# Spherical orbits around Kerr-Newman and Ghosh black holes

A. S. Alam<EMAIL_ADDRESS>L. C. Andaru<EMAIL_ADDRESS>B. N.
Jayawiguna111Present Address: High Energy Physics Theory Group, Department of
Physics, Faculty of Science, Chulalongkorn University, Bangkok 10330,
Thailand<EMAIL_ADDRESS>H. S. Ramadhan222Corresponding author.
<EMAIL_ADDRESS>Departemen Fisika, FMIPA, Universitas Indonesia, Depok,
16424, Indonesia.

###### Abstract

We conduct a comprehensive study on spherical orbits around two types of black
holes: Kerr-Newman black holes, which are charged, and Ghosh black holes,
which are nonsingular. In this work, we consider both null and timelike cases
of orbits. Utilizing the Mino formalism, all analytical solutions for the
geodesics governing these orbits can be obtained. It turns out that all
spherical photon orbits outside the black hole horizons are unstable. In the
extremal cases of both models, we obtain the photon boomerangs. The existence
of charge in the Kerr-Newman allows the orbits to transition between
retrograde and prograde motions, and its increase tends to force the orbits to
be more equatorial. On the other hand, the Ghosh black hole, characterized by
a regular core and a lack of horizons in certain conditions, presents the
possibility of observable stable spherical orbits in the so-called no-horizon
condition. As the Ghosh parameter $k$ increases, trajectories tend to exhibit
larger latitudinal oscillation amplitudes. We observe that as the Ghosh
parameter $k$ increases the trajectories tend to have larger latitudinal
oscillation amplitudes. Finally, we investigate the existence of innermost
stable spherical orbits (ISSOs). Both black holes demonstrate the appearance
of two branches of ISSO radii as a function of the Carter constant C. However,
there are notable differences in their behavior: in the case of the Kerr-
Newman black hole, the branches merge at a critical value, beyond which no
ISSO exists, while for the Ghosh black hole, the transcendental nature of the
metric function causes the branches to become complex at some finite distance.

##  I Introduction

It is well-known that any system possessing spherical symmetry and staticity
will preserve its angular momentum, and as a result, the corresponding
dynamics are confined on a plane. This is true in classical mechanics as well
as in general relativity (GR). What is genuine in GR, among other things, is
that not only massive particles but light can also orbit around a massive
object. Perhaps the ultimate massive object in the universe is a black hole.
Its visual image was first released in 2019 by the Event Horizon Telescope
(EHT) collaboration EventHorizonTelescope:2019dse  for M87*, and for Sgr A*
three years later EventHorizonTelescope:2022wkp . For Schwarzschild we know
that the photon circular orbit (photon sphere, $r_{ps}$) is $r_{ps}=3M$
Carroll:2004st , while for Reissner-Nordstrom (RN) there are
$r_{ps}^{\pm}=3/2\left(M\pm\sqrt{M^{2}-8Q^{2}/9}\right)$ Khoo:2016xqv ;
Pradhan:2010ws . Both the Schwarzschild and RN unstable photon spheres are
outside their corresponding shadow radii Bardeen0 ; Luminet:1979nyg ;
Falcke:1999pj ; Johannsen:2010ru ; Kumar:2018ple ; Afrin:2021wlj , thus can be
observed. In reality, however, all astrophysical objects including black holes
rotate. The appropriate metric solutions are the Kerr and the Kerr-Newman (KN)
Kerr:1963ud ; Newman:1965my . EHT observed the Sgr A* angular diameter
$d_{sh}=48.7\pm 7\mu as$, which exhibits consistency with theoretical
predictions of Kerr black hole EventHorizonTelescope:2022wkp . Moreover, the
observation confirms that the Sgr A* shows the viability of the Kerr model
with constrained charge (KN) Ghosh:2022kit ; Vagnozzi:2022moj ;
Khodadi:2022pqh ; KumarWalia:2022aop ; KumarWalia:2022ddq ; Uniyal:2022vdu .

The rotating metric introduces a departure from spherical symmetry, allowing
for the existence of non-planar orbits, the spherical orbits (SO) being one of
them. These orbits maintain a constant radius and trace the surface of a
sphere. Their discovery stemmed from Carter’s insight in 1968 Carter:1968rr ,
where he identified that rotation introduces an additional Killing tensor,
leading to the conservation of a quantity known as ”Carter’s constant” $C$
through Noether’s theorem. Building upon this foundation, Wilkins in 1972
showcased the existence of ”spherical timelike orbits” (STOs) in the extremal
Kerr metric. Subsequently, Johnston and Ruffini extended these findings to the
Kerr-Newman case Johnston:1974pn .

The exploration of ”spherical photon orbits” (SPO) began with Stuchlik’s
investigation of the Kerr model Stuchlik1981 , followed by Calvani, De Felice,
and Nobili’s study of the Kerr-Newman scenario Calvani:1980is . However, all
studies on SPOs suggest that orbits are stable only within the inner horizon;
beyond this region, they become inherently unstable. Nonetheless, SPOs play a
crucial role in the observation of black holes. It is the multiple orbits of
photon rays around black holes, terminated by their instability, that
contribute to the formation of the luminous ring detected by instruments like
the Event Horizon Telescope (EHT) EventHorizonTelescope:2019dse ;
EventHorizonTelescope:2022wkp . This discovery holds significant implications
for our understanding of black hole phenomena.

To determine the trajectory, one must solve geodesic equations with constant
radii. The first numerical solutions were obtained by Goldstein goldstein .
Mino demonstrated in Mino:2003yg  that the geodesic equations can be
decoupled, paving the way for Fujita and Hikida to achieve analytical
solutions to the timelike geodesic equations in terms of elliptic integrals
Fujita:2009bp . Further analytical investigations into Kerr(-Newman) geodesics
can be found in works such as Hackmann:2010zz ; Hackmann:2013pva ;
Lammerzahl:2015qps ; Wang:2022ouq . Teo leveraged this formalism to derive a
one-parameter class of analytical solutions for spherical photon orbits teo
and timelike orbits Teo:2020sey . These exact spherical orbits have been
extensively explored in subsequent studies (see Grossman:2011ps ;
Tavlayan:2020cso ; Tavlayan:2021ylq ; Hod:2013mgr ).

Additionally for null trajectories, a family of parameters exists that causes
the photon to return to the same point in the opposite direction after one
azimuthal oscillation, satisfying the condition $\Delta\phi=\pi$. This
phenomenon, termed the ”photon boomerang,” was introduced by Page Page:2021rhx
. In the case of Kerr black holes, it occurs in the near-extremal case,
$a\approx 0.994M$, with zero angular momentum. Page demonstrated that the
photon boomerang does not occur in naked singularities (NS) for pure Kerr.
However, in Anjum:2023axh , it was shown that a Kerr black hole surrounded by
perfect fluid dark matter (PFDM) enables the existence of the photon boomerang
under NS conditions. Related works on PFDM can be found in Das:2020yxw ;
Atamurotov:2021hoq ; Atamurotov:2021hck ; Hou:2018avu ; Haroon:2018ryd ;
Das:2021otl ; Narzilloev:2020qtd .

The singularity stands as one of the most generic and conceptually challenging
aspects of black holes. However, it is widely believed that the singularity
might be resolved or regularized within the framework of quantum gravity. The
concept of a regular black hole devoid of a singularity was initially proposed
by Sakharov Sakharov:1966aja  and later realized by Bardeen Bardeen . In the
Bardeen metric, the near-core region exhibits behavior similar to the de
Sitter space, and all invariants remain nonsingular throughout. Ayon-Beato and
Garcia (ABG) demonstrated that the Bardeen metric can be derived from coupling
Einstein’s equations with a nonlinear electrodynamics (NLED) source Ayon-
Beato:1998hmi ; Ayon-Beato:2000mjt . For an in-depth review of regular black
holes, readers are referred to Ansoldi:2008jw  and the accompanying
references.

The concept of static regular black holes was subsequently extended to the
rotating case by Bambi and Modesto Bambi:2013ufa , as well as by Toshmatov et
al. Toshmatov:2014nya  using the Newman-Janis algorithm. The resulting metrics
are characterized by singularity-free Petrov type D solutions, although they
suffer from violations of the weak energy condition. Ghosh Ghosh:2014pba
introduced a probability distribution function to mitigate these violations,
resulting in a class of three-parameter stationary, symmetric metrics
describing regular (nonsingular) rotating black holes. These metrics depend on
the mass ($M$), spin ($a$), and a free parameter ($k$) that quantifies the
deviation from Kerr and generalizes the Kerr-Newman solution. Extensive
literature discussing these solutions and their implications, including
constraints from observations by the Event Horizon Telescope (EHT), can be
found in Amir:2016cen ; Kumar:2020yem ; Kumar:2020cve ; Kumar:2020ltt ;
KumarWalia:2022aop .

This paper aims to extensively explore the spherical orbit dynamics around
Kerr-Newman and Ghosh black holes. In Section II, we delve into the behavior
of spherical orbits around the Kerr-Newman (KN) black hole. We provide
exhaustive analytical solutions and explicitly describe trajectories of the
corresponding spherical photon orbits (SPOs) in Section II.1, and spherical
timelike orbits (STOs) in Section II.2. Transitioning to Section III, we
investigate the spherical orbit solutions of the (regular) Ghosh black hole.
There, we present analytical solutions and conduct a thorough analysis of
various types of SPOs and STOs trajectories in Section III.1 and Section
III.2, respectively. Finally, in Section IV, we synthesize and summarize our
findings.

##  II Spherical Orbits around the Kerr-Newman Black Hole

The Kerr-Newman (KN) solutions in Boyer-Lindquist coordinate are Newman:1965my

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

and

$A_{\mu}=\frac{Qr}{\Sigma}\bigg{(}1,0,0,-a\sin^{2}\theta\bigg{)},$ (2)

where

$\begin{split}\Sigma&\equiv r^{2}+a^{2}\cos^{2}{\theta},\\\ \Delta&\equiv
r^{2}+a^{2}-2Mr+Q^{2},\\\ \end{split}$ (3)

and $M$, $Q$, $a$ denote the black hole’s mass, charge, and angular momentum
per unit mass, respectively. The allowed range of $a$ $0<a^{2}\leq
M^{2}-Q^{2},$ the upper and lower bounds of which are related to the extremal
KN and the RN, respectively. The event horizons are $r_{\pm}\equiv
M\pm\sqrt{M^{2}-a^{2}-Q^{2}}.$ By defining $u\equiv\cos\theta$ the geodesic
equations can be derived using the Hamilton-Jacobi formalism Carter:1968rr ;
Johnston:1974pn ; Wang:2022ouq :

$\displaystyle\Sigma\dot{t}$ $\displaystyle=$
$\displaystyle\left(\frac{r^{2}+a^{2}}{\Delta}\right)P(r)-a\left[aE(1-u^{2})-L_{z}\right],$
(4) $\displaystyle\Sigma\dot{r}$ $\displaystyle=$ $\displaystyle\sqrt{R(r)},$
(5) $\displaystyle\Sigma\dot{u}$ $\displaystyle=$
$\displaystyle\sqrt{\Theta(u)},$ (6) $\displaystyle\Sigma\dot{\phi}$
$\displaystyle=$
$\displaystyle\frac{a}{\Delta}P-\left(aE-\frac{L_{z}}{1-u^{2}}\right),$ (7)

with

$\displaystyle R(r)$ $\displaystyle\equiv$ $\displaystyle
P(r)-\Delta[\mathcal{C}+\mu^{2}r^{2}+(L_{z}-aE)^{2}],$
$\displaystyle\Theta(u)$ $\displaystyle\equiv$
$\displaystyle\mathcal{C}-\left[\mathcal{C}+L_{z}^{2}+a^{2}(\mu^{2}-E^{2})\right]u^{2}+a^{2}(\mu^{2}-E^{2})u^{4},$
(8) $\displaystyle P(r)$ $\displaystyle\equiv$
$\displaystyle\left[E(r^{2}+a^{2})-aL_{z}\right]^{2}.$

The photon (/timelike) case appears when $\mu=0$ $(/\mu=1)$. The dot refers to
the derivative wrt the Affine parameter $\tau$, $\mathcal{C}$ is the Carter’s
constant, $E$ and $L_{z}$ are the energy and angular momentum about
$\phi$-axis of the photon, respectively.

###  II.1 Spherical Photon Orbits

The SPO conditions are Eqs. (4)-(II) with $\mu=0$. The solutions for Eq. (6)
are physical whenever $\Theta(u)\geq 0$. The bounds of $u$ values are the
roots of $\Theta(u_{0})=0$. Since $\Theta(u)$ is not $Q-$dependent, the roots
are exactly the same as in the Kerr case teo ; Teo:2020sey . Defining

$\Phi\equiv\frac{L_{z}}{E},\ \ \ \ C\equiv\frac{\mathcal{C}}{E^{2}},$ (9)

we have, for $C>0$,

$u_{0}^{2}=\frac{(a^{2}-C-\Phi^{2})+\sqrt{(a^{2}-C-\Phi^{2})^{2}+4a^{2}C}}{2a^{2}}.$
(10)

When $C<0$, the following condition must be satisfied

$a^{2}-C-\Phi^{2}>0.$ (11)

The spherical orbit condition requires the corresponding radius ($\equiv
r_{SO}$) satisfies

$R(r=r_{SO})=\frac{dR(r)}{dr}\bigg{|}_{r=r_{SO}}=0.$ (12)

Solving the two equations simultaneously, we get two conditions:

$\displaystyle(i)~{}~{}\Phi$ $\displaystyle=$
$\displaystyle\frac{r_{SO}^{2}+a^{2}}{a},\ \ \ \ \ \ \ \ \ \
C=-\frac{r_{SO}^{4}}{a^{2}};$ (13) $\displaystyle(ii)~{}~{}\Phi$
$\displaystyle=$
$\displaystyle-\frac{a^{2}M+a^{2}r_{SO}-3Mr_{SO}^{2}+2Q^{2}r_{SO}+r_{SO}^{3}}{a(r_{SO}-M)},$
$\displaystyle C$ $\displaystyle=$
$\displaystyle-\frac{r_{SO}^{2}\left[4a^{2}\left(Q^{2}-Mr_{SO}\right)+\left(r_{SO}(r_{SO}-3M)+2Q^{2}\right)^{2}\right]}{a^{2}(r_{SO}-M)^{2}}.$
(14)

Note that both solutions (13)-(14) satisfy

$\frac{d^{2}R(r)}{dr^{2}}\bigg{|}_{r=r_{SO}}>0,$ (15)

implying that they are unstable under radial perturbation Wilkins:1972rs . The
first condition (13) is unphysical teo  since it does not satisfy condition
(11). In the second condition (14) $C$ must take a positive sign since $C<0$
violates the constraint (11).

Qualitatively, the plots of $C$ and $\Phi$ look similar to the Kerr case.
However, they shift as $Q$ varies, as shown in Fig. 1. As $Q$ increases, the
plot of $C$ shifts to the left while its maximum point decreases. Since
Carter’s constant $C$ represents the angular velocity of the photon in the
equatorial direction (as in the Kerr case), for highly-charged black holes the
orbits are forced to concentrate on the equatorial plane only.

|
---|---
Figure 1: Typical plots of $C$ and $\Phi$ as functions of $r_{SO}$ for several
values of $Q$: $Q=0$ (black), $Q=0.3M$ (purple), $Q=0,5M$ (green) dan $Q=0,7M$
(red).

####  II.1.1 Analytical Solutions

The geodesic Eqs. (4)-(II) can be solved analytically. Defining the Mino’s
parameter $\lambda$ Mino:2003yg

$\frac{d\tau}{d\lambda}\equiv\Sigma\ \ \ \rightarrow\ \ \ \
\Sigma\frac{dx^{\mu}}{d\tau}=\frac{dx^{\mu}}{d\lambda},$ (16)

the coordinates of $(\theta,\phi,t)$ can be solved in terms of the elliptic
integral and the Jacobi elliptic functions Grad Abram Byrd . The Eq. (6) can
be cast, by defining $w\equiv u^{2}$, as

$\frac{dw}{d\lambda}=\pm 2Y(w),$ (17)

where

$Y(w)^{2}\equiv-a^{2}w(w-w_{1})(w-w_{2}),$ (18)

and $w_{1,2}$ are the roots of $Y(w)^{2}$:

$w_{1,2}=\frac{a^{2}-C-\Phi^{2}\pm\sqrt{4a^{2}C\left(-C-\Phi^{2}+1\right)^{2}}}{2a^{2}}.$
(19)

Choosing the positive sign and taking the initial condition $w(\lambda=0)=0$,
Eq. (17) can be integrated

$\lambda=\frac{1}{2}\int_{0}^{w}\frac{dw}{Y(w)}=\frac{1}{a\sqrt{w_{1}-w_{2}}}\
F(\Psi,k),$ (20)

where

$\Psi\equiv\arcsin{\sqrt{\frac{w(w_{1}-w_{2})}{w_{1}(w-w_{2})}}},\ \ \
k\equiv\sqrt{\frac{w_{1}}{w_{1}-w_{2}}},$ (21)

and $F(\Psi,k)$ is the incomplete elliptic integral function of the first kind
Grad . From Eq. (20) it can be seen that $\lambda$ increases along with $w$.
The one full latitudinal oscillation for $\lambda$ is

$\Delta\lambda=\frac{4}{a\sqrt{w_{1}-w_{2}}}K(k),$ (22)

with $K(k)$ is the complete elliptic integral function of the first kind.

To obtain the solution for $\theta$, it is necessary to write the inverse of
Eq. (20) in the form of the Jacobi amplitude function:

$\Psi=\mathrm{am}(a\sqrt{w_{1}-w_{2}}\lambda,k).$ (23)

by writing the Eq. (23) above in terms of the elliptic sine $sn$ function,

$\sin{\Psi}=\mathrm{sn}(a\sqrt{w_{1}-w_{2}}\lambda,k),$ (24)

the analytical solution of $\theta$ can be written as

$\theta(\lambda)=\arccos\left[{\sqrt{-w_{2}}\ k\
\text{sd}\left(a\sqrt{w_{1}-w_{2}}\lambda,k\right)}\right],$ (25)

where $sd(x,y)\equiv sn(x,y)/dn(x,y)$ and $dn(x,y)$ is the so-called the delta
amplitude function defined as

$dn(x,y)\equiv\frac{d}{dx}\mathrm{am}(x,y).$ (26)

From Eq. (16), the geodesic equations for $\phi$ and $t$ can be written

$\displaystyle\frac{d\phi}{dw}$ $\displaystyle=$
$\displaystyle\frac{1}{2Y(w)}\left[\frac{a(r^{2}+a^{2}-a\Phi)}{\Delta}-a+\frac{\Phi}{1-w}\right],$
(27) $\displaystyle\frac{dt}{dw}$ $\displaystyle=$
$\displaystyle\frac{1}{2Y(w)}\left[\frac{(r^{2}+a^{2})^{2}-(r^{2}+a^{2})a\Phi}{\Delta}+a\Phi-a^{2}(1-w).\right]$
(28)

Assuming the initial condition $\phi(w=0)=t(w=0)=0$, the solutions are

$\displaystyle\phi(\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Phi}{a(1-w_{2})\sqrt{w_{1}-w_{2}}}[F(\Psi,k)-w_{2}\Pi(\Psi,k^{2}(1-w_{2}),k)]$
(29)
$\displaystyle+\left[\frac{a(r^{2}+a^{2}-a\Phi)}{\Delta}-a\right]\lambda,$
$\displaystyle t(\lambda)$ $\displaystyle=$
$\displaystyle-\frac{a}{\sqrt{w_{1}-w_{2}}}[(1-w_{2})F(\Psi,k)+w_{2}\Pi(\Psi,k^{2},k)]$
(30)
$\displaystyle+\left[\frac{(r^{2}+a^{2})^{2}-(r^{2}+a^{2})a\Phi}{\Delta}+a\Phi\right]\lambda,$

where $\Pi(\Psi,w_{1},k)$ is the incomplete elliptic integral function of the
third kind. The above two solutions are valid for all values of the $\lambda$
parameter. When $Q=0$ the solution reduces to the Teo’s teo . The changes in
$\phi$ and $t$ for a period of $\Delta\lambda$, i.e. one complete latitudinal
oscillation, are

$\displaystyle\Delta\phi$ $\displaystyle=$
$\displaystyle\frac{4}{\sqrt{w_{1}-w_{2}}}\left[\frac{\Phi}{a(1-w_{1})}\Pi(-\frac{-w_{1}}{1-w_{1}},k)+\left(\frac{r^{2}+a^{2}-a\Phi}{\Delta}-1\right)K(k)\right],$
(31) $\displaystyle\Delta t$ $\displaystyle=$
$\displaystyle-\frac{4}{\sqrt{w_{1}-w_{2}}}\bigg{\\{}-a[(1-w_{2})K(k)+w_{2}\Pi(k^{2},k)]$
(32)
$\displaystyle+\left[\frac{(r^{2}+a^{2})^{2}-(r^{2}+a^{2})a\Phi}{a\Delta}+\Phi\right]K(k)\bigg{\\}}.$

Eq. (31) determines the type of spherical orbits formed around the black hole.
When $\Delta\phi$ is positive the orbit is prograde, and when it is negative
it becomes retrograde.

The analytical solutions (25) and (29)-(30) completely specify the photon
trajectories. In the following subsections, we shall present and discuss
extensively their 3D plots based on the extremality of the black hole.

####  II.1.2 Non-extreme Kerr-Newman Orbits

The number of KN horizons is parametrized by (II). Here we discuss the non-
extremal KN, specifically, we work on the case where

$a=0.8\sqrt{M^{2}-Q^{2}}.$ (33)

The constraints (13)-(14) can be used to determine the orbit parameters. Once
both angular momenta for the source,$a$, and the test particle, $\Phi$, are
fixed, then the rest of the parameters are completely determined.

In Tables 1-2 we show orbit parameters for $\Phi=0,-M$ and several variations
of the charge $Q$.

Orbit |  $Q/M$ |  $r_{SO}/M$ |  $C/M^{2}$ |  $u_{0}$ |  $\Delta\phi$
---|---|---|---|---|---
(a) |  0 |  2.671 |  24.209 |  1 |  2.242
(b) |  0.6 |  2.477 |  21.546 |  1 |  2.050
(c) |  0.7 |  2.398 |  20.5271 |  1 |  1.941
Table 1: Parameters of the SPO around a non-extreme KN for $\Phi=0$.

In Fig. 2, we present examples of orbit trajectories for two latitudinal
oscillations when $\Phi=0$ in the $(x-y-z)$ plane, $x-y$, and $x-z$ planes,
respectively. Each plot depicts the orbit on an imaginary sphere with a fixed
radius $r_{SO}$. The photons start from the equator and progress northward in
each orbit, consistent with the direction of rotation of the black hole
itself, which is from west to east. The nine illustrations from Figs. 2 (a),
(b), and (c) correspond to cases where $Q=0$ (Kerr), $Q=0.6M$, and $Q=0.7M$,
respectively. When $Q=0.6M$ the orbit veers away from the pole. The presence
of spherical photon orbits (SPOs) appears to be hindered by the increasing
charge. As depicted in Fig. 2 (c), for $Q=0.7M$ the photon orbits
predominantly concentrate near the equatorial plane.

Figure 2: Two latitudinal oscillations of the photon orbits around the non-
extreme KN with $\Phi=0$ in the $(x-y-z)$ as well as the projective $x-y$ and
$x-z$ planes.. The orbit begins from the equator and heads northwards. The
direction of the black hole’s rotation is from west to east. Figures (a), (b),
and (c) are cases with $Q=0$, $Q=0.6M$, and $Q=0.7M$, respectively.

In Fig. 3 we illustrate three examples of orbits for $\Phi=-M$ in sixteen
latitudinal oscillations. Each orbit is plotted on an imaginary sphere with a
fixed radius $r_{SO}$ (see Table 2).

Orbit |  $Q/M$ |  $r_{SO}/M$ |  $C/M^{2}$ |  $u_{0}$ |  $\Delta\phi$
---|---|---|---|---|---
(a) |  0 |  2.882 |  26.611 |  0.9821 |  -0.1922
(b) |  0.3 |  2.833 |  25.854 |  0.9815 |  -0.0299
(c) |  0.6 |  2.669 |  23.45 |  0.97967 |  0.309725
Table 2: Parameters of the SPO around a non-extreme KN for $\Phi=-M$.

In Fig. 3, plots (a), (b), and (c) correspond to the cases of $Q=0$ (Kerr),
$Q=0.3M$, and $Q=0.6M$, respectively. Similar to the previous scenario, an
increase in charge causes the orbital trajectory to align more closely with
the equatorial plane. Of particular interest is the behavior observed for
$Q=0.6M$: the orbits densely cluster around a segment of solid angle but do
not complete a full azimuthal rotation. This phenomenon arises because
$Q=0.6M$ is the transition value between retrograde and prograde motions.

Figure 3: Sixteen latitudinal oscillations of the photon orbits around the
non-extreme KN with $\Phi=-M$ in the $(x-y-z)$ as well as the projective $x-y$
and $x-z$ planes. The orbit begins from the equator and heads northwards. The
direction of the black hole’s rotation is from west to east. Figures (a), (b),
and (c) are cases with $Q=0$, $Q=0.3M$, and $Q=0.6M$, respectively.

####  II.1.3 Extreme Kerr-Newman Orbit

The extremal KN happens when

$a=\sqrt{M^{2}-Q^{2}}.$ (34)

In this work, the orbit parameters for extremal KN are shown in Tables 3-4 for
$\Phi=0,-M$, respectively.

Orbit |  $Q/M$ |  $r/M$ |  $C/M^{2}$ |  $u_{0}$ |  $\Delta\phi$
---|---|---|---|---|---
(a) | 0 | 2.414 | 22.314 | 1 | 3.1761
(b) | 0.4 | 2.356 | 21.372 | 1 | 2.473
(c) | 0.8 | 2.166 | 18.410 | 1 | 0.596
Table 3: Parameters of the spherical photon orbit around extreme Kerr-Newman
black hole when $\Phi=0$.

In Fig. 4, we depict three examples of orbits for two latitudinal oscillations
when the angular momentum of the photon $\Phi=0$ in the $(x-y-z)$ plane, the
$x-y$ and $x-z$ planes respectively. Each plot showcases the orbit on an
imaginary sphere with a fixed radius (details regarding the radius and other
parameters can be found in Table 3). The photons initiate their movement from
the equator and progress northward in each orbit, mirroring the direction of
rotation of the black hole itself, which is from west to east. The nine
illustrations in Figs. 4 (a), (b), and (c) correspond to cases where $Q=0$,
$Q=0.4M$, and $Q=0.8M$, respectively. Notably, the orbit depicted in (a)
aligns with the one observed in Teo teo , where they touch one another at the
poles. However, for $Q=0.4M$ the orbits, previously tangent to each other,
begin to separate. The charge of the black hole begins to change the direction
of the photons’ orbits at the poles. Upon further increasing the charge to
$Q\sim 0.8M$, the two latitudinal oscillations are insufficient to complete an
orbit. Additionally, the increased charge concentration renders the orbit more
aligned with the equatorial plane, as evidenced in Fig. 4 (c). This trend is
attributed to the reduction in the black hole’s angular momentum. Furthermore,
we investigate the occurrence of a photon boomerang in this extremal case. The
boomerang condition,

$\Delta\phi=\pi,$ (35)

is satisfied when the charge value is small, $Q=0.02528M$ and $\Phi=0$. The
corresponding plot closely resembles Fig. 4 (a).

Figure 4: Two latitudinal oscillations of the photon orbit around the extremal
KN with $\Phi=0$ in the $(x-y-z)$ as well as the projective $x-y$ and $x-z$
planes. The orbit begins from the equator and heads northwards. The direction
of the black hole’s rotation is from west to east. Figures (a), (b), and (c)
are cases with $Q=0$, $Q=0.4M$, and $Q=0.8M$, respectively.

In Fig. 5 we illustrate three examples of SPO with $\Phi=-M$ for five latitude
oscillations. Each orbit is plotted on an imaginary sphere with a fixed radius
whose radius values, as well as other orbital parameters, can be found in
Table 4.

Orbit |  $Q/M$ |  $r/M$ |  $C/M^{2}$ |  $u_{0}$ |  $\Delta\phi$
---|---|---|---|---|---
(a) |  0 |  2.732 |  25.85 |  1 |  -3.714
(b) |  0.5 |  2.617 |  23.92 |  0.9803 |  -0.3959
(c) |  0.6 |  2.562 |  23.02 |  0.9794 |  0.0303
Table 4: Parameters of the spherical photon orbit around Kerr-Newman black
hole when $\Phi=-M$.

Figs. 5 (a)-(c) depict the scenarios where $Q=0$ (Kerr), $Q=0.5M$, and
$Q=0.6M$, respectively. The presence of charge alters the direction of the
photon, and its augmentation prompts the orbit to return to the equatorial
plane. Notably, for $Q=0.6M$ an intriguing phenomenon happens: the photon
undergoes a twist in its rotation direction about the $z$ axis. Initially
retrograde ($Q<0.6M$), it abruptly transitions to prograde ($Q\geq 0.6M$). The
charge can effectively steer photons to be in the direction of the black
hole’s rotation, resulting in a ribbon-shaped orbit, as evidenced in Fig. 5
(c).

Figure 5: Five latitudinal oscillations of the photon orbits around extremal
KN with $\Phi=-M$ in the $(x-y-z)$ as well as the projective $x-y$ and $x-z$
planes. Figures (a), (b), and (c) are cases with $Q=0$, $Q=0.5M$, and $Q=0.6M$
respectively.

###  II.2 Spherical Timelike Orbits

The timelike geodesics equations around a Kerr-Newman black hole remain
consistent with Eqs. (4)-(II), with $\mu=1$. Bound and unbound orbits manifest
when $E^{2}<1$ and $E^{2}>1$, respectively. Adhering to condition (12),
typically yields four solutions. The initial two solutions are described by:

$\displaystyle E_{a,b}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{X\pm
2ar_{SO}^{2}Y\left(-Mr_{SO}+Q^{2}\right)\left(a^{2}-2Mr_{SO}+Q^{2}+r_{SO}^{2}\right)}{\Gamma}},$
(36) $\displaystyle\Phi_{a,b}$ $\displaystyle=$
$\displaystyle\frac{ar_{SO}^{2}Y\left(Mr_{SO}-Q^{2}\right)\left[a^{2}+r_{SO}(r_{SO}-2M)+Q^{2}\right]+\alpha}{\beta}E_{a,b},$

where

$\displaystyle X$ $\displaystyle\equiv$ $\displaystyle
r_{SO}^{2}\bigg{\\{}-2a^{4}\mathcal{C}\left(Q^{2}-Mr\right)-a^{2}s+r_{SO}^{2}\left[r_{SO}(r_{SO}-3M)+2Q^{2}\right]\left[r(r_{SO}-2M)+Q^{2}\right]^{2}\bigg{\\}},$
$\displaystyle Y$ $\displaystyle\equiv$
$\displaystyle\sqrt{a^{2}C^{2}+r_{SO}^{2}\left[3CMr_{SO}-r_{SO}^{2}\left(C+Q^{2}\right)-2CQ^{2}+Mr_{SO}^{3}\right]},$
(38) $\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle
r_{SO}^{6}\\{{4a^{2}(Q^{2}-Mr_{SO})+[2Q^{2}+r_{SO}(-3M+r_{SO})]^{2}\\}}.$ (39)
$\displaystyle s$ $\displaystyle\equiv$
$\displaystyle\mathcal{C}\left[r_{SO}^{2}\left(5M^{2}-4Mr_{SO}+r_{SO}^{2}\right)+2Q^{2}r_{SO}(r_{SO}-3M)+2Q^{4}\right]$
(40)
$\displaystyle+r_{SO}^{2}\left(Mr-Q^{2}\right)\left[r_{SO}(3r_{SO}-5M)+2Q^{2}\right],$
$\displaystyle\alpha$ $\displaystyle\equiv$ $\displaystyle
a^{2}r_{SO}\left(Q^{2}-Mr_{SO}\right)\left[-Mr_{SO}^{2}\left(a^{2}+C+4Q^{2}\right)+a^{2}\mathcal{C}M+Q^{2}r_{SO}\left(a^{2}+C+Q^{2}\right)\right]$
(41)
$\displaystyle+\left[2r_{SO}^{3}\left(2M^{2}+Q^{2}\right)-3Mr_{SO}^{4}\right]a^{2}r_{SO}\left(Q^{2}-Mr_{SO}\right),$
$\displaystyle\beta$ $\displaystyle\equiv$ $\displaystyle a\
r_{SO}\left(Q^{2}-Mr_{SO}\right)\bigg{\\{}a^{2}\left[C(M-r_{SO})+r_{SO}\left(Q^{2}-Mr_{SO}\right)\right]$
(42) $\displaystyle+r_{SO}\left[r_{SO}(r_{SO}-2M)+Q^{2}\right]^{2}\bigg{\\}}.$

The third and fourth solutions are related to the first two as the following

$(E_{c,d},\Phi_{c,d})=-(E_{a,b},\Phi_{a,b}),$ (43)

just like the Kerr case Teo:2020sey . The first two are, however, the only
physical solution: $(E_{a,b},\Phi_{a,b})$.

####  II.2.1 Analytical Solutions

The roots of $\Theta(u)=0$ (Eq. (II)) with $w\equiv u^{2}$ can be written as

$w_{1,2}=\frac{\left[a^{2}\left(1-E^{2}\right)+\mathcal{C}+\Phi^{2}\mp\sqrt{\left[a^{2}\left(1-E^{2}\right)+\mathcal{C}+\Phi^{2}\right]^{2}-4a^{2}\mathcal{C}\left(1-E^{2}\right)}\right]}{2\left(1-E^{2}\right)}.$
(44)

The ranges of $w_{1,2}$ depend on the sign of $E^{2}<1$ or $E^{2}>1$, and the
sign of $C$. For bound orbit we consider $E^{2}<1$ and $0\leq w\leq w_{1}\leq
1<w_{2}$. The exact solutions for $(\theta,\phi,t)$ are

$\displaystyle\theta(\lambda)$ $\displaystyle=$
$\displaystyle\arccos\left(\sqrt{w_{1}}\
\text{sn}[a\sqrt{\left(1-E^{2}\right)w_{2}}\ \lambda,k]\right),$ (45)
$\displaystyle\phi(\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Phi}{a\sqrt{\left(1-E^{2}\right)w_{2}}}\Pi(\Psi,w_{1},k)+\frac{aP-a\Delta
F}{\Delta}\lambda,$ (46) $\displaystyle t(\lambda)$ $\displaystyle=$
$\displaystyle-\frac{aE}{\sqrt{(1-E^{2})w_{2}}}[(1-w_{2})F(\Psi,k)+w_{2}E(\Psi,k)]$
(47)
$\displaystyle+\left[\frac{(r^{2}+a^{2})^{2}-(r^{2}+a^{2})a\Phi}{\Delta}+a\Phi\right]\lambda,$

where $E(\psi,k)$ is the incomplete elliptic integral of the second kind, and

$\Psi\equiv\text{am}(a\sqrt{\left(1-E^{2}\right)w_{2}}\ \lambda,k),\ \ \
k\equiv\sqrt{\frac{w_{1}}{w_{2}}}.$ (48)

$\theta$ is a periodic function of $\lambda$, with period

$\Delta\lambda=\frac{4}{a\sqrt{\left(1-E^{2}\right)w_{2}}}K(k).$ (49)

The change in $\phi$ and $t$ for one period are

$\displaystyle\Delta\phi$ $\displaystyle=$
$\displaystyle\frac{4}{a\sqrt{\left(1-E^{2}\right)w_{2}}}\left[\frac{\Phi}{a}\Pi(w_{1},k)+\frac{a(P-E\Delta)}{\Delta}K(k)\right],$
(50) $\displaystyle\Delta t$ $\displaystyle=$
$\displaystyle-\frac{4}{a\sqrt{\left(1-E^{2}\right)w_{2}}}\bigg{\\{}-a[(1-w_{2})K(k)+w_{2}E(k)]$
(51)
$\displaystyle+\left[\frac{(r^{2}+a^{2})^{2}-(r^{2}+a^{2})a\Phi}{a\Delta}+\Phi\right]K(k)\bigg{\\}}.$

For $E^{2}>1$ the orbits become unbound, and the range of $w$ is $w_{2}<0\leq
w\leq w_{1}\leq 1$. The exact solutions for $(\theta,\phi,t)$ are similar to
the Eqs. (45)-(47), only that the elliptic functions have imaginary modulus
Teo:2020sey . The case for $E^{2}=1$ is known as the marginally bound orbits
Bardeen:1972fi .

####  II.2.2 Selected Orbit

The analytical solutions described by Eqs. (45)-(47) offer insight into the
behavior of STOs. We will focus on plotting the case of extreme Kerr-Newman
(KN) only, as the non-extreme scenario exhibits similar qualitative behavior.

Orbit |  $Q/M$ |  $r_{SO}/M$ |  E |  $u_{0}$ |  $\Delta\phi$
---|---|---|---|---|---
(a) |  0 |  10 |  0.9559 |  1 |  0.4869
(b) |  0.45 |  10.39 |  0.9570 |  0.8930 |  0.4624
(c) |  0.999 |  11.74 |  0.9606 |  0.045 |  0.3915
Table 5: Parameters for STO around extremal KN with $\Phi=0$ and $C=14.1M$.

In Fig. 6, we present three examples of orbits corresponding to $\Phi=0$. Each
plot depicts the orbit on an imaginary sphere with a fixed radius (its value
is provided in Table 5). The particle initiates its movement from the equator
and progresses northward in each orbit, in alignment with the rotation
direction of the black hole itself, which is from west to east.

Figure 6: Spherical timelike orbits of extremal KN with $\Phi=0$ and $C=14.1M$ in the $(x-y-z)$ as well as the projective $x-y$ and $x-z$ planes. The orbit begins from the equator and heads northwards. The direction of the black hole’s rotation is from west to east. Figures (a), (b), and (c) are cases with $Q=0$, $Q=0.45M$, and $Q=0.999M$, respectively. Orbit |  $Q/M$ |  $r_{SO}/M$ |  E |  $u_{0}$ |  $\Delta\phi$
---|---|---|---|---|---
(a) |  0 |  7 |  0.9500 |  0.9313 |  -46.4684
(b) |  0.45 |  8.5361 |  0.9526 |  0.8317 |  -14.3344
(c) |  0.999 |  11.35 |  0.9595 |  0.0416 |  -50.765
Table 6: STO parameters around the extremal Kerr-Newman black hole with
$\Phi=-1.350M$ and $C=12M$.

As the charge increases, the orbits progressively concentrate more around the
equator. When $Q=0.999M$ (approaching the limit $Q\rightarrow M$) the orbit
effectively becomes equatorial.

In Fig. 7 we depict STOs for $\Phi=-1.350M$, with the radii and other orbit
parameters detailed in Table6. These orbits are plotted for $Q=0$, $Q=0.45M$,
and $Q=0.999M$, respectively. As the charge $Q$ increases, it not only alters
the orbital direction of the test particle but also confines the orbit closer
to the equatorial plane. However, unlike photon orbits, we observe that in the
timelike case, there is no transition from retrograde to prograde motion with
increasing charge.

Figure 7: Spherical orbit for extremal KN with $\Phi=-1350M$ and $C=12M$ in
$(x-y-z)$ plane, the projective $x-y$ and $x-z$ plane. The orbit begins from
the equator and heads northwards. The direction of the black hole’s rotation
is from west to east. Label (a), (b), and (c) are cases when the charged value
is $Q=0$, $Q=0.45M$, and $Q=0.999M$ respectively.

####  II.2.3 Innermost Stable Spherical Orbits

From Eq.(5), it is evident that the function $R(r)$ in (II) serves as the
effective radial potential, $V_{eff}(r)\equiv-R(r)$. Analogous to static
cases, extrema typically exist for each specific choice of parameters,
representing stable or unstable spherical orbits. At certain critical
parameter values, these extrema merge. In the context of equatorial motion,
this phenomenon is referred to as the ”innermost stable circular orbit”
(ISCO), while for spherical motion, it is termed the ”innermost stable
spherical orbit” (ISSO) Druart:2023mba . The ISSO is defined as:

$V_{eff}(r)\bigg{|}_{r=r_{ISSO}}=\frac{dV_{eff}(r)}{dr}\bigg{|}_{r=r_{ISSO}}=\frac{d^{2}V_{eff}(r)}{dr^{2}}\bigg{|}_{r=r_{ISSO}}=0.$
(52)

This condition, supplemented with Eq. (36), yields

$\displaystyle\frac{-\gamma+\delta+f+6Mr\sqrt{\sigma}-8Q^{2}\sqrt{\sigma}}{rT\left(Mr-Q^{2}\right)}=0,$
(53)
$\displaystyle\frac{V\left(\gamma+\delta-f+6Mr\sqrt{\sigma}-8Q^{2}\sqrt{\sigma}\right)}{T}=0,$
(54)

for the $(\pm)$ conditions, respectively. Here,

$\displaystyle\delta$ $\displaystyle\equiv$ $\displaystyle
S\left(8r_{ISSO}^{2}\left(C\left(3M^{2}+Q^{2}\right)-Q^{4}\right)-26CMQ^{2}r_{ISSO}+8CQ^{4}+U\right),$
$\displaystyle\gamma$ $\displaystyle\equiv$ $\displaystyle
2a^{4}Cr_{ISSO}^{2}\left(4Q^{2}-5Mr_{ISSO}\right)\left(Q^{2}-Mr_{ISSO}\right),$
$\displaystyle V$ $\displaystyle=$ $\displaystyle
ar_{ISSO}\left(a^{2}\eta+r_{ISSO}\left(r_{ISSO}(r_{ISSO}-2M)+Q^{2}\right)^{2}\right),$
$\displaystyle T$ $\displaystyle\equiv$ $\displaystyle
4a^{2}\left(Q^{2}-Mr_{ISSO}\right)+\left(r_{ISSO}(r_{ISSO}-3M)+2Q^{2}\right)^{2},$
$\displaystyle f$ $\displaystyle\equiv$ $\displaystyle\xi
r_{ISSO}^{4}\left(Q^{2}-Mr_{ISSO}\right)\left(r_{ISSO}(r_{ISSO}-3M)+2Q^{2}\right),$
$\displaystyle\sigma$ $\displaystyle\equiv$ $\displaystyle
S^{2}\left(a^{2}+r_{ISSO}(r_{ISSO}-2M)+Q^{2}\right)^{2}\zeta,$ $\displaystyle
S$ $\displaystyle\equiv$ $\displaystyle
a^{2}r_{ISSO}^{2}\left(Q^{2}-Mr_{ISSO}\right),$ $\displaystyle U$
$\displaystyle\equiv$ $\displaystyle
r_{ISSO}^{4}\left(4C-7M^{2}+4Q^{2}\right)+2Mr_{ISSO}^{3}\left(7Q^{2}-9C\right)-3Mr_{ISSO}^{5},$
$\displaystyle\xi$ $\displaystyle\equiv$
$\displaystyle(-9MQ^{2}r_{ISSO}+Mr_{ISSO}^{2}(6M-r_{ISSO})+4Q^{4},$
$\displaystyle\zeta$ $\displaystyle\equiv$
$\displaystyle(a^{2}C^{2}+r_{ISSO}^{2}\left(3CMr_{ISSO}-r_{ISSO}^{2}\left(C+Q^{2}\right)-2CQ^{2}+Mr_{ISSO}^{3}\right),$
$\displaystyle\eta$ $\displaystyle\equiv$ $\displaystyle
C(M-r_{ISSO})+r_{ISSO}\left(Q^{2}-Mr_{ISSO}\right).$

Figure 8: Plot of $r_{isso}$ as a function of $C$ for $a=0.7M$ and $Q=0.1M$.
The dashed lines are the corresponding horizons. The blue (red) lines are the
negative (positive) roots of Eq. (53) (Eq. (54)), respectively.

We solve these equations to derive $r_{ISSO}$ as a function of Carter
constant. The resulting plot is depicted in Fig. 8, aligning with the findings
of Wang, Lee, and Lin Wang:2022ouq . For each fixed value of $a$ and $Q$, two
branches of observable $r_{ISSO}$($C$) exist outside the horizons. The upper
branch originates at $r_{ISSO}/M\sim 8.13$, while the lower branch begins at
$r_{ISSO}/M\sim 3.39$. These branches merge at a critical value
$r_{ISSO}/M\sim 6$, corresponding to $C_{crit}/M^{2}\sim 12.23$, beyond which
no $r_{ISSO}$ exists. The associated ISSO trajectory is illustrated in Fig. 9,
with $r_{ISSO}=1.00085M$. While the trajectory may initially appear dense
around the equator, it is important to note that the figure only depicts one-
thirtieth of a complete latitudinal oscillation. A simulation of a full
latitudinal oscillation reveals that the orbit fills the entire solid angle.

Figure 9: ISSO in one-thirtieth second latitude oscillation when $Q=0.45M$ and
$\Phi=0.192M$ for the extremal KN, having the same energy as in Table 6 (b).
The corresponding radius is $r_{ISSO}=1.00085M$. The plot will fill the
spherical coordinate as it reaches one latitudinal oscillation.

##  III Spherical Orbits around Regular Black Hole

In 1968, Bardeen introduced a regular metric that lacked a physical
singularity, exhibiting regularity in both the metric and all invariants as
$r\rightarrow 0$ Bardeen . However, Bardeen did not explicitly detail the
method used to derive this solution in his original work. Later, Ayon-Beato
and Garcia provided an explanation, demonstrating that such a solution can be
sourced by a nonlinear electrodynamics (NLED) charge Ayon-Beato:1998hmi ;
Ayon-Beato:2000mjt . Since then, numerous proposals have emerged regarding
nonsingular black holes coupled with NLED. Balart and Vagenas Balart:2014cga ,
followed by Culetu Culetu:2014lca , employed a distribution function inspired
by the shape of the probability density function to regularize the static
metric. The first to extend this regularization approach to the case of a
rotating metric in Boyer-Lindquist coordinates was Ghosh Ghosh:2014pba , who
derived a class of three-parameter axisymmetric metrics:

$\displaystyle
ds^{2}=-\left(1-\frac{2m(r)r}{\Sigma}\right){dt}^{2}-\frac{4am(r)r}{\Sigma}\sin^{2}\theta
dt\
d\phi+\Sigma\left(\frac{{dr}^{2}}{\Delta}+{d\theta}^{2}\right)+\frac{\mathcal{A}}{\Sigma}{\sin^{2}\theta
d\phi}^{2},$

where

$\displaystyle m(r)$ $\displaystyle\equiv$ $\displaystyle Me^{-k/r},$
$\displaystyle\Sigma$ $\displaystyle\equiv$ $\displaystyle
r^{2}+a^{2}\cos^{2}\theta,$ $\displaystyle\Delta$ $\displaystyle\equiv$
$\displaystyle r^{2}+a^{2}-2m(r)r,$ $\displaystyle\mathcal{A}$
$\displaystyle\equiv$ $\displaystyle\left(r^{2}+a^{2}\right)^{2}-{\Delta
a}^{2}\sin^{2}\theta.$ (57)

The metric depends on the mass $M$, the angular momentum $a$, and a new
parameter $k$ which controls its regularity. The range of $a$ is between
$0<a^{2}\leq M^{2}$, where the upper bound is related to the extreme condition
or even ”naked” (depending on the value of k) and the lower bound is
Schwarzchild black hole (for $k=0$). The event horizons $r_{\pm}$ satisfy

$\displaystyle\begin{aligned} r_{\pm}^{2}+a^{2}-2m(r_{\pm})=0.\end{aligned}$
(58)

Here the radii $r_{\pm}$ can only be obtained numerically.

Ghosh obtained the solution (III) by transforming the non-singular static
black holes Balart:2014cga ; Culetu:2014lca  to the rotating solutions via the
Newman-Janis algorithm Newman:1965tw . At the core ($r\rightarrow 0$) all the
invariants of this metric are regular everywhere. Specifically, the
Kretschmann scalar ($K$) approaches zero there, $K(r\rightarrow 0)\rightarrow
0$. For $k=0$ the metric reduces to Kerr (when $a\neq 0$) or Schwarzschild
(when $a=0$). In the non-rotating limit ($a\rightarrow 0$) the metric reduces
to the regular spherically-symmetric black hole Balart:2014cga ;
Culetu:2014lca

$ds^{2}=-\left(1-\frac{2Me^{-k/r}}{r}\right)dt^{2}+\left(1-\frac{2Me^{-k/r}}{r}\right)^{-1}dr^{2}+d\Omega_{2}^{2}.$
(59)

Interestingly, the spacetime structure at the core is (as long as $r\gg k$)
Minkowski/RS-like,

$\left(1-\frac{2Me^{-k/r}}{r}\right)\approx
1-\frac{2M}{r}+\frac{2Mk}{r^{2}}+\mathcal{O}\left(r^{-3}\right),$ (60)

rather than de Sitter-like, as in the Bardeen case Bardeen ; Ayon-
Beato:2000mjt ; Ramadhan:2023ogm .

###  III.1 Spherical Photon Orbits

####  III.1.1 Equation of Motion

The four geodesic equations of photons in regular rotating space-time are

$\displaystyle\Sigma\Delta\dot{t}$ $\displaystyle=$
$\displaystyle\mathcal{A}E-2am(r)rL_{z},$ (61)
$\displaystyle\Sigma\Delta\dot{\phi}$ $\displaystyle=$ $\displaystyle
2am(r)rE+\left(\Sigma-2m(r)r\right)\frac{L_{z}}{\sin^{2}\theta},$ (62)
$\displaystyle\Sigma^{2}\dot{\theta}$ $\displaystyle=$
$\displaystyle\mathcal{C}-\left[\frac{L_{z}}{\sin^{2}\theta}-E^{2}a^{2}\right]\cos^{2}\theta,$
(63) $\displaystyle\Sigma^{2}\dot{r}$ $\displaystyle=$ $\displaystyle
E^{2}r^{4}+\left(E^{2}a^{2}-L_{z}^{2}-\mathcal{C}\right)r^{2}+2m(r)r\left[\left(aE-
L_{z}\right)^{2}+\mathcal{C}\right]-a^{2}\mathcal{C}.$ (64)

The notation of $E$, $L_{z}$, and $\mathcal{C}$ are the same three constants
of motion-defined previously.

The geodesic equation of $\theta$ has a similar form with Kerr. Setting
$u=\cos\theta$, Eq. (63) can be rewritten as

$\left(\frac{\Sigma}{E}\right)^{2}\dot{u}^{2}=C-\left(C+\Phi^{2}-a^{2}\right)u^{2}-a^{2}u^{4},$
(65)

where the new parameters are

$\Phi\equiv\frac{L_{z}}{E}\>,\>C\equiv\frac{\mathcal{C}}{E}.$ (66)

The physical requirement is also similar to Kerr’s black hole. When $C>0$, the
positive root of (65) is

$u_{0}^{2}=\frac{(a^{2}-C-\Phi^{2})+\sqrt{(a^{2}-C-\Phi^{2})^{2}+4a^{2}C}}{2a^{2}}.$
(67)

For $C<0$,

$a^{2}-C-\Phi^{2}>0.$ (68)

####  III.1.2 Requirement for the Existence of SPO

Solving the two conditions resulting from Eq. (12) simultaneously we get

$\displaystyle(i)~{}~{}\Phi$ $\displaystyle=$
$\displaystyle\frac{r_{SO}^{2}+a^{2}}{a},~{}~{}C~{}~{}=-\frac{r_{SO}^{4}}{a^{2}};$
(69) $\displaystyle(ii)~{}~{}\Phi$ $\displaystyle=$
$\displaystyle\frac{a^{2}kM+a^{2}r_{SO}^{2}e^{k/r_{SO}}+a^{2}Mr_{SO}+kMr_{SO}^{2}+r_{SO}^{4}e^{k/r_{SO}}-3Mr_{SO}^{3}}{a\left(kM-
r_{SO}^{2}e^{k/r_{SO}}+Mr_{SO}\right)},$ $\displaystyle C$ $\displaystyle=$
$\displaystyle-\frac{r_{SO}^{4}\bigg{\\{}{4a^{2}Me^{k/r_{SO}}(k-r_{SO})+\left[r_{SO}\left(r_{SO}e^{k/r_{SO}}-3M\right)+kM\right]^{2}\bigg{\\}}}}{a^{2}\left[r_{SO}\left(M-r_{SO}e^{k/r_{SO}}\right)+kM\right]^{2}}.$
(70)

As in the KN case, both possibilities satisfy condition (15) for all
$r_{+}<r_{SO}$. Thus, all SPOs here are also unstable. Solution (69) can be
ruled out, and so the acceptable one is (70) with $C>0$ to satisfy condition
(68).

|
---|---
Figure 10: Typical plots of $C$ and $\Phi$ as functions of $r_{SO}$ for
several values of $k$: $k=0.1M$ (red), $k=0.3M$ (green), and $k=0.5M$ (black).

The behavior of $C(r_{SO})$ and $\Phi(r_{SO})$ for Ghosh BH are depicted in
Fig. 10. They look similar to the KN case, Fig. 1, but due to the nature of
$m(r)$ the plots are not linear but bent around a small value of $r_{SO}$.

####  III.1.3 Analytical Solutions

Once again we employ the Mino formalism (16) to solve Eqs. (62)-(64). The
geodesic equations for $\phi$ and $t$ can be written

$\displaystyle\frac{d\phi}{dw}$ $\displaystyle=$
$\displaystyle\frac{1}{2Y(w)}\left[\frac{2am(r)r-a^{2}\Phi}{\Delta}+\frac{\Phi}{\left(1-w\right)}\right],$
(71) $\displaystyle\frac{dt}{dw}$ $\displaystyle=$
$\displaystyle\frac{1}{2Y(w)}\left[\frac{(r^{2}+a^{2})^{2}-2a\Phi
m(r)r}{\Delta}-a^{2}(1-w),\right]$ (72)

where $Y(w)$ is the same as in Eq. (18). Assuming the initial condition
$\phi=t=0$ when $w=0$, the analytic solution of $\phi$ and $t$ can be written
as

$\displaystyle\phi(\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Phi}{a(1-w_{2})\sqrt{w_{1}-w_{2}}}[F(\Psi,\kappa)-w_{2}\Pi(\Psi,\kappa^{2}(1-w_{2}),\kappa)]$
$\displaystyle+\frac{a}{\Delta}\left(2m(r)r-a\Phi\right)\lambda,$
$\displaystyle t(\lambda)$ $\displaystyle=$
$\displaystyle-\frac{a}{\sqrt{w_{1}-w_{2}}}[(1-w_{2})F(\Psi,\kappa)+w_{2}\Pi(\Psi,\kappa^{2},\kappa)]$
$\displaystyle+\frac{1}{\Delta}\left[(r^{2}+a^{2})^{2}-2a\Phi
m(r)r\right]\lambda,$

where $\Psi$ and $\kappa$ are given by Eq. (21). The solution for $\theta$ is
the same as in Eq. (25).

The changes in $\phi$ and $t$ for one complete latitudinal oscillation are

$\displaystyle\Delta\phi$ $\displaystyle=$
$\displaystyle\frac{4}{\sqrt{w_{1}-w_{2}}}\left[\frac{\Phi}{a(1-w_{1})}\Pi(-\frac{w_{1}}{1-w_{1}},\kappa)+\frac{2m(r)r-a\Phi}{\Delta}K(\kappa)\right],$
(75) $\displaystyle\Delta t$ $\displaystyle=$
$\displaystyle\frac{4}{\sqrt{w_{1}-w_{2}}}[-a((1-w_{2})K(\kappa)+w_{2}\Pi(\kappa^{2},\kappa))$
(76) $\displaystyle+\frac{(r^{2}+a^{2})^{2}-2a\Phi m(r)r}{a\Delta}K(\kappa)].$

The equation $\Delta\phi$ determines the orbit direction. Positive
$\Delta\phi$ means prograde orbit, while negative one means retrograde orbit.

####  III.1.4 Selected Orbits

Using the analytical solutions $(\theta,\phi,t)$ we will show typical of
orbits for non-extreme, extreme, and naked conditions of regular rotating
black holes. Unlike the KN case, however, here we have an additional parameter
to consider, $k$, which appears non-linearly in the Eq. (58),

$\left(\frac{r_{\pm}}{M}\right)^{2}+\left(\frac{a}{M}\right)^{2}-\frac{2r_{\pm}}{M}e^{-Mk/r_{\pm}}=0.$
(77)

Consequently, the extremality can be achieved by either adjusting $a/M$ or
$Mk$.

Non-extremal

Orbit |  $Mk$ |  $r_{SO}/M$ |  $C/M^{2}$ |  $u_{0}$ |  $\Delta\phi$
---|---|---|---|---|---
(a) |  0 |  2.892 |  24.412 |  0.088 |  22.436
(b) |  0.1 |  2.743 |  22.524 |  0.088 |  23.698
(c) |  0.5 |  1.996 |  14.245 |  0.087 |  33.482
Table 7: SPO Parameters around a non-extremal Ghosh black hole with $\Phi=M$
and $a/M=0.3$.

The non-extremal condition is not compatible with $Mk>>1$ regime. There exists
no parameter $a/M$ that satisfies:

  1. 1.

$Mk>>1$,

  2. 2.

$r_{SO}>r_{+}\neq r_{-}$, and

  3. 3.

Eq (70),

simultaneously. For example at $a/M\sim 10^{-6}$ the maximum $Mk$ is $\sim
0.73$. In this work, we choose $a/M=0.3$, and the maximum $Mk$ we can have
here is $Mk\sim 0.5$. In Fig. 11, we depict three examples of orbits with
$\Phi=M$ in the $(x-y-z)$, the $x-y$, and $x-z$ planes, respectively. The
radius and other parameter values of the orbit are provided in Table7. The
SPOs appear predominantly equatorial, characterized by minimal latitudinal
oscillations with elongated oscillation periods. This observation is
consistent with expectations for the slowly rotating black hole regime
($0<a/M<1$).

Figure 11: Spherical photon orbits around non-extremal Ghosh black hole with
$\Phi=M$ in the $(x-y-z)$ as well as the projective $x-y$ and $x-z$ planes,
respectively. The direction of the black hole’s rotation is from west to east.
Figures (a), (b), and (c) are cases with $Mk=0$, $Mk=0.1$, and $Mk=0.5$,
respectively.

Extremal Condition

Orbit |  $k$ |  $a/M$ |  $r_{SO}/M$ |  $C/M^{2}$ |  $u_{0}$ |  $\Delta\phi$
---|---|---|---|---|---|---
(a) |  0 |  1 |  2.414 |  22.314 |  1 |  3.1761
(b) |  0.1 |  0.899 |  2.339 |  21.124 |  0.809 |  3.7483
(c) |  0.5 |  0.472 |  1.953 |  15.861 |  0.222 |  9.931
Table 8: SPO parameters around an extremal Ghosh black hole with $\Phi=0$. Orbit |  $k$ |  $a/M$ |  $r_{SO}/M$ |  $C/M^{2}$ |  $u_{0}$ |  $\Delta\phi$
---|---|---|---|---|---|---
(a) |  0 |  1 |  2 |  16 |  0.971 |  10.843
(b) |  0.1 |  0.899 |  1.946 |  15.182 |  0.785 |  8.417
(c) |  0.5 |  0.472 |  1.675 |  11.756 |  0.214 |  21.825
Table 9: SPO parameters around an extremal regular rotating black hole with
$\Phi=M$. Figure 12: Spherical photon orbits around extremal regular black
hole with $\Phi=0$ in the $(x-y-z)$ as well as the projective $x-y$ and $x-z$
planes. The direction of the black hole’s rotation is from west to east.
Figures (a), (b), and (c) are cases with $k=0$, $k=0.1$, and $k=0.5$,
respectively.

The extremal condition turns out to also be incompatible with $Mk>>1$ regime
due to the same reasons as observed in the non-extremal scenario. In this
study, we investigate the orbital behavior under the extremal case with
$Mk=0,\ 0.1,$ and $0.5$. These cases are illustrated in Figs. 12 (for
$\Phi=0$) and 13 (for $\Phi=M$), respectively, with the corresponding SPO
parameters detailed in Tables 8 and 9. For $k=0$, both SPOs ($\Phi=0,\ M$)
reduce to the static one previously obtained by Teo in teo . For each value of
fixed $\Phi$ it is evident that as $Mk$ increases, the orbits become
increasingly confined to the equatorial plane. Moreover, with higher values of
$\Phi$, the amplitude of the latitudinal oscillations diminishes, leading to
longer oscillation periods.

In the extremal Ghosh black hole case, we also attempted to identify the
photon boomerang, characterized by the condition where Eq. 75 equals $\pi$.
Through our numerical investigation, we found only an approximate photon
boomerang with $k=0.05$. We label it as “approximate” because the value of
$\Phi=0.07$ is non-zero. This is detailed in Table 10, and the corresponding
trajectory is depicted in Fig. 14.

$k$ |  $a/M$ |  $r_{SO}/M$ |  $C/M^{2}$ |  $u_{0}$ |  $\Delta\phi$
---|---|---|---|---|---
0.05 |  0.94998 |  2.402 |  22.0487 |  0.902 |  3.1415
Table 10: (Approximate) photon boomerang parameters around an extremal regular
rotating black hole with $\Phi=-0.07$. Figure 13: Spherical photon orbits
around extremal Ghosh black hole with $\Phi=M$ in the $(x-y-z)$ as well as the
projective $x-y$ and $x-z$ planes. The direction of the black hole’s rotation
is from west to east. Figures (a), (b), and (c) are cases with $k=0$, $k=0.1$,
and $k=0.5$, respectively. Figure 14: Photon boomerang around extremal Ghosh
black hole with $\Phi=-0.07$ in the $(x-y-z)$ as well as the projective $x-y$
and $x-z$ planes. The direction of black hole’s rotation is from west to east.

No-horizon Spacetime

Certain combinations of ${a/M,\ Mk}$ in Eq.(77) can give rise to complex
solutions for $r_{\pm}$. In the context of a conventional black hole, such
instances correspond to what is termed a ”naked singularity,” a phenomenon
deemed unstable and unphysical according to the cosmic censorship theorem
Penrose:1964wq . However, within the framework of a regular black hole, the
concept of singularity vanishes. This scenario is often referred to as ”the
no-horizon spacetime” in the literature. While discussing trajectories around
a naked singularity is nonsensical, the notion is perfectly acceptable in the
case of a regular no-horizon spacetime.

The notion of a regular solution devoid of singularity is not a new concept in
GR. Mazur and Mottola Mazur:2004fk ; Mazur:2001fv  discussed an alternative
final state of non-singular gravitational collapse by invoking the Bose-
Einstein condensate formalism in the interior. The consequence is a
horizonless, regular core reminiscent of de Sitter space, which they coined as
“gravastar”. More recently, there has been discourse surrounding the
correlation between regular black holes and horizonless ultracompact stars
Carballo-Rubio:2019fnb ; Carballo-Rubio:2022nuj ; Cadoni:2022chn . It is
pertinent to distinguish that the Ghosh no-horizon solution differs
conceptually from the gravastar, as the former features a Minkowski core while
the latter exhibits a de Sitter-like core.

Orbit |  $Mk$ |  $\Phi/M$ |  $a/M$ |  $r_{SO}/M$ |  $C/M^{2}$ |  $u_{0}$ |  $\Delta\phi$
---|---|---|---|---|---|---|---
(a) |  0.1 |  0 |  1 |  1.242 |  24.784 |  1 |  11.426
(b) |  0.1 |  -0.55 |  1 |  1.178 |  36.224 |  0.995 |  6.291
(c) |  0.3 |  -1.2 |  1 |  1.571 |  20.143 |  0.967 |  -0.487
(d) |  0.3 |  -2 |  1 |  1.347 |  26.21 |  0.933 |  0.861
(e) |  0.5 |  -2.3 |  1 |  1.82 |  17.377 |  0.879 |  -2.04
(f) |  0.5 |  -3.5 |  1 |  1.362 |  21.47 |  0.802 |  1.082
Table 11: Parameters of the stable SPO around naked regularity.

The absence of a horizon in the geometry opens avenues for stable SPOs (and
STOs) by circumventing Eq.(15). Here, we aim to explore the existence of
stable spherical orbits around it, adhering to the condition:

$\frac{d^{2}R(r)}{dr^{2}}<0.$ (78)

However, our investigation uncovers that conditions (78) and (70) harmonize
only when $\Phi\leq 0$.

Figure 15: Spherical photon orbits around no-horizon spacetime in the
$(x-y-z)$ as well as the projective $x-y$ and $x-z$ planes. The direction of
the black hole’s rotation is from west to east. (a) $\Phi=0$ and $Mk=0.1$, (b)
$\Phi=-0.55M$ and $Mk=0.1$, (c) $\Phi=-1.2M$ and $Mk=0.3$.

We illustrate orbit trajectories around the no-horizon solution in Figs.
15-16, delineating parameters detailed in Table 11. Each figure represents
distinct values of $\Phi$ and $k$, as the stability requirement imposes
stringent constraints on them. Given the negative $\Phi$ values, the
occurrence of the Lens-Thirring effect is inevitable. This phenomenon exerts a
pronounced influence on orbits near the equator compared to those near the
poles. Notably, orbits with negative $\Phi$ values maintain a prograde motion
around the equator, transitioning to a retrograde direction as they approach
the poles. This observation holds particular significance for orbits with
$\Phi\sim 0$, as the Lens-Thirring effect possesses the capability to
counteract the photon’s angular momentum.

Figure 16: Spherical photon orbits around no-horizon spacetime in the
$(x-y-z)$ as well as the projective $x-y$ and $x-z$ planes. The direction of
the black hole’s rotation is from west to east. (d) $\Phi=-2M$ and $Mk=0.3$,
(e) $\Phi=-2.3M$ and $Mk=0.5$, (f) $\Phi=-3.5M$ and $Mk=0.5$.

###  III.2 Spherical Timelike Orbits

The geodesics equations for an uncharged timelike particle are

$\displaystyle\Sigma\dot{t}$ $\displaystyle=$
$\displaystyle-a^{2}E\left(1-u^{2}\right)+\frac{1}{\Delta}\left[E\left(r^{2}+a^{2}\right)^{2}-2a\Phi
m(r)r\right],$ (79) $\displaystyle\Sigma\dot{\phi}$ $\displaystyle=$
$\displaystyle\frac{\Phi}{1-u^{2}}+\frac{a}{\Delta}\left(2Em(r)r-a\Phi\right),$
(80) $\displaystyle\Sigma\dot{u}$ $\displaystyle=$
$\displaystyle\pm\sqrt{V\left(u\right)},$ (81) $\displaystyle\Sigma\dot{r}$
$\displaystyle=$ $\displaystyle\pm\sqrt{R\left(r\right)},$ (82)

where

$\displaystyle R\left(r\right)$ $\displaystyle\equiv$
$\displaystyle\left(E^{2}-1\right)r^{4}+2m(r)r^{3}+\left[a^{2}\left(E^{2}-1\right)-\mathcal{C}-\Phi^{2}\right]r^{2}$
$\displaystyle+2m(r)rr\left[\left(aE-\Phi\right)^{2}+\mathcal{C}\right]re^{-k/r}-a^{2}\mathcal{C},$
$\displaystyle V\left(u\right)$ $\displaystyle\equiv$ $\displaystyle
a\left(1-E^{2}\right)u^{4}-\left[a^{2}\left(1-E^{2}\right)+\mathcal{C}+\Phi^{2}\right]+\mathcal{C}.$

As in the KN case above, for the timelike case here we only investigate the
stable ones; that is when (III.2) satisfies conditions (12) and (78). These
conditions yield four types of solutions, the first two of which are

$\displaystyle E_{a,b}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{e^{-2k/r_{SO}}\left[-2a^{4}M\mathcal{C}e^{3k/r_{SO}}\left(k-r_{SO}\right)-A+r_{SO}^{4}\left(B\pm
2\sqrt{N}\right)\right]}{D}},$ $\displaystyle\Phi_{a,b}$ $\displaystyle=$
$\displaystyle-\frac{e^{-k/r_{SO}}E_{a,b}}{aHM(k-r_{SO})}$
$\displaystyle\times\left[a^{4}M^{2}e^{2k/r_{SO}}(k-r_{SO})\left[k\left(c+r_{SO}^{2}\right)+r_{SO}\left(\mathcal{C}-r_{SO}^{2}\right)\right]+F+r_{SO}^{6}\sqrt{\frac{G}{r_{SO}^{8}}}\right],$

where

$\displaystyle A$ $\displaystyle\equiv$ $\displaystyle
a^{2}e^{2k/r_{SO}}\bigg{\\{}M^{2}\left[r_{SO}^{2}\left(k^{2}+5\mathcal{C}\right)+k^{2}\mathcal{C}-2k\mathcal{C}r_{SO}+4kr_{SO}^{3}-5r_{SO}^{4}\right]$
(85) $\displaystyle
Mr_{SO}^{3}e^{k/r_{SO}}[3r_{SO}(r_{SO}-k)-4\mathcal{C}]+\mathcal{C}r_{SO}^{4}e^{2k/r_{SO}}\bigg{\\}},$
$\displaystyle B$ $\displaystyle\equiv$ $\displaystyle
e^{k/r_{SO}}\left(r_{SO}e^{k/r_{SO}}-2M\right)^{2}\left[r_{SO}\left(r_{SO}e^{k/r_{SO}}-3M\right)+kM\right],$
(86) $\displaystyle N$ $\displaystyle\equiv$
$\displaystyle\frac{1}{r_{SO}^{8}}\bigg{\\{}a^{2}M^{2}e^{3k/r_{SO}}(k-r_{SO})^{2}\left[\left(a^{2}+r_{SO}^{2}\right)e^{k/r_{SO}}-2Mr_{SO}\right]^{2}\bigg{\\}}$
(87)
$\displaystyle\times\bigg{\\{}\mathcal{C}e^{k/r_{SO}}\left(a^{2}\mathcal{C}-r_{SO}^{4}\right)+Mr_{SO}^{2}\left[-k\left(\mathcal{C}+r_{SO}^{2}\right)+3\mathcal{C}r_{SO}+r_{SO}^{3}\right]\bigg{\\}},$
$\displaystyle D$ $\displaystyle\equiv$ $\displaystyle
r_{SO}^{4}\bigg{\\{}4a^{2}Me^{k/r_{SO}}(k-r_{SO})+\left[r_{SO}\left(r_{SO}e^{k/r_{SO}}-3M\right)+kM\right]^{2}\bigg{\\}},$
(88) $\displaystyle F$ $\displaystyle\equiv$ $\displaystyle
a^{2}M^{2}r_{SO}^{2}e^{k/r_{SO}}(k-r_{SO})\bigg{\\{}e^{k/r_{SO}}\left[k\left(\mathcal{C}+r_{SO}^{2}\right)-r_{SO}\left(\mathcal{C}+3r_{SO}^{2}\right)\right]+4Mr_{SO}^{2}\bigg{\\}},$
(89) $\displaystyle G$ $\displaystyle\equiv$ $\displaystyle
a^{2}M^{2}e^{3k/r_{SO}}(k-r_{SO})^{2}\left[\left(a^{2}+r_{SO}^{2}\right)e^{k/r_{SO}}-2Mr_{SO}\right]^{2}$
(90)
$\displaystyle\times\bigg{\\{}\mathcal{C}e^{k/r_{SO}}\left(a^{2}\mathcal{C}-r_{SO}^{4}\right)+Mr_{SO}^{2}\left[-k\left(\mathcal{C}+r_{SO}^{2}\right)+3\mathcal{C}r_{SO}+r_{SO}^{3}\right]\bigg{\\}},$
$\displaystyle H$ $\displaystyle\equiv$ $\displaystyle
a^{2}e^{k/r_{SO}}\left[r_{SO}\left(\mathcal{C}r_{SO}e^{k/r_{SO}}-M\mathcal{C}+Mr_{SO}^{2}\right)-kM\left(\mathcal{C}+r_{SO}^{2}\right)\right]-r_{SO}^{4}\left(r_{SO}e^{k/r_{SO}}-2M\right)^{2}.$

The third and fourth solutions are related through

$(E_{c,d},\Phi_{c,d})=-(E_{a,b},\Phi_{a,b}).$ (92)

####  III.2.1 Analytical Solutions

From $V(u^{2})=0$ the roots $w_{1,2}\equiv u^{2}$ can be found as

$w_{1,2}=\frac{a^{2}\left(1-E^{2}\right)+\mathcal{C}+\Phi^{2}\mp\sqrt{\left[a^{2}\left(1-E^{2}\right)+\mathcal{C}+\Phi^{2}\right]^{2}-4a^{2}\mathcal{C}\left(1-E^{2}\right)}}{2a^{2}\left(1-E^{2}\right)}.$
(93)

As before, for bound orbits we have $E^{2}<1$ and $0\leq w\leq w_{1}\leq
1<w_{2}$. The analytical solutions for
$\left\\{\theta(\lambda),\phi(\lambda),t(\lambda)\right\\}$ are

$\displaystyle\theta(\lambda)$ $\displaystyle=$
$\displaystyle\arccos\left[\sqrt{w_{1}}\
\text{sn}(a\sqrt{\left(1-E^{2}\right)w_{2}}\lambda,\kappa)\right],$ (94)
$\displaystyle\phi(\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Phi}{a\sqrt{\left(1-E^{2}\right)w_{2}}}\Pi(\Psi,w_{1},\kappa)+\frac{a}{\Delta}\left(2MrEe^{-k/r}-a\Phi\right)\lambda,$
(95) $\displaystyle t(\lambda)$ $\displaystyle=$
$\displaystyle-\frac{aE}{\sqrt{(1-E^{2})w_{2}}}\left[(1-w_{2})F(\Psi,\kappa)+w_{2}E(\Psi,\kappa)\right]$
(96) $\displaystyle+\frac{1}{\Delta}\left[E(r^{2}+a^{2})^{2}-2Mra\Phi
e^{-k/r}\right]\lambda,$

where

$\displaystyle\Psi$ $\displaystyle\equiv$
$\displaystyle\text{am}(a\sqrt{\left(1-E^{2}\right)w_{2}}\lambda,\kappa),$
(97) $\displaystyle\kappa$ $\displaystyle\equiv$
$\displaystyle\sqrt{\frac{w_{1}}{w_{2}}}.$

The period of $\theta(\lambda)$ is

$\Delta\lambda=\frac{4}{a\sqrt{\left(1-E^{2}\right)w_{2}}}K(\kappa).$ (98)

The change in $\phi$ and $t$ for one period are

$\displaystyle\Delta\phi$ $\displaystyle=$
$\displaystyle\frac{4}{a\sqrt{\left(1-E^{2}\right)w_{2}}}\left[\frac{\Phi}{a}\Pi(w_{1},\kappa)+\frac{2MrEe^{-k/r}-a\Phi}{\Delta}K(\kappa)\right],$
(99) $\displaystyle\Delta t$ $\displaystyle=$
$\displaystyle-\frac{4}{a\sqrt{\left(1-E^{2}\right)w_{2}}}\bigg{\\{}-a\left[(1-w_{2})K(\kappa)+w_{2}E(\kappa)\right]$
(100) $\displaystyle+\left[\frac{E(r^{2}+a^{2})^{2}-2Mra\Phi
e^{-k/r}}{a\Delta}\right]K(\kappa)\bigg{\\}}.$

####  III.2.2 Selected orbits

From the derived solutions (94)-(96), we can visualize certain timelike
orbits.

Orbit |  $Mk$ |  $a/M$ |  $r_{SO}/M$ |  $C/M^{2}$ |  $E$ |  $\Phi$/M |  $u_{0}$ |  $\Delta\phi$
---|---|---|---|---|---|---|---|---
(a) |  0 |  0.3 |  6 |  8 |  0.937 |  1.733 |  0.852 |  6.535
(b) |  0.1 |  0.3 |  6 |  8 |  0.936 |  1.576 |  0.873 |  6.535
(c) |  0.5 |  0.3 |  6 |  8 |  0.93 |  0.857 |  0.956 |  6.537
Table 12: Parameters of the STO around a non-extremal regular rotating black
hole. Figure 17: Spherical timelike orbits around non-extremal regular black
hole case in the $(x-y-z)$ as well as the projective $x-y$ and $x-z$ planes.
The direction of the black hole’s rotation is from west to east. (a)
$\Phi/M=1.733$ and $Mk=0$, (b) $\Phi/M=1.576$ and $Mk=0.1$, (c) $\Phi/M=0.857$
and $Mk=0.5$.

We present the STOs for non-extremal cases in Fig. 17, accompanied by
parameters detailed in Table 12. Analogous to the Kerr-Newman (KN) and regular
SPO scenarios, there is an inclination for larger values of $k$ to draw the
orbit trajectory nearer to the pole, resulting in a notably expanded
latitudinal oscillation amplitude. Distinctive plots for STOs encircling
extremal black holes are showcased in Fig. 18, with corresponding parameters
outlined in Table 13. Here, we observe a distinctive orbit morphology: with
increasing $k$, the trajectories depicted in the projective $x-y$ plane
exhibit diminishing ”leaf” shapes and augmented latitudinal oscillation
amplitudes. Lastly, we delineate the orbits surrounding naked or no-horizon
scenarios in Fig. 19, accompanied by parameters expounded in Table 14.

Orbit |  $Mk$ |  $a/M$ |  $r_{SO}/M$ |  $C/M^{2}$ |  $E$ |  $\Phi/M$ |  $u_{0}$ |  $\Delta\phi$
---|---|---|---|---|---|---|---|---
(a) |  0 |  1 |  4 |  8 |  0.918 |  0.918 |  0.95 |  7.78
(b) |  0.1 |  0.8998 |  4 |  8 |  0.918 |  0.819 |  0.96 |  7.645
(c) |  0.5 |  0.472 |  4 |  8 |  0.918 |  0.043 |  0.999 |  7.023
Table 13: Parameters of the STO around the extremal regular rotating black hole. Figure 18: Spherical timelike orbits around extremal regular black hole in the $(x-y-z)$ as well as the projective $x-y$ and $x-z$ planes. The direction of the black hole’s rotation is from west to east. (a) $\Phi/M=0.918$ and $k=0$, (b) $\Phi/M=0.819$ and $k=0.1$, (c) $\Phi/M=0.043$ and $k=0.5$. Orbit |  $Mk$ |  $a/M$ |  $r/M$ |  $C/M^{2}$ |  $E$ |  $\Phi/M$ |  $u_{0}$ |  $\Delta\phi$
---|---|---|---|---|---|---|---|---
(a) |  0.1 |  1 |  8 |  8 |  0.943 |  1.596 |  0.869 |  6.79
(b) |  0.5 |  1 |  8 |  8 |  0.941 |  1.211 |  0.918 |  6.798
Table 14: Parameters of the STO around naked regularity. Figure 19: Spherical
timelike orbits around no-horizon spacetime in the $(x-y-z)$ as well as the
projective $x-y$ and $x-z$ planes. The direction of the black hole’s rotation
is from west to east. (a) $\Phi/M=1.596$ and $k=0.1$, (b) $\Phi/M=1.211$ and
$k=0.5$.

####  III.2.3 Innermost Stable Spherical Orbits

Using a similar method with Kerr-Newman black holes, we can get the ISSO as a
function of Carter’s constant. The ISSO for the Ghosh BH presents some
interesting features: the branches are discontinuous. Some of them stop at
some finite values, as we shall see below. The form of the regular metric is
such that there exist some ranges of $C/M^{2}$ which have complex $r_{ISSO}$.

For a non-extremal case the result is shown in Fig. 20 with $a=0.7M$ and
$k=0.1M$. Both branches are observable (outside the horizons). The lower
branch starts at $r_{ISSO}/M\sim 2.92$ while the upper one starts
$r_{ISSO}/M\sim 8.12$. Unlike the KN case, here the two branches do not merge
and stop at $C_{crit}$, but they somehow cross as can be seen from the figure.
The crossing value is around $r_{ISSO}/M\sim 6.09$ at $C/M^{2}\sim 12$. A more
interesting case happens for the extremal BH. In Fig. 21 it can be seen that
the lower branch starts at the extremal horizon for $C/M^{2}=0$, then goes up
as usual. However, instead of merging or crossing with the upper branch, the
two branches do not merge at a single $C$ value. The upper branch stops being
real at around $r_{ISSO}/M\sim 6.55$. It only exists between $0\leq
C/M^{2}\lesssim 11$. Finally, the no-horizon case is the most interesting of
all. As can be seen from Fig. 22, the lower branch ceases to exist up until
$C/M^{2}\sim 4.4$, at $r_{ISSO}\sim 2.23$. On the other hand, the upper branch
starts with $r_{ISSO}/M\sim 8.8$ and it stops being real around
$r_{ISSO}\lesssim 5.98$ at $C/M^{2}\sim 11$. A typical ISSO trajectory is
shown in Fig. 23. This is an example of ISSO in a no-horizon case, with
$a/M=0.95$, $k/M=0.1$, and $C/M^{2}=10$. The corresponding ISSO radius is
$r_{ISSO}/M\sim 7.08$.

Figure 20: $r_{ISSO}$ as a function of Carter’s constant for a non-extremal
regular black hole when $a=0.7M$ and $k=0.1M$. The red and blue lines are the
$r_{isso}$ from the $\pm$ sign in eq. LABEL:timelikereqreg, and black dashed
lines are the event horizons. Figure 21: $r_{ISSO}$ as a function of Carter’s
constant for an extremal regular black hole when $a=0.8998M$ and $k=0.1M$. The
red and blue lines are the $r_{isso}$ from $\pm$ sign in eq.
LABEL:timelikereqreg, and the black dashed line is the event horizon. Figure
22: $r_{ISSO}$ as a function of Carter’s constant for a naked regular black
hole when $a=0.95M$ and $k=0.1M$. The red and blue lines are the $r_{isso}$
from the $\pm$ sign in eq. LABEL:timelikereqreg. Figure 23: The ISSO
trajectory around the no-horizon case of Ghosh BH (based on the data in Fig.
22). Here $C/M^{2}=10$, $a/M=0.95$, and $k/M=0.1$. The corresponding ISSO
radius is $r_{ISSO}\sim 7.075M$.

##  IV Conclusion

In this study we expand upon Teo’s seminal analysis of spherical orbits teo ;
Teo:2020sey , delving into the charged rotating black holes. Specifically, our
focus lies on two distinct models: the Kerr-Newman and the Ghosh black holes,
the latter representing a novel regular rotating black hole obtained by
conducting the Newman-Janis transformation of a spherically-symmetric regular
metric. Notably, the metric described by Eq. (III) reduces to Kerr at
$k\rightarrow 0$ and to the Kerr-Newman at $r>>k$. Leveraging exact analytical
solutions in terms of Mino parameters, we explore the presence of both
spherical photon orbits (SPOs) and spherical timelike orbits (STOs) in these
models. Noteworthy is our assumption that the direction of rotation for the
black hole is from west to east, with all orbits originating from the
equatorial plane and progressing northward.

In the case of Kerr-Newman black holes we found that, similar to the Kerr
case, all observable SPOs prove to be unstable under radial perturbation. This
instability is evident from the effective potential $V_{eff}\equiv-R(r)$,
where all SPOs satisfy Eq. (15). With the introduction of charge, the $C-r$
and $\Phi-r$ plots undergo a shift to the right, while the maximum value of
the former decreases compared to Kerr. This shift indicates a higher
concentration of photon trajectories around the equatorial plane. This
concentration is further confirmed by the directional velocity $\theta$ and
angular momentum $\Phi$ which decrease and increase, respectively, as the
charge $Q$ increases. Additionally, it requires more latitudinal oscillations
to form a closed trajectory. Notably, the charge $Q$ can reorient the photon’s
motion about the z-axis of the black hole. For $\Phi=-M$, the initially
retrograde orbit transitions to prograde motion. This transition occurs at
$Q/M=0.6$ for both non-extremal and extremal black holes. Furthermore, we
observe a photon boomerang trajectory, $\Delta\phi=\pi$, in the extremal case.
This happens in the extremal case, occurring at low charge values
$Q=0.02528M$, and $\Phi=0$.

Exploring geodesics around Ghosh black holes raises the intriguing question of
whether stable SPOs exist outside the horizon. This is a nonsingular rotating
black hole, parametrized by an additional parameter $k$. However, our
investigation yielded disappointing results, as all SPOs around regular black
holes prove to be unstable. Additionally, we observed that the condition
$Mk>>1$ is incompatible with both non-zero $a/M$ and observability criteria
$r_{SO}>r_{h}$. In the non-extremal case, this circumstance renders photon
orbits predominantly equatorial. However, in the extremal case, where
$Mk\lesssim 0.5$, a complete SPO exists. Note that in this BH the extremality
can be achieved by either $a/M$ or $Mk$. We identified an approximate photon
boomerang in this extremal case, with $k=0.05$ and $\Phi=-0.07$. Since regular
black holes lack naked singularities, the absence of a horizon in the metric
poses no physical paradox. This so-called ”no-horizon” condition may
conceptually align with that of a horizonless ultracompact star. Digging
further into our investigation around no-horizon condition, we discovered that
photons can orbit spherically under stable conditions, albeit only when
$\Phi\leq 0$. This finding highlights the presence of the Lens-Thirring
effect.

For stable spherical timelike orbits (STOs), both KN and Ghosh BHs allow
orbits to exist outside their horizons. In the KN scenario, bound orbits occur
when $E^{2}<1$, while $E^{2}=1$ signifies a marginally bound orbit. Unlike
null orbits, timelike orbits do not exhibit retrograde-prograde switching. In
the Ghosh case, higher values of $k$ drive trajectories closer to the pole,
resulting in larger latitudinal oscillation amplitudes. Notably, the concept
of innermost stable spherical orbits (ISSOs) in spherical trajectory parallels
the innermost stable circular orbits (ISCOs) in equitorial motion. The ISSO
radius in KN can be determined by solving Eqs. (53)-(54). These solutions,
depicted in Fig. 8, manifest as two branches of $r_{ISSO}\left(C\right)$. A
typical ISSO trajectory in the KN BH, illustrated in Fig. 9, appears densely
concentrated around the equator initially, gradually filling the entire solid
angle as it completes one latitudinal oscillation. In contrast, ISSO behavior
in the Ghosh BH presents intriguing nuances. Unlike the KN case, the
nonsingular nature of the Ghosh metric causes ISSO branches to terminate at
finite values, as depicted in Figs. 20-22. A representative ISSO trajectory is
shown in Fig. 23. As a closing statement in this paragraph we should briefly
comment on this ISSO issue. It is evident that in both types of black holes
under examination, the radial potential $R(r)$ is at least quartic in $r$.
This leads to two roots for each solution of $R^{\prime\prime}=0$, hence the
presence of two branches. The lower branch in each $r_{ISSO}$ vs $C$ plot
should be the one that represents the ”true” ISSO, as the name by itself
stands.

Finally, we should also comment on the phenomenological aspects of our
results. Our theoretical findings hold significant potential for astrophysical
applications. Firstly, the prevalence of unstable spherical photon orbits
(SPOs) in our study is advantageous for detection purposes. This diversity of
unstable SPOs around various rotating black hole models contributes to a more
comprehensive understanding and visualization of black holes. In particular,
the calculation of the photon ring’s two-point correlation function, reliant
on unstable SPOs as demonstrated for the Kerr black hole in Hadar:2020fda ,
holds promise for future validation through experiments like the upcoming
Event Horizon Telescope (EHT) project Anjum:2023axh . Secondly, our study
sheds light on the crucial information regarding stable spherical timelike
orbits (STOs). Being the asymptotic trajectories of homoclinic orbits
Levin:2008yp ; Li:2023bgn , STOs carry fundamental significance, for example
in the context of the gravitational waves analysis from extreme mass-ratio
inspirals (EMRIs)Teo:2020sey ; Ryan:1995zm ; Barack:2006pq ; Amaro-
Seoane:2014ela . Additionally, our results pave the way for further
investigations, such as exploring the gravitational redshift/blueshift
emission from STOs around Ghosh black holes, a research avenue previously
explored for Kerr and Kerr-de Sitter black holes inKraniotis:2019ked .

###### Acknowledgements.

We thank Edward Teo for the discussion on his analytical spherical orbit
solutions. We also thank Leonardo B. Putra for the discussion on ISSO. BNJ is
supported by the Second Century Fund (C2F), Chulalongkorn University,
Thailand. HSR is funded by the Hibah Riset FMIPA UI No.
PKS-026/UN2.F3.D/PPM.00.02/2023.

## Statement of the Conflict of Interest

All authors hereby declare that we do not have any conflict of interest.

## Data Availability Statement

Data sharing is not applicable to this article as no data sets were generated
or analyzed during the current study.

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