# Shadow of regular black hole in scalar-tensor-vector gravity theory

Subhadip<EMAIL_ADDRESS>1,2 ID and John W.
<EMAIL_ADDRESS>3,4
1Department of Physics, Jhargram Raj College, Jhargram, West Bengal-721507
2School of Physical Sciences, Indian Association for the Cultivation of
Science,
2A & 2B Raja S. C. Mullick Road, Kolkata-700032,India
3Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5,
Canada
4Department of Physics and Astronomy, University of Waterloo, Waterloo,
Ontario N2L 3G1, Canada

###### Abstract

We investigate the shadow cast by a regular black hole in scalar-tensor-vector
mOdified gravity theory. This black hole differs from a Schwarzschild-Kerr
black hole by the dimensionless parameter $\beta$. The size of the shadow
depends on this parameter. Increasing the value of the parameter $\beta$
shrinks the shadow. A critical value of the parameter $\beta$ is found to be
$\beta_{\rm crit}=0.40263$. The shadow for the horizonless dark compact object
has been analysed for the static, spherically symmetric case and compared with
M87* and Sgr A* data. Shadow observables have been determined in the context
of the regular black hole and used for obtaining the energy emission rate. The
peak of the energy emission rate shifts to lower frequency for the increasing
value of the parameter $\beta$.

##  1 Introduction

One of the most remarkable predictions of the general theory of relativity is
the occurrence of black holes. The recent observations by the Event Horizon
Telescope (EHT) collaboration[1, 2, 3, 4, 5, 6, 7] and the detection of
gravitational wave signals by the Laser-Interferometer Gravitational Wave-
Observatory (LIGO) and Virgo [8, 9, 10], corroborate the existence of these
celestial objects. Despite its success, the theory of general relativity is
not flawless. The two major drawbacks of this theory are the presence of
singularities[11, 12] in the theory and the lack of observational data
verifying the existence of the dark sector[13]. The research community is
divided into two groups regarding this issue[14, 15, 16, 17, 18, 19]. Either
dark matter exists or alternatively Einstein’s gravitational theory has to be
modified. Despite numerous attempts to find the existence of the dark sector,
in particular, the dark matter, all experimental attempts to detect dark
matter have until now failed[20, 21]. This motivates us to explore the nature
of black holes in a theory where these above mentioned ambiguities are
removed. One of the successful approaches towards this goal has been developed
by one of the authors[22]. The theory is popularly known in the literature as
the scalar-tensor-vector gravity (STVG) theory and MOdified gravity (MOG). The
solar system observations[23], cosmological observations[24], galaxy rotation
curves[25, 26, 27] and the dynamics of galaxy clusters[28, 29] have all been
satisfactorily explained by the MOG. It has also been successful in describing
structure growth, the matter power spectrum, and cosmic microwave background
(CMB) acoustical and angular power spectrum data[30, 31, 32, 33].
Observational signatures and constrains of the black holes and other compact
objects as appearing in MOG theory have been discussed in the literature[34,
35, 36]. To distinguish the MOG theory from general relativity, EHT
observational data have been used to study the shadow cast by the supermassive
MOG black holes Sgr A* and M87*[37].

As a result of lensing phenomena[38, 39, 40, 41], the black hole scatters the
higher angular momentum photons from the source, sending them to the distant
observer, while the photons with less angular momentum fall into the black
hole and create a shadow zone and a possible light ring. The black hole
shadow, which develops next to the event horizon, gives us a general notion of
the fundamental geometrical structure of horizons[42]. A review of these
developments can be found in [43]. Sagittarius A*, the supermassive black hole
at the heart of our galaxy, and M87* at the galactic centre of M87 have both
been confirmed by the EHT astronomical observations[1, 2, 3, 4, 5, 6, 7]. A
two-dimensional dark disc encircled by bright rings is the black hole’s
observable appearance. The light rings are photon orbits, while the dark area
represents the black hole shadow. The accreted matter around the black hole
has an impact on how the shadow is shaped. Since the black hole’s shadow
carries the geometry of the surrounding region in its shape and size, it is
considered a helpful tool for determining the black hole’s spin and other
deformation characteristics and parameters[44, 45, 46, 47]. This in turn can
help to distinguish and test general relativity and other alternative
theories[48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64,
65]. The black hole candidates display a significant rotation. Our main goal
will be to explore the rotating black holes in MOG/STVG theory. To evade the
problem of a singularity, we will focus our study on regular solutions in the
STVG/MOG theory of gravity. One of the crucial methods for obtaining
information about a black hole is the study of its shadow[66, 67, 68, 69, 70,
71]. Earlier attempts have been made to analyse the shadows of regular black
holes[72, 73, 74, 75, 76, 77, 78, 54]. Many of these regular black holes arise
from gravity coupled to non-linear electrodynamics. However, the electrical
charge of black holes is expected to have negligible effect on the geometry of
spacetime[79]. In STVG/MOG theory, the regular black hole solutions are
obtained from a purely gravitational theory and can be potential candidates
for astrophysical black holes.

The paper is organized as follows: In 2 a brief introduction to the STVG/MOG
gravitational action and field equations is presented. The static regular MOG
compact object is discussed in 3. In 4, we investigate the regular MOG
spherically symmetric solution and analytically derive the critical value of
the parameter $\beta$. We also derive the parameter dependence of the black
hole horizon, photon sphere and shadow. 5 is dedicated to a study of the
regular MOG rotating solution. In 6, we have determined the shape and size of
the black hole shadow for the regular MOG rotating solution along with the
observables associated with it. Finally we have calculated the energy emission
rate for the concerned black hole with the help of associated observables in
7.

Throughout the paper, we will use mostly the positive metric convention
assuming the velocity of light to be unity ($c=1$).

##  2 STVG action and field Equations

The action for MOG/ STVG theory is

$\displaystyle S=S_{\rm GR}+S_{\phi}+S_{S}+S_{M}$ (1)

where

$\displaystyle S_{\rm GR}$ $\displaystyle=\dfrac{1}{16\pi}\int
d^{4}x\sqrt{-g}\dfrac{1}{G}R$ (2) $\displaystyle S_{\phi}$
$\displaystyle=-\int
d^{4}x\sqrt{-g}\left(\dfrac{1}{4}B^{\mu\nu}B_{\mu\nu}-\dfrac{1}{2}{\mu}^{2}\phi^{\mu}\phi_{\mu}-J^{\mu}\phi_{\mu}\right)$
(3) $\displaystyle S_{S}$ $\displaystyle=\int
d^{4}x\sqrt{-g}\dfrac{1}{G^{3}}\left(\dfrac{1}{2}g^{\mu\nu}\nabla_{\mu}G\nabla_{\nu}G-V(G)-JG\right)+\int
d^{4}x\sqrt{-g}\dfrac{1}{{\mu}^{2}G}\left(\dfrac{1}{2}g^{\mu\nu}\nabla_{\mu}{\mu}\nabla_{\nu}{\mu}-V(\mu)\right)$
(4)

Here $g_{\mu\nu}$ is the spacetime metric, $g$ is the determinant of the
metric, $R$ is the Ricci scalar, $\phi^{\mu}$ is a proca-type massive vector
field such that
$B_{\mu\nu}=\partial_{\mu}\phi_{\nu}-\partial_{\nu}\phi_{\mu}$, $G(x)$ and
$\mu(x)$ are scalar fields and $V(G)$ and $V(\mu)$ are the corresponding
potentials. $S_{M}$ is the matter action. The energy-momentum tensor for the
gravitational source can be written as

$\displaystyle T_{\mu\nu}=T_{\mu\nu}^{M}+T_{\mu\nu}^{\phi}+T_{\mu\nu}^{S}$ (5)

where

$\displaystyle T_{\mu\nu}^{M}=-\dfrac{2}{\sqrt{-g}}\dfrac{\partial
S_{M}}{\delta g^{\mu\nu}}$ (6a) $\displaystyle
T_{\mu\nu}^{\phi}=-\dfrac{2}{\sqrt{-g}}\dfrac{\partial S_{\phi}}{\delta
g^{\mu\nu}}$ (6b) $\displaystyle
T_{\mu\nu}^{S}=-\dfrac{2}{\sqrt{-g}}\dfrac{\partial S_{S}}{\delta g^{\mu\nu}}$
(6c)

Here, $T_{\mu\nu}^{M}$ is the ordinary matter energy-momentum tensor,
$T_{\mu\nu}^{\phi}$ is the energy-momentum tensor for the field $\phi^{\mu}$
and the scalar contribution to the energy-momentum tensor is denoted by
$T_{\mu\nu}^{S}$. Moreover, $J^{\mu}$ and $J$ are the vector and scalar field
currents, respectively.

The Schwarzschild-MOG and Kerr-MOG black hole solutions can be found with the
following assumptions:

  * •

It is assumed that the matter energy-momentum tensor $T_{\mu\nu}^{M}$ and the
vector and scalar field currents $J^{\mu}$ and $J$ are zero.

  * •

Since the effects of the vector field $\phi_{\mu}$ mass $\mu$ becomes
prominent at kiloparsec distances from the source, the mass of the vector
field is disregarded when solving the field equations for compact objects like
black holes.

  * •

The constant $G$ depends on the parameter $\beta=\alpha/(1+\alpha)$ by
$G=G_{N}(1+\alpha)=\dfrac{G_{N}}{1-\beta}$. Here, $G_{N}$ is Newton’s
gravitational constant and we assume that $\partial_{\mu}G\approx 0$. The
range of the dimensionless parameter $\beta$ is $0\leq\beta\leq 1$.

The action in 1 assumes the following form:

$\displaystyle S=\frac{1}{16\pi
G}\int\differential^{4}x\sqrt{-g}\left(R-\dfrac{1}{4}B^{\mu\nu}B_{\mu\nu}\right)$
(7)

Varying this action with respect to $g_{\mu\nu}$, we get the following field
equations:

$\displaystyle G_{\mu\nu}=8\pi GT_{\mu\nu}^{\phi}$ (8)

Here $G_{\mu\nu}$ is the Einstein tensor $R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$.
The energy-momentum tensor associated with vector field $\phi_{\mu}$ is given
by

$\displaystyle
T_{\mu\nu}^{\phi}=\frac{1}{4\pi}\left(B_{\mu}^{~{}~{}\rho}B_{\nu\rho}-\dfrac{1}{4}g_{\mu\nu}B^{\alpha\beta}B_{\alpha\beta}\right)$
(9)

To obtain the dynamical equation for the vector field, we need to vary the
action in 7 with respect to the vector field $\phi_{\mu}$. Such a variation
leads to the following dynamical equation:

$\displaystyle\nabla_{\nu}B^{\mu\nu}=\dfrac{1}{\sqrt{-g}}\partial_{\nu}\left(\sqrt{-g}B^{\mu\nu}\right)=0$
(10)

One should note here that the gravitational charge $Q_{g}$ associated with the
MOG vector field is proportional to the mass of the gravitational source
as[80]

$\displaystyle Q_{g}=\sqrt{\alpha G_{N}}M=\sqrt{\beta(1-\beta)G_{N}}M_{\beta}$
(11)

where $M_{\beta}=(1+\alpha)M$. The gravitational charge $Q_{g}$ results in the
modified Newtonian acceleration for weak gravitational fields and slow
particle motion:

$\displaystyle a(r)=-\frac{G_{N}M}{r^{2}}[1+\alpha-\alpha\exp(-\mu r)(1+\mu
r)]$ (12)

For small scale objects and weak gravitational fields $\mu r<<1$ and the
parameter $\alpha$ cancels, reducing the acceleration to Newtonian gravity.
With parameter-post-Newtonian corrections this guarantees that MOG is
consistent with accurate solar system experiments.

##  3 Static regular MOG compact object

The gravitational action for the matter-free MOG theory using non-linear field
equations for the gravitational spin 1 vector field $\phi_{\mu}$ is given by
[81]

$\displaystyle S_{\rm MOG}=\dfrac{1}{16\pi G}\int
d^{4}x\sqrt{-g}\left[R-L(B)\right]$ (13)

where $R$ is the Ricci scalar, $L(B)$ describes the non-linear contribution of
$B_{\mu\nu}=\partial_{\mu}\phi_{\nu}-\partial_{\nu}\phi_{\mu}$ with
$B=\dfrac{1}{4}B_{\mu\nu}B^{\mu\nu}$. The associated field equations are

$\displaystyle G_{\mu\nu}=8\pi GT^{\phi}_{\mu\nu}$ (14a)
$\displaystyle\nabla_{\nu}\left(\dfrac{\partial L}{\partial
B}B^{\mu\nu}\right)=0$ (14b)
$\displaystyle\nabla_{\mu}\left({}^{\star}B^{\mu\nu}\right)=0$ (14c)

where ${}^{\star}B^{\mu\nu}=\epsilon^{\mu\nu\rho\sigma}B_{\rho\sigma}$ is the
Hodge-dual of $B^{\mu\nu}$. The energy-momentum tensor associated with the
theory is given by

$\displaystyle T_{\mu\nu}^{\phi}=\dfrac{1}{4\pi}\left[\dfrac{\partial
L}{\partial B}g^{\rho\sigma}B_{\mu\rho}B_{\nu\sigma}-g_{\mu\nu}L(B)\right]$
(15)

In this theory, the gravitational constant is enhanced by $G=G_{N}(1+\alpha)$.
The gravitational source charge associated with the vector field $\phi_{\mu}$
is given by

$\displaystyle Q_{g}=\sqrt{\alpha G_{N}}M$ (16)

where $M$ is the mass parameter of the theory. The gravi-electric field is
given by

$\displaystyle E_{\rm grav}(r)=B_{01}(r)=-B_{10}(r)$ (17)

The energy-momentum tensor components are given by

$\displaystyle T^{\phi 0}_{0}=T^{\phi 1}_{1}=-\dfrac{1}{4\pi}\left(E_{\rm
grav}^{2}\dfrac{\partial L}{\partial B}+L(B)\right)$ (18)

To describe the non-linear system in an alternative way, one can consider the
function $H$ obtained from the Legendre transformation. The function $H$ is
given by

$\displaystyle H=2B\dfrac{\partial L}{\partial B}-L(B)$ (19)

We assume

$\displaystyle P_{\mu\nu}=\dfrac{\partial L}{\partial B}B_{\mu\nu}$ (20)

and

$\displaystyle P=\dfrac{1}{4}P_{\mu\nu}P^{\mu\nu}=\left(\dfrac{\partial
L}{\partial B}\right)^{2}B$ (21)

Now, $H$ can be expressed as the function of $P$. For the regular spacetime
metric solution the form of the function $H(P)$ is given by

$\displaystyle
H(P)=P\dfrac{\left(1-3\sqrt{-2\alpha(1+\alpha)M^{2}P}\right)}{\left(1+\sqrt{-2\alpha(1+\alpha)M^{2}P}\right)^{3}}-\dfrac{3}{2\alpha(1+\alpha)M^{2}b}\left(\dfrac{\sqrt{-2\alpha(1+\alpha)M^{2}P}}{1+\sqrt{-2\alpha(1+\alpha)M^{2}P}}\right)$
(22)

where $b=\dfrac{\sqrt{\alpha}M}{2}$ and
$P=-\dfrac{\alpha}{(1+\alpha)}\dfrac{M^{2}}{2r^{4}}$ and we have set the
gravitational constant $G_{N}=1$. The associated Lagrangian L is provided by

$\displaystyle
L(P)=P\dfrac{\left(1-8\sqrt{-2\alpha(1+\alpha)M^{2}P}-6\alpha(1+\alpha)M^{2}P\right)}{\left(1+\sqrt{-2\alpha(1+\alpha)M^{2}P}\right)^{4}}-\dfrac{3\left(-2\alpha(1+\alpha)M^{2}P\right)^{5/4}\left(3-\sqrt{-2\alpha(1+\alpha)M^{2}P}\right)}{4\alpha(1+\alpha)M^{2}b\left(1+\sqrt{-2\alpha(1+\alpha)M^{2}P}\right)^{7/2}}$
(23)

##  4 Regular MOG static spherically symmetric spacetime

The MOG regular, static spherically symmetric solution can be written as[81,
82]

$\displaystyle
ds^{2}=-f(r)dt^{2}+\dfrac{1}{f(r)}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)$ (24)

with

$\displaystyle
f(r)=1-\dfrac{2(1+\alpha)Mr^{2}}{\left(r^{2}+\alpha(1+\alpha)M^{2}\right)^{3/2}}+\dfrac{\alpha(1+\alpha)M^{2}r^{2}}{\left(r^{2}+\alpha(1+\alpha)M^{2}\right)^{2}}$
(25)

Here $M$ is the mass parameter of the gravitating object. The associated
gravi-electric field is given by

$\displaystyle E_{\rm
grav}(r)=\sqrt{\alpha}Mr^{4}\left[\dfrac{r^{2}-5\alpha(1+\alpha)M^{2}}{\left\\{r^{2}+\alpha(1+\alpha)M^{2}\right\\}^{4}}+\dfrac{15}{2}\dfrac{(1+\alpha)M}{\left\\{r^{2}+\alpha(1+\alpha)M^{2}\right\\}^{7/2}}\right]$
(26)

For a convenient way of studying the theory, we introduce the alternative
parameter $\beta$ as

$\displaystyle\beta=\dfrac{\alpha}{1+\alpha}$ (27)

The ADM mass of the gravitating object is

$\displaystyle M_{\rm ADM}=(1+\alpha)M=\dfrac{M}{1-\beta}\equiv M_{\beta}$
(28)

We can express the metric in 24 in terms of the ADM mass with

$\displaystyle f(r)=1-\dfrac{2M_{\beta}r^{2}}{\left(r^{2}+\beta
M_{\beta}^{2}\right)^{3/2}}+\dfrac{\beta M_{\beta}^{2}r^{2}}{\left(r^{2}+\beta
M_{\beta}^{2}\right)^{2}}$ (29)

Here $M_{\beta}$ is the ADM mass of the spacetime and $\beta$ is the
enhancement parameter. The gravitational source charge in terms of the ADM
mass is given by

$\displaystyle Q_{g}=\sqrt{\beta(1-\beta)G_{N}}M_{\beta}$ (30)

The horizon of the spacetime depends on the zeros of the function $f(r)$ and
that can be used to determine the critical value of the parameter $\beta$. Let
us assume $\dfrac{r^{2}}{M_{\beta}^{2}}+\beta=x^{2}$, then zeros of $f(r)$ can
be determined by the equation

$\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0$ (31)

where, $a=1,b=-2,c=\beta,d=2\beta,e=-\beta^{2}$. The discriminant of the
quartic equation is

$\displaystyle\Delta=$ $\displaystyle
256a^{3}e^{3}-192a^{2}bde^{2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e-27a^{2}d^{4}$
$\displaystyle+144ab^{2}ce^{2}-6ab^{2}d^{2}e-80abc^{2}de+18abcd^{3}+16ac^{4}e$
$\displaystyle-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde-4b^{3}d^{3}-4b^{2}c^{3}e+b^{2}c^{2}d^{3}$
$\displaystyle=$
$\displaystyle-4\beta^{2}\left(27+4\beta\left[-16+\beta\left\\{20+\beta(-28+25\beta)\right\\}\right]\right)$
(32)

As the discriminant satisfies $\Delta\leq 0$, there will be two distinct real
roots of the quartic equation. The critical value at which there will be only
one horizon is given by the solution of the equation

$\displaystyle 10800\beta^{3}-12096\beta^{2}+20304\beta-6912=0$ (33)

The solution of this equation is $\beta=0.402186=\beta_{\rm crit}$. For
$\beta<\beta_{\rm crit}$ there will be a black hole with two horizons[80] and
there will be no horizon for $\beta>\beta_{\rm crit}$. This result is
displayed in 1.

Figure 1: The zeros of the function $f(r)$ have been shown to confirm that
either two horizons or no horizon are possible for the regular MOG spherically
symmetric spacetime. For $\beta=0$ the spacetime becomes Schwarzschild and has
a singularity at $r=0$. One horizon solution is possible for the critical
value of the parameter $\beta_{\rm crit}\approx 0.40263$. It is easy to check
that there is no singularity for non-zero values of parameter $\beta$.

###  4.1 Motion of photons in MOG spherically symmetric spacetime

The Lagrangian for the photon motion is given by

$\displaystyle\mathcal{L}=\dfrac{1}{2}\left[-f(r)\dot{t}^{2}+\dfrac{1}{f(r)}\dot{r}^{2}+r^{2}\dot{\theta}^{2}+r^{2}\sin^{2}\theta\dot{\phi}^{2}\right]$
(34)

For a spherically symmetric spacetime, we can always choose without loss of
generality $\theta=\pi/2$ and $\dot{\theta}=0$. The equations for $\dot{t}$
and $\dot{\phi}$ can be deduced using the symmetries of the MOG regular,
static spherically symmetric spacetime. The associated equations are

$\displaystyle f(r)\,\dot{t}$ $\displaystyle=E$ (35) $\displaystyle
r^{2}\dot{\phi}$ $\displaystyle=L$ (36)

where $E$ and $L$ are, respectively, the energy and angular momentum of the
photon. The radial equation can be written as

$\displaystyle\dot{r}^{2}+V(r)=E^{2}$ (37)

where, $V(r)=L^{2}\dfrac{f(r)}{r^{2}}$. The structure of the potential helps
to determine the presence of stable or/and unstable circular orbits. From 2,
we conclude that for the whole parameter space, there exists a stable circular
orbit and an unstable circular orbit. This special situation arises for
$\beta_{\rm crit}<\beta\lesssim 0.5$. However, with close inspection and from
3a for $\beta<\beta_{\rm crit}$, we only have unstable circular orbits.

(a) Minima of the potential have been shown here. This corresponds to stable
circular orbits. However, the relevance of the stable circular orbit is valid
only for $\beta_{\rm crit}<\beta\lesssim 0.5$ (b) Maxima of the potential have
been shown here. This corresponds to the existence of the unstable circular
orbits and is valid for the range $0<\beta\lesssim 0.5$

Figure 2: The variation of the potential for the photon particle has been
shown in these plots (a) minima of the potential has been shown and (b) maxima
of the potential has been shown . For $\beta<\beta_{\rm crit}$, there exists a
stable circular orbit along with an unstable circular orbit. However, for the
MOG static spherically symmetric solution only unstable circular orbits exist
for $\beta<\beta_{\rm crit}$. The existence of both stable and unstable
circular orbits is possible for $\beta_{\rm crit}<\beta\lesssim 0.5$.

As assumed earlier, $r^{2}/M_{\beta}^{2}+\beta=x^{2}$ can be used to find the
position of the photon sphere. The position of the photon sphere can be found
by the real greatest solution of the following equation

$\displaystyle x^{4}-3x^{3}+2\beta x^{2}+5\beta x-3\beta^{2}=0$ (38)

For $\beta<\beta_{\rm crit}$, there will be a photon sphere and a gradual
increase in the enhancement parameter $\beta$ causes the decrease of both the
horizon and shadow radius. For the dark compact object with $\beta_{\rm
crit}<\beta\lesssim 0.5$, although there is no horizon, we still have a photon
sphere. In spherically symmetric spacetimes, the shadow of the black hole is
circular in structure. For the regular MOG black holes, the shadow has been
shown with the variation in the parameter $\beta$ in 3b. In the figures, $A$
and $B$ are the celestial coordinates.

(a) Variation of various radii with variation of the parameter $\beta$.

(b) Variation of the shadow radius with variation of parameter $\beta$.

Figure 3: (a)The variation of the horizon radius, photon sphere and the radius
of the shadow are depicted as a function of the parameter $\beta$. It is
interesting to note that in the range $0.4\lesssim\beta\lesssim 0.5$ there is
no event horizon. However, this does not hinder us in defining the photon
sphere and shadow for the compact object. Also, for the range of parameter
space, we have both stable and unstable circular orbits. (b) The circular
shadow structures have been depicted.

###  4.2 Parameter estimation using M87* and Sgr A* data

Although astrophysical black holes are rotating in nature, for a first-hand
estimation of black hole parameters, one can use the shadow of the spherically
symmetric black holes. As the shadow for spherically symmetric black holes
does not depend on the inclination angle to obtain the initial estimation of
the parameter $\beta$, we can work with the shadow of the regular MOG black
hole solution. Apart from this, the observed shadows for M87* and Sgr A* are
more or less circular in nature. This motivates us to find the observational
signatures of the regular MOG black hole or compact objects in M87* and Sgr A*
using EHT data.

We have calculated how the radius of the photon sphere and the shadow affected
the parameter $\beta$ in the previous section. We can determine the values of
$\beta$ based on the size of the angular diameter, which is defined as

$\displaystyle\tan\alpha\approx\alpha=\dfrac{r_{sh}}{D}$ (39)

Where $r_{sh}$ is the radius of the black hole shadow, $D$ is the distance of
the centre of the black hole from the observer, $2\alpha$ is the angular
diameter. As the distance between the black hole and the observer is much
greater than the radius of the black hole shadow, the small angle
approximation is justified.

The mass and distance of M87* needs to be independently measured. The mass of
M87* has been reported to be $M=3.5_{-0.3}^{+0.9}\times 10^{9}M_{\odot}$ from
model gas dynamics mass measurements[83]. However, based on model stellar
dynamics mass measurements, the mass is reported to be
$M=6.2_{-0.5}^{+1.1}\times 10^{9}M_{\odot}$[84, 85]. The distance of the
gravitating source is reported to be $D=(16.8\pm 0.8)\rm Mpc$. Having the
information of mass and distance of the black hole one can define the angular
gravitational radius $\theta={GM}/{c^{2}D}$. The angular gravitational radius
$\theta_{dyn}$ as measured by stellar-dynamics process and the angular
gravitational radius $\theta_{g}$ as reported by EHT are more or less
consitent[86]. Theoretical bounds on the shadow diameter has been discussed by
Kocherlakota et. al. [87] Based on M87* shadow size they have implied
restrictions on the physical charges of several different spinning or non-
rotating black holes. We use the stellar dynamics mass measurement to
theoretically deduce the shadow radius of the black hole. The supermassive
black hole M87* in the core of the galaxy M87 has an angular diameter of
$(42\pm 3)\mu as$, according to the Event Horizon Telescope (EHT)
collaboration[2]. In the plots shown in 4, the central value $42~{}\mu as$
second has been shown with a grey line and the error bar has been shown with
the dashed grey line. There is an error in the mass estimation of M87* around
the central value. The variation of angular diameter, taking the central value
of mass, has been shown with a blue line. Taking the errors, we can also plot
the angular diameter. This has been shown with dot-dashed blue lines. The
central value of mass of M87* is $6.2\times 10^{9}M_{\odot}$.

Considering the error bars both for angular diameter measurement and mass
measurement, there is a possibility that M87* could be a regular MOG black
hole. For the angular diameter $(42\pm 3)\mu as$ the value of the parameter
$\beta$ can be as high as approximately $\beta=0.3$. This has been shown with
a vertical orange line in 4a. So, in this case we can say that the M87* is a
regular MOG black hole. With the angular diameter $(42\pm 3)\mu as$, the
possibility that M87* is a horizonless compact object can be rejected.
However, if one considers a $10\%$ offset value of the angular diameter, the
parameter $\beta$ can be as high as approximately $\beta=0.45$, and in this
case M87* can be a horizonless compact object. In 4b, the theoretical range of
$\beta$ has been shown by a grey shaded region and the region enclosed by the
two orange lines in grey shaded represents the observationally allowed range
of the parameter $\beta$ for M87*.

(a)  Angular diameter versus $\beta$ with observed values $(42\pm 3)\mu as$
marked in grey

(b)  Angular diameter versus $\beta$ with observed values $(37.8\pm 2.7)\mu
as$ marked in grey

Figure 4: (a)The variation of the ring diameter has been shown with the error
bar. (b) The variation of the shadow diameter has been shown.

According to the EHT collaboration, the angular diameter of the Sgr A* shadow
is $(48.7\pm 7)\mu as$[88, 89, 90, 91, 92, 93]. The angular diameter of the
Sgr A* shadow depends on the determined mass and distance of Sgr A*. Several
groups have reported the mass and distance of Sgr A*. From the Keck team,
keeping the redshift parameter free the mass and distance of Sgr A* have been
reported to $(3.975\pm 0.058\pm 0.026)\times 10^{6}M_{\odot}$ and $(7959\pm
59\pm 32)\rm pc$, respectively [94]. The same group has also reported the mass
and distance assuming the redshift parameter to be unity and these are
$(3.951\pm 0.047)\times 10^{6}M_{\odot}$ and $(7935\pm 50)\rm pc$,
respectively[94]. The mass and distance, according to the Gravity
collaboration are, respectively, $(4.261\pm 0.012)\times 10^{6}M_{\odot}$ and
$(8246.7\pm 9.3)\rm pc$[95, 96]. The Gravity Collaboration further limited the
BH mass $(4.297\pm 0.012\pm 0.040)\times 10^{6}M_{\odot}$ and the distance
$(8277\pm 9\pm 33)\rm pc$ by accounting for optical aberrations. In 5, we have
plotted the angular diameter as a function of the parameter $\beta$ with mass
and distance as given by the above teams. From the plot, using the Keck team
data, one can constrain the parameter to be $0<\beta\lesssim 0.4$. With the
Keck team data, it is almost impossible to say that Sgr A* is a horizonless
compact object. However, using the Gravity collaboration data, we can say that
there is a possibility that Sgr A* is a horizonless compact object, because
with the Gravity collaboration data the parameter range is $0<\beta\lesssim
0.46$.

Figure 5: Theoretical angular diameter for Sgr A* has been shown.

##  5 Regular MOG rotating compact object

The regular rotating MOG solution can be obtained with the help of the
modified Newman-Janis algorithm. The associated line element of the spacetime
in Boyer-Lindquist coordinates is given by[81]

$\displaystyle ds^{2}=$
$\displaystyle-f(r,\theta)dt^{2}-2a\sin^{2}\theta\left\\{1-f(r,\theta)\right\\}d\phi
dt$ $\displaystyle+\dfrac{\Sigma}{\Delta}dr^{2}+\Sigma
d\theta^{2}+\sin^{2}\theta\left[\Sigma-a^{2}\left\\{f(r,\theta)-2\right\\}\sin^{2}\theta\right]d\phi^{2}$
(40)

where

$\displaystyle f(r,\theta)$
$\displaystyle=1-\dfrac{2M_{\beta}r\sqrt{\Sigma}}{\left[\Sigma+\beta
M_{\beta}^{2}\right]^{3/2}}+\dfrac{\beta
M_{\beta}^{2}\Sigma}{\left[\Sigma+\beta M_{\beta}^{2}\right]^{2}}$ (41a)
$\displaystyle\Delta$ $\displaystyle=\Sigma f(r,\theta)+a^{2}\sin^{2}\theta$
(41b) $\displaystyle\Sigma$ $\displaystyle=r^{2}+a^{2}\cos^{2}\theta$ (41c)

Here, $M_{\beta}$ is the ADM mass of the spacetime, $\beta$ is the enhancement
parameter and $a$ is the spin parameter. A certain portion of the full
parameter space of $\beta-a$ is available for the existence of the regular
rotating MOG black hole. The parameter space has been shown in 6. From the
figure, it is noticeable that the highly spinning regular MOG black hole has a
relatively low value of the parameter $\beta$.

Figure 6: Parameter space of $(\beta-a)$ plane for the regular rotating MOG
solution is displayed here. The reddish region represents the black hole
solution and the boundary denotes the occurrence of an extremal black hole.

The location and the structure of the static limit surface are obtained by
setting the prefactor of $dt^{2}$ to zero. The SLS can be determined by
solving the following equation

$\displaystyle\left(\Sigma+\beta
M_{\beta}^{2}\right)^{2}-2M_{\beta}r\sqrt{\Sigma(\Sigma+\beta
M_{\beta}^{2})}+\beta M_{\beta}^{2}\Sigma=0$ (42)

For $\beta=0$, we have the usual Kerr scenario. The variation and existence of
the static limit surface for rotating regular MOG black holes has been
displayed in 7.

(a)  Variation of $g_{tt}$ with respect to $r$, when $\theta=0$ and
$\beta=0.1$ (b) Variation of $g_{tt}$ with respect to $r$, when $\theta=0$ and
$\beta=0.3$ (c) Variation of $g_{tt}$ with respect to $r$, when $\theta=\pi/4$
and $\beta=0.1$ (d) Variation of $g_{tt}$ with respect to $r$, when
$\theta=\pi/4$ and $\beta=0.3$

Figure 7: The nature of the static limit surface (SLS) has been depicted as a
function of coordinate $r$ for various values of spin parameter $a$. The
variation of the SLS with respect to the enhancement parameter $\beta$ can be
seen along the row of the figure matrix [i.e (a)-(b) and (c)-(d)]

The location of the horizon is determined by setting the $g^{rr}$ to be zero,
which in turn gives

$\displaystyle\Delta=\Sigma f(r,\theta)+a^{2}\sin^{2}\theta=0$ (43)

The existence of the horizon in the regular rotating MOG solution has been
depicted in 8.

(a) Variation of $g^{rr}$ with respect to $r$ when $\beta=0.1$ (b) Variation
of $g^{rr}$ with respect to $r$, when $\beta=0.3$

Figure 8: A set of parameter values are allowed for black hole solutions. In
these plots, the set of parameters has been shown for the extremal black hole.
The horizonless compact object can result for an increase in spin keeping the
enhancement parameter $\beta$ constant. (a) The enhancement parameter $\beta$
is $0.1$. Whereas in (b) the enhancement parameter is $0.3$.

.

##  6 Analysis of the black hole shadow

To investigate the black hole shadow, we must determine the photon geodesic
equations for the metric given in 40. When using the Hamilton-Jacobi
formulation for the rotating MOG regular solution, it is particularly
challenging to separate the equations, since the function $f(r,\theta)$ has a
highly complex structure. Consequently, to overcome this issue, we consider an
approximation for $\theta$, such that $\theta\approx\pi/2+\epsilon$.[97] Note
that although we are focusing on photon orbits that are close to the equator,
unstable photon circular orbits are not only limited to this region. This fact
does not invalidate the calculations that follow, because the major goal of
this work is to calculate the shadow of a black hole cast by an observer at
infinity, which can be done using the approximations indicated above. The
trigonometric functions here have the following form: $\sin\theta\approx 1$
and $\cos\theta\approx-\epsilon$. With these approximations the function
$f(r,\theta)$ becomes $f(r)$, which is given by

$\displaystyle f(r)=1-\dfrac{2M_{\beta}r^{2}}{\left(r^{2}+\beta
M_{\beta}^{2}\right)^{3/2}}+\dfrac{\beta M_{\beta}^{2}r^{2}}{\left(r^{2}+\beta
M_{\beta}^{2}\right)^{2}}$ (44)

###  6.1 Null geodesics

For a general stationary, axisymmetric metric the Lagrangian $\mathcal{L}$ can
be written as

$\displaystyle
g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=g_{tt}\dot{t}^{2}+2g_{t\phi}\dot{t}\dot{\phi}+g_{\phi\phi}\dot{\phi}^{2}+g_{rr}\dot{r}^{2}+g_{\theta\theta}\dot{\theta}^{2}=2\mathcal{L}$
(45)

For massive particles and massless particles, the Lagrangian is equal to unity
and zero, respectively. The associated Hamiltonian is provided by

$\displaystyle\mathcal{H}=p_{\mu}\dot{x}^{\mu}-\mathcal{L}=\dfrac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu}=\dfrac{k}{2}$
(46)

with $k$, which in this instance is zero and represents the test particle’s
rest mass. By utilising the Hamilton-Jacobi technique, we may connect the
Hamiltonian to the action S by

$\displaystyle\mathcal{H}(x^{\mu},p^{\mu})+\dfrac{\partial
S}{\partial\lambda}=0\hskip 28.45274pt{\rm with}\hskip
28.45274ptp_{\mu}=\dfrac{\partial S}{\partial x^{\mu}}$ (47)

The metric described in 45 is independent of $t$ and $\phi$, whereby the
(specific) energy E and the (specific) angular momentum L are conserved
quantities. These are supplied by

$\displaystyle E=g_{tt}\dot{t}+g_{t\phi}\dot{\phi}$ (48a) $\displaystyle
L=g_{t\phi}\dot{t}+g_{\phi\phi}\dot{\phi}$ (48b)

From the aforementioned context, the action may be expressed as

$\displaystyle S=-Et+L\phi+S(r,\epsilon)$ (49)

Now, for the metric given in 40, 49 is separable such that
$S(r,\theta)=S_{r}(r)+S_{\epsilon}(\epsilon)$. Substituting 49 in equation 46
we get

$\displaystyle g^{rr}\left(\dfrac{\partial S^{r}}{\partial
r}\right)^{2}+g^{\theta\theta}\left(\dfrac{\partial
S^{\theta}}{\partial\theta}\right)^{2}+g^{tt}(-E)^{2}+2g^{t\phi}(-E)L+g^{\phi\phi}L^{2}=0$
(50)

The metric in 40 causes the equation above to take the form:

$\displaystyle\Delta\left(\dfrac{dS^{r}}{dr}\right)^{2}+\left(\dfrac{dS^{\epsilon}}{d\epsilon}\right)^{2}-\left\\{\dfrac{1}{\Delta}\left[r^{2}+a^{2}\right]^{2}-a^{2}\right\\}E^{2}+\dfrac{2ar^{2}}{\Delta}\left\\{1-f(r)\right\\}EL+\dfrac{r^{2}f(r)}{\Delta}L^{2}=0$
(51)

It’s interesting to note that the $r$ and $\theta$ components of the preceding
equation may be split up so that

$\displaystyle\Delta\left(\dfrac{dS^{r}}{dr}\right)^{2}-\dfrac{1}{\Delta}\left[r^{2}+a^{2}\right]^{2}E^{2}+\dfrac{2ar^{2}}{\Delta}\left\\{1-f(r)\right\\}EL+\dfrac{r^{2}f(r)}{\Delta}L^{2}+a^{2}E^{2}=-\left(\dfrac{dS^{\epsilon}}{d\epsilon}\right)^{2}=-C$
(52)

The Carter constant is represented by C. The left-hand side of 52 is only a
function of $r$, whereas the right-hand side is a function of $\theta$ alone.
The radial component of 52 may be expressed as

$\displaystyle\left[\dfrac{dS^{r}}{dr}\right]^{2}=\dfrac{R(r)}{\Delta^{2}}$
(53)

where

$\displaystyle
R(r)=-\Delta\left[C+(L-aE)^{2}\right]+\left\\{\left[r^{2}+a^{2}\right]E-aL\right\\}^{2}$
(54)

The angular part can be written as

$\displaystyle\left(\dfrac{dS^{\epsilon}}{d\epsilon}\right)^{2}=C$ (55)

Consequently, the action adopts the form

$\displaystyle
S=-Et+L\phi+\int\dfrac{\sqrt{R(r)}}{\Delta}dr+\int\sqrt{C}d\epsilon$ (56)

The equation of motion for $r$ and $\epsilon$ is given by

$\displaystyle\dot{r}=\pm\dfrac{\sqrt{R}}{r^{2}}$ (57)
$\displaystyle\dot{\epsilon}=\pm\dfrac{\sqrt{C}}{r^{2}}$ (58)

For determining the unstable circular orbits, one needs to introduce
$\chi=\dfrac{C}{E^{2}}$ and $\eta=\dfrac{L}{E}$. The unstable circular orbit
can be obtained by setting $R(r)=0=\dfrac{dR(r)}{dr}$. So, using 54 with
aforementioned conditions one obtains

$\displaystyle\left[r^{2}f(r)+a^{2}\right]\left(\chi+\eta^{2}+a^{2}-2\eta
a\right)=\left[r^{2}+a^{2}-a\eta\right]^{2}$ (59)
$\displaystyle{\chi+\eta^{2}+a^{2}-2\eta
a}=\dfrac{4}{\left[2f(r)+rf^{\prime}(r)\right]}\left[r^{2}+a^{2}-a\eta\right]$
(60)

These two equations can be solved to get two one-parameter classes of
solutions parametrized in terms of $r$, which is the radius of unstable
circular orbits:

  1. (i)

$\displaystyle\chi$ $\displaystyle=-\dfrac{r^{4}}{a^{2}}$ (61)
$\displaystyle\eta$ $\displaystyle=\frac{a^{2}+r^{2}}{a}$ (62)

  2. (ii)

$\displaystyle\chi$
$\displaystyle=\frac{r^{3}\left[8a^{2}f^{\prime}(r)-r\left\\{rf^{\prime}(r)-2f(r)\right\\}^{2}\right]}{a^{2}\left\\{rf^{\prime}(r)+2f(r)\right\\}^{2}}$
(63) $\displaystyle\eta$
$\displaystyle=\dfrac{1}{a}\left[r^{2}+a^{2}-\dfrac{4(r^{2}f(r)+a^{2})}{2f(r)+rf^{\prime}(r)}\right]$
(64)

The solution of the first kind is not a physical solution, but the second
solution helps to determine the contour of the shadow in the $(\eta,\chi)$
plane. Further, this solution satisfies the following condition for the
critical curve:

$\displaystyle
a^{2}-\chi-\eta^{2}=\frac{8\left(a^{2}+r^{2}f(r)\right)}{rf^{\prime}(r)+2f(r)}-\frac{16\left(a^{2}+r^{2}f(r)\right)}{\left(rf^{\prime}(r)+2f(r)\right)^{2}}-2r^{2}$
(65)

When we consider the non-rotating case i.e the regular MOG static spherically
symmetric solution, we have

$\displaystyle\chi+\eta^{2}=\dfrac{2r_{ph}^{2}\left[4f(r_{ph})^{2}-8f(r_{ph})-r_{ph}^{2}f^{\prime}(r_{ph})^{2}\right]}{\left\\{r_{ph}f^{\prime}(r_{ph})+2f(r_{ph})\right\\}^{2}}$
(66)

Here, $r_{ph}$ is the radius of the photon sphere. The above equation helps to
find the shadow of the regular MOG static, spherically symmetric solution.

###  6.2 Celestial coordinates and shadow structure

We now want to find out how the rotating MOG regular black hole shadow appears
to be shaped. For a clearer depiction, we locate the shadow using the
celestial coordinates $A_{i}$ and $B_{i}$. These coordinates are introduced as

$\displaystyle A_{i}$
$\displaystyle=\displaystyle\lim_{r_{0}\to\infty}\left(-r_{0}^{2}\sin\theta_{0}\dfrac{d\phi}{dr}\right)$
(67) $\displaystyle B_{i}$
$\displaystyle=\displaystyle\lim_{r_{0}\to\infty}\left(r_{0}^{2}\dfrac{d\epsilon}{dr}\right)$
(68)

where $r_{0}$ is the distance between the black hole and the distant observer
and $\theta_{0}$ is the inclination angle i.e the angle between the line of
sight and the rotation axis of the black hole. From further calculations and
considering the limit, one can arrive at the following:

$\displaystyle A_{i}=-\eta$ (69) $\displaystyle B_{i}=\sqrt{\chi}$ (70)

Figure 9: Schematic diagram of lensing and formation of shadow

The photons are now parametrized by the conserved quantities $(\eta,\chi)$.
All the light rays, coming from the source placed behind the black hole, will
not be able to reach the observer as the black hole will hinder a portion of
light rays due to gravitational lensing as shown in 9. The dark patch that
appears to the observer is known as the black hole shadow and the boundary of
this shadow can be determined allowing the parameters $(\eta,\chi)$ all
possible values. The coordinates $A_{i}$ and $B_{i}$ are known as celestial
coordinates. For a spherically symmetric scenario, these coordinates do not
depend on the inclination angle and the shadow appears to be a perfect circle.
However, the inclusion of the rotational effect of compact objects and the
inclination angle make the shadow dented and not a perfect circle. The shape
and size of the shadow can be analysed to determine the parameters including
the spin parameter of the black hole. The variation of the shape and size of
the black hole shadow has been depicted in 10.

(a)  (b)  (c)  (d)

Figure 10: Shadow of the regular MOG rotating black hole is situated at the
origin of the coordinate system. The inclination angle is $\theta_{0}=\pi/2$.
Each image represents the shadow for a fixed value of the spin parameter a.
One can note that just like in the Kerr scenario, here also the shadow gets
dented with an increase in spin parameter $a$. An increase in the parameter
$\beta$ causes a shrinking of the shadow for a fixed value of the spin
parameter.

###  6.3 Observables

For further analysis of the shape of the critical curve, we are going to
define two new observables as prescribed by Hioki and Maeda [98]. To define
these observables, we need to characterize a few points of the critical curve
while fitting it with a circular outline. Consider a circle in 11 that passes
through the three extreme points of the shadow curve. The points are:

  * •

extreme right of the shadow i.e $U(A_{r},0)$, at which the shadow intersects
the $A-\rm axis$.

  * •

top-most point of the shadow i.e $V(A_{t},B_{t})$

  * •

bottom-most point of the shadow i.e $W(A_{b},B_{b})$

Figure 11: Characteristic points have been shown in the schematic diagram of
the black hole shadow. A solid blue curve is the outline of the black hole
shadow. The dot-dashed green curve is associated with the fitting circle. The
associated circle passes through the three points of the shadow outline. The
top-most and bottom-most points of the shadow are, respectively, $V$ and $W$.
The left-most and right-most points of the shadow outline are $P$ and $U$. A
separation distance of points $P$ and $Q$ measures the distortion parameter
$\delta_{s}$.

As the shape of the shadow is not circular, the extreme left point of the
shadow does not coincide with the extreme left point of the associated circle.
This characterizes the distortion of the shape of the shadow from a circular
shape. The extreme left point of the shadow is $P(A_{l},0)$ and the extreme
point of the associated circle is $Q(A_{L},0)$. Now we define the two
observables associated with the shadow curve, which are

  1. (i)

The characteristic radius $R_{s}$, which can be defined as

$\displaystyle R_{s}=\dfrac{(A_{t}-A_{r})^{2}+B_{t}^{2}}{2|A_{t}-A_{r}|}$ (71)

  2. (ii)

Distortion parameter $\delta_{s}$, which is defined as

$\displaystyle\delta_{s}=\dfrac{D_{s}}{R_{s}}=\dfrac{|A_{L}-A_{l}|}{R_{S}}$
(72)

For a non-rotating scenario, the distortion parameter becomes zero as the
shape of the shadow for such a case is always zero. Similarly, the
characteristic radius reduces to the radius of the circle for the non-rotating
scenario. So these two observables measure the deviation from the circularity
of the shape of the shadow. It has been illustrated in 12 how the parameter
$\beta$ affects the characteristic radius $R_{s}$ and distortion parameter
$\delta_{s}$ of the spinning regular MOG black hole. The change in the
observables has been shown for two fixed values of the spin parameters $a$.

(a) Variation of $R_{s}$ with $\beta$ (b) Variation of $\delta_{s}$ with
$\beta$

Figure 12: The variation of characteristic radius $R_{s}$ and distortion
parameter $\delta_{s}$ of the regular rotating MOG black hole as a function of
parameter $\beta$ has been shown. The variation has been shown for a fixed
value of spin parameter $a$. From the plots, one notices how the observables
get changed with the spin parameter.

##  7 Energy emission rate for the rotating regular MOG black hole

For the regular rotating MOG solution, the observers see the large energy
absorption cross-section is caused by the shadows of black holes. At high
energies, the black hole absorption cross sections exhibit a little modulation
close to a limiting constant value. We may use the absorption cross-section
limiting constant value for a nearly spherically symmetric black hole as a
decent approximation, which is given by

$\displaystyle\sigma_{lim}\approx\pi R_{s}^{2}$ (73)

The energy emission rate of the concerned black hole is given as [99]:

$\displaystyle\dfrac{d^{2}E(\omega)}{d\omega
dt}=\dfrac{2\pi^{3}R_{s}^{2}\omega^{3}}{e^{\omega/T}-1}$ (74)

where $\omega$ is the frequency of the photon and $T$ is the Hawking
temperature, which can be defined as [99]:

$\displaystyle T=\displaystyle\lim_{\theta=0,r\to
r_{+}}\dfrac{\partial_{r}\sqrt{-g_{tt}}}{2\pi\sqrt{g_{rr}}}$ (75)

Here, $r_{+}$ is the outer event horizon of the regular rotating MOG black
hole. In 13, we have plotted the energy emission rate of the black hole with
the frequency of the photon $\omega$ for different values of the parameter
$\beta$. One notices from the figure that the peak of the energy emission rate
shifts towards a lower frequency as the parameter $\beta$ increases.

(a) Energy emission rate for the value of spin parameter $a/M_{\beta}=0.20$
(b) Energy emission rate for the value of the spin parameter
$a/M_{\beta}=0.80$

Figure 13: Energy emission rate as a function of frequency has been shown. For
a fixed value of the spin parameter, the increase in the parameter $\beta$
causes a shift of the peak of the spectrum to a lower frequency.

##  8 Conclusions

In this paper, we have explored the regular black hole solution in STVG/MOG
theory. In early papers on this theory, the parameter space is from zero to
infinity. We have compactified the range of the parameter space with a
modification of the form of the parameter $\beta$. At first, we have focused
on the regular solution of STVG/ MOG theory in the static, spherically
symmetric scenario and also determined analytically the critical value of the
parameter $\beta$. For the dimensionless parameter $\beta\lesssim 0.4$, we
have a black hole solution with two horizons in the spherically symmetric
case. For $0.4\lesssim\beta\lesssim 0.5$, there is no black hole solution as
there exists no horizon. For the critical value of the parameter $\beta\cong
0.40263$, a single horizon black hole solution can be obtained. We have also
studied the null geodesics in this spacetime as it is a prerequisite to
analyse the shadow of the black hole. For $\beta<\beta_{\rm crit}$, only
unstable circular orbits exist. However, for $\beta_{\rm crit}<\beta\lesssim
0.5$, there exists a stable circular orbit. It is also noticeable that the
radius of the photon sphere, the radii of the shadow and the event horizon
decrease as the parameter $\beta$ increases. Thus, the circular shadow shrinks
as the parameter $\beta$ is increased. Furthermore, as the shadows of M87* and
Sgr A* are more or less circular in shape, we have tried to compare the
theoretical outcomes with the observational data. For this purpose, we have
used independent mass measurements to calculate the theoretical angular
diameter. The regular MOG black holes and the possibility of horizonless
compact objects are compatible with the EHT data and mass measurements.

We have considered the regular rotating MOG black hole by studying the
behaviours of the horizon and static limit surface for a change in the
parameter $\beta$. Just like the spherically symmetric case, here also a
critical value of parameter $\beta$ exists for a fixed value of the spin
parameter $a$. We have determined the parameter space for which a rotating
regular black hole exists. As a chief goal of this paper, a special emphasis
has been placed on the black hole shadow. However, we have considered only
equatorial approximations. How the shape and size of the associated shadow
transforms have been determined. As the spin parameter, $a$, increases the
shape gets more deformed from a circular shape. The increasing value of
parameter $\beta$ causes the size of the shadow to become smaller. To analyse
the shadow, the required observables have been defined and plotted. One of the
observables has been used to evaluate the energy emission rate for the
rotating regular MOG black hole. From this, we have concluded that the peak of
the energy emission rate shifts to a lower frequency for a relatively large
value of the parameter $\beta$.

It is hard to decouple the differential equations in terms of $r$ and $\theta$
without an equatorial approximation. We have reported the analysis of shadows
using numerical techniques, using the observables as introduced by Hioki and
Maeda. However, one can introduce new observables or can use other existing
observables to study the shadows.

In this work, we have demonstrated that classical regular black holes and
regular horizonless dark compact objects, generally considered to be distinct
families of astrophysical objects, are a family of connected astrophysical
objects continuously deformed into one another, depending on the range and
value of the parameter $\beta$. Our work illustrates that different strong
gravity geometries describe alternative states of black holes and compact
astrophysical objects in their lifetime. It is expected that at the small
scale reached at the central value of the compact object when $r\rightarrow
0$, quantum gravity will take over[100]. The regular and horizonless compact
objects derived from the MOG field equations in this work are classical in
nature. The stability of photon orbits around the black hole and dark compact
object shadows will produce viability issues for the existence of these
astrophysical objects. For the MOG-Schwarzschild and MOG-Kerr solutions with
two horizons, the inner Cauchy horizon can lead to instability problems.

In future work, the gravitational collapse of stars will be investigated,
assuming a form of matter and stress-energy, by solving the time dependent MOG
field equations. Moreover, the merging of the regular and horizonless dark
compact objects, producing gravitational waves and the subsequent ringdown
phase, will be investigated. Singularity-resolving physics in photon rings can
further be studied in context of MOG theory[101].

Acknowledgements
Research at the Perimeter Institute is supported by the Government of Canada
through the Department of Innovation, Science and Economic Development Canada
and by the Province of Ontario through the Ministry of Research, Innovation
and Science.

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