Eur. Phys. J. C
(2023) 83:814
https://doi.org/10.1140/epjc/s10052-023-11980-3
Regular Article - Theoretical Physics
Astrophysical and electromagnetic emissivity properties of black
holes surrounded by a quintessence type exotic fluid in the
scalar–vector–tensor modified gravity
Haidar Sheikhahmadi1,2,3,a , Saheb Soroushfar4,b , S. N. Sajadi5,c , Tiberiu Harko6,7,8,d
1
School of Astronomy, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran
Center for Space Research, North-West University, Potchefstroom, South Africa
3 Canadian Quantum Research Center, 204-3002 32 Avenue, Vernon, BC V1T 2L7, Canada
4 Department of Physics, College of Sciences, Yasouj University, Yasouj 75918-74934, Iran
5 School of Physics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran
6 Faculty of Physics, Babes-Bolyai University, Kogalniceanu Street, 400084 Cluj-Napoca, Romania
7 Department of Theoretical Physics, National Institute of Physics and Nuclear Engineering (IFIN-HH), 077125 Bucharest, Romania
8 Astronomical Observatory, 19 Ciresilor Street, 400487 Cluj-Napoca, Romania
2
Received: 12 April 2023 / Accepted: 28 August 2023
© The Author(s) 2023
Abstract The astrophysical consequences of the presence
of a quintessence scalar field on the evolution of the horizon
and on the accretion disk surrounding a static black hole, in
the scalar–vector–tensor version of modified gravity (MOG),
are investigated. The positions of the stable circular orbits of
the massive test particles, moving around the central object,
are obtained from the extremum of the effective potential.
Detailed calculations are also presented to investigate the
light deflection, shadow and Shapiro effect for such a black
hole. The electromagnetic properties of the accretion disks
that form around such black holes are considered in detail.
The energy flux and efficiency parameter are estimated analytically and numerically. A comparison with the disk properties in Schwarzschild geometry is also performed. The quantum properties of the black hole are also considered, and the
Hawking temperature and the mass loss rate due to the Hawking radiation are considered. The obtained results may lead
to the possibility of direct astrophysical tests of black hole
type objects in modified gravity theories.
2 Kiselev type S–V–T black hole in the presence of an
exotic fluid . . . . . . . . . . . . . . . . . . . . . .
2.1 General properties of the metric, and the structure of the horizons . . . . . . . . . . . . . . .
2.2 Horizons of the QMOG black hole . . . . . . .
2.3 Equations of motion, and the effective potential
2.4 Event horizons, and stable circular orbits of the
QMOG black hole . . . . . . . . . . . . . . . .
3 Astrophysical properties of the QMOG black holesdeflection of light, shadow and the Shapiro delay . .
3.1 The deflection of light . . . . . . . . . . . . . .
3.2 The shadow of the QMOG black hole . . . . .
3.3 The Shapiro time delay . . . . . . . . . . . . .
4 Electromagnetic emissivity of the thin accretion disks
around QMOG black holes . . . . . . . . . . . . . .
5 Hawking radiation of the QMOG black holes . . . .
6 Concluding remarks . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . .
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1 Introduction
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . .
a e-mails:
h.sh.ahmadi@gmail.com; h.sheikhahmadi@ipm.ir
b e-mails:
saheb.soroushfar@gmail.com; soroush@yu.ac.ir (corresponding author)
c e-mail:
naseh.sajadi@gmail.com
d e-mail:
tiberiu.harko@aira.astro.ro
0123456789().: V,-vol
According to the cold dark matter, CDM, model, the
material content of the Universe mainly consists of two basic
components, dark energy, and dark matter, respectively [1,2].
To explain the observed late-time cosmic acceleration of
the Universe [3–5], one can consider either a fundamental cosmological constant , which, for example, could be
interpreted as an intrinsic geometric property of the spacetime, or a dark energy, an ambiguous fluid component, of
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yet unknown physical origin, which spreads throughout the
whole Universe, and which would mimic a cosmological
constant. Currently, one of the main dark energy scenarios
is based on the so-called quintessence fields [6–9]. In this
model, the cosmological constant, , is considered as a variable term, and it’s evolution and properties can be explained
by considering a scalar field with negative pressure, representing the dark energy [10].
In cosmological studies the quintessence model faces
some drawbacks for providing a good concordance between
theory and observations, especially in local scales. These theoretical problems led to the introduction of some extended
scalar-tensor models, as chameleon, and Brans-Dicke chameleon
models [11–17]. Additionally, the explanation of the accelerated expansion phase itself in a consistent quantum model of
gravity is still a challenge [18]. The origin of this drawback
goes back to the so-called outer horizon problem. In other
word, this problem is related to the impossibility of introducing an observable and physical S-matrix for describing
the asymptotic in and out states.
In the following we will not investigate the cosmological
properties, with their advantages and disadvantages, of the
quintessence models. Instead of the large scale cosmological
studies we will consider the effects of the quintessence fields
on the behaviour of the compact objects, as, for instance, the
black holes.
Interestingly enough, in [18] it was shown that in the presence of the scalar field and under some simple assumptions
for the energy–momentum tensor, an exact black hole solution for the Einstein equation can be obtained. One interesting property of this solution is that when one considers the
interval −1 < ω < −1/3 for the equation of state parameter
ω = p/ρ, then an outer horizon is formed.
Before these investigations, in [19,20] a solution based on
a free quintessence model was found, which leads to a bare
singularity, with no hair and no horizon, with the cosmic censorship conjecture not satisfied. Therefore this solution does
not represents a black hole solution. Recently, many different
classes of black holes models in the presence of quintessence
fields have been investigated. For the rotating charged black
holes one can see [21]. For the black hole solutions originating from string theory we refer the readers to [22]. The
thermodynamics properties of the charged black holes in the
presence of quintessence have been investigated in [23]. In
this work, different conditions for the heat conductivity of
the black holes’ three horizons have been discussed in detail.
Entropy calculations of the black holes in the presence of
quintessence can be found in [24]. For more details on the
black holes in the presence of quintessence fields see [25–27],
and references therein. It deserves to also note that there are
some models of black hole solutions that are able to remove
the central singularity, namely, the regular black holes, and
the non-commutative black holes [33–35], respectively.
123
Eur. Phys. J. C
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Let us now briefly review the astrophysical and physical
properties of the accretion disks. Usually, in astrophysics the
term accretion is associated to the process of the attraction of
matter, especially gaseous, by a central massive object. When
this ingoing process happens, gravitational energy, and even
momentum, are extracted from the central object. Obviously,
the amount of such energy should be proportional to the ratio
M/R, where M denotes the mass, and R denotes the radius
of the central object, respectively. Therefore it can be concluded that the more massive and compact the central mass
is, the larger amount of energy will flow in. Hence, from a
phenomenological point of view, the accretion into a black
hole is one of the important and interesting phenomena in
the astrophysical context. Its importance goes back to the
fact that it indicates the transfer of gravitational energy into
radiation, and leads to an explanation of the properties of
the most luminous objects in the Universe, which recently
have been observed directly by the Event Horizon telescope,
EHT [28]. Obviously, the spherical symmetric accretion on a
cental compact object is the simplest case one can visualize.
For the first time such a system was studied in [29], and it is
well-known as the Bondi accretion. Bondi did show that this
solution is unique, and consequently the flow becomes transonic crossing the sonic radius. Observations have shown
that accretion processes are present in many astrophysical
objects.
An important consequence of the accretion processes is
that many compact objects are surrounded by accretion disk.
Astrophysical phenomena related to accretion disks have
been studied extensively. Here we give some simple example
of accretion disks. Usually, the protostellar clouds formed of
a molecular gas can form a protoplanetary disk around the
host newly formed star. As the second example, we can refer
to the binary systems, in which mass can be accreted to form
an accretion disk due to the presence of strong stellar winds.
The third possibility for the formation of an accretion disk
is perhaps the more interesting one, and recently it has been
confirmed by direct observations [28]. It is related to the
super massive black holes in active galactic nuclei, which
can acquire mass by accreting matter from their vicinity. It
is important to emphasize that in almost all high luminosity
astrophysical objects there is a central mass surrounded by an
accretion disk. For a valuable discussion of the accretion disk
properties of compact objects we refer the reader to [30]. The
study of the properties and of the physics of the Newtonian
accretion disks was initiated in the important and seminal
paper [31].
There are many different proposals advanced to explain
the rotation curves of galaxies, the dynamics of galaxy clusters, and other astrophysical and cosmological phenomena.
In one hand, as we have already mentioned above, the dark
matter model, for the first time hypothesized by Zwicky [32],
has become an essential part of the standard CDM cosmo-
Eur. Phys. J. C
(2023) 83:814
logical paradigm. On the other hand, to explain the dark matter phenomenology, models based on modified gravity have
also been proposed. Fore example, some models, based on
imposing a modification in Newtonian version of gravity, the
so called Modified Newtonian dynamics, MOND [36] have
been considered as alternatives to the standard dark matter
model.
Another proposal for a modified gravity theory is the
scalar–vector–tensor (S–V–T) modified gravity theory, or
MOG [37]. This theoretical model adequately explains Solar
System observations [38], the rotation curves of galaxies
[39], the dynamics of galaxy clusters [40], and the Cosmic Microwave Background predictions [41]. Recently, the
physics of some black hole solutions in the MOG version of
gravity has been investigated, and it has been shown that the
well known exact solutions of general relativity can be reproduced properly [42,43]. For instance, although the ReissnerNordtröm solution in the standard version of gravity theory (general relativity) corresponds to an electrically charged
object, in MOG it can be considered as describing electrically
neutral objects. In the MOG theory, a√gravitational repulsive
force, with an effective charge Q = αG N M, can be introduced. The parameter α = −1 + G/G N is known as the
MOG parameter, and it has to be determined from the observational constraints. The parameter G N is Newton’s constant,
and G is the enhanced gravitational constant. Moreover, by
M we have denoted the total mass of the central object, a
black hole, for example.
This physical solution for a MOG black hole motivated us
to extend our investigation to study the properties of the black
holes in the presence of a scalar field, that is, a quintessence
field, and to consider the properties of the compact objects
in such an extension of the MOG theory. In the following we
will cal, this solution of the MOG theory as a QMOG black
hole. More exactly, we consider first the positions of the stable circular orbits of the massive test particles, moving around
the central MOG type black hole, which are obtained from the
extremum of the effective potential. Several important astrophysical effects are investigated in detail, including the light
deflection, the shadow and the Shapiro effect, respectively.
The electromagnetic properties of the accretion disks are also
analyzed in detail, and the energy flux, the luminosity, and
the efficiency parameter are obtained by using both analytical and numerical methods. For each quantities we perform a
comparison with the disk properties in Schwarzschild geometry. The important quantum properties of the MOG black
holes are analyzed, and the Hawking temperature, and the
mass loss rate due to the Hawking radiation is obtained. The
results presented in this study may lead to the possibility of
the astrophysical tests of the black hole type objects in modified gravity theories, and of the modified gravity theories
themselves.
Page 3 of 18
814
This work is organised as follows. In Sect. 2, we write
down the Reissner–Nordström type black hole solution in the
presence of a quintessence field in the S–V–T theory. Moreover, the dynamical properties of the motion of the massive
particles in this geometry (effective potential, positions of the
event horizon, and the radii of the marginally stable orbits)
are analyzed in detail The astrophysical properties of the
QMOG black holes, like the deflection of light, the shadow
of the black holes, and the Shapiro effects are studied in
Sect. 3. Furthermore, in Sect. 4, the flux, temperature and the
efficiency of the accretion disks that form around QMOG
black hole are obtained both analytically, and numerically.
Finally, in Sect. 6 we discuss and conclude our results.
2 Kiselev type S–V–T black hole in the presence of an
exotic fluid
In the following section we will first write down the metric
of a static black hole in the S–V–T (MOG) version of gravity in the presence of a quintessence scalar field, aiming at
investigating the implications of this modified gravity theory
on the behaviour and properties of the massive astrophysical
objects. Moreover, the horizon properties and the equations
of motion of the massive test particles are analyzed in detail.
In the next step, the dynamical properties of this compact,
black hole type object, will be analyzed as well.
2.1 General properties of the metric, and the structure of
the horizons
In this section, we consider a static and spherically symmetric black hole, surrounded by a quintessence scalar field, in
the MOG theory. Generally, the geometry of the static, spherically symmetric black hole is given by [18,42–46]
ds 2 = − f (r )dt 2 + f (r )−1 dr 2 + r 2 (dθ 2 + sin2 θ dϕ 2 ).
(2.1)
In the following we consider a Kiselev type black hole in
MOG, with the metric function given by
γG
2G M
G Q2
(2.2)
+ 2 − 3ǫ+1 ,
r
r
r
where we have denoted
√ G = G N (1 + α), and M is the black
hole mass, Q = αG N M is the effective (non-electric)
charge, while α is the MOG parameter, γ is the quintessence
parameter, and ǫ is the equation of state parameter of the
quintessence field. The allowed values for ǫ are in the range
−1 < ǫ < −1/3. In the following we will call the black hole
described by the metric (2.2) as the QMOG black hole.
In the case of ǫ = −1, the metric (2.2) reduces to
the general relativistic black hole space-times, in the presence of a cosmological constant, i.e., to the anti de-Sitter
f (r ) = 1 −
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Eur. Phys. J. C
(2023) 83:814
and
1
φ
Tθθ = Tφ = − (1 + 3ε) ρ,
2
(2.5)
where ε is the parameter of the equation of state. By taking
the isotropic average over the angles of the components of the
energy–momentum tensor we obtain the barotropic equation
of state p = ερ. Hence, the interpretation of the Kiselev black
hole as describing a black hole solution in the presence of a
quintessence field is problematic, and this solution may be
better interpreted as describing a black hole in the presence
of an exotic matter source.
2.2 Horizons of the QMOG black hole
The horizon of the spacetime given by the metric (2.1) can
be determined by setting the condition g00 (r ) = 0, i.e.,
1−
Fig. 1 The plots of f (r ) for M = 1, ǫ = −2/3, G N = 1, α = 0.5
and γ = 0.1, 0.045, 0.058
Schwarzschild geometry (for more details see [42]). Moreover, in the limit of γ = 0, the metric reduces to an effective Reissner–Nordström type black hole, with a non-electric
charge. For γ = 0, and Q = 0, it reduces to the well-known
Schwarzschild black hole. For the case ǫ = −2/3, Eq. (2.1)
describes an effective Reissner–Nordström-like black hole,
surrounded by quintessence field in the S–V–T (MOG) version of gravity theories. In Fig. 1 the behaviour of the metric
coefficient is illustrated, for various numerical values of the
metric parameters.
It deserves here to note that Moffat [42] obtained his MOG
version of the black holes by neglecting the energy momenm = 0. Following
tum tensor of the matter sector, i.e., Tμν
[18], and by introducing the matter sector, as a quintessence
scalar field, the metric (2.1) obtained accordingly. To obtain
the metric one must decompose the energy momentum tensor, in the Einstein field equation G μν = −8π Tμν , in two
distinct components, i.e.,
q
V
Tμν = Tμν
+ Tμν
,
(2.3)
q
2G M
Gγ
G Q2
+ 2 − 3ǫ+1 = 0.
r
r
r
(2.6)
Consequently, we obtain for the position of the horizon the
algebraic equation
r 2 − 2G Mr + G Q 2 − Gγ r 1−3ǫ = 0.
(2.7)
For our purposes, and for mathematical convenience, we
introduce the following dimensionless parameters, defined
as,
r⋆ =
r
r̃
=
,
(1 + α)
(1 + α)G N M
Q⋆ = √
Q̃
=√
Q
,
G N (1 + α)M
1+α
γ⋆ = (1 + α)2 γ̃ = (1 + α)2 G 2N γ M.
(2.8)
For ǫ = −2/3, and by rearranging the parameters, Eq.
(2.7), interestingly, takes the following form,
γ⋆r⋆3 − r⋆2 + 2r⋆ − Q 2⋆ = 0.
(2.9)
Another interesting case corresponds to ǫ = − 21 . We have to
note that for this value of ǫ, the dimensionless parameter γ̃
takes the form γ̃ = G 3N γ 2 M. Accordingly, Eq. (2.7) can be
rearranged as follows,
√ 25
γ⋆r⋆ − r⋆2 + 2r⋆ − Q 2⋆ = 0.
(2.10)
V
where Tμν stands for the quintessence part [18], and Tμν
describes the vector field [42]. For the derivation of the metric
one can find more details in [47–49] as well.
It is important here to note that the Kiselev black holes can
be interpreted as a solution of the Einstein gravitational field
equations with the matter energy-momentum tensor given by
[50],
In the following we would like to investigate the effects of
the MOG parameter α on the properties of the black holes. To
this end, by taking into account the definitions of the effective
charge, and of the enhanced gravitational constant, one can
rewrite Eqs. (2.9) and (2.10), respectively, in the alternative
forms, namely,
Ttt
γ̃ r̃ 3 −
=
Trr
= ρ(r ),
123
(2.4)
r̃ 2
+ 2r̃ − α = 0,
(1 + α)
(2.11)
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(2023) 83:814
Page 5 of 18
and
5
γ̃ (1 + α)r̃ 2 − r̃ 2 + 2(1 + α)r̃ − α(1 + α) = 0,
(2.12)
γ⋆r⋆−3 − Q 2⋆ r⋆−2 + 2r⋆−1 − 1 = 0,
(2.13)
respectively.
It is interesting to notice that there is another possibility of
extending the range of ǫ by also considering the case ǫ > 0.
From the asymptotically flat behaviour of such a configuration it can be concluded that, obviously, it would not obey
the properties required for a quintessence field anymore. To
make these things clearer, as an example, let’s consider the
value ǫ = 2/3. Then, the dimensionless equation (2.7) can
be reformulated as,
where, for the positive values of ǫ, i.e., ǫ = 2/3, one has
γ̄ = G 2N γ M 3 . Now, from Eq. (2.8), and from the definition
of Q, one can rewrite Eq. (2.13) as follows,
γ̄ (1 + α)3r̃ −3 −
α −2
2
1
r̃ +
r̃ −1 −
= 0.
1+α
1+α
(1 + α)2
(2.14)
We will return to this matter, and discuss it in detail, in the
next sections.
2.3 Equations of motion, and the effective potential
In this Section, to investigate the physical properties of the
MOG black hole in the presence of a quintessence field (or an
exotic matter source), we are going to study first the equations
of motion of a massive test particle, and to derive the effective
potential as well. It is well known that for a point, massive
test particle, the Lagrangian L of the motion in the spacetime
given by Eq. (2.1), can be written as,
dxμ dxν
1
,
L = gμν
2
ds ds
(2.15)
Accordingly, in the equatorial plane, the conserved energy E
and the angular momentum L can be obtained according to,
dt
2G M
G Q2
dt
Gγ
= 1−
+ 2 − 3ǫ+1
, (2.16)
E = gtt
ds
r
r
r
ds
and
L = gϕϕ
dϕ
dϕ
= r2 ,
ds
ds
(2.17)
respectively. Plugging the above equations all together, the
geodesic equations are expressed in the form,
2
dr
G Q2
2G M
Gγ
+ 2 − 3ǫ+1
= E2 − 1 −
ds
r
r
r
2
L
(2.18)
× 1+ 2 ,
r
814
2
G Q2
r4
Gγ
2G M
2
+ 2 − 3ǫ+1
= 2 E − 1−
L
r
r
r
L2
,
(2.19)
× 1+ 2
r
2
2
dr
2G M
G Q2
1
Gγ
+ 2 − 3ǫ+1
= 2 1−
dt
E
r
r
r
G Q2
2G M
Gγ
2
+ 2 − 3ǫ+1
× E − 1−
r
r
r
2
L
×(1 + 2 ) .
(2.20)
r
dr
dϕ
See [51–56] for more details on the derivation of these equations.
Equations (2.18)–(2.20) give a complete description of
the dynamics of the massive test particles moving around the
MOG black hole. By considering Eq. (2.18), we can define
an effective potential of the motion as,
L2
G Q2
Gγ
2G M
+ 2 − 3ǫ+1
1 + 2 . (2.21)
Ve f f = 1 −
r
r
r
r
These definition allows us to compare our results with the
standard form of the equations of motion in other geometries,
and black hole solutions.
Now we are prepared to discuss the properties of the accretion disks that form around the central mass. We will begin
this discussion in the next Subsection by considering the
properties of the event horizons, and of the stable circular
orbits for the QMOG black holes.
2.4 Event horizons, and stable circular orbits of the QMOG
black hole
The accretion disks usually form through a simple astrophysical mechanism. In an accretion disk, hot gas particles, carrying an electric charge, are moving in stable circular orbits
around the central compact object, which in the present case
we consider to be a black hole. For the case ǫ = −2/3, the
specific energy Ẽ, the specific angular momentum L̃, and the
˜ of the particles that move in a circular
angular velocity ,
orbit can be written as,
Ẽ =
=
g̃tt
˜2
g̃tt − g̃φφ
√
3
2
2 α(α + 1) − (α + 1)γ̃ r̃ + r̃ − 2(α + 1)r̃
(α + 1)r̃ 2
2
α+1
−
6r̃ −4α
r̃ 2
,
− γ̃ r̃
(2.22)
L̃ =
˜
g̃φφ
˜2
g̃tt − g̃φφ
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=
Page 6 of 18
Eur. Phys. J. C
−2α − γ r̃ 3 + 2r̃
2
α+1
−
,
(2.23)
−2α − γ̃ r̃ 3 + 2r̃
,
√
2r̃ 2
(2.24)
6r̃ −4α
r̃ 2
− γ̃ r̃
and,
˜ =
g̃tt,r̃
=
g̃φφ,r̃
respectively.
Using Eqs. (2.8), (2.21) and (2.23), for the effective potential, and its derivatives, we obtain,
Ṽe f f
1
r̃ 2
L̃ 2
3
− 2r̃ + α 1 + 2
= 2 − γ̃ r̃ +
r̃
(1 + α)
r̃
,
(2.25)
d Ṽe f f
1
=−
− L̃ 2 α γ̃ r 3 + α γ̃ r 5
d r̃
(1 + α)r 5
− L̃ 2 γ̃ r 3 + γ̃ r 5 + 4 L̃ 2 α 2
− 6L 2 αr + 2 L̃ 2 r 2 + 2α 2 r 2 − 2αr 3
2
2
2
3
+ 4 L̃ α − 6 L̃ r + 2αr − 2r ,
(2023) 83:814
observer should be located in the domain of outer communication, which is between the event horizon and cosmological
horizon, similar to the de Sitter case. For convenience, the
observer in our paper is set to be near the cosmological horizon.
To visualize the effects of the MOG parameter we present
the Fig. 2, in which the parameters α̃ and γ̃ are plotted as
a function of the radius of the event horizon, for different
values of Q̃ and L̃. It is clear from Fig. 2 that as long as the
values of Q̃ increase, the peak values of γ̃ become larger.
Moreover, considering Fig. 2, it can be seen that the increase
in the values of γ̃ leads to a decrease of the effective potential.
The location of the innermost stable circular orbits is represented, for different values of the parameters of the black
hole solution, in Fig. 3.
As one can see from Fig. 2, the maximum value of the
parameter γ̃ corresponds to r̃ ≃ 8. By increasing the value
of α, the maximum value of the parameter γ̃ decreases, and
the effects of the maximum values of the scalar field do appear
far from the MOG black holes.
(2.26)
3 Astrophysical properties of the QMOG black
holes-deflection of light, shadow and the Shapiro delay
and
d 2 Ṽe f f
2
× − L̃ 2 γ̃ r̃ 3 (α + 1) + 10 L̃ 2 α 2
=
d r̃ 2
1 + α r̃ 6
In the present section, for the sake of completeness, we will
investigate in detail some of the basic astrophysical tests for
the QMOG black hole. In particular, we will consider the
deflection of light, the shadow of the black hole, and the
Shapiro time delay effect.
For more details on the derivation of the above equations
one can see [68–71,73,74]. It should be emphasised here
that the equation d 2 Ṽe f f /d r̃ 2 = 0 is employed as a necessary condition for the existence of innermost stable circular
orbits. In addition to this, the sign of d 2 Ṽe f f /d r̃ 2 , usually is
a criterion to show the stability of the orbits of the black hole.
By employing all these above mentioned conditions, the
location of the event horizons, and of the stable circular
orbits, for a S–V–T (QMOG) black hole, surrounded by a
quintessence scalar field, or an exotic matter source, are presented in Tables 1, and 2, respectively.
In Table 1, we did investigate the role of the quintessence
parameter γ̃ on the black hole properties, when the MOG
parameter α is fixed at 0 and 0.25. Then, in Table 2, we
considered the effects of changes in the parameter α, when
the parameter γ̃ takes the values 0 and 0.003.
In addition to the points mentioned above, the location of
the observer is also important. Usually, the observer is considered to be at the infinite boundary for the asymptotically
flat spacetimes. However, for the model of quintessence black
hole, the cosmological horizon does matter. Physically, the
3.1 The deflection of light
−12 L̃ 2 αr̃ + 3 L̃ 2 r̃ 2 + 3(αr̃ )2 − 2αr̃ 3 + 10 L̃ 2 α
2
2
3
(2.27)
−12 L̃ r̃ + 2αr̃ − 2r̃ .
123
The deflection angle of a photon as it moves from infinity to
rm and off to infinity for the metric (2.1) can be expressed as
[76–78],
∞
2dr
δϕ =
− π̄ = I − π̄ ,
(3.1)
4
rm
r
2
− f (r )r
b2
where b = rm2 / f (rm ) is the impact parameter of the null
ray, and rm is the coordinate distance of the closest approach.
Here π̄ is the change in the angle ϕ for the straight line
motion, and is therefore subtracted out from the total deflection angle.
We now calculate the integral in Eq. (3.1). Writing theterm
in the denominator of (3.1) as f (r )r 2 r 2 /b2 f (r ) − 1 , for
M ≪ r one has
⎡
1−
f (rm ) r 2
−1 = ⎣
f (r ) rm2
1−
2G M
rm
2G M
r
+
G Q2
rm2
+
G Q2
r2
−
−
Gγ
rm3ǫ+1
Gγ
r 3ǫ+1
⎤
⎦
r2
rm2
−1
Eur. Phys. J. C
(2023) 83:814
Page 7 of 18
Table 1 Location of the event horizons, and of the stable circular orbits
in a S–V–T (MOG) black hole for α = 0, 0.25, when the quintessence
parameter γ̃ takes different values, including γ̃ = 0 to γ̃ = 0.004. Here
for the left side of the table one notice r− stands for Cauchy radius, r+
814
refers event horizon and rc is cosmological horizon. Also considering
the right panel r I SC O stands for ISCO radius, r S refers stable radii and
r O SC O refers the most outer stable circular orbit
γ
r−
r+
rc
r I SC O
rS
r O SC O
α=0
0
×
2
6
0.1320
2.368
×
×
×
α = 0.25
0.001
0.1319
2.3755
0.003
0.1319
0.004
0.1319
0
Table 2 The location of the
event horizons, and of the stable
circular orbits in a S–V–T
(MOG) black hole for
γ̃ = 0, 0.003, and for different
values of the MOG parameter α,
indicated in the Table
γ̃ = 0
×
7.1106
797.49
7.3186
2.3907
264.14
8.0783
12.542
794.6
2.3984
197.469
493.59
×
×
r
rm
2
1 + 2G M
2394.7
r+
rc
r I SC O
rS
r O SC O
0
×
2
×
6
331.3
6.451
×
×
0.1319
2.3907
264.14
8.078
12.542
794.68
0
×
2.012
13.34
995.7
0.5
0.1340
1.877
331.3
5.961
13.47
995.8
0.5
0.2751
2.7628
219.18
660.27
0.75
0.4269
3.1325
186.91
563.952
×
×
1
1
−
r
rm
1
1
1
1
−1
−
+
Gγ
−
−G Q 2
r2
rm2
r 3ǫ+1
rm3ǫ+1
2
G Q2
r
2G Mr
+ 2
=
−1 1−
2
rm
rm (r + rm )
rm
1
Gγ r 2
.
(3.2)
+ 2
r − rm2 r 3ǫ+1 − rm3ǫ+1
×
r−
Fig. 2 Right panel: the variation of the quintessence parameter γ̃ as a
function of the radial coordinate of the event horizon, r̃ , for different
values of the MOG parameter α. When α = 0, the model reduces to the
Schwarzschild black hole in the presence of the quintessence field. Left
=
24.29
α
0.25
γ̃ = 0.003
×
×
×
panel: the variation of the effective potential for the parameter α = 0.25,
and for different values of γ̃ and L̃. The red dots indicate the location
of the innermost stable circular orbits, ISCO
Upon expanding in powers of M/r , M/rm , and Gγ , the
integrand in Eq. (3.1) then becomes,
∞
2dr
∞
1
r4
1
− r12
− f (r )r 2 rm
rm2
2
b
r2
G Q2 1
GM
1
1+
−
× 1+
+
r
rm (r + rm )
2
r2
rm2
rm
=
123
814
Page 8 of 18
Eur. Phys. J. C
(2023) 83:814
Fig. 3 The innermost stable circular orbit (solid line) and the angular momentum of the inflection point (dashed line) in terms of the coupling
constant of theory α (γ̃ = 0) (left panel) and γ̃ (α = 0) (right panel)
Gγ
+
2
1
r 3+1
r2
−
2
2 r − rm2
1
r 3+1
−
1
rm3+1
dr
.
r2
After making the substitution sin(θ ) = rm /r , the integral
can be expressed as follows,
∞
2dr
rm
shadow of the black hole we follow up the null geodesics, on
which the motion of the photons take place.
The angular radius of the shadow of the black hole, as seen
by an observer located at r0 , is defined as [80],
r4
sin2 (Ŵ) =
f (r )r 2
−
π
2
1
GM
sin(θ) +
=
dθ 1 +
rm
1 + sin(θ)
0
G Q2
− 2 1 + sin2 (θ)
2rm
3ǫ+1
Gγ
1
sin
+ 3ǫ+1 sin3ǫ+1 (θ) −
(θ)
−
1
2 cos2 (θ)
2rm
b2
2G M
3π G Q 2
Gγ
π
+
−
+ 3ǫ+1
2
rm
8rm2
4rm
π
2 cos(2θ ) sin3ǫ+1 (θ) + 1
dθ.
×
cos2 (θ)
0
(3.3)
Gathering together all the results above, the deflection
angle is obtained in the form,
4G M
3π G Q 2
Gγ
−
+ 3ǫ+1
2
rm
4rm
2rm
π
3ǫ+1
2 cos(2θ ) sin
(θ ) + 1
dθ,
×
2 (θ )
cos
0
(3.4)
a relation that is valid for large r (r → ∞).
3.2 The shadow of the QMOG black hole
Now we are in the position to consider the shadow of QMOG
classes of black holes [57,58,79]. In fact, to obtain the
123
r02 f (r ph )
(3.5)
,
where for small values of Ŵ, sin(Ŵ) ≈ Ŵ, r ph denotes the
radius of the photon sphere, and Ŵ is the angle subtended
by the radius of the shadow, as seen by a typical observer
located at r0 .
By using the expression of f (r ), as introduced in the metric (2.1), one gets,
=
ϕ =
r 2ph f (r0 )
⎤
2G M
Gγ
G Q2
1
−
−
+
⎥
⎢
r 2ph ⎢
r 2ph f (r0 )
r0
r02
r03ǫ+1 ⎥
⎥
⎢
=
⎥
2
r02 f (r ph )
r02 ⎢
⎣ 1 − 2G M + G Q − Gγ ⎦
r ph
r 2ph
r 3ǫ+1
ph
2
2
r ph
Gγ
GQ
2G M
= 2
+ 2 − 3ǫ+1
1−
r0
r0
r0
r0
2G M
Gγ
G Q2
× 1+
− 2 + 3ǫ+1
r ph
r ph
r ph
2
r ph
1
1
−
= 2 1 + 2G M
r ph
r0
r0
⎡
−G Q
2
1
r 2ph
1
− 2
r0
+ Gγ
1
r 3ǫ+1
ph
−
1
r03ǫ+1
,
Eur. Phys. J. C
(2023) 83:814
Page 9 of 18
yielding in turn
sin(Ŵ) =
(r 2 − r 2ph )
r ph
G M(r0 − r ph )
2 0
+
−
G
Q
r0
r02
2r02 r 2ph
+
Gγ
2
1
r 3ǫ+1
ph
−
1
r03ǫ+1
,
(3.6)
To get the shape of shadow, we obtain first the unstable
circular orbits of the null geodesics. They are determined by
the equations,
Ve f f |r ph = Ve′ f f |r ph = 0.
(3.7)
The impact parameters are now related as,
f (r ph )
κ + L 2 − E 2 = 0,
2
r ph
r ph f ′ − 2 f (r ph )
(κ + L 2 ) = 0,
r 3ph
(3.8)
where κ is the Carter constant, with r ph obtained as a solution
of the constraint equation,
r ph f ′ (r ph ) − 2 f (r ph ) = 0.
(3.9)
Defining the impact parameters ξ and η, which are functions of the energy E and of the angular momentum L, and
of the Carter constant κ, as [81]
ξ :=
κ
L
η := 2 ,
E
E
η + ξ2 =
r 2ph f (r0 )
f (r ph )
Dθ
.
M
3.3 The Shapiro time delay
Finally, we consider the Shapiro time-delay to obtain a bound
on the coupling constant α. The general expression for the
time delay for a photon moving in the metric of the QMOG
black hole reads [59],
r
dr
.
(3.13)
t (r0 , r ) =
r0
r02 f (r )
f (r ) 1 − 2
r f (r0 )
In order to evaluate the integral, we expand the metric in
the asymptotic regime. Similar manipulations as before yield
for the integral,
r
dr
r0
2
,
= Rsh
(3.11)
(3.12)
r02 f (r )
r 2 f (r0 )
r0
3G Q 2
GM
1+
−
1+
r
(r + r0 )
2r 2
f (r ) 1 −
=
2 refers the radius of the shadow, and needless to
where Rsh
say by using the conserved quantity E and L one can plot
the shadow of black holes satisfactorily.
The shape of the shadow of the QMOG black hole is presented in Fig. 4. Since, the black hole is static, the shadow
is circular. As reported in [82], for the M87 galaxy, the
angular diameter of the shadow of the central black hole
is θ M87 = 42 ± 3 µas as, the distance of the M87 from
the Earth is D = 16.8 Mpc, and the mass of the M87
is M M87 = 6.5 ± 0.9 × 109 M⊙ . Similarly, for Sagittarius A*, the observational data are provided in the recent
EHT paper [83]. The angular diameter of the shadow is
θ Sgr.A∗ = 48.7 ± 7 µas as (EHT), the distance of the Sgr.
A* from the Earth is D = 8277 ± 33 pc, and mass of the
black hole is M Sgr.A∗ = 4.3 ± 0.013 × 106 M⊙ [83].
Now, once we have the above data about the black holes,
we can calculate the diameter of the shadow size in units of
mass, by using the following expression,
dsh =
Hence, the theoretical shadow diameter, however, can be
θ = 2R . Therefore, by using the above
obtained via dsh
sh
expression, we obtain the diameter of the shadow image of
M87 = (11±1.5)M, and for Sgr. A* d Sgr.A∗ = (9.5±
M87 dsh
sh
1.4)M. The variation of the diameter of the shadow image
with coupling parameter α and γ for M87 and for Sgr. A is
shown in Fig. 5. We observe that there is a lower bounds for
α and γ .
(3.10)
therefore, one can get
814
1
r2
1 − r02
+Gγ
r02
1
2(r 2 − r02 )
r03ǫ+1
−
1
r 3ǫ+1
+
1
r 3ǫ+1
.
(3.14)
Working in the asymptotic regime (to 2nd order in the
continued fraction expansion), approximately the for the time
delay we obtain the expression,
t (r, r0 ) = t S R (r, r0 ) + t (r, r0 )
(3.15)
where t S R = r 2 − r02 is the special relativistic contribution
of the propagation of light in the flat spacetime.
Hence, the maximum round-trip excess time delay is given
by,
t (r, r0 ) = 2 t (r2 , r0 )+t (r1 , r0 )−
r12
− r02
−
r22
− r02
.
(3.16)
Here r0 is the distance of the closest approach of the radar
wave to the center of the Sun, r1 is the distance along the line
of sight from the Earth to the point of closest approach to
the Sun, and r2 represents the distance along the path from
this point to the planet, where r1,2 ≫ r0 . In the case of
r1 = r2 = r , this becomes,
123
814
Page 10 of 18
Eur. Phys. J. C
(2023) 83:814
Fig. 4 The shadow of the QMOG black hole for M = 1, ǫ =
−2/3, G N = 1, α = 0.5 and γ = 0.045, 0.04, 0.035, 0.03, 0.025 and
0.02(left panel), and for γ = 0.04 and α = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
(Middle panel), and for α = 0.1, γ = 0.04 and M = 1, 1.5, 2
(right panel). Here the dashed line presents the shadow of the pure
Schwarzschild solution with α = γ = 0. It deserves to note that the
order of the numbers is from the outside to the inside
Fig. 5 The plots of diameter of the shadow of the black hole
in terms of α for M = 1, ǫ = −2/3, G N = 1, γ =
0.001, 0.002, 0.003, 0.004, 0.005 (left panel) and in terms of γ for α =
0.001, 0.002, 0.003, 0.005, 0.01, 0.05, 0.1 (right panel). It deserves to
note that the order of the numbers is from the bottom to the upward
⎛
t (r, r0 ) = 4G M ln ⎝
r+
r 2 − r02
r0
⎞
⎠
3π G Q 2
r − r0
−
r + r0
r0
⎛
⎞⎤
r
2
0
⎠⎦
× ⎣1 − tan−1 ⎝
π
2
r − r2
+ 4G M
⎡
0
+ 2Gγ
123
r
r0
r (r0−3ǫ+1
− 3r02 r −3ǫ−1
3
(r 2 − r02 ) 2
+ 2r −3ǫ+1 )
dr.
(3.17)
Here it is supposed that r1 = r2 ≈ 1011 m is the distance
from the Earth to the point of the closest approach to the Sun,
and r0 ≈ 108 m is the distance of closest approach of the
signal to the center of the Sun. After integration, and some
manipulations, for Q = 0, and ǫ = −2/3, one gets,
⎞
⎛
r + r 2 − r02
r − r0
⎠
⎝
+ 4G M
t (r, r0 ) = 4G M ln
r0
r + r0
√
r0 2 + rr0 + r 2 (r − r0 )
+ 2γ G
.
√
r + r0
(3.18)
Eur. Phys. J. C
(2023) 83:814
The deviation of the Shapiro time delay from the prediction of the Einstein’s general relativity is smaller than
1.2 × 10−5 s [60,61]. By using the Solar System data in SI
units, one can constrain the parameter of the QMOG black
hole model as γ < 10−17 . To obtain this upper bound, we
have assumed that the parameter α is much less than unity in
the Solar System tests. This bound was predictable, since in
the local regions the effects of the quintessence scalar field is
usually very small, as compared to the cosmological scales.
4 Electromagnetic emissivity of the thin accretion disks
around QMOG black holes
The first systematic Newtonian theory of the thin accretion
disks existing around black holes dates back to the seminal work by Shakura and Sunyaev in 1973 [62]. In their
paper, they showed that by considering the effects of the friction between different layers of the Keplerian orbits around
the black hole, the motion around inward spirals resulted in
the loss of angular momentum. They concluded that due to
such a mechanism, the kinetic energy of the disk increases.
Moreover, the release of the gravitational energy also takes
place. The warming up of the disk leads to the possibility
of the emission of a significant fraction of electromagnetic
energy.
The generalization to the general relativistic case has
been done by Novikov, Thorne, and Page and Thorne, in
1973 and 1974, [63,64], respectively. Therefore, more realistic astrophysical disk models can be constructed in this
framework. For their analysis they considered a stationary
black hole, which obeys axially or spherical symmetry, like,
for example, the Schwarzschild black hole. In the NovikovThorne model one considers that the central plane of the disk
coincides with the equatorial plane of the black hole. This
assumption significantly decreases the mathematical complexity of the analysis, and leads to the result that all the
diagonal and off-diagonal components of the metric depend
only on the radial coordinate r [64]. Recently, in [65,66]
it was shown that for a Schwarzschild background metric,
for a specific type of accretion disk that contains force free
magnetic field [67], the form of the perturbed metric resulted
in some modifications not only in diagonal components of
the background metric, but also in the appearance of some
off-diagonal components, with all of them showing a radial
coordinate dependence behaviour. One may consider these
results as a confirmation for the assumptions made by the
authors of [64].
By taking into account all the points discussed above, we
are now in a position to study the electromagnetic emissivity
properties, and the physical properties for the QMOG black
hole, and of its proposed disk. The flux F of the radiant
Page 11 of 18
814
energy over the disk can be written as [63,68–74]
,r̃
− Ṁ0
F(r̃) =
√
4π −g ( Ẽ − L̃)2
r
r I sco
( Ẽ − L̃) L̃ ,r̃ d r̃ , (4.1)
where Ṁ0 denotes the time averaged value of the accreted rest
mass, which has no dependency to the radial coordinate, and
is taken as a constant [64]. Substituting Eqs. (2.22), (2.23)
and (2.24) into (4.1) one can obtain for the radiation flux the
expression
F(r̃) = −
γ̃ r̃ 3
8(α+1) Ṁ0 α− 3r̃
4 + 8
π r̃ 4 (2α−2r̃ +γ̃ r̃ 3 )(4α 2 +4α−6(α−1)r̃+2r̃ 2 +γ̃ (−α−1)r̃ 3 )
⎤
α 2 (α + 1) − 49 α 2 + α r̃ + 23 (α + 1) r̃ 2
⎦.
× ⎣ + 81 11α 2 γ̃ + 11α γ̃ − 2 r̃ 3
3
1
3
4
5
2
6
− 2 (α + 1)γ̃ r̃ + 8 γ̃ r̃ − 8 (α + 1)γ̃ r̃
2
⎡
(4.2)
Using the above expression for F(r̃ ), the variation of the
energy flux from the disk is presented in Fig. 6. In these plots
different values for the MOG and quintessence parameters
are considered. Obviously, when both α and γ̃ tends to zero,
the configuration goes back to the usual Schwarzschild solution. As an another interesting choice, one can let γ̃ = 0, but
α not, which we call it a Reissner–Nordström like solution.
From these plots, it is immediately seen that by increasing
the values of these parameters the peak of the flux, of the is
decreasing.
If one assumes that the radiation flux emitted by the disk’s
surface is in thermodynamical equilibrium, from the Stefan–
Boltzmann law [73,74],
F(r ) = σ T 4 (r ),
(4.3)
where σ is the Stefan–Boltzmann constant, one can introduce
the temperature T of the accretion disk, see Fig. 7.
Another important physical parameter of the accretion
disks is the efficiency parameter ζ , which measures the
amount of the accreting mass transformed into radiation, in
the presence of the central compact object [72–74]. The efficiency is measured at infinity, and it is defined as the ratio
of two rates: the rate of energy of the photons emitted from
the disk surface, and the rate with which the mass-energy is
transported to the central body [72].
If all photons reach infinity, an estimate of the efficiency
is given by the specific energy of the particles in the disk
measured at the marginally stable orbit [72–74]
ζ = 1 − Ẽ(r I sco ).
(4.4)
The behaviour of the efficiency of the disk in the QMOG
geometry is provided in Table 3.
123
814
Page 12 of 18
Eur. Phys. J. C
Fig. 6 The electromagnetic energy flux versus r̃ , for different values
of the free parameters of the QMOG black hole model α and γ̃ . In the
left panel the Schwarzschild case is also presented. In the right panel a
(2023) 83:814
comparison with the Reissner–Nordstrom type black hole, with γ = 0,
is depicted. In all cases we have considered Ṁ0 = 1.5 × 1017 kg s−1
Fig. 7 The behaviour of the disk temperature against r̃ , for different values of α and γ̃ . In the two panels the results based on the Schwarzschild
and Reissner–Nordstrom type black holes are also presented. For the Stefan–Boltzmann constant we have adopted the value σ = 5.67 × 10−8 W
m−2 K−4
5 Hawking radiation of the QMOG black holes
Table 3 By introducing the amount of r I sco provided in Table 1, the
behaviour of efficiency parameter is investigated
In the present section we would like to study the mass loss
rate due to the Hawking radiation, that is Ṁ, with the aim of
determining the lifetime of the black hole [75]. It is an already
known result that if the mass of a black hole is sufficiently
large, then the temperature will be low. Accordingly, without affecting the generality, one can suppose that the mass
loss phenomenon that appears due to the Hawking radiation
mechanism originates from the emission of massless particles. In a similar approach used to determine the relation
between the flux parameter and the temperature, the mass
loss rate is given by Stefan–Boltzmann law as follows [84],
Ṁ = −
21
π2
(σg + σγ + σν )T 4 ,
60
8
(5.1)
where σi , are the thermally averaged cross sections of the
black hole for gravitons, g, photons, γ , and neutrinos, ν. In
the following we define,
1
21
,
(5.2)
ρ = σg + σγ + σν
8
σ0
123
γ
r I SC O
ζ
α=0
0
6
0.0571
0
7.1106
0.15927
α = 0.25
0.001
7.3186
0.16694
0.003
8.0783
0.18313
0.004
493.59
×
where σ0 is the geometrical optics cross section of the black
hole. The cross sections for neutrinos, photons and gravitons
can be estimated as,
σν ∼ 0.67σ0 , σγ ∼ 0.24σ0 , σg ∼ 0.03σ0 .
(5.3)
To obtain the above values we have assumed ρ ∼ 2.02,
and σ0 should be determined as follows. Since the emitted
particles move along null geodesics, from Eq. (2.18), it fol-
Eur. Phys. J. C
(2023) 83:814
Page 13 of 18
lows that their motion is governed by the equation,
L2
2
2
ṙ = E + f (r ) μ − 2 ,
r
Finally, using this definition for the temperature, the mass
loss rate reads,
(5.4)
where, as previously discussed, E and L refer to the energy,
and the angular momentum, respectively.
For the emitted particles to reach the infinity, rather than
to fall back into the black hole horizon, one has to impose
the following condition,
1
E2
μ
1
≡
≥
f
−
.
(5.5)
l2
L2
r2
L2
where by μ we have denoted the mass of the test particles.
The geometrical optical cross section of the black hole is
given by,
σ0 = πlc2 =
πrc2
f (rc )
1
.
μr 2
1 − 2c
L
(5.6)
Using Eq. (5.5), one can obtain an equation for rc as follows,
6G Mrc − 4G Q
2
+ 3Gǫγ rc−3ǫ+1
+ 3Gγ rc−3ǫ+1
− 2rc2
= 0.
(5.7)
For Q = 0, and ǫ = −2/3, we have,
1 − 1 − 6G 2 γ M
rc =
.
Gγ
(5.8)
For small γ values, we obtain
rc ∼ 3G M +
9G 3 M 2 γ
+ O(γ 2 ).
2
(5.9)
By substituting Eq. (5.9) into the Eq. (5.6) for massless
particles, i.e., for μ = 0, one gets,
σ0 ∼ 27π G 2 M 2 + 243π G 4 M 3 γ + O(γ 2 ).
(5.10)
Hence, the radius of the event horizon is obtained as,
1 − 1 − 8G 2 γ M
.
(5.11)
r+ =
2Gγ
For γ small, satisfying the condition γ ≪ 1, we obtain
r+ ∼ 2G M + 4G 3 M 2 γ + O(γ 2 ) ∼
2
Gγ 2
rc +
r + O(γ 2 ).
3
9 c
(5.12)
Considering the above expression of the event horizon, the
temperature of the black hole is given as follows,
T =
1
3Gγ
f ′ (r )
=
−
+ O(γ 2 ),
4π
8π G M
4π
814
(5.13)
where ′ denotes differentiation with respect to r . The
behaviour of T against r+ is shown in Fig. 8.
27
ρς
405 ρς γ
dM
∼
+ O(γ 2 ).
−
dt
4096 π 3 G 2 M 2
4096 π 3 M
(5.14)
where ς = π 2 /60. The behaviour of the mass loss rate for
different situations are illustrated in Fig. 9.
By integrating the above expression one obtains the lifetime of the black hole as,
t=
5120 G 4 π 3 M 4 γ
4096 G 2 π 3 M 3
+
+ O(γ 2 )
81
ςρ
9
ρς
(5.15)
Here one should notice that, G = G N (1 + α). Beside this
modification, the second term appears due to the presence
of the quintessence scalar field surrounding the black hole.
Here to realize the behaviour of the lifetime of the black hole
one can consider the plots appear in the Fig. 10.
6 Concluding remarks
In the present work we have investigated in detail some astrophysical effects related to the possible presence of exotic
forms of matter around black hole type cosmic objects. We
have adopted, as a particular black hole model, the vacuum
solution of the field equations in the MOG theory proposed,
and developed, in [37]. This theory is a scalar–vector–tensor
type theory, and its black-hole solutions has remarkable similarities with both the classic Reissner–Nordström solution of
general relativity, as well as with the Kiselev black hole solutions [18], which initially interpreted as a black hole solution
in the presence of a quintessence field. However, this interpretation is problematic [50], but the Kiselev solution, interpreted as a solution of the field equations in the presence of
an exotic fluid, still has many interesting physical and astrophysical features that could lead to a better understanding of
the properties of black holes embedded in an exotic cosmic
environment. The solution of the field equations in the MOG
gravity contains a term similar to the one appearing in the
Reissner–Nordström solution, describing the properties of a
charged black hole, of the form Q 2 /r 2 , but with Q unrelated
to the electric charge, but proportional to the mass and to the
effective gravitational constant of the theory. Still, in order to
point out to the similarities to the charged general relativistic black holes we call this class of vacuum solutions of the
MOG theory as QMOG black holes.
The first important property of black holes is the position
of their event horizon. The radius of the event horizon essentially depends on the parameter ε of the equation of state of
the exotic matter. In general, the position of the event horizon
can be obtained for the QMOG black hole as a solution of
a nonlinear algebraic equation. However, for some particu-
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814
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Eur. Phys. J. C
(2023) 83:814
Fig. 8 The behaviour of the black hole temperature against r+ , for different values of α and γ̃ . In the two panels the results based on the
Schwarzschild type black hole are also presented. Here G N = 1 is assumed. For the left panel γ = 0.25 and for the right panel α = 0.5
Fig. 9 The behaviour of the mass loss against r+ , for different values of α and γ̃ . In the two panels the results based on the Schwarzschild type
black hole are also presented. Here G N = 1 is assumed. For the left panel γ = 0.25 and for the right panel α = 0.5
lar values of the parameter of the equation of state ε, some
polynomial equations can be obtained.
The dynamical characteristics of the motion of massive
test particles around a QMOG black hole can be obtained
from the properties of the effective potential, which can be
defined through the geodesic equations of motion. The effective potential also allows the determination of another important characteristic of the black hole geometry, the location of
the innermost stable circular orbits. For the QMOG black
hole, the positions of the event horizons and of the stable
circular orbits depend on the solution parameters α and γ ,
as summarized in Tables 1 and 2, respectively. The effective
123
potential is also strongly dependent on the solution parameters. for example, as one can see from Fig. 2, the increase in
the value of γ̃ , leads to a decrease of the magnitude of the
effective potential.
Three important effects that could help to discriminate
between different black hole types are the light deflection,
the Shapiro delay and the shadow of the black hole, respectively. We have investigated in detail these effects, and we
have explicitly obtained, by using some algebraic approximations, the explicit expressions for the deflection angle, the
angular radius and the diameter of the shadow, as well as the
time delay due to the presence of the black hole. With the help
Eur. Phys. J. C
(2023) 83:814
Page 15 of 18
814
Fig. 10 The behaviour of the evaporation time against r+ , for different values of α and γ̃ . In the two panels the results corresponding to the
Schwarzschild type black hole are also presented. Here G N = 1 is assumed. For the left panel γ = 0.25, while for the right panel α = 0.5
of each of these observations, one can obtain some constraints
on the parameters of the black hole solution. As an interesting result, we found an upper bound on the quintessence
parameter, γ < 10−17 . To obtain this upper bound we have
supposed that the parameter α is much smaller than one in
the Solar System tests. The light deflection angle for a black
hole in this configuration was also obtained. Important information on the nature of the central compact object can be
found through the study of its shadow. We have studied in
detail the shadow of the QMOG black hole, and obtained the
angular radius, diameter, and shape of the lack hole. These
important information may provide the theoretical basis for
constraining the parameters of the modified gravity model,
and discriminate between the different types of black holes.
Many astrophysical objects interact gravitationally with
the cosmic environment, and grow through matter accretion.
The Universe is filled with interstellar matter, whose density
was much higher in the early stages of cosmological evolution. The presence of interstellar matter determines the formation of accretion disks, located around compact objects.
Accretion disks are highly dynamical system, and in particular they are an important source of energy, emitted mostly
in the electromagnetic spectrum. Neutrino emissivity of the
disks is also possible. The emission of the electromagnetic
radiation by the disk is mainly a consequence of the presence
and influence of the external gravitational field of the central
compact object. The gravitational field of the massive star
around which the disk is formed is determined by its nature
– black hole, neutron or quark star, naked singularity etc.
Thus, the direct observations of the electromagnetic emission spectra from astrophysical accretion disks give us the
chance of obtaining essential information on the physical and
astrophysical properties of the massive object around which
the accretion disk was formed due to the interaction with
the cosmic environment. Hence, modified gravity theories,
can be tested and constrained by using the properties of their
accretion disks. Moreover, one can obtain a large amount of
relevant astrophysical information from the observation of
the motion of matter around the compact object.
In the present study we have performed a detailed analysis
of the emissivity properties of the accretion disks that form
around the QMOG black holes. By using the explicit form of
the metric, the expression of the electromagnetic flux can be
obtained in an exact analytical form. The flux is dependent
on the parameters of the metric, and, for certain values of the
parameters, it shows a significant difference as compared to
the general relativistic Schwarzschild case. The differences
do appear at the level of the maximum values of the flux, and
in the position of the maximum. For the QMOG black holes,
there is a slight displacement of the maximum for higher r
values. A similar pattern can be observed in the temperature
distribution on the surface of the disk, with the temperature
maximum displaced towards the interior of the disk. Similar
differences do appear in the case of the efficiency parameter.
The quantum properties of the black holes are important
for both theoretical, and astrophysical point of view. Even
they are not directly detectable at the present moment, on a
long run they can influence essentially the properties of the
black holes. We have obtained the corrections to the Hawking temperature due to the extra terms present in the QMOG
black hole solutions, and we have investigated its dependence
on the model parameters. The mass loss rate and the evaporation time have also been obtained, and their dependencies on
the model parameters have been investigated. Interestingly,
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814
Page 16 of 18
an increase in the lifetime of the black hole is obtained for
the QMOG black holes. This is due to the presence of sup5120 G 4 π 3 M 4 γ
plementary terms of the form
, which do
9
ρς
appear besides the usual general relativistic terms.
In the present study we have considered some basic astrophysical properties of a particular modified gravity theory
black hole. The obtained results may open some new perspectives in the observational testing of this type of objects, and in
discriminating between different types of compact objects.
Investigating these corrections for other type of black holes,
for example, rotating black holes, can be the subject of future
work.
Acknowledgements H.S. is grateful to H. Firouzjahi for the constructive discussions they had on black holes and singularities. He also thanks
J. Moffat for the discussions on a primary version of the paper. The work
of TH is supported by a grant of the Romanian Ministry of Education
and Research, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE2020-2255 (PNCDI III).
Data Availability Statement This manuscript has no associated data
or the data will not be deposited. [Authors’ comment: There are no
external data associated with the manuscript.]
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation,
distribution and reproduction in any medium or format, as long as you
give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article
are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not
included in the article’s Creative Commons licence and your intended
use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm
ons.org/licenses/by/4.0/.
Funded by SCOAP3 . SCOAP3 supports the goals of the International
Year of Basic Sciences for Sustainable Development.
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