# Circular geodesics in the field of double-charged dilatonic black holes

Kuantay Boshkayev<EMAIL_ADDRESS>Al-Farabi Kazakh National University, Al-
Farabi av. 71, 050040 Almaty, Kazakhstan.  National Nanotechnology Laboratory
of Open Type, Al-Farabi av. 71, Almaty 050040, Kazakhstan.  International
University of Information Technology, Manas st. 34/1, 050040 Almaty,
Kazakhstan.  Gulnara Suliyeva<EMAIL_ADDRESS>Al-Farabi Kazakh National
University, Al-Farabi av. 71, 050040 Almaty, Kazakhstan.  Fesenkov
Astrophysical Institute, Observatory 23, 050020 Almaty, Kazakhstan.  Vladimir
Ivashchuk<EMAIL_ADDRESS>Center for Gravitation and Fundamental Metrology,
VNIIMS,
46 Ozyornaya St., Moscow 119361, Russian Federation.  Institute of Gravitation
and Cosmology, Peoples’ Friendship University of Russia (RUDN University),
6 Miklukho-Maklaya St., Moscow 117198, Russian Federation  Ainur Urazalina
<EMAIL_ADDRESS>Al-Farabi Kazakh National University, Al-Farabi av. 71,
050040 Almaty, Kazakhstan.  National Nanotechnology Laboratory of Open Type,
Al-Farabi av. 71, Almaty 050040, Kazakhstan.

###### Abstract

A non-extreme dilatonic charged (by two “color electric” charges) black hole
solution is considered in a 4-dimensional gravity model including two scalar
(dilaton) fields and two Abelian vector fields. Scalar and vector fields
interact through exponential terms which contain two dilatonic coupling
vectors. The solution is governed by a dimensionless parameter $a$ $(0<a<2)$,
which is a certain function of dilatonic coupling vectors. The paper presents
solutions for timelike and null circular geodesics which may play an important
role in different astrophysical problems, including quasinormal modes of
various test fields in the eikonal approximation. For $a=1/2,1,3/2,2$ the
innermost stable circular orbit (ISCO) radii are presented and analysed.

black holes, timelike geodesics, null geodesics, scalar fields, Abelian vector
fields, ISCO.

##  1 Introduction

After the discovery of gravitational waves in 2016, the interest in the
physics of stellar-mass black holes has begun to grow rapidly Abbott et al.
(2016a). Gravitational waves were originally postulated by Einstein over a
century ago. Their long search led to the creation of huge super-sensitive
laser interferometers LIGO, VIRGO and others, and the development of the
latest highly precise methods of detection and data analysis Abbott et al.
(2016b).

Observations and studies of the motion of stars at the heart of the Milky Way
galaxy have confirmed the existence of a supermassive black hole Ghez et al.
(1998, 2000); Gravity Collaboration et al. (2018). These surveys provide
information about the star clusters near the galactic center and the black
hole mass which is used by researches to test the predictions of general
relativity GRAVITY Collaboration et al. (2020).

In addition, the imaging process of the shadow of the supermassive black holes
at the center of the M87 galaxy and in the Milky Way galaxy has required
remarkable ingenuity and cohesion among scientists around the world Event
Horizon Telescope Collaboration et al. (2019, 2022a). For these purposes, our
entire planet was used as one giant radio telescope, consisting of separate
groups of telescopes scattered in all continents. The presence of the image of
a black hole shadow and its analysis verify the correctness and reliability of
general relativity. All discoveries over the past 60 years testify to the
significance and relevance of research in this direction.

At the same time, in addition to general relativity, there are various
modified and extended theories of gravity, where alternative black holes with
extra parameters exist Sotiriou (2006); Clifton et al. (2012); Astashenok et
al. (2015, 2013, 2017); Capozziello et al. (2019); Astashenok et al. (2020).
However, to date, within the range of observational errors, those black holes
cannot be for 100$\%$ distinguished from ordinary Schwarzschild, Reissner-
Nordström, and Kerr black holes Event Horizon Telescope Collaboration et al.
(2022b).

Commonly observational data is used as an important tool for constraining
black hole parameters in those theories Pani and Cardoso (2009);
Shankaranarayanan and Johnson (2022); Antoniou et al. (2023). This fact makes
it possible to examine black holes with “colored charges” when there are
scalar fields. Classical Schwarzschild and Reissner-Nordström black holes, as
well as various models of dilatonic black holes with two colored electric
charges are considered here.

The dilaton is a hypothetical scalar field particle. It appears in the metric
through the introduction of coupling vectors (constants), which, in turn,
define a new particular subclass of black holes. In limiting cases, dilatonic
solutions reduce to the Schwarzschild and Reissner-Nordström solutions,
depending on the values of the coupling vectors. Since the dilaton has not
been discovered yet, it would be interesting to investigate effects associated
with its presence.

In this paper we consider the possibility of distinguishing ordinary
astrophysical black holes from dilatonic ones. In this regard, the main
objective of the paper is to study the motion of test particles and photons in
circular orbits in the gravitational field of astrophysical and dilatonic
black holes with different color charges and coupling vectors.

To achieve this goal, the following problems are posed:

  * •

Obtaining geodesic equations in the field of dilatonic black holes using the
Lagrange formalism.

  * •

Calculation of the angular momentum, the energy, the effective potential of
test particles and photons, and the radii of the innermost stable circular
orbits in the field of dilatonic black holes.

The solutions of dilatonic dyonic black holes with an arbitrary coupling
constant and a canonical scalar field have been considered in Refs. Abishev et
al. (2015, 2017); Belissarova et al. (2020); Malybayev et al. (2021). The
solutions were obtained up to the solutions of two master equations for moduli
functions. Some physical parameters of the solutions were also obtained:
gravitational mass, scalar charge, Hawking temperature, black hole area
entropy, and parameterized post-Newtonian (PPN) parameters. While in Ref.
Abishev et al. (2015, 2017); Belissarova et al. (2020) only the physical
characteristics of black holes were studied, in Ref. Malybayev et al. (2021)
quasi-normal modes of a massless test scalar field were studied in the
background of a gravitational field for a non-extremal dilatonic dyonic black
hole.

Timelike and null geodesics, for example circular ones, play an important role
in various astrophysical context, including accretion disks, quasiperiodic
oscillations, quasinormal modes of various test fields in the eikonal
approximation Konoplya et al. (2019) and shadows of supermassive black holes
Uniyal et al. (2023); Lambiase et al. (2023). These results and studies will
allow one to distinguish ordinary black holes from dilatonic ones Pani and
Cardoso (2009).

The novelty of the work resides in the fact that the geodesics of test
particles in the gravitational field of dilatonic black holes with two scalar
fields and two-color electric charges will be studied for the first time. In
the past, some of us have investigated geodesics in the context of standart
black holes, naked singularities, and spinning deformed relativistic compact
objects Boshkayev et al. (2021); Boshkayev et al. (2016a); Bini et al. (2013);
Boshkayev et al. (2016b).

In Ref. Li et al. (2022), geodesics around rotating black hole mimickers are
studied, given by the exact solution of the stationary and axially symmetric
field equations in vacuum, known as the $\delta$-Kerr metric. The authors
study its optical properties based on a ray-tracing code for photon motion,
characterize the apparent shape of the shadow of a compact object, and compare
it with a Kerr black hole. In order to obtain qualitative estimates related to
the observed shadow of a supermassive compact object in the M87 galaxy, the
authors are guided by the values of the object’s spin a and observation
inclination angle close to the measured values. It has been shown that, based
on only one set of shadow edge observations, it is not possible to rule out
the $\delta$-Kerr solution as a viable source of geometry outside the compact
object.

It was shown in Turimov et al. (2022) that from a geometric point of view it
is impossible to distinguish electrically and magnetically charged Reissner-
Nordström black holes. One of the ways to describe the differences between
these solutions is to study the dynamical motion of charged test particles
near a charged black hole and study the effect of charge coupling parameters
on the stability of circular orbits. The authors investigate the synchrotron
radiation of charged particles accelerated by a charged black hole and
estimate the intensity of the relativistic radiation of charged particles.

Recently in Ref. Cvetič et al. (2016) the authors have studied closed photon
orbits in spherically-symmetric static solutions of supergravity theories, a
Horndeski theory, and a theory of quintessence. These orbits lie in what is
called a photon sphere (anti-photon sphere) if the orbit is unstable (stable).
It was shown that in all the asymptotically flat solutions examined that admit
a regular event horizon, and whose energy-momentum tensor satisfies the strong
energy condition, there is one and only one photon sphere outside the event
horizon.

Note that while in Huang and Zhang (2022, 2020); Cadoni et al. (2010); Chen
(2008) the features of dilatonic black holes are studied, in Li et al. (2022);
Sarikulov et al. (2022); Turimov et al. (2022) various solutions to the field
equation are considered, which in the limiting case describe ordinary black
holes. Refs. Li et al. (2022); Sarikulov et al. (2022); Turimov et al. (2022);
Cvetič et al. (2016) also study geodesics, which may differ in the field of
ordinary black holes. There are some works studying the motion of test
particles and photons around static Zeng et al. (2006); Blaga (2015); Sarkar
et al. (2018); Blaga et al. (2023); Heydari-Fard et al. (2022) and rotating
Soroushfar et al. (2016); Flathmann and Grunau (2015) dilatonic black holes in
different theories of gravity. To the knowledge of the authors, geodesics
around dilatonic black holes with two scalar fields and two vector fields have
not been studied elsewhere in the literature.

In this paper, a solution for a charged dilaton black hole (BH) (it is S-dual
to dyonic-like solution from Malybayev et al. (2021)) and a particular
solutions for null geodesics are considered. It should be noted that there is
some interest in spherically symmetric solutions see Refs. Bronnikov and
Shikin (1977); Gibbons and Maeda (1988); Garfinkle et al. (1991, 1992) and
references there. These solutions appear in gravitational models with scalar
fields and antisymmetric forms. In our analyses, we follow the methodology of
Pugliese et al. (2011), where the motion of neutral test particles is studied
in the field of a Reissner-Nordström black hole.

The paper is organised as follows: in Section 2 we present charged black
solutions with two scalar (dilaton) fields and two Abelian vector fields and
consider their main characteristics. In Section 3 we examine the geodesics of
neutral test particles and photons, derive the expressions for energy, angular
momentum, effective etc, study their behavior and analyze the stability of
circular geodesics. In Section 4 we summarise our conclusion and discuss
perspectives.

##  2 Charged black hole solution

The action of a model containing two scalar fields, 2-form and dilatonic
coupling vectors is given by

$\displaystyle S=\frac{1}{16\pi G}\int
d^{4}x\sqrt{|g|}\biggl{\\{}R[g]-g^{\mu\nu}\partial_{\mu}\vec{\varphi}\partial_{\nu}\vec{\varphi}\qquad\qquad$
$\displaystyle-\frac{1}{2}e^{2\vec{\lambda}_{1}\vec{\varphi}}F^{(1)}_{\mu\nu}F^{(1)\mu\nu}-\frac{1}{2}e^{2\vec{\lambda}_{2}\vec{\varphi}}F^{(2)}_{\mu\nu}F^{(2)\mu\nu}\biggr{\\}},\quad$
(2.1)

where $g=g_{\mu\nu}(x)dx^{\mu}\otimes dx^{\nu}$ is the metric,
$|g|=|\det(g_{\mu\nu})|$, $\vec{\varphi}=(\varphi^{1},\varphi^{2})$ is the
vector of scalar fields belonging to ${{\mathbb{R}}}^{2}$,
$F^{(i)}=dA^{(i)}=\frac{1}{2}F^{(i)}_{\mu\nu}dx^{\mu}\wedge dx^{\nu}$ is the
$2$-form with $A^{(i)}=A^{(i)}_{\mu}dx^{\mu}$, $i=1,2$; $G$ is the
gravitational constant, $\vec{\lambda}_{1}=(\lambda_{1i})\neq\vec{0}$,
$\vec{\lambda}_{2}=(\lambda_{2i})\neq\vec{0}$ are the dilatonic coupling
vectors obeying

$\vec{\lambda_{1}}\neq\vec{\lambda_{2}},$ (2.2)

and $R[g]$ is the Ricci scalar. Here and in what follows we set $G=c=1$ (where
$c$ is the speed of light in vacuum.)

We consider a so-called double-charged black hole solution to the field
equations corresponding to the action (2) which is defined on the (oriented)
manifold

${\cal M}={\mathbb{R}}\times(2\mu,+\infty)\times S^{2},$ (2.3)

and has the following form

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
H^{a}\biggl{\\{}-H^{-2a}\left(1-\frac{2\mu}{R}\right)dt^{2}$ (2.4)
$\displaystyle\color[rgb]{1,1,1}+$
$\displaystyle\quad\qquad\qquad\color[rgb]{0,0,0}+\frac{dR^{2}}{1-\frac{2\mu}{R}}+R^{2}d\Omega^{2}\biggr{\\}},$
$\displaystyle\varphi^{i}$ $\displaystyle=$ $\displaystyle\nu^{i}\ln H,$ (2.5)

with the 2-form defined by

$F^{(1)}=\frac{Q_{1}}{H^{2}R^{2}}dt\wedge dR,\quad
F^{(2)}=\frac{Q_{2}}{H^{2}R^{2}}dt\wedge dR,$ (2.6)

where $Q_{1}$ and $Q_{2}$ are the (color) electric charges, $\mu>0$ is the
extremality parameter, $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ is
the canonical metric on the unit sphere $S^{2}$ ($0<\theta<\pi$,
$0<\phi<2\pi$), $\tau=\sin\theta d\theta\wedge d\phi$ is the standard volume
form on $S^{2}$ and the moduli function is adopted as

$H=1+\frac{P}{R},$ (2.7)

with $P>0$ obeying

$P(P+2\mu)=\frac{1}{2}Q^{2},$ (2.8)

or equivalently

$P=-\mu+\sqrt{\mu^{2}+\frac{1}{2}Q^{2}}.$ (2.9)

All the remaining parameters of the solution are defined as follows 111It
should be emphasized that for vanishing $a\rightarrow 0$, $\nu^{i}\rightarrow
0$, $Q_{i}\rightarrow 0$ and $\varphi^{i}\rightarrow 0$ and the line element
Eq. (2.4) reduces to the Schwarzschild metric.

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{(\vec{\lambda}_{1}-\vec{\lambda}_{2})^{2}}{\Delta},$
(2.10) $\displaystyle\Delta$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}(\vec{\lambda}_{1}-\vec{\lambda}_{2})^{2}+\vec{\lambda}_{1}^{2}\vec{\lambda}_{2}^{2}-(\vec{\lambda}_{1}\vec{\lambda}_{2})^{2},$
(2.11) $\displaystyle\nu^{i}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{1i}\vec{\lambda}_{2}(\vec{\lambda}_{2}-\vec{\lambda}_{1})+\lambda_{2i}\vec{\lambda}_{1}(\vec{\lambda}_{1}-\vec{\lambda}_{2})}{\Delta},\qquad$
(2.12)

$i=1,2$ and

$Q_{1}^{2}=\frac{\vec{\lambda}_{2}(\vec{\lambda}_{2}-\vec{\lambda}_{1})}{2\Delta}Q^{2},\quad
Q_{2}^{2}=\frac{\vec{\lambda}_{1}(\vec{\lambda}_{1}-\vec{\lambda}_{2})}{2\Delta}Q^{2}.$
(2.13)

Here the following additional restrictions on dilatonic coupling vectors are
imposed

$\vec{\lambda}_{1}(\vec{\lambda}_{1}-\vec{\lambda}_{2})>0,\qquad\vec{\lambda}_{2}(\vec{\lambda}_{2}-\vec{\lambda}_{1})>0.$
(2.14)

We note that

$\Delta>0,$ (2.15)

is valid for $\vec{\lambda}_{1}\neq\vec{\lambda}_{2}$.

Due to relations (2.14) and (2.15) the $Q_{s}^{2}$ are well-defined. Note that
the restrictions (2.14) imply relations $\vec{\lambda}_{s}\neq\vec{0}$,
$s=1,2$, and (2.2).

Indeed, in this case we have the sum of two non-negative terms in (2.12):
$(\vec{\lambda}_{1}-\vec{\lambda}_{2})^{2}>0$ and

$C=\vec{\lambda}_{1}^{2}\vec{\lambda}_{2}^{2}-(\vec{\lambda}_{1}\vec{\lambda}_{2})^{2}\geq
0,$ (2.16)

due to the Cauchy-Schwarz inequality. Moreover, $C=0$ if and only if vectors
$\vec{\lambda}_{1}$ and $\vec{\lambda}_{2}$ are collinear. Relation (2.16)
implies

$0<a\leq 2.$ (2.17)

For non-collinear vectors $\vec{\lambda}_{1}$ and $\vec{\lambda}_{2}$ we get
$0<a<2$ while $a=2$ for collinear ones.

This solution may be verified just by a straightforward substitution into the
equations of motion.

In addition, in Ref. Malybayev et al. (2021), the definition of gravitational
mass was obtained in relation to $\mu$ and $P$ parameters:

$M=\mu+\frac{a}{2}P.$ (2.18)

The following relations can be found/verified from Eqs. (2.10)-(2.13)

$\vec{\nu}^{2}=\frac{(\vec{\lambda}_{1}-\vec{\lambda}_{2})^{2}(\vec{\lambda}_{1}^{2}\vec{\lambda}_{2}^{2}-(\vec{\lambda}_{1}\vec{\lambda}_{2})^{2})}{\Delta^{2}}=\frac{a(2-a)}{2},\qquad$
(2.19)

and

$Q_{1}^{2}+Q_{2}^{2}=\frac{a}{2}Q^{2}.$ (2.20)

The calculation of scalar curvature for the metric
$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$ in (2.4) yields Malybayev et al. (2021)

$R[g]=\frac{a(2-a)P^{2}(R-2\mu)}{2R^{4-a}(R+P)^{1+a}}.$ (2.21)

For the Schwarzschild $(a=0)$ and Reissner-Nordström $(a=2)$ solutions one
immediately gets $R[g]=0$.

##  3 Geodesic motion

Geodesics is a fundamental tool for understanding the motion of test particles
in the field of dilatonic black holes, and can provide valuable insights into
the nature of these objects and their effects on surrounding spacetime.

Geodesics can be derived from the Lagrangian

$\mathcal{L}=\frac{1}{2}g_{\alpha\beta}(x)\dot{x}^{\alpha}\dot{x}^{\beta},$
(3.1)

using the Euler-Lagrange equations:

$\frac{d}{d\tau}\left(\frac{\partial\mathcal{L}}{\partial\dot{x}^{\alpha}}\right)-\frac{\partial\mathcal{L}}{\partial
x^{\alpha}}=0,$ (3.2)

where $\dot{x}^{\alpha}=dx^{\alpha}/d\tau=u^{\alpha}$ is the 4-velocity
vector, e.g. of a test particle, moving along the curve $x^{\alpha}(\tau)$,
$\tau$ is the proper time for massive particles moving along timelike
geodesics and affine parameter in case of null geodesics, respectively, and
$\alpha=0,1,2,3$. The generalized momentum
$p_{\alpha}=g_{\alpha\beta}(x)\dot{x}^{\beta}$ for Lagrangian (3.1) is
normalized as :

$g^{\alpha\beta}(x)p_{\alpha}p_{\beta}=g_{\alpha\beta}(x)u^{\alpha}u^{\beta}=-k,$
(3.3)

where $k=-1,0,1$ for spacelike, null and timelike geodesics, correspondingly.
For general $k$ the quantity $(-k/2)$ is the energy integral of motion for the
Lagrangian (3.1). For simplicity we do not consider spacelike geodesics in the
paper.

We consider null or timelike geodesics in the equatorial plane
($\theta=\pi/2$). In this case, the Lagrangian for the metric given by Eq.
(2.4) reads

$\mathcal{L}=\frac{1}{2}H^{a}\left[-H^{-2a}\left(1-\frac{2\mu}{R}\right)\dot{t}^{2}+\frac{\dot{R}^{2}}{1-\frac{2\mu}{R}}+R^{2}\dot{\phi}^{2}\right].$
(3.4)

For cyclic coordinates $t$ and $\phi$ one obtains the integrals of motion

$\tilde{E}=H^{-a}\left(1-\frac{2\mu}{R}\right)\dot{t},\quad\tilde{L}=H^{a}R^{2}\dot{\phi},$
(3.5)

associated for $k=1$ with the total energy $E=\tilde{E}m$ and angular momentum
$L=\tilde{L}m$ of a test (neutral point-like) particle of mass $m$, which are
the constants of motion. In order to find the equation for the $R$-coordinate
for $\dot{R}\neq 0$ one has to use Eq. (3.3), which has the following form for
the line element from Eq. (2.4)

$-H^{-a}\left(1-\frac{2\mu}{R}\right)\dot{t}^{2}+\frac{H^{a}\dot{R}^{2}}{1-\frac{2\mu}{R}}+H^{a}R^{2}\dot{\phi}^{2}=-k.$
(3.6)

This relation reduces to the following differential equation by virtue of Eq.
(3.5):

$-\frac{H^{a}E^{2}}{m^{2}\left(1-\frac{2\mu}{R}\right)}+\frac{H^{a}\dot{R}^{2}}{1-\frac{2\mu}{R}}+\frac{H^{-a}L^{2}}{m^{2}R^{2}}=-k.$
(3.7)

Eq. (3.7) can be presented in terms of the effective potential:

$\dot{R}^{2}+V^{2}=\frac{E^{2}}{m^{2}},$ (3.8)

which is explicitly given by

$V=\sqrt{H^{-2a}\left(1-\frac{2\mu}{R}\right)\left(H^{a}k+\frac{L^{2}}{m^{2}R^{2}}\right)}.$
(3.9)

It may be readily verified that the Lagrange equation for the radial
coordinate $R$ is equivalent to the following one

$\ddot{R}+V\frac{\partial V}{\partial R}=0.$ (3.10)

Figure 1: The effective potential of test particles for $L/(\mu m)$=0, 3, 4,
5, 6 with different $a$=0, 0.5, 1, 1.5, 2 as a function of the
dimensionless/normalized radial coordinate $R/(2\mu)$. As one may notice for
decreasing $L$ the minimum of the effective potential shifts from right to
left.

For $\dot{R}\neq 0$ it can be obtained just by differentiating Eq. (3.8) with
respect to parameter $\tau$ and then dividing the result by $\dot{R}$. In case
$\dot{R}=0$ the radial equation reads

$\frac{\partial V}{\partial R}=0.$ (3.11)

It does not follow from Eq. (3.8) and should be considered separately.

Now we focus on timelike geodesics $(k=1)$. The behavior of the effective
potential as a function of $R/\mu$ for the fixed value of $Q/\mu=0.6$ (which
is roughly $P/\mu=0.0863$) and for different values of $L/(\mu m)$ is
illustrated in Fig. 1. The cases of $a=0$, $a=1$ and $a=2$ formally correspond
to the Schwarzschild, Sen Sen (1992) and Reissner-Nordström solutions,
respectively. It is clearly seen that the maximum of the effective potential
for $a=0$ exceeds the maximums for other cases of $a$.

Figure 2: The effective potential of test particles with $L_{ISCO}$ for
different $a$=0, 0.5, 1, 1.5, 2 given in Table 1 as a function of the
dimensionless radial coordinate $R/(2\mu)$. Dots indicate the inflection
points ($R_{ISCO}/(2\mu)$, $V_{ISCO}=V(R_{ISCO}/(2\mu))$).

In Fig. 2 the effective potential at fixed $L_{ISCO}$ is illustrated as a
function of the dimensionless radial coordinate $R/(2\mu)$ for different
$a$=0, 0.5, 1, 1.5, 2. Dots show the inflection points. As one may see in
terms of $R$ and $\mu$ these points differ from the standard Schwarzschild and
Reissner-Nordström black holes.

Moreover, three dimensional plot of the effective potential with fixed value
of $Q/\mu=0.6$ is shown in Fig. 3 as a function of $R/\mu$ and $L/(\mu m)$.

$a$ | $\frac{R_{ISCO}}{2\mu}$ | $\frac{L_{ISCO}}{\mu m}$ | $\frac{E_{ISCO}}{m}$ | $V_{ISCO}$
---|---|---|---|---
0.0 | 3.0000 | 3.4641 | 0.9428 | 0.9428
0.5 | 3.0211 | 3.5176 | 0.9422 | 0.9422
1.0 | 3.0427 | 3.5716 | 0.9417 | 0.9417
1.5 | 3.0650 | 3.6260 | 0.9412 | 0.9412
2.0 | 3.0879 | 3.6809 | 0.9407 | 0.9407
Table 1: Values of parameters $x_{ISCO}=R_{ISCO}/(2\mu)$, $L_{ISCO}/(\mu m)$,
$E_{ISCO}/m$ and $V_{ISCO}=V(R_{ISCO}/(2\mu))$ for different $a$. Note, the
net charge here is fixed as $Q/\mu=0.6$ which is equivalent to $P/\mu=0.0863$
according to Eq. (2.9). $x_{ISCO}$ has been calculated accounting for Eqs.
(3.17)-(3.21).

In Table 1 we present the radii of ISCO, corresponding orbital angular
momentum, energy and effective potential of test particles for fixed
$Q/\mu=0.6$ as a function of parameter $a$. The graphical representation of
these values (points) are shown in Figs. 2-8.

Figure 3: Three dimensional plot of the effective potential $V$ as a function
of $L/(\mu m)$ and $R/\mu$ according to Eq. (3.9). The black solid curve
indicates $V$ when $L/(\mu m)$ is defined from Eq. (3.13) for circular
geodesics. Dot shows the radius of ISCO with corresponding $L/(\mu m)$ and
$V$. Here we only consider the case with $a$=1 and $Q/\mu=0.6$.

###  3.1 Circular geodesics

Here we contemplate circular motions, which are characterized by condition:
$\dot{R}=0$, so $V=E/m$.

The derivative of the effective potential with respect to $R$ is given by:

$\displaystyle\frac{\partial V}{\partial R}$ $\displaystyle=$
$\displaystyle\left[2R^{3}H^{2+a}V\right]^{-1}\Big{[}H^{a}(aP(R-2\mu)$ (3.12)
$\displaystyle+$ $\displaystyle
2\mu(P+R))+\frac{2L^{2}}{m^{2}R^{2}}\Big{\\{}R(3\mu-R)$ $\displaystyle+$
$\displaystyle P(R(a-1)+\mu(3-2a))\Big{\\}}\Big{]}.$

The radial equation (3.11) implies

$\frac{L^{2}}{m^{2}}=\frac{H^{a}R^{2}\left[aP(R-2\mu)+2\mu(P+R)\right]}{2\left[R(R-3\mu)+P(R(1-a)+\mu(2a-3))\right]},$
(3.13)

which is substituted to Eq. (3.9) to obtain:

$\frac{E^{2}}{m^{2}}=\frac{H^{-a}(R-2\mu)^{2}\left[2R-P(a-2)\right]}{2R\left[R(R-3\mu)+P(R(1-a)+\mu(2a-3))\right]}.$
(3.14)

Figs. 4 and 5 show $E/m$ and $L/(\mu m)$ as functions of $R/\mu$ for fixed
$Q/\mu=0.6$. Here, the distinctions among all the curves with various $a$ will
be larger for larger values of $Q/\mu$.

Figure 4: Energy $E/m$ of test particles as a function of $R/\mu$ for cases
$a$=0.0, 0.5, 1.0, 1.5, 2.0.when $Q/\mu=0.6$. Dots indicate $R_{ISCO}/\mu$ and
corresponding $E_{ISCO}/\mu$ see Table 1. Figure 5: Orbital angular momentum
$L/(\mu m)$ of test particles as a function of $R/\mu$ for cases $a$=0.0, 0.5,
1.0, 1.5, 2.0. when $Q/\mu=0.6$. Dots indicate $R_{ISCO}/\mu$ and
corresponding $L_{ISCO}/(\mu m)$ see Table 1.

From Eqs. (3.13) and (3.14), it can be seen, that for timelike geodesics the
motion is possible only for for $R(R-3\mu)+P(R(1-a)+\mu(2a-3))>0$, with
limiting radius:

$\displaystyle R_{\gamma\pm}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\Big{[}P(a-1)+3\mu$ (3.15) $\displaystyle\pm$
$\displaystyle\sqrt{(P(1-a)-3\mu)^{2}-4P\mu(2a-3)}\Big{]},\qquad\quad$

where $R_{\gamma+}=R_{0}$, in fact, is the radius of the photon sphere and
$R_{\gamma-}$ is not physical since it is less than $2\mu$. The dependence of
the normalized radius of photon sphere $R_{0}/\mu$ on $Q/\mu$ is depicted in
Fig. 6. As one can see, the larger $Q/\mu$ the larger $R_{0}/\mu$, at least in
this representation of parameters.

Figure 6: The location of normalized radii $R_{\gamma+}/\mu=R_{0}/\mu$
depending on $Q/\mu$.  Figure 7: The location of normalized radii
$R_{ISCO}/\mu$ as a function of $Q/\mu$.

###  3.2 Innermost stable circular orbits

Innermost stable circular orbit (ISCO) is an important quantity in the study
of geodesics in the field of black holes. For example, it plays a crucial role
in the physics of accretion disks as it defines the inner radius of the disk.
Any massive particle that goes beyond the ISCO will fall onto a black hole, so
the disk is supposed to have a radius larger than the ISCO to maintain its
structure. The location of the ISCO, and the properties of the accretion disk
(such as its size, temperature, and brightness) can be used to study the
features of black holes and their surrounding environment.

Using the condition

$\frac{\partial^{2}V}{\partial R^{2}}=0,$ (3.16)

or equivalently by equating to zero the derivatives of Eq. (3.13) or Eq.
(3.14) with respect to $R$ one can find $R_{ISCO}$. For selected values of
$a=0,0.5,1,1.5,2$ we have the following expressions of $R_{ISCO}$.

For $a=0$, which corresponds to the Schwarzschild black hole, the ISCO radius
is given by

$x_{isco}(a=0)=3,$ (3.17)

where $x_{isco}=R_{isco}/(2\mu)$.

In Fig. 7 the radii of ISCOs $R_{ISCO}/\mu$ are presented versus $Q/\mu$. As
one can see, in the representation for vanishing $Q/\mu$ all radii reduce to 6
and for increasing $Q/\mu$, apart from $a$=0 case, $R_{ISCO}/\mu$ grows non-
linearly.

For $a=1/2$ the ISCO radius is

$\displaystyle x_{isco}$ $\displaystyle=$
$\displaystyle\frac{1}{8}(6-3p)+\frac{\sqrt{X_{1/2}}}{8(p+2)}$ (3.18)
$\displaystyle+$
$\displaystyle\frac{1}{2}\sqrt{\frac{A_{1/2}}{\sqrt{X_{1/2}}}-\frac{X_{1/2}}{16(p+2)^{2}}+T_{1/2}},$
$\displaystyle X_{1/2}$ $\displaystyle=$ $\displaystyle
16(2+p)^{2}Y_{1/2}^{\frac{1}{3}}$ $\displaystyle+$
$\displaystyle(p+2)(72+92p+22p^{2}+p^{3})$ $\displaystyle+$ $\displaystyle
p^{2}(1+p)(8+16p+7p^{2}+p^{3})Y_{1/2}^{-\frac{1}{3}},$ $\displaystyle Y_{1/2}$
$\displaystyle=$
$\displaystyle\frac{p^{3}(1+p)\sqrt{R_{1/2}}}{64(p+2)^{2}}+Z_{1/2},$
$\displaystyle R_{1/2}$ $\displaystyle=$
$\displaystyle(2+p)^{-1}(1+p)(2+3p)\Big{[}3968+12288p$ $\displaystyle+$
$\displaystyle 12192p^{2}+4392p^{3}+563p^{4}+51p^{5}\Big{]},$ $\displaystyle
Z_{1/2}$ $\displaystyle=$ $\displaystyle 4^{-3}(p+2)^{-3}p^{3}(1+p)^{2}$
$\displaystyle\times$
$\displaystyle\Big{[}128+264p+114p^{2}+10p^{3}+p^{4}\Big{]},$ $\displaystyle
A_{1/2}$ $\displaystyle=$
$\displaystyle\frac{1}{8}(432+720p+280p^{2}+9p^{4}),$ $\displaystyle T_{1/2}$
$\displaystyle=$ $\displaystyle\frac{3(72+92p+22p^{2}+p^{3})}{16(2+p)},$

where $p=P/2\mu$.

For $a=1$ which is formally identical to the Sen black hole case, the result
is straightforward

$x_{isco}(a=1)=1+(1+p)^{\frac{1}{3}}+(1+p)^{\frac{2}{3}}.$ (3.19)

For $a=3/2$ the ISCO radius is

$\displaystyle x_{isco}$ $\displaystyle=$ $\displaystyle
1+\frac{p}{2}+X_{3/2}^{\frac{1}{3}}+\frac{Q_{3/2}}{X_{3/2}^{\frac{1}{3}}},$
(3.20) $\displaystyle X_{3/2}$ $\displaystyle=$ $\displaystyle
T_{3/2}+\frac{p(1+p)}{16(2+3p)}\sqrt{R_{2/3}},$ $\displaystyle T_{3/2}$
$\displaystyle=$
$\displaystyle\frac{32+96p+108p^{2}+51p^{3}+9p^{4}}{16(2+3p)},$ $\displaystyle
R_{3/2}$ $\displaystyle=$
$\displaystyle\frac{(2+p)\Big{[}384+896p+544p^{2}+88p^{3}-13p^{4}\Big{]}}{2+3p},$
$\displaystyle Q_{3/2}$ $\displaystyle=$
$\displaystyle\frac{8+20p+15p^{2}+4p^{3}}{4(2+3p)}.$

(a) $a=0.5$ (b) $a=1.0$ (c) $a=1.5$ (d) $a=2.0$

Figure 8: The normalized by $2\mu$ radius of ISCO as a function of $Q/(2\mu)$
for different $a$=0.5, 1, 1.5, 2. Numbers near the points imply the values of
energy $E/m$ and angular momentum $L/(\mu m)$ (underlined numbers)
corresponding to a specific ISCO.

For $a=2$, which corresponds to the Reissner-Nordström black hole case, the
ISCO radius is

$x_{isco}(a=2)=1+p+X_{2}^{\frac{1}{3}}+\frac{1+p+p^{2}}{X_{2}^{\frac{1}{3}}},$
(3.21)

where

$X_{2}=\frac{2+p(1+p)\Big{[}7+4p(1+p)+\sqrt{5+4p(1+p)}\Big{]}}{2(1+2p)}.$

Figure 9: Left: The normalized by $2\mu$ photon sphere radius as a function of
$p=P/(2\mu)$ for different values of $a$. Right: The radius of ISCOs as a
function of $p=P/(2\mu)$ for different values of $a$. Figure 10: The location
of normalized by gravitational mass $M$ photon sphere radii as a function net
charge over gravitational mass $Q/M$.

Figure 11: Left: The normalized by gravitational mass $M$ radii of ISCOs as a
function of $Q/M$ for different values of $a$. Right: The efficiency as a
function of $Q/M$ for different values of $a$.

Using expressions (2.9) and (2.18) allows one to represent Eqs. (3.17) -
(3.21) in terms of the gravitational mass $M$ and net charge $Q$.
Correspondingly, in Table 2 we present $\mu$ and $P$ in terms of $M$ and $Q$
depending upon $a$.

$a$ | $\mu$ | $P$
---|---|---
0.0 | $M$ | –
0.5 | $\frac{3M}{2}-\frac{1}{4}\sqrt{4M^{2}+Q^{2}}$ | $-2M+\sqrt{4M^{2}+Q^{2}}$
1.0 | $M-\frac{Q^{2}}{8M}$ | $\frac{Q^{2}}{4M}$
1.5 | $-\frac{M}{2}+\frac{3}{4}\sqrt{4M^{2}-Q^{2}}$ | $2M-\sqrt{4M^{2}-Q^{2}}$
2.0 | $\sqrt{M^{2}-\frac{Q^{2}}{2}}$ | $M-\sqrt{M^{2}-\frac{Q^{2}}{2}}$
Table 2: Values of parameters $\mu$ and $P$ in terms of $M$ and $Q$ depending
on $a$. Note, for vanishing $Q$ one gets $P=0$ and $\mu=M$. At the same time
for vanishing $a$ one recovers the Schwarzschild metric, $\mu=M$ and $P$
disappears in the metric.

Constraints for $Q/M$ can be obtained from Table 2, by requiring $\mu>0$.
Correspondingly, one finds $-2\sqrt{2}/a<Q/M<2\sqrt{2}/a$ and the case $a=2$
will give $-\sqrt{2}<Q/M<\sqrt{2}$.

In Ref Malybayev et al. (2021) it was shown that for $a=2$ case under the
radial coordinate transformation, $R=r_{RN}-P$, along with $M_{RN}=\mu+P$, the
metric (2.4) coincides with the Reissner-Nordström metric. Taking this into
account, one can rewrite Eq. (3.21) in the following form:

$\displaystyle\frac{r_{isco}^{(RN)}}{M_{RN}}$ $\displaystyle=$ $\displaystyle
2+F_{RN}^{1/3}+F_{RN}^{-1/3}\Big{[}4-\frac{3Q_{RN}^{2}}{M_{RN}^{2}}\Big{]},$
(3.22) $\displaystyle F_{RN}$ $\displaystyle=$ $\displaystyle
8+\frac{2Q_{RN}^{4}}{M_{RN}^{4}}+\frac{Q_{RN}^{2}}{M_{RN}^{2}}\left(-9+\sqrt{G_{RN}}\right),$
$\displaystyle G_{RN}$ $\displaystyle=$ $\displaystyle
5-\frac{9Q_{RN}^{2}}{M_{RN}^{2}}+\frac{4Q_{RN}^{4}}{M_{RN}^{4}},$

which matches with the equation for $r_{isco}$ in Ref. Pugliese et al. (2011).
Here $M_{RN}$ and $Q_{RN}$ are the mass and charge, respectively, of the
Reissner-Nordström solution. Besides, the relationship between the charges is
given by $Q_{RN}^{2}=Q^{2}/2$, whereas the masses are equal $M_{RN}=M$.

As expected, in the limiting case $Q\rightarrow 0$, $R_{ISCO}$ corresponds to
the Schwarzschild value $6M$. In the cases of $0<a\leq 2$ the values of
$R_{ISCO}$ depend upon the ratio of $Q/\mu$. This behaviour is illustrated in
Fig. 8. The values of $R_{ISCO}$ at $Q=\mu$ can also been seen in Fig. 7 which
are equal to $R_{ISCO}/\mu$ = 6.10, 6.22, 6.34, 6.46 for cases $a$=0.5, 1.0,
1.5, 2.0, respectively.

The energy and the angular momentum in the last stable circular orbit can be
found numerically after plugging expressions (3.18), (3.19), (3.20) and (3.21)
into Eqs. (3.13) and (3.14). The results are reported in Table 1.

In Fig. 9 the normalized photon sphere radius $R_{0}/(2\mu)$ (left panel) and
$R_{ISCO}/(2\mu)$ (right panel) are given versus $p=P/(2\mu)$. As one can see,
also in these representations $R_{0}/(2\mu)$ and $R_{ISCO}/(2\mu)$ increase
with increasing $p=P/(2\mu)$.

Figure 12: Representation of the second derivative of the effective potential
as a function of $R/\mu$ for $a=0$.

(a) a=1 (b) a=2

Figure 13: Second derivative of the effective potential as a function of
$R/\mu$. Left: for $a=1$. Right: for $a=2$. In both plots, black, red and blue
lines are related to values $Q/\mu$ = 0, 0.6, 1.0, respectively. Orbits with
$\partial^{2}V/\partial R^{2}>0$ are stable, and orbits with
$\partial^{2}V/\partial R^{2}<0$ are unstable.

In Fig. 10 the normalized by gravitational mass $M$ photon sphere radius is
shown versus $Q/M$ for different $a$. As one may notice, the Schwarzschild
case with $a=0$ possess $R_{0}/M=3$ and for other cases $R_{0}/M<3$. The
Reissner-Nordström case with $a=2$ has smallest radius $R_{0}/M=1$ for the
extreme case $Q/M=\sqrt{2}$.

In Fig. 11 (left panel) we present the normalized by gravitational mass $M$
radii of ISCOs as a function of net charge over gravitational mass $Q/M$. As
one may see that for increasing charge the radii of ISCOs decrease as one
expects in analogy with the Reissner-Nordström solution Pugliese et al.
(2011). In this representation all physical quantities can be measured and
used to distinguish charged black holes with different $a$ i.e. different
coupling constant vectors $\vec{\lambda}_{1}$ and $\vec{\lambda}_{2}$. It is
interesting to note, that a similar result was obtained in Fig. 3 (left panel)
of Ref. Narzilloev et al. (2020) for a neutral test particle in the Sen
spacetime. It is shown that for increasing charge over gravitational mass
ratio the normalized by gravitational mass $M$ radius $r_{ISCO}$ decreases. In
order to compare this result with our findings one should find the
relationship between the net charge $Q$ of the present paper and the one in
Ref. Narzilloev et al. (2020), which we denote as $Q_{Sen}$. To do so, one has
to give a close look at the line element Eq. (1) in Ref. Narzilloev et al.
(2020) and compare it with the one considered here when $a=1$. The two
solutions, apart from notations, physical origin and interpretation, are
identical. Thus, by comparing the two solutions one finds that $Q=2Q_{Sen}$
and the value $Q/M=1.4$ in Fig. 11 left panel for $a=1$ is equal to
$Q_{Sen}/M=0.7$ in in Fig. 3 (left panel) of Ref. Narzilloev et al. (2020).

Furthermore, one of the quantities which is of great interest is the
efficiency of converting matter into radiation (see for details page 662 of
Ref. Misner et al. (1973))

$\eta=[1-\tilde{E}(R_{ISCO})]\times 100\%$ (3.23)

In Fig. 11 (right panel) the efficiency is shown versus $Q/M$. As we expected,
the efficiency for different values of $a=0.0,0.5,1.0,1.5,2.0$ is always
larger than the $a=0$ case.

In general, the circular orbits are allowed in the region $R_{0}<R<R_{ISCO}$,
however, all those orbits are unstable. For stability of circular orbits, the
following condition must be fulfilled:

$\frac{\partial^{2}V}{\partial R^{2}}>0.$ (3.24)

The behavior of the second derivative of the effective potential is depicted
in Figs. 12 and 13. From these plots, it can be seen where orbits become
stable. The location of ISCO is determined by finding the radius at which the
second derivative of the effective potential changes sign from negative to
positive.

##  4 Conclusion

In this work we considered the solution for a double-charged dilatonic black
hole. We studied circular geodesics of massive neutral test particles and
photons for different values of parameters $a$ and $P$ (the latter is directly
connected with the net charge $Q$). We calculated the energy, angular momentum
and radius of ISCO of test particles in terms of $a$, $P$ and $\mu$ and
investigated in detail their behavior for $a=0,1/2,1,3/2,2$ cases,
corresponding to various configurations of the double-charged dilatonic black
holes.

Our analysis is mainly based on the study of the behavior of an effective
potential which defines the position and stability properties of circular
orbits. The stability of circular geodesics was analyzed by examining the sign
of the second derivative of the effective potential with respect to the radial
coordinate. For neutral test particles stable orbits exist only starting from
$R_{ISCO}$ up to infinity.

The ISCO radius and photon sphere radius have been calculated for particular
values of $a$. It turned out that, in this representation of the parameters of
the line element, for larger $a$ with increasing $Q/\mu$ the radius of the
ISCO and photon sphere radius increase. The photon sphere radius and the ISCO
radius have their minimum values $R_{0}/\mu=3$ and $R_{ISCO}/\mu=6$,
respectively, in the Schwarzschild limiting case ($a$=0).

However, in the representation of the net charge over gravitational mass the
situation is utterly opposite. For the Schwarzschild case ($a=0$) one gets the
largest values for the photon sphere and ISCO radii $3M$ and $6M$,
respectively, and for the Reissner-Nordström case ($a=2$) one obtains $M$ and
$3M$, respectively. Taking into consideration Table 2 for $a=2$ and
$Q/M=\sqrt{2}$ the parameter $P$ will be equal to $M$. Correspondingly, the
photon sphere and ISCO radii will be equal to $2M$ and $4M$, respectively, if
one uses the coordinate transformation $r_{RN}=R+P$ and the relation between
charges $Q_{RN}=Q/\sqrt{2}$. These results are in agreement with the ones
reported in Ref. Pugliese et al. (2011).

The efficiency of converting matter into radiation as expected is larger for
the Reissner-Nordström case $\eta=8.14\%$ and smaller for the Schwarzschild
case $\eta=5.72\%$ Bokhari et al. (2020). All other configurations are
restricted within these two cases.

This peculiarity can be used to distinguish ordinary or astrophysical black
holes from the double-charged dilatonic black holes.

It would be interesting to extend the analyses of the paper in future studies
related to the quasinormal modes at final moments of black hole mergers in
binaries, quasiperiodic oscillations in the X-ray systems, radiative flux and
spectral luminosity of accretion disks around astrophysical black holes.

## Acknowledgements

KB thanks professors Daniele Malafarina and Hernando Quevedo for fruitful
discussions during the preparation of this paper. KB acknowledges partial
support from the Science Committee of the Ministry of Science and Higher
Education of the Republic of Kazakhstan (Grant No. AP19680128). For VI the
research was funded by RUDN University, scientific project number
FSSF-2023-0003.

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