour
$C$. (b) The sketch of the direction of the vector $v$ along the contour $C$
for prograde case. The black circular arrow indicates the direction of the
contour we concerned. The red arrows represent the direction of the vector.

Next, we aim to obtain $\Delta\Omega$ by considering the asymptotic behavior
of $v$ at these four segments ($C_{0,\infty,\pi,h}$). After some algebra
calculations, we have following results:

  * •

Near the event horizon boundary,

$\left.v_{r}^{\pm}\right|_{r\rightarrow
r_{h}}=\pm\frac{\sqrt{r_{h}^{2}+\left(n+a\cos\theta\right)^{2}}}{\left(r_{h}^{2}+a^{2}+n^{2}\right)^{2}\sin\theta}\left(r-r_{h}\right)+\mathcal{O}\left(\sqrt{r-r_{h}}\right),\quad\left.v_{\theta}^{\pm}\right|_{r\rightarrow
r_{h}}=\mathcal{O}\left(\sqrt{r-r_{h}}\right).$ (23)

  * •

Near the infinity boundary,

$\left.v_{r}^{\pm}\right|_{r\rightarrow\infty}=\mp\frac{1}{r^{2}\sin\theta}+\mathcal{O}\left(1/r^{3}\right),\quad\left.v_{\theta}^{\pm}\right|_{r\rightarrow\infty}=\mp\frac{\cos\theta}{r^{2}\sin^{2}\theta}+\mathcal{O}\left(1/r^{3}\right).$
(24)

  * •

Near the $\theta=0$ boundary,

$\left.v_{r}^{\pm}\right|_{\theta\rightarrow
0}=\mathcal{O}\left(\theta\right),\quad\left.v_{\theta}^{\pm}\right|_{\theta\rightarrow
0}=\mp\frac{\sqrt{r^{2}+(a+n)^{2}}}{4n^{2}\sqrt{r^{2}-2mr+a^{2}+q^{2}-n^{2}}}+\mathcal{O}\left(\theta\right).$
(25)

  * •

Near the $\theta=\pi$ boundary,

$\left.v_{r}^{\pm}\right|_{\theta\rightarrow\pi}=\mathcal{O}\left(\pi-\theta\right),\quad\left.v_{\theta}^{\pm}\right|_{\theta\rightarrow\pi}=\pm\frac{\sqrt{r^{2}+(a-n)^{2}}}{4n^{2}\sqrt{r^{2}-2mr+a^{2}+q^{2}-n^{2}}}+\mathcal{O}\left(\pi-\theta\right).$
(26)

From above analysis, we can find the direction of the vector on these four
segments. For the prograde orbit case marked with “+”, $v$ is downward, left,
up, and right for $C_{0,\infty,\pi,h}$. While for the retrograde case marked
with “-”, the direction of the vector reveres. Nevertheless, their topological
number keeps the same

$W^{\pm}=-1,$ (27)

which means that there is at least one unstable light ring outside the black
hole horizon. As a supplement, the sketch of boundary vector field for
prograde light rings is shown in Fig. 1(b). It is worth noting that when
$n\rightarrow\infty$, the values of (25) and (26) seem to become imaginary.
However, this is not true because $r$ must be greater than $r_{h}$, which
ensures that the formula in the square root is positive. Further comparing
with the Kerr-Newman black hole, a direct conclusion is that although the NUT
charge $n$ affects the vector field $v$ on the boundary, the topological
number corresponding to the light rings does not change. We also must be
careful that the current discussion is only valid for non-extreme black hole,
which is because that the leading term of
$\left.v_{r}^{\pm}\right|_{r\rightarrow r_{h}}$ will change for the extremal
black hole. And more specific analysis will be shown in section V.

Reviewing the line element (1), $\mathbb{Z}_{2}$ symmetry of
$\theta\rightarrow\pi-\theta$ is breaking for the Kerr-Newman Taub-NUT black
hole. But it fails to affect the topological number of the vector $v$, leading
to the fact that Kerr-Newman Taub-NUT black hole and Kerr-Newman black hole
are in the same topological class for the light rings. This is mainly because
that the topological number represents a global property rather than a local
one. In order to uncover the influence of $\mathbb{Z}_{2}$ symmetry on the
topological number, we define following two contour integrals

$\displaystyle W_{u}$
$\displaystyle=\frac{1}{2\pi}\int_{C^{u}}d\Omega=\frac{1}{2\pi}\left(\int_{C_{\pi/2}^{u}}d\Omega+\int_{C_{\infty}^{u}}d\Omega+\int_{C_{\pi}^{u}}d\Omega+\int_{C_{h}^{u}}d\Omega\right),$
(28) $\displaystyle W_{d}$
$\displaystyle=\frac{1}{2\pi}\int_{C^{d}}d\Omega=\frac{1}{2\pi}\left(\int_{C_{0}^{d}}d\Omega+\int_{C_{\infty}^{d}}d\Omega+\int_{C_{\pi/2}^{d}}d\Omega+\int_{C_{h}^{d}}d\Omega\right).$
(29)

The contours $C^{u}$ and $C^{d}$ are shown in Fig. 2(a), which are for the
upper half-plane and lower half-plane, respectively. The corresponding
topological numbers $W_{u}$ and $W_{d}$ are for the regions

$\displaystyle X_{u}$
$\displaystyle=\\{(r,\theta)|r_{h}<r<\infty,\pi/2\leqslant\theta<\pi\\},$ (30)
$\displaystyle X_{d}$
$\displaystyle=\\{(r,\theta)|r_{h}<r<\infty,0<\theta\leqslant\pi/2\\},$ (31)

respectively. This further requires us to consider the behavior of vector $v$
on the equatorial plane with $\theta=\pi/2$, which are

$\displaystyle\left.v_{r}^{\pm}\right|_{\theta\rightarrow\pi/2}$
$\displaystyle=\pm\frac{(r-m)\left(r^{2}+n^{2}\right)-2r\sqrt{\Delta}\left(\pm
a+\sqrt{\Delta}\right)}{\sqrt{r^{2}+n^{2}}\left(r^{2}+n^{2}+a^{2}\pm
a\sqrt{\Delta}\right)^{2}}+\mathcal{O}\left(\theta-\frac{\pi}{2}\right),$ (32)
$\displaystyle\left.v_{\theta}^{\pm}\right|_{\theta\rightarrow\pi/2}$
$\displaystyle=-\frac{2n\sqrt{\Delta}\left(\pm
a+\sqrt{\Delta}\right)}{\sqrt{r^{2}+n^{2}}\left(r^{2}+n^{2}+a^{2}\pm
a\sqrt{\Delta}\right)^{2}}+\mathcal{O}\left(\theta-\frac{\pi}{2}\right),$ (33)

for $n\neq 0$. When $n$ vanishes, Eq. (33) reduces to

$\left.v_{\theta}^{\pm}\right|_{\theta\rightarrow\pi/2}=\pm\frac{\sqrt{\Delta}\left(r^{2}+2a^{2}\pm
2a\sqrt{\Delta}\right)}{2r\left(r^{2}+a^{2}\pm
a\sqrt{\Delta}\right)^{2}}(\theta-\frac{\pi}{2})+\mathcal{O}\left((\theta-\frac{\pi}{2})^{2}\right).$
(34)

By using $r>m$ and $m^{2}>a^{2}+q^{2}$, it is easy to find $r^{2}+2a^{2}\pm
2a\sqrt{\Delta}>0$.

To clarify this issue, we take the prograde light ring and NUT charge $n>0$ as
an example. Other cases would be the similar. Note that in both prograde and
retrograde cases, the black hole spin $a$ is set to be positive in priority.
In Fig. 2(b), we show the direction of the vector $v^{+}$ on the segments of
$X_{u}$ and $X_{d}$, where the negative $v_{\theta}^{+}$ has been considered.
Accordingly, we easily obtain $W_{u}=-1$ and $W_{d}=0$. Besides, since
$C=C^{u}\cup C^{d}$, one easily gets $W=W_{u}+W_{d}$.

In other cases, we summarize the results in Table 1. Note that when $n$
vanishes, it reduces to the Kerr-Newman case, and our above analysis fails
because its light ring always on the equatorial plane, while we list the total
topological number for a comparison. From the table, it is clear that $W_{d}$
and $W_{u}$ take different values for each case. However their sum is always
-1, further confirming the relation $W=W_{d}+W_{u}$. This result also
indicates that the unstable light ring is shifted off the equatorial plane by
the NUT charge, but the number of the light ring keeps unchanged. Therefore,
the NUT charge does not alter the topological class of the light ring even
when the $\mathbb{Z}_{2}$ symmetry is broken.

(a) (b)

Figure 2: Representations of the contour $C^{u}$ and $C^{d}$ which encloses
$X_{u}=\left\\{\left(r,\theta\right)\middle|r_{h}<r<\infty,\pi/2\leqslant\theta<\pi\right\\}$
and
$X_{d}=\left\\{\left(r,\theta\right)\middle|r_{h}<r<\infty,0<\theta\leqslant\pi/2\right\\}$,
respectively. Four blue line segments (upper side of $\theta=\pi/2$) represent
curve $C^{u}=C^{u}_{\pi/2}\cup C^{u}_{\infty}\cup C^{u}_{\pi}\cup C^{u}_{h}$
and four red line segments (under side of $\theta=\pi/2$) represent curve
$C^{d}=C^{d}_{\pi/2}\cup C^{d}_{h}\cup C^{d}_{0}\cup C^{d}_{\infty}$. (a) The
arrows indicate the direction of contours $C^{u}$ and $C^{d}$. (b) The sketch
of vector direction along these line segments for prograde light rings in the
case of $n>0$ and $a>0$. The red and blue circular arrows indicate the
directions of the contours $C^{u}$ and $C^{d}$, respectively. The black arrows
indicate the direction of vector along the boundary.

To provide an more intuitive understanding, we give some examples on the
behaviors of the vector $v$ in $X$ space in Appendix B. The deflection angle
$\Delta\Omega$ is also shown, from which one can easily obtains the value of
the topological number corresponding to the light rings.

“$+$” |  $X_{d}=\\{(r,\theta)|r_{h}<r<\infty,0<\theta\leqslant\pi/2\\}$ |  $X_{u}=\\{(r,\theta)|r_{h}<r<\infty,\pi/2\leqslant\theta<\pi\\}$ |  $X=\\{(r,\theta)|r_{h}<r<\infty,0<\theta<\pi\\}$
---|---|---|---
$n>0$ | $W_{d}=0$ | $W_{u}=-1$ | $W=-1$
$n<0$ | $W_{d}=-1$ | $W_{u}=0$ | $W=-1$
$n=0$ | - | - | $W=-1$
“$-$” |  $X_{d}=\\{(r,\theta)|r_{h}<r<\infty,0<\theta\leqslant\pi/2\\}$ |  $X_{u}=\\{(r,\theta)|r_{h}<r<\infty,\pi/2\leqslant\theta<\pi\\}$ |  $X=\\{(r,\theta)|r_{h}<r<\infty,0<\theta<\pi\\}$
$n>0$ | $W_{d}=-1$ | $W_{u}=0$ | $W=-1$
$n<0$ | $W_{d}=0$ | $W_{u}=-1$ | $W=-1$
$n=0$ | - | - | $W=-1$

Table 1: Topological numbers corresponding to different regions for prograde
(“+”) and retrograde (“-”) light rings.

##  IV Solutions of light rings

In the previous section, one can draw a conclusion that Kerr-Newman Taub-NUT
black hole has at least one light ring. Due to the NUT charge, the light ring
will deviate from the equatorial plane. On the other hand, there may be a
possibility that the contribution of a saddle point and a extremal point to
the topological number cancels, so the specific number of the light ring could
not be uniquely determined by the topological number. In order to show the
detailed information of the light ring, we shall examine the zero point of the
vector $v$.

The equation of $v=0$ for prograde light ring can be reduced to

$\displaystyle\partial_{r}H_{+}$
$\displaystyle=\frac{f\left(r,\theta\right)\partial_{r}\Delta\sin\theta}{2\sqrt{\Delta}\left(\left(r^{2}+a^{2}+n^{2}\right)\sin\theta+\left(a\sin^{2}\theta-2n\cos\theta\right)\sqrt{\Delta}\right)^{2}}=0,$
(35) $\displaystyle\partial_{\theta}H_{+}$
$\displaystyle=\frac{-g\left(r,\theta\right)\sqrt{\Delta}}{\left(\left(r^{2}+a^{2}+n^{2}\right)\sin\theta+\left(a\sin^{2}\theta-2n\cos\theta\right)\sqrt{\Delta}\right)^{2}}=0,$
(36)

where

$\displaystyle
f\left(r,\theta\right)=-4r\frac{(a\sin\theta+\sqrt{\Delta})\sqrt{\Delta}}{\partial_{r}\Delta}+r^{2}+(n+a\cos\theta)^{2},$
(37) $\displaystyle
g\left(r,\theta\right)=-a^{2}\cos^{3}\theta+\left(r^{2}+n^{2}+2a^{2}\right)\cos\theta+2an+2\left(n+a\cos\theta\right)\sin\theta\sqrt{\Delta}\
.$ (38)

The retrograde case is not shown, which can be obtained by some relations with
the prograde case. For example, changing both sign of $a$ and $n$ in Eqs. (35)
and (36), one shall obtain these corresponding equations for the retrograde
case. So for simplicity, we only focus on the prograde case marked with “$+$”.
In addition, the black hole discussed in this section is non-extremal, and the
region we concerned is
$X=\left\\{(r,\theta)|r_{h}<r<\infty,0<\theta<\pi\right\\}$. Note that, Eqs.
(35) and (36) is equivalent to $f(r,\theta)=0$ and $g(r,\theta)=0$. Now let us
turn to the discussion about the number of roots for Eqs. (35) and (36).
First, Eq. (35) gives the equation $f\left(r,\theta\right)=0$. Here we attempt
to examine the partial derivative of $f\left(r,\theta\right)$

$\partial_{r}f\left(r,\theta\right)=-\frac{2\left(r(r-m)^{2}-m\Delta\right)\left(a\sin\theta+\sqrt{\Delta}\right)}{(r-m)^{2}\sqrt{\Delta}}.$
(39)

For the case of $a\geqslant 0$, we always have
$\partial_{r}f\left(r,\theta\right)<0$, thus $f\left(r,\theta\right)$ is
monotonically decreasing with $r$. While when $a<0$, there will be a root
$r=r_{0}$ outside the event horizon such that
$\left.\partial_{r}f\left(r,\theta\right)\right|_{r_{h}<r<r_{0}}>0$ and
$\left.\partial_{r}f\left(r,\theta\right)\right|_{r_{0}<r<\infty}<0$. Thus,
$f\left(r,\theta\right)$ increases with $r$ in interval $(r_{h},r_{0})$ and
decreases in interval $(r_{0},\infty)$. Combining with these boundary
behaviors

$f\left(r_{h},\theta\right)=r_{h}^{2}+(n+a\cos\theta)^{2}>0,\
f\left(\infty,\theta\right)=\lim_{r\rightarrow\infty}-r^{2}<0,$ (40)

and the continuity of function, we arrive an interesting conclusion: for any
$0<\theta<\pi$, there is only one root $\hat{r}\in(r_{h},\infty)$ such that
$f(\hat{r},\theta)=0$ and $\left.\partial_{r}f(r,\theta)\right|_{r=\hat{r}}<0$
hold. For the convenience of discussion, we denote $r=r(\theta)$ as the
solution of $f(r,\theta)=0$. Then, we turn to the remaining Eq. (36).
Substituting $r=r(\theta)$ into it, we obtain

$\left.\partial_{\theta}H_{+}\left(r,\theta\right)\right|_{r=r(\theta)}=0,$
(41)

which is an equation with only one variable $\theta$. Solving it, we shall
obtain the light rings determined by Eqs. (35) and (36). To explore the root
of Eq. (41), we focus on

$\displaystyle\frac{d}{d\theta}\left(\left.\partial_{\theta}H_{+}\left(r,\theta\right)\right|_{r=r(\theta)}\right)$
$\displaystyle=\left.\frac{dr}{d\theta}\partial_{r}\partial_{\theta}H_{+}\right|_{r=r(\theta)}+\left.\partial_{\theta}^{2}H_{+}\right|_{r=r(\theta)}$
(42)
$\displaystyle=\left.\frac{1}{\partial_{r}^{2}H_{+}\left(r,\theta\right)}\left(-\left(\partial_{r}\partial_{\theta}H_{+}\right)^{2}+\partial_{r}^{2}H_{+}\left(r,\theta\right)\partial_{\theta}^{2}H_{+}\left(r,\theta\right)\right)\right|_{f(r,\theta)=0}.$
(43)

If one substitutes the form (13) of $H_{+}$, this equation will become very
complex and hard to analyze. However, we have

$\left.\frac{1}{\partial_{r}^{2}H_{+}\left(r,\theta\right)}\left(-\left(\partial_{r}\partial_{\theta}H_{+}\right)^{2}+\partial_{r}^{2}H_{+}\left(r,\theta\right)\partial_{\theta}^{2}H_{+}\left(r,\theta\right)\right)\right|_{f(r,\theta)=0,\
g(r,\theta)=0}>0,$ (44)

for the light rings. This is mainly because that, with the conditions (35) and
(36) or $f(r,\theta)=0$ and $g(r,\theta)=0$, we have

$\partial_{r}^{2}H_{+}\propto\partial_{r}f(r,\theta)<0,\
\partial_{\theta}^{2}H_{+}\propto-\partial_{\theta}g(r,\theta)>0,$ (45)

where $\partial_{\theta}g(r,\theta)<0$ can be found in Appendix C. It must be
noted that compared to formula (43), Eq. (44) has an additional condition
$g(r,\theta)=0$. That means that Eq. (44) is the behavior of formula (43) at
any zero points of $g(r,\theta)=0$. Thus, based on the zero point theorem of
continuous functions and

$\left.\partial_{\theta}H_{+}\left(r,\theta\right)\right|_{r=r(\theta=0),\theta=0}<0,\
\left.\partial_{\theta}H_{+}\left(r,\theta\right)\right|_{r=r(\theta=\pi),\theta=\pi}>0,$
(46)

Eq. (43) is positive at any zero points of the Eq. (41), implying that it has
only one root. Therefore, we arrive a conclusion: Kerr-Newman Taub-NUT black
hole has only one prograde light ring. Moreover, employing with the relation
between the prograde and retrograde cases, this conclusion is also valid for
the retrograde light ring. Besides, due to inequality (45), these light rings
are stable in $\theta$ direction and unstable in $r$ direction. This means
that the light ring corresponds to the saddle point, which is consistent with
the result of winding number.

Although we know from the above analysis that there is only one light ring
outside the black hole horizon, the analytical solution is still hard to
obtain. However, when we consider the case of $n\rightarrow\pm\infty$, the
analytical solution reads

$r=\sqrt{3}n,\ \theta=\arccos\left(\mp\frac{1}{\sqrt{3}}\right).$ (47)

Moreover, we also numerically solve the light rings, and exhibit them in Fig.
3 for the prograde light rings. From Fig. 3(a), it is easy to see that there
always exists one and only one light ring for an arbitrary value of $n$. The
deviation angle is also bounded by $\mp\arccos(1/\sqrt{3})$. With the increase
of NUT charge $n$, we also display the light ring in Fig. 3(b). The asymptotic
behaviors of $\theta$ at large $n$ are also explicitly shown. Meanwhile, in
the separated ranges ($0$, $\pi/2$) and ($\pi/2$, $\pi$), $\theta$ is a
monotonic function with the radial radius of the light ring. This result is
also consistent with our above analysis. From the perspective of topology, we
always have a conserved topological number $W=-1$ indicating the existence of
an unstable light ring. This result is also independent of the black hole
parameters. So, one can obtain some interesting properties of the light ring
without solving the corresponding equations.

(a) (b)

Figure 3: Numerical solutions for prograde light ring with $m=3$, $q=1$,
$a=1$. The blue curves are for the light rings, and the red dashed lines
represent $\theta=\arccos(-1/\sqrt{3})$ and $\theta=\arccos(1/\sqrt{3})$. (a)
The angular location of the light rings as a function of the NUT change $n$.
(b) The location of the light rings in $r$-$\theta$ plane with varying $n$.
The arrows represent the increasing $n$ from $-\infty$ to $\infty$. In
addition, (7.50, $\pi/2$) denotes the light ring when $n$ vanishes.

##  V Topology for extremal black holes

In above sections, we have studied the topological properties of the light
rings for a non-extremal Kerr-Newman Taub-NUT black hole. However, for an
extremal black hole, the topological properties is hard to study due to the
fact that the behavior of vector $v$ near the degenerate event horizon is
relatively dependent on the specific metric. Therefore, exploring the
corresponding topology has potential enlightening significance on
understanding the light rings for the extremal black holes. Recent study on
this issue in spherically symmetric black hole backgrounds can be found in
Refs. Hod:2022EBH_LR ; Bargueno:2022EBL_S_LR ; Peng:2022EFHBH_NCG .

Now, we attempt to examine the topology of light rings for the extremal
spinning Kerr-Newman Taub-NUT black hole. The metric function $\Delta$ reads

$\Delta=\left(r-m\right)^{2},$ (48)

due to the constraint $m^{2}+n^{2}=a^{2}+q^{2}$. This means that these two
horizons of the black hole coincide exactly at $r_{h}=r_{+}=r_{-}=m$. Thus,
the vector $v$ (15) near the horizon shall be significantly changed, whereas
slightly modified on other boundaries. For clarity, we show the vector
behavior near the horizon

$\displaystyle\left.v_{r}^{\pm}\right|_{r\rightarrow r_{h}}$
$\displaystyle=\pm\frac{\left(n+a\cos\theta\right)^{2}+m^{2}\mp
2ma\sin\theta}{\sin\theta\left(m^{2}+n^{2}+a^{2}\right)^{2}\sqrt{m^{2}+\left(n+a\cos\theta\right)^{2}}}(r-r_{h})+\mathcal{O}\left((r-r_{h})^{2}\right),$
(49) $\displaystyle\left.v_{\theta}^{\pm}\right|_{r\rightarrow r_{h}}$
$\displaystyle=\pm\frac{a^{2}\cos^{3}\theta-\left(m^{2}+n^{2}+2a^{2}\right)\cos\theta-2an}{\sin^{2}\theta\left(m^{2}+n^{2}+a^{2}\right)^{2}\sqrt{m^{2}+\left(n+a\cos\theta\right)^{2}}}(r-r_{h})+\mathcal{O}\left((r-r_{h})^{2}\right).$
(50)

Since the concerned region is outside the black hole horizon, we need to
examine the leading terms of Eqs. (49) and (50) near the event horizon.
Although these equations are more complicated than that of the non-extremal
black holes, we are fortunate enough to solve $v_{\theta}^{\pm}=0$ to the
first order as $\theta=\theta_{c}$ given by

$\cos\theta_{c}=-\frac{2\sqrt{\left(m^{2}+n^{2}+2a^{2}\right)}}{\sqrt{3}a}\sin\left(\frac{\pi}{6}-\frac{1}{3}\arccos\left(\frac{3\sqrt{3}a^{2}n}{\sqrt{\left(m^{2}+n^{2}+2a^{2}\right)^{3}}}\right)\right).$
(51)

The simple result indicates that, $\left.v_{\theta}^{+}\right|_{r\rightarrow
r_{h}}<0$ and $\left.v_{\theta}^{-}\right|_{r\rightarrow r_{h}}>0$ for
$0<\theta<\theta_{c}$. If $\theta_{c}<\theta<\pi$, their values reverse.
Further combining with the fact that
$\left.v_{\theta}^{+}\right|_{\theta\rightarrow 0}<0$,
$\left.v_{\theta}^{+}\right|_{\theta\rightarrow\pi}>0$,
$\left.v_{\theta}^{-}\right|_{\theta\rightarrow 0}>0$, and
$\left.v_{\theta}^{-}\right|_{\theta\rightarrow\pi}<0$, we find

$V_{c}^{\pm}\equiv\text{the leading term of}\
\left.v_{r}^{\pm}\right|_{\theta=\theta_{c},r\rightarrow r_{h}}$ (52)

is critical to the topological number,

$\Delta\Omega_{h}=\int_{C_{h}}d\Omega=\mp\pi\operatorname{sgn}\left(V_{c}^{\pm}\right).$
(53)

Counting the contributions from other segments of the boundary, we reach the
following conclusion:

  * •

For the prograde case, the topological number is 0 for $V_{c}^{+}<0$ (shown in
Fig. 4(a)), and -1 for $V_{c}^{+}>0$ (shown in Fig. 4(b)).

  * •

For the retrograde case, the topological number is -1 and 0 for $V_{c}^{-}<0$
and $V_{c}^{-}>0$, respectively.

(a) (b)

Figure 4: The sketch of vector on the boundary for prograde case. Four black
line segments represent the contour $C$ which is the boundary of
$X=\left\\{\left(r,\theta\right)\middle|r_{h}<r<\infty,0<\theta<\pi\right\\}$.
The black circular arrows indicate the directions of contours. The red arrows
represent the direction of vector on the boundary. “$\theta_{c}$” in blue
color of ($r_{h}$, $\theta_{c}$) corresponds is given in formula (51). (a)
Case for $V_{c}<0$. (b) Case for $V_{c}>0$.

Therefore, quite different from the non-extremal black holes, all the extremal
black hole can not be classified into the same topological class simply. In
particular, as shown above, when $V_{c}^{\pm}$ changes its sign, a topological
phase transition must occur. Some examples will be given in the following
contents, and some supplementary materials can also be found in Appendix D.

It is also worth to note that outside the black hole horizon, the leading term
of $V_{c}^{\pm}$ does not vanish for a spinning black hole, which can ensure
that $\Delta\Omega_{h}$ for horizon boundary is clear and formula (53) is
correct. The reason is given in followings. Let us focus on the expansion

$\left.v_{r}^{\pm}\right|_{\theta=\theta_{c},r\rightarrow
r_{h}}=V_{c}^{\pm(1)}(r-r_{h})+V_{c}^{\pm(2)}(r-r_{h})^{2}+\mathcal{O}\left((r-r_{h})^{3}\right),$
(54)

where $V_{c}^{\pm(1)}$ are

$V_{c}^{\pm(1)}=\pm\frac{\left(n+a\cos\theta_{c}\right)^{2}+m^{2}\mp
2ma\sin\theta_{c}}{\sin\theta_{c}\left(m^{2}+n^{2}+a^{2}\right)^{2}\sqrt{m^{2}+\left(n+a\cos\theta_{c}\right)^{2}}}.$
(55)

When $V_{c}^{+(1)}$ vanishes, one has $a>0$ and
$V_{c}^{+}=V_{c}^{+(2)}\propto-a<0$, which ensures that $V_{c}^{+}$ does not
vanish. In the same way, $V_{c}^{-}$ can also be analyzed. Thus, the
topological number can be obtained and expressed as

$W^{\pm}=\left\\{\begin{matrix}-\frac{1}{2}\left(1\pm\operatorname{sgn}\left(V_{c}^{\pm\left(1\right)}\right)\right),&\text{if}\,\,V_{c}^{\pm\left(1\right)}\neq
0.\\\ 0,&\text{if}\,\,V_{c}^{\pm\left(1\right)}=0.\\\ \end{matrix}\right.$
(56)

If $a>0$ is fixed, it can be found that $V_{c}^{-(1)}$ is always less than
zero, which means that the retrograde light rings always exist; but for
$V_{c}^{+(1)}$, it is not always greater than zero, implying that prograde
light ring might not exist.

###  V.1 Example 1: Extremal Kerr-Newman black holes

If we take $n=0$, the solution will reduce to the extremal Kerr-Newman black
hole with $m^{2}=a^{2}+q^{2}$. Eq. (51) will be greatly simplified and it
gives $\theta_{c}=\pi/2$. Therefore, the functions $V_{c}^{\pm(1)}$ are

$V_{c}^{\pm(1)}=\pm\frac{(m\mp 2a)}{\left(m^{2}+a^{2}\right)^{2}}.$ (57)

Employing with this result, we can obtain the corresponding topological number
for different black hole spin. The results are summarized in Table 2. For the
prograde light ring, we see that there exists a topological phase transition
at $a=m/2$, below or above which the topological number $W$=-1 or 0,
respectively. This indicates that there exists one unstable prograde light
ring for the lowly spinning extremal Kerr-Newman black holes, while no for the
rapidly spinning black holes. Of particular interest is that there always
exists one retrograde light ring for the extremal Kerr-Newman black holes. So
we conclude that the topological phase transition exists for the prograde
light ring, while absent for the retrograde light ring.

Range |  $V_{c}^{+}$ |  “$+$” direction |  $V_{c}^{-}$ |  “$-$” direction
---|---|---|---|---
$0<a<m/2$ | $V_{c}^{+}>0$ | $W=-1$ | $V_{c}^{-}<0$ | $W=-1$
$a\geqslant m/2$ | $V_{c}^{+}<0$ | $W=0$ | $V_{c}^{-}<0$ | $W=-1$

Table 2: Topological number for prograde and retrograde light rings outside
the extreme Kerr-Newman black hole.

On the other hand, for the extreme Kerr-Newman black hole, the location of
prograde light rings is determined

$\displaystyle a^{4}\cos^{2}\theta-r(2a\sin\theta-2m+r)=0,$ (58)
$\displaystyle\left(r^{2}+a^{2}+a^{2}\left(1-\cos\theta\right)+2a\sin\theta(r-m)\right)\cos\theta=0,$
(59)

solving which gives

$\theta^{+}_{LR}=\pi/2,\quad r^{+}_{LR}=2(m-a).$ (60)

Since we require a light ring outside the black hole horizon $r^{+}_{LR}>m$,
this exactly gives our above result that the prograde light ring only exists
for $a<m/2$.

For the retrograde case, we only need to change $a$ to $-a$ in Eqs. (58) and
(59). The solution is

$\theta^{-}_{LR}=\pi/2,\quad r^{-}_{LR}=2(m+a).$ (61)

Obviously, the radius of the retrograde light rings is always larger than the
radius of the extremal black hole horizon. So one is expected to get
topological number $W=-1$ regardless of the black hole spin.

###  V.2 Example 2: Extremal Kerr-Newman Taub-NUT black holes

Now, let us turn to the case with $n\neq 0$. For the extremal Kerr-Newman
Taub-NUT black hole, one requires $m^{2}+n^{2}=a^{2}+q^{2}$ on these black
hole parameters. Without loss of generality, we take $m=3$ and $n=1$ as an
example. Then we have $0\leqslant a=\sqrt{10-n^{2}}\leqslant\sqrt{10}$.

For simplicity, we plot the first order term $V_{c}^{\pm(1)}$ in Fig. 5 for
both prograde and retrograde cases. It is clear that $V_{c}^{+(1)}$ changes
its sign at $a\approx 1.57$. So there exists a topological phase transition
for the prograde light ring. Below and above that value, there exists one or
no light ring, and the topological number $W$ changes from -1 to 0. While for
the retrograde case, $V_{c}^{-(1)}$ always takes negative values. So no
topological phase transition occurs, and $W=-1$ for an arbitrary black hole
spin $a$. This results are also consistent with that of Kerr-Newman black
hole.

Figure 5: Quantities $V_{c}^{\pm(1)}$ as a function of black hole spin $a$
with $m=3$ and $n=1$.

On the other hand, we focus on the location of light rings for the extremal
Kerr-Newman Taub-NUT black holes. The prograde light rings are determined by

$\displaystyle
a^{4}\cos^{2}\theta+2a^{2}n\cos\theta+2\left(m-a\sin\theta\right)r-r^{2}+n^{2}=0,$
(62)
$\displaystyle-a^{2}\cos^{3}\theta+\left(r^{2}+n^{2}+2a^{2}\right)\cos\theta+2(r-m)(n+a\cos\theta)\sin\theta+2an=0.$
(63)

For the retrograde case, the equations can be obtained by $n\rightarrow-n$ and
$a\rightarrow-a$,

$\displaystyle
a^{4}\cos^{2}\theta-2a^{2}n\cos\theta+2\left(m+a\sin\theta\right)r-r^{2}+n^{2}=0,$
(64)
$\displaystyle-a^{2}\cos^{3}\theta+\left(r^{2}+n^{2}+2a^{2}\right)\cos\theta+2(r-m)(n-a\cos\theta)\sin\theta+2an=0.$
(65)

Taking $m=3$ and $n=1$, we plot these curves determined by Eqs. (62) and (63)
in Fig. 6 near $a$=1.57. Then the solutions or the locations of the light
rings are at the intersection points marked with black dots of these curves.
For the prograde case, the results are listed in Figs. 6(a) and 6(b). With the
increase of the black hole spin, these intersection points are shifted towards
to small $r$. Of particular interest is that at $a\approx 1.57$, the
interaction point exactly coincides with the horizon. Further increasing $a$,
the point will possess smaller $r$, which indicates that the light rings for
these cases will hide behind the black hole horizon. Since we only concern
these light rings outside the black hole horizon, the pattern exhibits the
tendency that the light rings disappear with the increase of the black hole
spin. This also results in the topological phase transition near $a\approx
1.57$, where the number of the light rings changes. A minor phenomenon is
that, the light rings represented by these black dots slightly deviate from
the equatorial plane, which is mainly caused by the breaking of the
$\mathbb{Z}_{2}$ symmetry.

For the retrograde case, the results are also shown in Figs. 6(c) and 6(d). It
is obvious that these light rings marked with black dots are all outside the
black hole horizon near $a=1.57$. This also holds for other allowed black hole
spin. Therefore, we always observe one unstable light ring, indicating
topological number $W=-1$ and no topological phase transition occurs.

Note that we here discuss the case for varying black hole spin, while the case
considered for varying the NUT charge can be found in Appendix D.

(a) (b)

(c) (d)

Figure 6: The schematic diagram for light ring solutions in ($r$, $\theta$)
space. The intersection points marked with black dots are for the light rings.
These curves are given via Eqs. (62)-(65). The black vertical lines denote the
locations of the black hole horizons. (a) and (b) are for the prograde case,
whereas (c) and (d) are retrograde case. (b) and (d) are the zooming diagrams
of (a) and (c) near the light rings.

##  VI Conclusions and discussions

In this paper, we considered the topology of light rings in the backgrounds of
the Kerr-Newman Taub-NUT black holes. Our study extends previous work to a
black hole without the $\mathbb{Z}_{2}$ symmetry. The corresponding light
rings are found to deviate from the equatorial plane with non-vanished NUT
charge $n$. Moreover, the topology is also examined for the extremal spinning
black holes.

Our first concern focuses on the non-extremal Kerr-Newman Taub-NUT black
holes. By considering the direction of vector on the four segments, at
$r=r_{h}$ and $\infty$, $\theta=0$ and $\pi$, of the boundary, we observed the
topological number $W=-1$ for both the prograde and retrograde light ring
cases. This implies that at least one unstable light ring exists. Such result
is the same as that of the non-extremal Kerr-like black hole with
$\mathbb{Z}_{2}$ symmetry. Therefore, the $\mathbb{Z}_{2}$ symmetry has no
influence on the number of the light rings, while it indeed affects their
locations. Furthermore, we also confirmed that there is one and only one light
ring for the non-extremal Kerr-Newman Taub-NUT black holes.

In order to uncover the detailed influence of the $\mathbb{Z}_{2}$ symmetry on
the light ring topology. We further divided the considered whole parameter
space $X$ in $\theta-r$ plane into the upper half-plane $X_{u}$ and lower
half-plane $X_{d}$. Accordingly, the corresponding boundary is separated to
$C=C^{u}\cup C^{d}$. Further considering the direction of the vector along
these segments of the boundary, we found that, for the prograde light ring,
$W_{d}=0$ and $W_{u}=-1$ for positive NUT charge, while $W_{d}=-1$ and
$W_{u}=0$ for negative NUT charge. This also uncovers that the positive or
negative NUT charge shifts the light ring above or below the equatorial plane.
On the other hand, for the retrograde light ring, the results reverse.
Nevertheless, the total topological number $W=W_{d}+W_{u}$ keeps unchanged.

Moreover, we also performed our topology study to the extremal Kerr-Newman
Taub-NUT black holes. Although it is known that there must exist one unstable
light ring for the retrograde case, whether it still holds for the prograde
case is still unclear. Here we expanded the vector near the black hole
degenerate horizon. Further combining with the analysis, we clearly showed
that the topological pattern is critically dependent of the leading term of
$v_{r}^{\pm}$ at the zero point of $v_{\theta}^{\pm}$ on the black hole
horizon. Different signs of the leading term gives different topological
number. Meanwhile, the zero point of the first-order term represents a phase
transition.

Based on these results, we took the extremal Kerr-Newman black hole as an
first example. For the prograde case, the topological number $W=-1$ or 0 for a
slowly spinning ($0<a<m/2$) or rapidly spinning ($a\geq m/2$) extremal black
hole, which indicates that the light ring only exists for the slowly spinning
black holes. And a topological phase transition occurs at $a=m/2$. However,
for the retrograde case, the unstable light ring always exists. When we
applied this approach to the extremal Kerr-Newman Taub-NUT black holes, we
observed the similar results. The retrograde light ring exists for an
arbitrary allowed spin, while the prograde light ring encounters a topological
phase transition at a certain black hole spin. One significant difference is
that the locations of these light rings are off the equatorial plane.

As a brief summary, we found several characterized topological properties of
the light rings in this paper. First, the $\mathbb{Z}_{2}$ symmetry affects
the location of the light ring, while has no contribute to the topological
number. Second, the topological properties are similar for both the prograde
and retrograde light rings in non-extremal spinning black hole backgrounds.
Third, there is potential topological phase transition for the prograde light
ring in the extremal spinning black hole backgrounds. While the retrograde
light ring always exists. Although we mainly dealt with the Kerr-Newman Taub-
NUT black holes, our study still uncovers some universal topological
properties for the light rings even without $\mathbb{Z}_{2}$ symmetry. These
shall have enlightening significance on understanding the light rings for
other spinning black holes.

## Acknowledgments

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

##  Appendix A The light rings with $g_{\phi\phi}=0$

In this appendix, we would like to discuss the case of $g_{\phi\phi}=0$
appearing in Eqs. (10) or (12). After a simple calculation, we will show that
$g_{\phi\phi}=0$ is contained in Eq. (14). We begin with Eq. (12)

$V=-\frac{\Phi^{2}}{D}g_{\phi\phi}(E/\Phi-H_{+})(E/\Phi-H_{-}).$ (66)

Employing Kerr-Newman Taub-NUT black hole metric (1), we can easily get

$\displaystyle
g_{\phi\phi}=\frac{\varLambda_{+}\left(r,\theta\right)\varLambda_{-}\left(r,\theta\right)}{r^{2}+\left(n+a\cos\theta\right)^{2}},$
(67) $\displaystyle
H_{\pm}=\frac{a\sin\theta\pm\sqrt{\Delta}}{\varLambda_{\pm}\left(r,\theta\right)},$
(68)

where

$\varLambda_{\pm}\left(r,\theta\right)=\left(r^{2}+n^{2}+a^{2}\right)\sin\theta\pm\left(a\sin^{2}\theta-2n\cos\theta\right)\sqrt{\Delta}.$
(69)

It is easy to find that $g_{\phi\phi}=0$ corresponds to

$\varLambda_{+}(r,\theta)=0\quad\text{or}\quad\varLambda_{-}(r,\theta)=0.$
(70)

Their roots of ($r$, $\theta$) are different if they exist. Assuming
$(r,\theta)=(\hat{r},\hat{\theta})$ is a solution of
$\varLambda_{+}(r,\theta)=0$, the Eq. (66) can be simplified as

$\left.V\right|_{r=\hat{r},\theta=\hat{\theta}}=2\frac{\Phi^{2}}{D}\frac{\left(\hat{r}^{2}+n^{2}+a^{2}\right)\sin\hat{\theta}}{\hat{r}^{2}+(n+a\cos\hat{\theta})^{2}}\left(a\sin\hat{\theta}+\sqrt{\Delta}\right)\left.(E/\Phi-
H_{-})\right|_{r=\hat{r},\theta=\hat{\theta}}.$ (71)

Moreover, due to $\varLambda_{+}(\hat{r},\hat{\theta})=0$, we have

$\sqrt{\Delta}=\frac{\left(\hat{r}^{2}+n^{2}+a^{2}\right)\sin\hat{\theta}}{2n\cos\hat{\theta}-a\sin^{2}\hat{\theta}}.$
(72)

Thus we get

$a\sin\hat{\theta}+\sqrt{\Delta}=\frac{\hat{r}^{2}+(n+a\cos\hat{\theta})^{2}}{2n\cos\hat{\theta}-a\sin^{2}\hat{\theta}}\sin\hat{\theta}>0,$
(73)

which implies that the coefficient of Eq. (71) is greater than zero.
Therefore, the conditions of the light ring $V=0$ and $\partial_{\mu}V=0$
indicate that $\varLambda_{+}(r,\theta)=0$ corresponds to “-” light ring, i.e,

$E/\Phi=H_{-}\quad\text{and}\quad\partial_{\mu}H_{-}=0,\quad\mu=r,\theta.$
(74)

Furthermore, $\partial_{\mu}^{2}V>0$ (stable) and $\partial_{\mu}^{2}V<0$
(unstable) for light ring correspond to $\partial_{\mu}^{2}H_{-}<0$ and
$\partial_{\mu}^{2}H_{-}>0$, respectively. These show that
$\varLambda_{+}(r,\theta)=0$ can be determined by the retrograde case.
Similarly, $\varLambda_{-}(r,\theta)=0$ can also be determined by the prograde
case. As a result, one can conclude that the case $g_{\phi\phi}=0$ for light
ring is contained in Eq. (14).

##  Appendix B Vector and topological number for $n\neq 0$

We aim to show the pattern of vector, and the corresponding topological number
for certain NUT charge $n$.

The behaviors of the vector (normalized to unity) are shown in Fig. 7 for the
prograde light ring with fixing $m=3$, $a=q=1$, but varying $n$ = 0, 1, 2, and
-1, respectively. From the figures, we can clearly see that the vector is
outwards at $\theta=0$ and $\pi$. In particular, for the NUT charge $n$=0, the
zero points marked with black dot is on the equatorial plane with
$\theta=\pi/2$, see Fig. 7(a). Moreover, the zero points are shifted upwards
for positive $n$ and downwards for negative $n$.

For positive and negative $n$, we exhibit the the vector (normalized to
unity), and the contours $C^{u}$ and $C^{d}$ in Figs. 8(a) and 8(b)
represented by the blue solid and red dashed lines. We also calculate the
deflection angle $\Delta\Omega$ as the function of $\lambda$, and the results
are shown in Figs. 8(c) and 8(d). If $\lambda$ varies from 0 to 1, one shall
make one loop counterclockwise along the contour. Note that the point relating
$\lambda=0$ is at $(r_{h},\pi/2)$. The topological number shall be obtained
via $W=\Delta\Omega(\lambda=1)/2\pi$. Therefore, it is clear that $W^{u}=-1$
and $W^{d}=$0 for positive $n$, and $W^{u}=$0 and $W^{d}=-1$ for negative one.
For the retrograde case, the results will reverse. Interestingly,
$W=W^{u}+W^{d}$ is also obvious. Therefore, we confirm the results given in
Table 1 by taking the specific examples.

(a) $(m,q,a,n)=(3,1,1,0)$ (b) $(m,q,a,n)=(3,1,1,1)$

(c) $(m,q,a,n)=(3,1,1,2)$ (d) $(m,q,a,n)=(3,1,1,-1)$

Figure 7: The red arrows represent $v$ (normalized to unity) on a portion of
the ($r$, $\theta$) plane with $m=3,q=1,a=1$ for the prograde case. Black dots
denote the zero points of $v$, which correspond to the prograde light rings.
(a) $n=0$. (b) $n=1$. (c) $n=2$. (d) $n=-1$.

(a) $(m,q,a,n)=(3,1,2,1)$ (b) $(m,q,a,n)=(3,1,2,-1)$

(c) $(m,q,a,n)=(3,1,2,1)$ (d) $(m,q,a,n)=(3,1,2,-1)$

Figure 8: Vector $v$ (normalized to unity) on a portion of the ($r$, $\theta$)
plane and the deflection angle $\Delta\Omega$ along the boundary. We set
$m$=3, $q$=1, and $a=2$. (a) $n$=1. (b) $n$=-1. (c) $n$=1. (d) $n$=-1. Black
dots denote the zero points of $v$, which correspond to the prograde light
rings. $\lambda$ is a length parameter of the boundaries shown in (a) and (b).
The winding number is given by $w=\Delta\Omega/2\pi$. Note that there exists
no zero points for large $r$.

##  Appendix C Proof for inequality $\partial_{\theta}g$($r$, $\theta$)$<0$ at
$g$($r$, $\theta$)=0

For convenience, we introduce $x=\cos\theta$. Then the Eq. (38) becomes

$g(r,x)=-a^{2}x^{3}+(r^{2}+n^{2}+2a^{2})x+2an+2(n+ax)\sqrt{1-x^{2}}\sqrt{\Delta},$
(75)

where $x\in(-1,1)$. To simplify the discussion, we denote $g(x)\equiv g(r,x)$.
Then the inequality $\partial_{\theta}g(r,\theta)<0$ at $g(x)=0$ turns to

$\partial_{x}g(x)>0.$ (76)

The case of $n=0$ corresponds to the Kerr-Newman black hole whose uniqueness
of the light ring (for both the prograde and retrograde cases) solution is
obvious. Therefore, we mainly focus on the case of $n\neq 0$, and for
simplicity, we consider the following two cases: $a=0$ and $a>0$.

###  C.1 Case: $a=0$

When $a$ vanishes, $g(x)$ will be simplified as

$g\left(x\right)=x\left(n^{2}+r^{2}\right)+2n\sqrt{1-x^{2}}\sqrt{\Delta}.$
(77)

At the boundaries $x=-1$ and $x=1$, we respectively have

$\displaystyle g\left(x=-1\right)$ $\displaystyle=$
$\displaystyle-\left(r^{2}+n^{2}\right),$ (78) $\displaystyle
g\left(x=1\right)$ $\displaystyle=$ $\displaystyle r^{2}+n^{2}.$ (79)

Obviously, $g\left(x=-1\right)$ and $g\left(x=1\right)$ are negative and
positive, respectively. The first and second derivatives of $g(x)$ are given
by

$\displaystyle\partial_{x}g\left(x\right)$
$\displaystyle=n^{2}+r^{2}-\frac{2nx\sqrt{\Delta}}{\sqrt{1-x^{2}}},$ (80)
$\displaystyle\partial_{x}^{2}g\left(x\right)$
$\displaystyle=-\frac{2n\sqrt{\Delta}}{\left(1-x^{2}\right)^{3/2}}.$ (81)

For further discussion, we divide the NUT charge into the following two cases:
$n>0$ and $n<0$.

  * •

$n>0$. In this case, $\partial_{x}^{2}g(x)<0$ is obvious, which indicates that
$\partial_{x}g(x)$ monotonically decreasing with $x$. On the other hand, we
can find $\left.\partial_{x}g(x)\right|_{x\rightarrow-1}>0$ and
$\left.\partial_{x}g(x)\right|_{x\rightarrow 1}<0$. Thus, $g(x)$ increases at
first and then decreases with $x$. Further considering Eqs. (78) and (79),
$g(x)=0$ has only one root and $\partial_{x}g\left(x\right)$ must be positive
at the zero point of $g(x)=0$.

  * •

$n<0$. In this case, $\partial_{x}^{2}g(x)>0$ is obvious, which indicates that
$\partial_{x}g(x)$ monotonically increasing with $x$. On the other hand, we
can find $\left.\partial_{x}g(x)\right|_{x\rightarrow-1}<0$ and
$\left.\partial_{x}g(x)\right|_{x\rightarrow 1}>0$. Thus, $g(x)$ decreases at
first and then increases with $x$. Further combining with Eqs. (78) and (79),
$g(x)=0$ has only one root and $\partial_{x}g\left(x\right)$ must be positive
at the zero point of $g(x)=0$.

Therefore, one can obtain the inequality (76) for $a=0$.

###  C.2 Case: $a>0$

For positive black hole spin, we can re-scale the function $g(x)$ with $a$. In
order to preserve its expression, we need to set $r/a\rightarrow r$,
$n/a\rightarrow n$, $q/a\rightarrow q$, and $m/a\rightarrow m$. Or
equivalently, we can set $a=1$. Then the Eq. (75) shall be

$g(x)=-x^{3}+(r^{2}+n^{2}+2)x+2n+2(n+x)\sqrt{1-x^{2}}\sqrt{\Delta}.$ (82)

From this formula, we can obtain

$\displaystyle g(-1)$
$\displaystyle=-\left(r^{2}+\left(n-1\right)^{2}\right),$ (83) $\displaystyle
g(1)$ $\displaystyle=r^{2}+\left(n+1\right)^{2},$ (84) $\displaystyle g(0)$
$\displaystyle=2n\left(1+\sqrt{\Delta}\right).$ (85)

The corresponding derivatives of $g(x)$ are

$\displaystyle\partial_{x}g(x)$
$\displaystyle=-3x^{2}+r^{2}+n^{2}+2-\frac{2\left(2x^{2}+nx-1\right)\sqrt{\Delta}}{\sqrt{1-x^{2}}},$
(86) $\displaystyle\partial_{x}^{2}g(x)$
$\displaystyle=-6x-\frac{2\left(-2x^{3}+3x+n\right)\sqrt{\Delta}}{\left(1-x^{2}\right)^{3/2}},$
(87) $\displaystyle\partial_{x}^{3}g(x)$
$\displaystyle=-6-\frac{6(nx+1)\sqrt{\Delta}}{\left(1-x^{2}\right)^{5/2}}.$
(88)

For further discussion, we divide the NUT charge into the following three
cases: $n>1$, $0<n\leq 1$, and $n<0$. The discussion is as follows.

  * •

$0<n\leq 1$. We have $\partial_{x}^{3}g(x)<$ 0,
$\left.\partial_{x}^{2}g(x)\right|_{x\rightarrow-1}>0$, and
$\left.\partial_{x}^{2}g(x)\right|_{x\rightarrow 1}<0$, which implies that
$\partial_{x}^{2}g\left(x\right)$ has one zero point and $\partial_{x}g(x)$
first increases and then decreases. Furthermore, we also have
$\left.\partial_{x}g(x)\right|_{x\rightarrow 0}>0$,
$\left.\partial_{x}g(x)\right|_{x\rightarrow-1}\leqslant 0$ and
$\left.\partial_{x}g(x)\right|_{x\rightarrow 1}<0$, which indicate that $g(x)$
decreases first222If $\left.\partial_{x}g(x)\right|_{x\rightarrow-1}=0$ holds,
the decreasing behavior will no longer exist., then increases, and finally
decreases as $x$ varies from $-1$ to $1$. Finally, combining with $g(-1)<0$
and $g(1)>0$, we get $\partial_{x}g(x)>0$ when $g(x)=0$.

  * •

$n>1$. When $-1<x\leqslant 0$, one has $\partial_{x}g(x)>0$, while when
$0<x<1$, $\partial_{x}^{2}g(x)<0$. We also have
$\left.\partial_{x}g(x)\right|_{x\rightarrow 0}>0$ and
$\left.\partial_{x}g(x)\right|_{x\rightarrow 1}<0$. These mean that $g(x)$
will increase first and then decrease as $x$ varies from $-1$ to $1$. By
making use the condition $g(-1)<0$ and $g(1)>0$, we can obtain
$\partial_{x}g(x)>0$ when $g(x)=0$.

  * •

$n<0$. As shown above, we have confirmed $\partial_{x}g(x)>0$ at $g(x)=0$ for
$n>0$. The case with $n<0$ is straightforward by using the relation
$g(x,n)=-g(-x,-n)$. For example, for negative $n$, we have
$\partial_{-x}g(-x,-n)>0$ from above result. Further considering
$\partial_{-x}g(-x,-n)=-\partial_{-x}g(x,n)=\partial_{x}g(x,n)$, one easily
has $\partial_{x}g(x)>0$ as expected.

In summary, we have checked $\partial_{x}g(x)>0$ at $g(x)=0$ for the non-
negative black hole spin.

##  Appendix D Topological phase transition with varying $n$

Here, we would like to consider the topological number and phase transition
for extremal black hole with varying the NUT charge $n$. For simplicity, we
set $m=2$, $a=2$, and $\left|q\right|_{max}=10$. Due to the constraint
$m^{2}+n^{2}=a^{2}+q^{2}$, we need vary $n$ from -10 to 10 as expected.

The quantity $V_{c}^{+(1)}$ is plotted in Fig. 9, but $V_{c}^{-(1)}$ is not
plotted for the reason that it is always less than zero and no phase
transition exists. From this figure, we can find that there will be a phase
transition when $|n|$ is around $3.96$. A simple calculation shows that, the
topological number is -1 for $\left|n\right|>3.96$, and zero for
$\left|n\right|\leq 3.96$.

Figure 9: Quantities $V_{c}^{+(1)}$ (55) as a function of $n$ with $m=2$ and
$a=2$.

On the other hand, the prograde light ring can be obtained by the
intersections of these two kinds curves determined by Eqs. (62) and (63).
Taking $n=$3.70, 3.96, 4.22 and $n=$-3.70, -3.96, 4.22, we show these curves
in Fig. 10. The light rings are marked with the black dots. It is clear that
when $\left|n\right|>3.96$, there are light rings outside the black hole
horizon. While when $\left|n\right|<3.96$, the light rings are behind the
event horizon, leading to the disappearance of these light rings. This clearly
exhibits that there is a topological phase transition near $n\approx 3.96$
characterizing whether the light ring exists or not.

(a) (b)

(c) (d)

Figure 10: The schematic diagram of light ring solutions in ($r$, $\theta$)
plane for the prograde light ring. The intersection points of these curves
marked with black dots are for the prograde light rings. These curves are
given by Eqs. (62) and (63). The black vertical lines represent the locations
of the black hole horizons.

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