# The Mukai conjecture for Fano quiver moduli

Markus Reineke  Markus Reineke
Ruhr-University Bochum
Faculty of Mathematics
Universitätsstrasse 150
44780 Bochum (Germany<EMAIL_ADDRESS>
###### Abstract.

We verify the Mukai conjecture for Fano quiver moduli spaces associated to
dimension vectors in the interior of the fundamental domain.

For a Fano variety $X$, that is, a smooth projective algebraic variety with
ample anticanonical bundle, a famous conjecture of Mukai [7] asserts that the
inequality

$\dim X\geq r_{X}\cdot(i_{X}-1),$

where $r_{X}$ denotes the Picard rank of $X$ and $i_{X}$ the index of $X$ (the
maximal integer dividing the anticanonical class) holds, with equality only
for powers of projective space.

In [2], the moduli spaces $M_{\bf d}^{\rm st}(Q)$ of stable quiver
representations [9] which are Fano varieties are identified, forming a rather
special (for example, always being rational [11] and of algebraic cohomology
[6]), but arbitrarily high-dimensional, class of Fano varieties.

In this note, we verify the Mukai conjecture for Fano quiver moduli spaces (in
the spirit of such a verification for toric varieties [1], horospherical
varieties [8] and symmetric varieties [3]), under a reasonable genericity
assumption on the dimension vector ${\bf d}$, to be discussed below.
So let $Q$ be a connected finite acyclic quiver with set of vertices $Q_{0}$
and arrows written $\alpha:i\rightarrow j$ for $i,j\in Q_{0}$, and let

$\langle{\bf d},{\bf e}\rangle=\sum_{i\in
Q_{0}}d_{i}e_{i}-\sum_{\alpha:i\rightarrow j}d_{i}e_{j},$

for ${\bf d},{\bf e}\in\mathbb{Z}Q_{0}$, its Euler form; we also consider its
symmetrization $({\bf d},{\bf e})$ as well as its antisymmetrization $\\{{\bf
d},{\bf e}\\}$. For a dimension vector ${\bf d}=\sum_{i}d_{i}{\bf
i}\in\mathbb{Z}_{>0}Q_{0}$, the linear form $\\{{\bf d},\\_\\}$ defines a
stability in the sense of [5], for which we assume in the following that there
exists a stable representation of dimension vector ${\bf d}$, that ${\bf d}$
is coprime (that is, $\\{{\bf d},{\bf e}\\}\not=0$ for all $0\not={\bf
e}\lneqq{\bf d}$), and that ${\bf d}$ is amply stable (there are no divisors
in the unstable locus of the representation space, see [2]). The main result
of [2] then asserts:

######  Theorem 1 ([2]).

The moduli space $M_{\bf d}^{\rm st}(Q)$ parametrizing isomorphism classes of
$\\{{\bf d},\\_\\}$-stable complex representations of $Q$ of dimension vector
${\bf d}$ is a complex Fano variety of dimension $1-\langle{\bf d},{\bf
d}\rangle$, Picard rank $|Q_{0}|-1$, and index $\gcd(\\{{\bf d},{\bf
i}\\}:i\in Q_{0})$.

That ${\bf d}$ admits a stable representation can, in general, only be
verified using a highly recursive criterion [10], except if ${\bf d}$ belongs
to the so-called fundamental domain [4], that is, $({\bf d},{\bf i})\leq 0$
for all $i\in Q_{0}$. We have to slightly strengthen this reasonable
assumption to obtain our main result:

######  Theorem 2.

In addition to the previous assumptions, assume that $({\bf d},{\bf i})\leq-2$
for all $i\in Q_{0}$. Then $M_{\bf d}^{\rm st}(Q)$ fulfills the Mukai
conjecture.

We will reduce the statement to the following elementary counting argument:

######  Lemma 3.

Let $a_{1},\ldots,a_{k},b_{1},\ldots,b_{l}$ be positive integers. Assume that
no proper partial sums coincide, that is, if $\sum_{i\in I}a_{i}=\sum_{j\in
J}b_{j}$ for subsets $I\subset\\{1,\ldots,k\\}$ and
$J\subset\\{1,\ldots,l\\}$, then $I=J=\emptyset$ or $I=\\{1,\ldots,k\\}$,
$J=\\{1,\ldots,l\\}$. Then

$\sum_{i}a_{i}+\sum_{j}b_{j}\geq 2(k+l-1),$

with equality only if $l=1$ and all $a_{i}$ equal to one, or $k=1$ and all
$b_{j}$ equal to one.

###### Proof.

Consider the numbers

$d_{i,j}=a_{1}+\ldots+a_{i}-b_{1}-\ldots-b_{j}$

for $0\leq i\leq k$, $0\leq j\leq l$. Their mutual differences are sums or
differences of partial sums of the $a_{i}$ and the $b_{j}$, thus they are
either manifestly positive or negative, or non-zero by the assumption of the
lemma, except possibly for $d_{0,0}$, $d_{k,l}$. So the $d_{i,j}$ form at
least $(k+1)(l+1)-1$ mutually different integers in the interval
$[-\sum_{j}b_{j},\sum_{i}a_{i}]$, and thus

$\sum_{i}a_{i}+\sum_{j}b_{j}\geq(k+1)(l+1)-2=(k-1)(l-1)+2(k+l-1)\geq
2(k+l-1).$

Moreover, if equality holds, then $k=1$ or $l=1$ by the second inequality, and
the first inequality forces $d_{0,0}=d_{k,l}$, which reads
$\sum_{i}a_{i}=\sum_{j}b_{j}$. This in turn implies (assuming $l=1$ without
loss of generality) $\sum_{i}a_{i}=k+l-1=k$, and thus all $a_{i}=1$.∎

In light of Theorem 1, the claim of Theorem 2 reduces to the following
statement, since, by [2, Section 5.3], for thin dimension vectors of
$m$-thickened subspace quivers the moduli spaces realize powers of projective
space:

######  Proposition 4.

Assume that ${\bf d}$ is $\\{{\bf d},\\_\\}$-coprime and that $({\bf d},{\bf
i})\leq-2$ for all $i\in Q_{0}$. Then

$1-\langle{\bf d},{\bf d}\rangle\geq(|Q_{0}|-1)\cdot(\gcd(\\{{\bf d},{\bf
i}\\}:i\in Q_{0})-1),$

with equality only if $Q$ is an $m$-thickened subspace quiver or its opposite,
and ${\bf d}$ is thin.

###### Proof.

For all $i\in Q_{0}$, define nonnegative integers

$\alpha_{i}=d_{i}-\langle{\bf i},{\bf d}\rangle,\;\beta_{i}=d_{i}-\langle{\bf
d},{\bf i}\rangle,$

so that, by assumption,

$-2\geq({\bf d},{\bf i})=2d_{i}-\alpha_{i}-\beta_{i}\mbox{ and
}\alpha_{i}-\beta_{i}=\\{{\bf d},{\bf i}\\}.$

We can then estimate $1-\langle{\bf d},{\bf d}\rangle=$

$=1+\sum_{i}((\alpha_{i}+\beta_{i})/2-d_{i})d_{i}\geq
1+\sum_{i}((\alpha_{i}+\beta_{i})/2-1)=\sum_{i}(\alpha_{i}+\beta_{i})/2-(|Q_{0}|-1).$

The claim thus follows if

$\sum_{i}(\alpha_{i}+\beta_{i})\geq 2(|Q_{0}|-1)\cdot m$

for $m=\gcd(\\{{\bf d},{\bf i}\\}:i\in Q_{0})$. We can thus write

$\alpha_{i}-\beta_{i}=\varepsilon_{i}\gamma_{i}\cdot m$

for a sign $\varepsilon_{i}$ and nonnegative integral $\gamma_{i}$. The
estimate

$\alpha_{i}+\beta_{i}\geq|\alpha_{i}-\beta_{i}|$

then reduces the claim to

$\sum_{i}\gamma_{i}\geq 2(|Q_{0}|-1).$

The coprimality assumption reads

$0\not=\sum_{i}\varepsilon_{i}\gamma_{i}e_{i}$

for all $0\not={\bf e}\lneqq{\bf d}$. In particular, $\gamma_{i}$ is positive
for all $i\in Q_{0}$. Denoting the $\gamma_{i}$ for positive $\varepsilon_{i}$
by $a_{1},\ldots,a_{k}$, and those for negative $\varepsilon_{i}$ by
$b_{1},\ldots,b_{l}$, coprimality implies that the assumptions of the previous
lemma are satisfied, and the first claim there yields the desired estimate.
Now assume that equality holds in all estimates. By the lemma we thus find
(without loss of generality) a unique vertex $i_{0}\in Q_{0}$ such that

$\alpha_{i}-\beta_{i}=m\mbox{ for all }i\not=i_{0}\mbox{, whereas
}\alpha_{i_{0}}-\beta_{i_{0}}=(1-|Q_{0}|)\cdot m.$

The equalities $\alpha_{i}+\beta_{i}=|\alpha_{i}-\beta_{i}|$ then imply
$\beta_{i}=0$, and thus $\alpha_{i}=m$, for all $i\not=i_{0}$, and
$\alpha_{i_{0}}=0$, and thus $\beta_{i_{0}}=(|Q_{0}|-1)\cdot m$. By definition
of $\alpha_{i},\beta_{i}$, we find that $Q$ is the $(m/d_{i_{0}})$-thickened
$s$-subspace quiver, with unique sink $i_{0}$ and $s=|Q_{0}|-1$, and that

$d_{i_{0}}=\sum_{i\not=i_{0}}d_{i}/s.$

Furthermore, the first step in the chain of inequalities shows that $d_{i}=1$
or $d_{i}=m/2-1$ for all $i\not=i_{0}$, and that $d_{i_{0}}=1$ or
$d_{i_{0}}=sm/2-1$. A quick estimate shows that the latter is impossible, thus
${\bf d}$ is thin. ∎

Acknowledgments: The author would like to thank H. Franzen and S. Sabatini for
helpful discussions on the topic. This work is supported by the DFG CRC-TRR
191 “Symplectic structures in geometry, algebra and dynamics” (281071066).

## References

  * [1] C. Casagrande, The number of vertices of a Fano polytope. Ann. Inst. Fourier (Grenoble) 56 (2006), no. 1, 121–130.
  * [2] H. Franzen, M. Reineke, S. Sabatini, Fano quiver moduli spaces. Can. Math. Bull. 64 (2021) 984–1000.
  * [3] G. Gagliardi, J. Hofscheier, The generalized Mukai conjecture for symmetric varieties. Trans. Amer. Math. Soc. 369 (2017), 2615–2649.
  * [4] V. G. Kac. Root systems, representations of quivers and invariant theory. Invariant theory, Lecture Notes in Math. 996, pages 74–108. Springer, 1983.
  * [5] A. King, Moduli of representations of finite-dimensional algebras. Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530.
  * [6] A. King, C. Walter, On Chow rings of fine moduli spaces of modules. J. Reine Angew. Math. 461 (1995), 179–187.
  * [7] S. Mukai, Problems on characterization of the complex projective space. Birational Geometry of Algebraic Varieties, Open Problems, Katata, 1988, 23rd Symposium, Taniguchi Foundation (1988), 57–60.
  * [8] B. Pasquier, The pseudo-index of horospherical Fano varieties. Internat. J. Math. 21 (2010), no. 9, 1147–1156.
  * [9] M. Reineke, Moduli of representations of quivers. Trends in representation theory of algebras and related topics, EMS Series of Congress Reports, European Mathematical Society, Zürich, 2008, 589–637.
  * [10] A. Schofield, General representations of quivers. Proc. London Math. Soc. (3) 65 (1992), no. 1, 46–64.
  * [11] A. Schofield, Birational classification of moduli spaces of representations of quivers. Indag. Math. (N.S.) 12 (2001), no. 3, 407–432.


# Do shadows of Sgr A* and M87* indicate black holes with a magnetic monopole
charge?

Indrani<EMAIL_ADDRESS>1, Subhadip
<EMAIL_ADDRESS>2,3 and Soumitra<EMAIL_ADDRESS>3
1Department of Physics and Astronomy, National Institute of Technology,
Rourkela
2Department of Physics, Jhargram Raj College, Jhargram, West Bengal-721507
3School of Physical Sciences, Indian Association for the Cultivation of
Science, Kolkata-700032

###### Abstract

We study the prospect of Bardeen black holes in explaining the observed shadow
of Sgr A* and M87*. Bardeen black holes are regular black holes endowed with a
magnetic monopole charge that arise in Einstein gravity coupled to non-linear
electrodynamics. These black holes are interesting as they can evade the $r=0$
curvature singularity arising in general relativity. It is therefore
worthwhile to look for signatures of Bardeen black holes in astrophysical
observations. With two successive release of black hole images by the Event
Horizon Telescope (EHT) collaboration, the scope to test the nature of strong
gravity has substantially increased. We compare the theoretically computed
shadow observables with the observed image of Sgr A* and M87*. Our analysis
reveals that while the observed angular diameter of M87* favors the Kerr
scenario, the shadow of Sgr A* can be better explained by the Bardeen
background. This indicates that although rare, certain black holes exhibit a
preference towards regular black holes like the Bardeen spacetime.

##  1 Introduction

Black holes which mark a one-way passage to the world of the unknown are the
most enigmatic predictions of general relativity. These objects are surrounded
by an event horizon from where nothing can escape to the outside world. The
horizon represents the characteristic radius within which if a mass
distriution is compressed, no signals from it can reach the external observer.
The existence of black holes have received ample observational confirmation
since the last fifty years with the launch of X-ray satellites like RXTE,
Chandra, etc. However, these are indirect detections of black holes. With two
successive ground breaking discoveries, namely, the gravitational waves from
the merger of binary compact objects [1, 2] and the image of the black hole
[3, 4, 5, 6, 7, 8, 9], direct detection of these esoteric objects have become
a reality. Yet the discovery of black holes mark at the same time the greatest
feat and fiasco of general relativity. This is because these objects have a
curvature singularity at the origin where the physical observables diverge and
the theory loses all its predictability. The celebrated theorems of Hawking
and Penrose state that black holes in general relativity eventually end up in
singularities [10, 11, 12, 13]. It is believed that a suitable quantum theory
of gravity may cure these problems and various quantum gravity models have
been proposed in this regard [14, 15, 16, 17, 18, 19].

However, there exist a classical route to address the singularity issue in GR
which involves investigating black holes in Einstein gravity coupled to non-
linear electrodynamics [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. These
models arise in the low energy limit of various quantum gravity theories where
quantum fluctuations during the last stages of a stellar collapse can
potentially avoid the unbounded increase in the spacetime curvature thereby
giving rise to regular cores. For these black holes the metric and the
curvature invariants assume finite values at all spacetime points. The regular
core can be either de Sitter or Minkowski type [31, 32, 33, 34, 35, 36]. In
this work we shall consider regular black holes which are endowed with a
repulsive de Sitter core. Inspired by the ideas of Sakharov [37] and Gliner
[35] such black holes were first proposed by Bardeen as solution of Einstein
equations coupled to non-linear electrodynamics. Further investigations by
Ayon-Beato and Garcia [20, 21, 22, 26] revealed that the Bardeen black holes
are associated with the gravitational field of a non-linear magnetic monopole
of a self-gravitating magnetic field. Since then there has been a lot of work
in the direction of black holes in non-linear electrodynamics [34, 35, 36, 38,
39, 40, 41, 42] and several other regular black hole solutions have been put
forward [26, 20, 43, 23, 34, 44, 45, 46, 47, 30, 48]. Since astrophysical
black holes are generally rotating, we shall consider the axisymmetric
counterpart of the Bardeen black holes in this work. The rotating solution is
obtained by applying the Newman-Janis algorithm to the static spherically
symmetric spacetime [49, 50, 51, 52].

Observational signatures of regular black holes have been explored extensively
[46, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64]. In this work we shall
study the signatures of Bardeen black holes in the recently observed shadow of
Sgr A* and M87*. The imaging of black holes by the Event Horizon Telescope
collaboration has opened up a new and unique window to test the nature of
gravity in the strong field regime. The shadow of a black hole represents the
directions in the local sky from where light just escapes the event horizon
and reaches the observer on earth [65, 66, 67, 68, 69]. As a result there is
total absence of radiation from certain directions in the observer’s sky
epitomized by a dark patch which is known as the black hole shadow. Since the
boundary of the shadow nearly traces the event horizon, the shape and size of
the shadow bears the imprints of the background metric [67, 70, 71, 72, 73,
74]. As a result the black hole image turns out to be a useful observational
tool to look for signatures of regular black holes, which is the primary goal
of the present work.

The paper is organized as follows: In 2 we have discussed the non-linear
electrodynamics model which gives rise to Bardeen black holes. 3 includes
derivation of the shadow outline for Bardeen black holes. 4 is dedicated in
discussing the comparison of our theoretical findings with the observed shadow
of M87* and Sgr A*. We conclude with a summary of our results with some scope
for future work in 5.

##  2 Brief description of Bardeen black holes

In this section we discuss the theoretical background related to Bardeen black
holes, the first ever regular black hole proposed in the literature. Such
black holes arise in Einstein gravity coupled to non-linear electrodynamics
such that the associated action is given by [46, 20, 75, 36, 44, 48],

$\displaystyle S=\dfrac{1}{16\pi G}\int
d^{4}x\sqrt{-h}\left[R-4\mathcal{L}(F)\right]$ (1)

where $G$ is gravitational constant, $h$ is the determinant of metric tensor,
$R$ is Ricci scalar, $\mathcal{L}$ is Lagrangian density for non-linear
electrodynamics and the Faraday invariant is
$F=F_{\alpha\beta}F^{\alpha\beta}$. Electromagnetic field strength tensor is
$F_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}$, where
$A^{\alpha}$ is the associated vector potential. The Lagrangian density
corresponding to Bardeen black holes is given by [46],

$\displaystyle\mathcal{L}(F)=\frac{3\mathcal{M}}{|\tilde{g}|\tilde{g}^{2}}\bigg{[}\frac{\sqrt{2\tilde{g}^{2}F}}{1+\sqrt{2\tilde{g}^{2}F}}\bigg{]}^{5/2}$
(2)

where $\tilde{g}$ is the magnetic monopole charge associated with a self-
gravitating magnetic field. Variation of the action with respect to the metric
yields the gravitational field equations,

$\displaystyle\mathcal{G}^{\nu}_{\mu}=2\bigg{[}\mathcal{L}_{F}({F}_{\mu\lambda}{F}^{\nu\lambda})-\delta^{\nu}_{\mu}\mathcal{L}(F)\bigg{]}$
(3)

while variation with respect to the gauge field $A_{\mu}$ leads to the
equation of motion for the electromagnetic field,

$\displaystyle\nabla_{\mu}(\mathcal{L}_{F}\mathcal{F}^{\beta\mu})=0$ (4)

with $\mathcal{L}_{F}=\frac{\partial\mathcal{L}}{\partial F}$. The static,
spherically symmetric black hole solution of 3 assumes the form,

$\displaystyle
ds^{2}=-\bigg{(}1-\frac{2m(r)}{r}\bigg{)}dt^{2}+\bigg{(}1-\frac{2m(r)}{r}\bigg{)}^{-1}dr^{2}+r^{2}d\Omega^{2}$
(5)

where the mass function is denoted by $m(r)$. When the black hole possesses
pure magnetic charge the solution of the gauge field is given by
$dA=-gcos\theta d\phi$ while from 3 and 4 we note that the mass function
satisfies the following two equations:

$\displaystyle-\frac{2m^{\prime}(r)}{r^{2}}+2\mathcal{L}(F)=0$
$\displaystyle-\frac{m^{\prime\prime}(r)}{r}+2\mathcal{L}(F)-\frac{2g^{2}}{r^{4}}\mathcal{L}_{F}=0$
(6)

For the Lagrangian density given by 2, solution of 2 gives rise to the
following form of the mass function,

$\displaystyle
m(r)=M\bigg{\\{}\frac{r^{2}}{r^{2}+\tilde{g}^{2}}\bigg{\\}}^{3/2}$ (7)

where we denote the magnetic monopole charge parameter by $g=\tilde{g}^{2}$ in
the rest of the paper.

Since astrophysical black holes are generally rotating one needs to find the
rotating counterpart of 5. This is accomplished by applying the Newman-Janis
algorithm to the static, spherically symmetric seed metric [49, 50, 51, 52].
The rotating counterpart of the Bardeen metric in Boyer-Lindquist coordinates
correspond to,

$\displaystyle ds^{2}=$
$\displaystyle-\left(1-\dfrac{2m(r)r}{\Sigma}\right)dt^{2}+\dfrac{\Sigma}{\Delta}dr^{2}+\Sigma~{}d\theta^{2}-\dfrac{4m(r)ar\sin^{2}\theta}{\Sigma}dtd\phi$
$\displaystyle\hskip
142.26378pt+\left[{(r^{2}+a^{2})+\frac{2m(r)r}{\Sigma}a^{2}\sin^{2}\theta}\right]\sin^{2}\theta
d\phi^{2}$ (8)

where

$\displaystyle\Delta$ $\displaystyle=r^{2}-2m(r)r+a^{2}$ (9)
$\displaystyle\Sigma$ $\displaystyle=r^{2}+a^{2}\cos^{2}\theta$ (10)

and the mass function $\lim_{r\to\infty}m(r)=M$ ($M$ is ADM mass of black
hole) and $a$ is the spin parameter of the black hole. From 8 and in the rest
of the paper, we use dimensionless quantities, e.g. $r\equiv r/M$, $a\equiv
a/M$ and $g\equiv g/M^{2}$.

Since we are interested in black hole solutions, the horizon radius/radii must
be real and positive. As the horizon comprises of a null surface with
spherical topology, the corresponding radii are obtained by solving for the
roots of $g^{rr}=\Delta=0$,

$\displaystyle r^{2}+a^{2}-2rm(r)=0$ (11)

Requirement of a real, positive horizon imposes bounds on the magnitude of
$g$: $0\lesssim g\lesssim 0.55$, the upper limit being obtained by putting
$a=0$ [76]. Moreover, for a given $g$ in the allowed range, it restricts the
allowed values of $a$ [76]. In the next section we derive the shape and size
of the black hole shadow for the rotating Bardeen metric given by 8. The
theoretically obtained shadow is then compared with the observed image of Sgr
A* and M87* to establish constraints on the metric parameters $g$ and $a$.

##  3 Nature of shadow for Kerr-like solutions

In this section we derive the nature of the shadow cast by Bardeen black
holes. This requires investigating the geodesic motion of photons around black
holes which may originate from a distant astrophysical object or from the
accretion disk surrounding the black hole. When these photons arrive close to
the event horizon a part of it gets trapped inside the horizon while the
remaining photons escape to infinity. This creates a dark patch in the
observer’s sky which is known as the black hole shadow [65, 66, 67, 68, 69].
The outline of the shadow bears imprints of strong gravitational lensing of
nearby radiation such that the shape and size of the shadow can unravel useful
information regarding the nature of the background metric [67, 70, 71, 72, 73,
74].

In order to compute the outline of the shadow we need to adopt the Hamilton-
Jacobi approach. This approach allows us to easily identify the constants of
motion arising from the symmetries of the metric. The derivation of the shadow
outline has been discussed in detail in [74], therefore here we present only
the main results. We start with the Lagrangian for particle motion
$\mathcal{L}(x^{\mu},\dot{x}^{\mu})=\dfrac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}$,
from where it can be shown that the Hamiltonian satisfies the Hamilton-Jacobi
equation,

$\displaystyle\mathcal{H}\left(x^{\mu},\dfrac{\partial S}{\partial
x^{\mu}}\right)+\dfrac{\partial S}{\partial\lambda}=0;\quad\dfrac{\partial
S}{\partial x^{\mu}}=p_{\mu}$ (12)

where the four momentum $p_{\mu}$ is connected with the Hamiltonian as
$\mathcal{H}=\dfrac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu}=\dfrac{k}{2}$, $k$ being the
rest mass of the test particle. In 12 $S$ represents the action and $\lambda$
the curve parameter. Further, due to the symmetry of given space-time (in our
case a stationary and axisymmetric space-time) we note that the specific
energy E and the angular momentum L are the constants of motion given by,

$\displaystyle p_{t}$ $\displaystyle=-E=\rm constant$ (13) $\displaystyle
p_{\phi}$ $\displaystyle=L=\rm constant$ (14)

Using 12 in the on-shell condition $p^{\mu}p_{\mu}=0$ we note that the action
is completely separable in all the four coordinates

$\displaystyle S=-Et+L\phi+S^{(r)}(r)+S^{\theta}(\theta)$ (15)

This is attributed to the presence of Killing vectors ($\partial_{t}$,
$\partial_{\phi}$) and a Killing tensor corresponding to the spacetime in 8.
The two unknown functions $S^{r}$ and $S^{\theta}$ can be obtained by solving
the following differential equations,

$\displaystyle\left(\dfrac{dS^{r}}{dr}\right)^{2}=\dfrac{R(r)}{\Delta^{2}}$
(16)
$\displaystyle\left(\dfrac{dS^{\theta}}{d\theta}\right)^{2}=\Theta(\theta)$
(17)

where

$\displaystyle R(r)=$
$\displaystyle\Delta\left[-C-(L-aE)^{2}\right]+\left\\{\left(r^{2}+a^{2}\right)E-aL\right\\}^{2}$
(18) $\displaystyle\Theta(\theta)=$ $\displaystyle
C+\cos^{2}\theta\left[E^{2}a^{2}-\dfrac{L^{2}}{\sin^{2}\theta}\right]$ (19)

In 19 $C$ refers to the third constant of motion and is known as the Carter
constant. Therefore, the solution of the Hamilton-Jacobi equation has the form

$\displaystyle
S=-Et+L\phi+\int\dfrac{\sqrt{R(r)}}{\Delta}dr+\int\sqrt{\Theta(\theta)}d\theta$
(20)

from where one can show that the equations of motion for $r$ and $\theta$ are
given by

$\displaystyle\dot{r}$ $\displaystyle=\dfrac{\sqrt{R(r)}}{\Sigma}$ (21a)
$\displaystyle\dot{\theta}$ $\displaystyle=\dfrac{\sqrt{\Theta}}{\Sigma}$
(21b)

From the constants of motion we can construct the two impact parameters

$\displaystyle\chi=\dfrac{C}{E^{2}}\quad\textit{\rm
and}\quad\eta=\dfrac{L}{E}$ (22)

which represents the distance from the equatorial plane and the axis ratio
respectively. In the next section we briefly discuss the nature of motion of
photons in the radial and vertical direction.

###  3.1 Motion of photons in the vertical and radial direction

At first we will deal with the angular part and hence will solve 21b. For
solving the same we introduce the new variable $u=\cos\theta$. In terms of
this new variable the differential equation for angular part becomes

$\displaystyle\left(\dfrac{\Sigma}{E}\right)^{2}\dot{u}^{2}=\chi-u^{2}(\chi+\eta^{2}-a^{2})-a^{2}u^{4}\equiv
F(u(\theta))$ (23)

Above equation (23) indicates that for physically realisable solution, $F(u)$
cannot be negative. However, when $u=1$ i.e $\theta=0^{\circ}$ the function
$F(u)$ becomes negative which suggests that the photon cannot access the full
angular space. If one wish to find the maximum angular space available i.e
$\theta_{max}$ one need to set $F(u_{0})=0$. One can solve this equation and
get

$\displaystyle
u_{0}^{2}=\dfrac{-(\chi+\eta^{2}-a^{2})\pm\sqrt{(\chi+\eta^{2}-a^{2})^{2}+4a^{2}\chi}}{2a^{2}}$
(24)

Depending on the signature of $\chi$ the photons can access certain regions of
angular space. For example, when $\chi>0$, only the positive root of 24 is
relevant. In this case the physically allowed range of $u$ lies between
$\pm|u_{0}|$ such that the orbits cross the equatorial plane. When $\chi=0$,
we can have two roots,

$\displaystyle
u_{1}^{2}=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\rm{and}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}u_{2}^{2}=1-\frac{\eta^{2}}{a^{2}}$
(25)

and when $\eta^{2}>a^{2}$ only the first root is allowed. When $\chi$ is
negative, 24 can be written as,

$\displaystyle
u_{0}^{2}=\frac{|\chi|-\eta^{2}+a^{2}\pm\sqrt{(-|\chi|+\eta^{2}-a^{2})^{2}-4a^{2}|\chi|})}{2a^{2}}$
(26)

where $\chi=-|\chi|$. It is evident from 26 that $u_{0}$ has a physically
realizable solution when the following condition is satisfied,

$\displaystyle a^{2}+|\chi|-\eta^{2}>0$ (27)

Differential equation associated to the radial part is given by

$\displaystyle\left(\dfrac{\Sigma}{E}\right)^{2}\dot{r}^{2}=\Delta\left[-\chi+\dfrac{k}{E^{2}}r^{2}-\left(\dfrac{L}{E}-a\right)^{2}\right]+\left[r^{2}+a^{2}-a\eta\right]^{2}\equiv
V(r)$ (28)

Spherical photon orbits are realised with the conditions
$V(r)=0=V^{\prime}(r)$. These conditions further determine two parametric
equations

$\Delta\left(\chi+\eta^{2}+a^{2}-2\eta
a\right)=\left[r^{2}+a^{2}-a\eta\right]^{2}$ (29a) ${\chi+\eta^{2}+a^{2}-2\eta
a}=\dfrac{2}{\left[1-\dfrac{m(r)}{r}-m^{\prime}(r)\right]}\left[r^{2}+a^{2}-a\eta\right]$
(29b)

Solving 29b we immediately land with two sets of solutions for the impact
parameters as given below

  1. 1.

$\chi=-\dfrac{r^{4}}{a^{2}}$ (30a) $\eta=\frac{a^{2}+r^{2}}{a}$ (30b)

  2. 2.

$\displaystyle\chi$
$\displaystyle=-\frac{r^{3}\left(4a^{2}rm^{\prime}(r)-4a^{2}m(r)+r^{3}m^{\prime}(r)^{2}+2r^{3}m^{\prime}(r)-6r^{2}m(r)m^{\prime}(r)-6r^{2}m(r)+9rm(r)^{2}+r^{3}\right)}{a^{2}\left(rm^{\prime}(r)+m(r)-r\right)^{2}}$
(31a) $\displaystyle\eta$
$\displaystyle=\frac{a^{2}rm^{\prime}(r)+a^{2}m(r)+a^{2}r+r^{3}m^{\prime}(r)-3r^{2}m(r)+r^{3}}{a\left(rm^{\prime}(r)+m(r)-r\right)}$
(31b)

First solution is physically not allowed because it does not satisfy 27. For
the second solution $\chi$ may take either sign depending upon the value of
$r$ (where $r$ is associated with the radius of the spherical photon orbits).
But in general

$\displaystyle
a^{2}-\chi-\eta^{2}=-\frac{2r\left(\left\\{rm^{\prime}(r)+m(r)\right\\}\left\\{2a^{2}+r^{2}m^{\prime}(r)-3rm(r)\right\\}+r^{3}\right)}{\left\\{rm^{\prime}(r)-r+m(r)\right\\}^{2}}$
(32)

and $\chi=0$ refers to case of motion in the equatorial plane.

###  3.2 Shape of the black hole shadow

Figure 1: In this figure geometry of gravitational lens has been shown. Any
observer far away from black hole can imagine a reference coordinate system
$(x,y,z)$ with the black hole at its origin. At infinity the Boyer-Lindquist
coordinates coincides with the reference coordinate system and as seen from
infinity the black hole rotates around $z$-axis. The line connecting the
origin and the observer is normal to the $\alpha-\beta$ plane.

As the space-time that we have considered in our present scenario is
asymptotically flat, a distant observer can imagine a reference Euclidean
coordinate system $(x,y,z)$ with black hole at its origin. One should note
that the Boyer-Lindquist coordinate becomes identical to the reference
Euclidean coordinate system for large values of $r$. Further we have also
assumed that as seen from infinity the axis of rotation of black hole is along
$z-$axis. Without loss of generality, let us assume the coordinate of the
distant observer is $(r_{0},\theta_{0},0)$. In a similar way
$(r_{s},\theta_{s},\phi_{s})$ is assumed to be the position of source [77].

Considering the observer’s frame, an incoming light ray can be described by a
parametric curve ${x(r),y(r),z(r)}$ such that $r^{2}=x^{2}+y^{2}+z^{2}$. One
can note that this is just the usual radial coordinate in BL coordinate system
if $r$ is considerably large. At the location of the observer, the tangent
vector $\left(\vec{v}\big{|}_{(r_{0},\theta_{0},0)}\right)$ to the parametric
curve can be provided as follows

$\displaystyle\vec{v}\big{|}_{{(r_{0},\theta_{0},0)}}=\dfrac{dx}{dr}\bigg{|}_{{(r_{0},\theta_{0},0)}}\hat{x}+\dfrac{dy}{dr}\bigg{|}_{{(r_{0},\theta_{0},0)}}\hat{y}+\dfrac{dz}{dr}\bigg{|}_{{(r_{0},\theta_{0},0)}}\hat{z}$
(33)

This tangent vector represents a straight line which intersects the
$\alpha-\beta$ plane at $(\alpha_{i},\beta_{i})$. One should note that the
line connecting the origin and the observer is normal to this $\alpha-\beta$
plane. This $\alpha-\beta$ plane is known as observer’s (view) plane. This is
schematically shown in 1.

Using usual coordinate transformation rule and for the observer considering
$\phi=0$, we have the position of observer as follows

$\displaystyle x_{o}$ $\displaystyle=r_{o}\sin\theta_{o}$ $\displaystyle
y_{o}$ $\displaystyle=0$ $\displaystyle z_{o}$
$\displaystyle=r_{o}\cos\theta_{o}$

Using coordinate transformation one can easily find the components of the
tangent vector at the position of the observer. These are given by

$\displaystyle\dfrac{dx}{dr}\bigg{|}_{\vec{r}_{0}}$
$\displaystyle=\sin\theta_{0}+r_{0}\cos\theta_{0}\dfrac{d\theta}{dr}\bigg{|}_{\vec{r}_{0}}$
(34a) $\displaystyle\dfrac{dy}{dr}\bigg{|}_{\vec{r}_{0}}$
$\displaystyle=r_{0}\sin\theta_{0}\dfrac{d\phi}{dr}\bigg{|}_{\vec{r}_{0}}$
(34b) $\displaystyle\dfrac{dz}{dr}\bigg{|}_{\vec{r}_{0}}$
$\displaystyle=\cos\theta_{0}-r_{0}\sin\theta_{0}\dfrac{d\theta}{dr}\bigg{|}_{\vec{r}_{0}}$
(34c)

(a) Variation of BH shadow with metric parameter $g$. The inclination angle
and spin parameter are taken to be respectively $\theta=45^{\circ}$ and
$a=0.2$.

(b) Variation of shadow with metric parameter $g$.The inclination angle and
spin parameter are taken to be respectively $\theta=60^{\circ}$ and $a=0.5$.

(c) Variation of shadow with spin-parameter $a$. Here the inclination angle is
$\theta=60^{\circ}$ and $g=0.1$.

(d) Variation of shadow structure with different inclination angle
$\theta$.Here the spin is $a=0.8$ and $g=0.1$.

Figure 2: Variation of shape and size of black hole shadow with magnetic
monopole charge parameter $g$, rotation parameter $a$ and inclination angle
$\theta$.

This vector describes a straight line which intersect the $\alpha-\beta$
plane.The point $(\alpha_{i},\beta_{i})$ in the plane $(\alpha,\beta)$ in
$(x,y,z)$ system is given by

$\displaystyle x$ $\displaystyle=-\beta_{i}\cos\theta_{0}$ $\displaystyle y$
$\displaystyle=\alpha_{i}$ $\displaystyle z$
$\displaystyle=\beta_{i}\sin\theta_{0}$

Now

$\displaystyle\vec{r}_{0}-r_{0}\vec{v}\bigg{|}_{\vec{r}_{0}}=\left(r_{0}\sin\theta_{0}-r_{0}\dfrac{dx}{dr}\bigg{|}_{\vec{r}_{0}},-r_{0}\dfrac{dy}{dr}\bigg{|}_{\vec{r}_{0}},r_{0}\cos\theta_{0}-r_{0}\dfrac{dz}{dr}\bigg{|}_{\vec{r}_{0}}\right)$
(35)

Hence we have

$\displaystyle-\beta_{i}\cos\theta_{0}$
$\displaystyle=r_{0}\sin\theta_{0}-r_{0}\dfrac{dx}{dr}\bigg{|}_{\vec{r}_{0}}=r_{0}\sin\theta_{0}-r_{0}\left\\{\sin\theta_{0}+r_{0}\cos\theta_{0}\dfrac{d\theta}{dr}\bigg{|}_{\vec{r}_{0}}\right\\}$
$\displaystyle\implies\beta_{i}$
$\displaystyle=r_{0}^{2}\dfrac{d\theta}{dr}\bigg{|}_{\vec{r}_{0}}$ (36)

Similarly we have

$\displaystyle\alpha_{i}=-r_{0}\dfrac{dy}{dr}\bigg{|}_{\vec{r}_{0}}=-r_{0}^{2}\sin\theta_{0}\dfrac{d\phi}{dr}\bigg{|}_{\vec{r}_{0}}$
(37)

For the sake of visualization of black hole shadow, let us consider a distant
observer at position $(r_{0},\theta_{0})$. If $\\{\alpha_{i},\beta_{i}\\}$ are
the celestial coordinates then the stereographic projection of the shadow from
celestial body to the image plane can be defined as follows [77]

$\displaystyle\alpha=$
$\displaystyle\lim_{r_{0}\to\infty}\left(-r_{0}^{2}\sin\theta_{0}\dfrac{d\phi}{dr}\right)$
(38a) $\displaystyle\beta=$
$\displaystyle\lim_{r_{0}\to\infty}\left(r_{0}^{2}\dfrac{d\theta}{dr}\right)$
(38b)

In terms of impact parameters, as defined in 22 these coordinates are given by

$\displaystyle\alpha_{i}$ $\displaystyle=-\eta\csc\theta_{o}$ (39)
$\displaystyle\beta_{i}$
$\displaystyle=\sqrt{\chi+a^{2}\cos^{2}\theta_{0}-\eta^{2}\cot^{2}\theta_{0}}$
(40)

In 2 we have plotted the variation of size and shape of black hole shadow with
change in the charge parameter $g$, the rotation parameter $a$ and the
inclination angle $\theta$. From 2a and 2b it can be noted that for a fixed
value of inclination angle the size of the black hole shadow gradually
decreases for an increase in non-linear electrodynamics charge parameter $g$.
From 2c and 2d one can notice that the shape of the black hole shadow becomes
more dented and deviates from circular shape for both an increase in spin
parameter $a$ and inclination angle $\theta$.

##  4 Comparison with the observed shadow of M87* and Sgr A*

In this section we compare the theoretically derived shadow with the observed
image of M87* and Sgr A*. In order to accomplish this we need to define a few
observables associated with black hole shadow. The boundary of the shadow is
given by the curve $\beta(\alpha)$. As the shadow is not circular in shape,
one need to define one major axis and one minor axis associated with two
diameters as shown in 3.

The major axis is introduced by $\Delta\beta=\beta_{t}-\beta_{b}$, where
$\beta_{t}$ and $\beta_{b}$ are the top-most and the bottom-most points of the
shadow respectively in $\alpha-\beta$ plane. In a similar manner the minor
axis is given by $\Delta\alpha=\alpha_{r}-\alpha_{l}$, where $\alpha_{r}$ and
$\alpha_{l}$ are the right-most and left-most points of the shadow
respectively.

One can note from 2 that these two axes i.e $\Delta\beta$ and $\Delta\alpha$
can also be expressed as a function of metric parameters ($g$ and $a$) and the
inclination angle. The shadow is symmetric in nature about the minor axis. We
next discuss some of the observables which are defined to gain a theoretical
understanding regarding the effect of modification in the gravity theory:

  1. (i)

Angular diameter which is given by

$\displaystyle\Phi=\dfrac{GM}{c^{2}D}\Delta\beta$ (41)

It is important to note that the theoretical angular diameter depends on the
mass $M$ of the black hole and the distance $D$ of the black hole from the
observer. One therefore requires independent measurement of the mass and
distance to evaluate the theoretical angular diameter which is then compared
with the observations.

  2. (ii)

Axis ratio that is given by

$\displaystyle\Delta A=\dfrac{\Delta\beta}{\Delta\alpha}$ (42)

Figure 3: Major axis and minor axis of shadow structure

  3. (iii)

Deviation from circularity: Another important observable associated with the
black hole shadow is the deviation from circularity. Unlike shadow of
spherically symmetric metric, the shadow structure for Kerr-like solution has
an appearance resembling the dented circular shape and that is why one needs
to define the deviation from the circular shape of the shadow. The deviation
from circularity is defined as

$\displaystyle\Delta C=\dfrac{1}{R_{\rm
avg}}\sqrt{\dfrac{1}{2\pi}\int_{0}^{2\pi}d\phi\left\\{\ell(\phi)-R_{\rm
avg}\right\\}}$ (43)

where
$\ell(\phi)=\sqrt{\left\\{\alpha(\phi)-\alpha_{c}\right\\}^{2}+\beta^{2}(\phi)}$
corresponds to the distance between the geometrical centre and any point
$(\alpha,\beta)$ on the shadow curve with azimuthal angle $\phi$. The
geometric centre for shadow is given by $\alpha_{c}=\int\alpha
d\mathcal{A}/\int d\mathcal{A}$ and $\beta_{c}=0$ (due to reflection symmetry
around minor axis), where $d\mathcal{A}$ is the area element while the average
radius $R_{\rm avg}$ assumes the form,

$\displaystyle R_{\rm
avg}=\sqrt{\dfrac{1}{2\pi}\int_{0}^{2\pi}d\phi\,\ell^{2}(\phi)}$ (44)

###  4.1 Constrains on the magnetic monopole charge from the image of M87*

In this section, we compare the theoretically computed shadow outline with the
observed image of M87*. Observational results of the Event Horizon Telescope
(EHT) collaboration [4, 8, 9] has imposed bounds on the angular diameter, the
deviation from circularity and the axis ratio. These bounds are:

  1. 1.

Angular diameter: $\Phi=(42\pm 3)\mu as$

  2. 2.

Deviation from circularity: $\Delta C\lesssim 10\%$

  3. 3.

Axis ratio: $\Delta A\lesssim 4/3$

(a) Contours represent the angular diameter of the shadow of M87* calculated
with mass $M\simeq 3.5\times 10^{9}M_{\odot}$ and distance $D\simeq 16.8\rm
Mpc$

(b) Contours represent the angular diameter of the shadow of M87* calculated
with mass $M\simeq 6.2\times 10^{9}M_{\odot}$ and distance $D\simeq 16.8\rm
Mpc$

Figure 4: Variation of the angular diameter of M87* in $(a,g)$ plane has been
depicted.  Figure 5: The variation of angular diameter of shadow for M87* with
$a$ and $g$ for mass $M\simeq 6.5\times 10^{9}M_{\odot}$ and distance
$D=16.8\rm Mpc$

Analysing the motion of stars and gas clouds moving near the M87* super
massive black hole, the mass of the source has been measured. Study of gas
dynamics points out the mass of M87* to be $M\simeq 3.5_{-0.3}^{+0.9}\times
10^{9}M_{\odot}$ [78], whereas the stellar dynamics measurements suggest the
mass to be $M\simeq 6.2_{-0.5}^{+1.1}\times 10^{9}M_{\odot}$ [79]. The
distance of M87* has been reported $D=(16.8\pm 0.8)\rm Mpc$ (from stellar
population measurement) [80, 81, 82]. Assuming the jet axis to coincide with
the rotation axis, the inclination angle has been estimated to be $17^{0}$ [4,
8, 9]. Additionally, from the observed angular diameter of M87* EHT
collaboration reports the mass of the black hole to be $M\approx 6.5\times
10^{9}M_{\odot}$.

(a) The above figure presents the variation of deviation from circularity
$\Delta C$ with $g$ and $a$ assuming an inclination angle of $17^{\circ}$
corresponding to M87*

(b) The above figure presents the variation of axis ratio $\Delta A$ with $g$
and $a$ assuming an inclination angle of $17^{\circ}$ corresponding to M87*

Figure 6: In above two figures we have shown the variation of axis ratio
$\Delta A$ and deviation from circularity $\Delta C$ as a function of metric
parameter $g$ and spin parameter $a$. In these cases we have taken the
inclination angle $17^{\circ}$ for M87*

We recall at this point that independent measurements of mass and distance are
required to obtain the theoretical angular diameter (see 41) whereas the
dependence of $\Phi$ on the background metric is encoded in $\Delta\beta$
(where $\Delta\beta=\beta_{t}-\beta_{b}$ as shown in 3). In 4 and 5 the
variation of the angular diameter has been shown with the change in the
magnetic monopole charge parameter $g$ and the spin parameter $a$. In 4a the
angular diameter is calculated with mass $3.5\times 10^{9}M_{\odot}$ while in
4b the angular diameter is evaluated with mass $6.2\times 10^{9}M_{\odot}$. In
both cases the distance is taken to be $16.8$ Mpc. The observed angular
diameter of M87* reported by the Event Horizon Telescope collaboration turns
out to be $(42\pm 3)\mu as$. The EHT collaboration also states that the
observed image can have a maximum offset of $10\%$ which implies that the
shadow of M87* can be as small as $(37.8\pm 2.7)\mu as$. From 4a it is clear
that with mass $3.5\times 10^{9}M_{\odot}$ the observed angular diameter
cannot be reproduced. When $M\simeq 6.2\times 10^{9}M_{\odot}$ is used, the
maximum value of $g$ that can explain the observed angular diameter with
$10\%$ offset and upto 1-$\sigma$ (37.8-2.7=35.1) corresponds to $g\simeq 0.3$
(denoted by the red dashed line in 4b). The red solid line corresponds to the
centroid value of the angular diameter with $10\%$ offset. We however do not
emphasize on this result because the spin corresponding to the red solid line
is $a<0.5$. In this regard it is important to mention that numerical
simulations indicate the spin of M87* to be either $a=0.5$ or $a=0.94$ [8].
This result is obtained by fitting the numerical models with the observed jet
power of M87*. This is the reason we draw vertical lines in 4 and 5
corresponding to $a=0.5$.

For completeness we mention that when $M\simeq 6.5\times 10^{9}M_{\odot}$ is
used to calculate the theoretical angular diameter $g\lesssim 0.02$ is
required to explain the observed angular diameter within 1-$\sigma$ (blue
dashed line in 5. When $10\%$ offset in the observed angular diameter is
considered then $g\lesssim 0.16$ is necessary (red solid line in 5) to explain
the observations. We do not emphasize much on this result because in this case
the mass itself is derived from the observed shadow. With this mass therefore
we cannot establish any constrains on $g$. We further note that a non-zero $g$
causes a decrease in the shadow size (2), and as a result a greater mass is
required (as reported by the EHT collaboration) to explain the observed
angular diameter (41). Since a Kerr black hole casts a larger shadow than the
Bardeen black hole, the observed image of M87* can be better explained by the
Kerr scenario in general relativity.

Another important and independent constraint on the non-linear electrodynamics
charge parameter can be established with the help of the observable deviation
from circularity $\Delta C$. According to the EHT collaboration the deviation
from circularity for M87* is $\Delta C\lesssim 10\%$ [4]. This leads to the
constraint $g\lesssim 0.2$ from 6a. which is deduced from the fact that for
M87* $|a|\gtrsim 0.5$. The third observable $\Delta A$ cannot provide any
bound on $g$ because the maximum value of $\Delta A$ from 6b is 1.025 which is
less than the EHT observation of $\Delta A\lesssim 4/3$ [4].

###  4.2 Constrains on the magnetic monopole charge from the image of Sgr A*

In this section we compare the theoretical angular diameter with the observed
image of Sgr A*. According to the EHT collaboration the angular diameter of
the emission ring of Sgr A* turns out to be $(51.8\pm 2.3)\mu as$ [83, 84, 85,
86, 87, 88]. However, the angular diameter of the shadow is $(48.7\pm 7)\mu
as$[83, 84, 85, 86, 87, 88]. Since computation of the theoretical angular
diameter depends on the mass and distance of the source, we first discuss the
previously estimated distance and mass of Sgr A*.

(a) Contours represent variation of angular diameter of Sgr A* with $a$ and
$g$ for mass $M\simeq 3.951\times 10^{6}M_{\odot}$ and distance $D\simeq
7.935\rm pc$

(b) Contours represent variation of angular diameter of Sgr A* with $a$ and
$g$ for mass $M\simeq 3.975\times 10^{6}M_{\odot}$ and distance $D\simeq
7.959\rm Kpc$

(c) Contours represent variation of angular diameter of Sgr A* with $a$ and
$g$ for mass $M\simeq 4.261\times 10^{6}M_{\odot}$ and distance $D\simeq
8.277\rm Kpc$

(d) Contours represent variation of angular diameter of Sgr A* with $a$ and
$g$ for mass $M\simeq 4.297\times 10^{6}M_{\odot}$ and distance $D\simeq
8.2467\rm Kpc$

Figure 7: Variation of angular diameter with $g$ and $a$ with various mass and
distance estimations for Sgr A*. The inclination angle is taken to be
$134^{\circ}$.

The mass and distance of Sgr A* has been reported by several groups. According
to the measurements of the Keck team Sgr A* has a distance of $D=(7959\pm
59\pm 32)\rm pc$ and mass $M=(3.975\pm 0.058\pm 0.026)\times 10^{6}M_{\odot}$,
which is obtained by keeping the red-shift parameter free [89]. The same group
has reported distance $D=(7935\pm 50)\rm pc$ and mass $M=(3.951\pm
0.047)\times 10^{6}M_{\odot}$, assuming the red-shift parameter to be unity
[89]. Estimates by the Gravity collaboration suggest the mass of Sgr A* to be
$M=(4.261\pm 0.012)\rm pc$ and distance $D=(8246.7\pm 9.3)\rm pc$ [90, 91]. By
taking into account the optical aberrations the Gravity collaboration further
constrained the BH mass $M=(4.297\pm 0.012\pm 0.040)\times 10^{6}M_{\odot}$
and the distance $D=8277\pm 9\pm 33\rm pc$. For obtaining the theoretical
angular diameter of Sgr A* one needs the inclination angle of the BH, which in
our case is $i=134^{\circ}$ [92]. Using this inclination angle we plot the
variation of the angular diameter with the charge parameter $g$ and the
rotation paameter $a$ in 7 for various combinations of masses and distances as
discussed above.

(a) Above figure shows the variation of the deviation from circularity $\Delta
C$ with $g$ and $a$ for Sgr A* with an inclination angle $134^{\circ}$

(b) Above figure shows the variation of the axis ratio $\Delta A$ with $g$ and
$a$ for Sgr A* with an inclination angle $134^{\circ}$

Figure 8: We have plotted the variation of axis ratio $\Delta A$ and deviation
from circularity $\Delta C$ in the $(g-a)$ plane for Sgr A* with an
inclination angle $134^{\circ}$

When mass and distance reported by the Keck team (7a and 7b) is used,
$0.15\lesssim g\lesssim 0.2$ is required to reproduce the centroid value of
the observed shadow diameter ($48.7\mu as$, denoted by the red solid line in
7a and 7b). We further note that $0.04\lesssim g\lesssim 0.1$ can explain the
observed image diameter within 1-$\sigma$ (denoted by the blue dashed line in
7a and 7b). Like M87*, for Sgr A* as well numerical models that best explain
the observed emission ring correspond to $|a|=0.5$ and $|a|=0.94$ [87, 88].
This is consistent with the spin estimates of Sgr A* from its radio spectrum
[93, 94, 95] which reports $a\sim 0.9$ [94] and $a\sim 0.5$ [95]. Therefore in
7 the vertical dashed lines are drawn for $|a|=0.5$.

When mass and distance reported by the Gravity collaboration is used
$0.34\lesssim g\lesssim 0.36$ is required to explain the observed shadow
diameter (which is $48.7\mu as$ denoted by the red solid line in 7c and 7d).
However, when one intends to explain the diameter of the centroid value of the
emission ring (51.8 $\mu as$ denoted by the blue solid line in 7c and 7d ) $g$
can be as high as $0.06$ for $M=4.261\times 10^{6}M_{\odot}$ and $D=8.277$ kpc
and $0.08$ for $M=4.297\times 10^{6}M_{\odot}$ and $D=8.2467$ kpc. When the
1-$\sigma$ interval in the emission ring is considered a higher value of the
magnetic monopole charge parameter $0.25\lesssim g\lesssim 0.3$ is allowed.
This contour is represented by the blue dashed line in 7c and 7d. From the
above discussion it is clear that a small but non-zero value of $g$ can
explain the observed shadow of Sgr A* better than the Kerr scenario. This is
attributed to the fact that increase in $g$ leads to a decrease in the shadow
size (2)

The Event Horizon Telescope collaboration does not impose any constrains on
$\Delta C$ and $\Delta A$ for the image of Sgr A*. We however present plots
corresponding to the two aforesaid observables assuming the inclination angle
$i=134^{\circ}$ which can be used for future work once these bounds are
reported by the EHT collaboration. This is illustrated in 8 for completeness.

##  5 Conclusion

In this work we investigate the role of the magnetic monopole charge arising
in the Bardeen black hole scenario in explaining the observed shadow of M87*
and Sgr A*. Bardeen black holes are interesting as they have a regular core
and hence can potentially avoid the $r=0$ curvature singularity arising in
general relativity. In the absence of a well accepted quantum theory of
gravity regular black holes provide a classical route to address the
singularity issue in GR thereby making the theory finite at all length scales.
Such black holes generally arise in Einstein gravity coupled to non-linear
electrodynamics which becomes relevant when the electromagnetic field is very
strong. Therefore, it is worthwhile to explore the signatures of regular black
holes in astrophysical observations. With two successive release of black hole
images by the EHT collaboration, the scope to test the nature of srong gravity
has enhanced considerably.

We study the nature of the shadow cast by Bardeen black holes and explore the
effect of the magnetic monopole charge parameter on the shape and size of the
shadow. Such a study reveals that with an increase in the inclination angle
and the black hole spin the shadow becomes more dented. Further, an increase
in the magnetic monopole charge parameter leads to a decrease in the shadow
size. This feature plays a pivotal role in explaining the observed angular
diameter of the image of Sgr A* and M87*.

For M87*, when the theoretical angular diameter is calculated with mass
estimated from gas dynamics measurements, the observed shadow diameter of
$42\pm 3\mu as$ cannot be reproduced. When mass reported from stellar dynamics
measurements is used, the observed shadow diameter can be explained within
1-$\sigma$ only when the maximum offset of 10% is considered in the image
diameter. Since a non-zero magnetic monopole charge $g$ shrinks the shadow,
the centroid value of the observed shadow of M87* can be better explained by
the Kerr scenario.

For Sgr A*, the mass and distance is very well constrained. When the distance
and mass estimated by the Keck team is used, $0.15\lesssim g\lesssim 0.2$ is
required to reproduce the centroid value of the observed angular diameter of
Sgr A*. Similarly, $0.34\lesssim g\lesssim 0.36$ explains the central value of
the shadow diameter ($48.7\mu as$) of Sgr A* when mass and distance reported
by the Gravity collaboration is used. Therefore, for Sgr A* the Bardeen black
hole scenario better explains its observed shadow.

We have previously studied the prospect of Bardeen black holes in explaining
the optical observations of quasars [63] and found that quasar optical data
favors the Kerr scenario. A similar conclusion has been reached when we
studied the possibility of Bardeen black holes in explaining the observed
quasi-periodic oscillations [62]. Finally in the present work we note that the
shadow of M87* can be better explained by the Kerr scenario while the image of
Sgr A* favors the Bardeen black hole framework. Thus, although rare, black
holes do exhibit signatures of magnetic monopole charge in certain cases. The
study of the nature of spacetime with respect to different black holes can be
better accomplished in the near future with the availability of more images of
black holes with improved resolution.

## Acknowledgements

## References

  * [1] LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116 no. 6, (2016) 061102, arXiv:1602.03837 [gr-qc].
  * [2] LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., “GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral,” Phys. Rev. Lett. 119 no. 16, (2017) 161101, arXiv:1710.05832 [gr-qc].
  * [3] Event Horizon Telescope Collaboration, V. L. Fish, K. Akiyama, K. L. Bouman, A. A. Chael, M. D. Johnson, S. S. Doeleman, L. Blackburn, J. F. C. Wardle, and W. T. Freeman, “Observing — and Imaging — Active Galactic Nuclei with the Event Horizon Telescope,” Galaxies 4 no. 4, (2016) 54, arXiv:1607.03034 [astro-ph.IM].
  * [4] Event Horizon Telescope Collaboration, K. Akiyama et al., “First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,” Astrophys. J. 875 no. 1, (2019) L1, arXiv:1906.11238 [astro-ph.GA].
  * [5] Event Horizon Telescope Collaboration, K. Akiyama et al., “First M87 Event Horizon Telescope Results. II. Array and Instrumentation,” Astrophys. J. 875 no. 1, (2019) L2, arXiv:1906.11239 [astro-ph.IM].
  * [6] Event Horizon Telescope Collaboration, K. Akiyama et al., “First M87 Event Horizon Telescope Results. III. Data Processing and Calibration,” Astrophys. J. 875 no. 1, (2019) L3, arXiv:1906.11240 [astro-ph.GA].
  * [7] Event Horizon Telescope Collaboration, K. Akiyama et al., “First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole,” Astrophys. J. 875 no. 1, (2019) L4, arXiv:1906.11241 [astro-ph.GA].
  * [8] Event Horizon Telescope Collaboration, K. Akiyama et al., “First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring,” Astrophys. J. 875 no. 1, (2019) L5, arXiv:1906.11242 [astro-ph.GA].
  * [9] Event Horizon Telescope Collaboration, K. Akiyama et al., “First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole,” Astrophys. J. 875 no. 1, (2019) L6, arXiv:1906.11243 [astro-ph.GA].
  * [10] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2, 2011.
  * [11] R. Penrose, “Gravitational collapse and space-time singularities,” Phys. Rev. Lett. 14 (1965) 57–59.
  * [12] S. W. Hawking, “Breakdown of Predictability in Gravitational Collapse,” Phys. Rev. D14 (1976) 2460–2473.
  * [13] D. Christodoulou, “The formation of black holes and singularities in spherically symmetric gravitational collapse,” Commun. Pure Appl. Math. 44 no. 3, (1991) 339–373.
  * [14] P. Horava and E. Witten, “Heterotic and type I string dynamics from eleven-dimensions,” Nucl.Phys. B460 (1996) 506–524, arXiv:hep-th/9510209 [hep-th].
  * [15] P. Horava and E. Witten, “Eleven-dimensional supergravity on a manifold with boundary,” Nucl.Phys. B475 (1996) 94–114, arXiv:hep-th/9603142 [hep-th].
  * [16] J. Polchinski, “String theory. Vol. 1: An introduction to the bosonic string,”.
  * [17] J. Polchinski, “String theory. Vol. 2: Superstring theory and beyond,”.
  * [18] A. Ashtekar, T. Pawlowski, and P. Singh, “Quantum nature of the big bang,” Phys. Rev. Lett. 96 (2006) 141301, arXiv:gr-qc/0602086 [gr-qc].
  * [19] D. Kothawala, “Minimal Length and Small Scale Structure of Spacetime,” Phys. Rev. D88 no. 10, (2013) 104029, arXiv:1307.5618 [gr-qc].
  * [20] E. Ayon-Beato and A. Garcia, “The Bardeen model as a nonlinear magnetic monopole,” Phys. Lett. B 493 (2000) 149–152, arXiv:gr-qc/0009077.
  * [21] E. Ayon-Beato and A. Garcia, “Regular black hole in general relativity coupled to nonlinear electrodynamics,” Phys. Rev. Lett. 80 (1998) 5056–5059, arXiv:gr-qc/9911046.
  * [22] E. Ayon-Beato and A. Garcia, “Four parametric regular black hole solution,” Gen. Rel. Grav. 37 (2005) 635, arXiv:hep-th/0403229.
  * [23] K. A. Bronnikov, “Regular magnetic black holes and monopoles from nonlinear electrodynamics,” Phys. Rev. D 63 (2001) 044005, arXiv:gr-qc/0006014.
  * [24] A. Borde, “Regular black holes and topology change,” Phys. Rev. D 55 (1997) 7615–7617, arXiv:gr-qc/9612057.
  * [25] C. Barrabes and V. P. Frolov, “How many new worlds are inside a black hole?,” Phys. Rev. D 53 (1996) 3215–3223, arXiv:hep-th/9511136.
  * [26] E. Ayon-Beato and A. Garcia, “Nonsingular charged black hole solution for nonlinear source,” Gen. Rel. Grav. 31 (1999) 629–633, arXiv:gr-qc/9911084.
  * [27] A. Bonanno and M. Reuter, “Renormalization group improved black hole space-times,” Phys. Rev. D 62 (2000) 043008, arXiv:hep-th/0002196.
  * [28] P. Nicolini, A. Smailagic, and E. Spallucci, “Noncommutative geometry inspired Schwarzschild black hole,” Phys. Lett. B 632 (2006) 547–551, arXiv:gr-qc/0510112.
  * [29] Y. S. Myung, Y.-W. Kim, and Y.-J. Park, “Quantum Cooling Evaporation Process in Regular Black Holes,” Phys. Lett. B 656 (2007) 221–225, arXiv:gr-qc/0702145.
  * [30] S. A. Hayward, “Formation and evaporation of nonsingular black holes,” Phys. Rev. Lett. 96 (Jan, 2006) 031103. https://link.aps.org/doi/10.1103/PhysRevLett.96.031103.
  * [31] V. P. Frolov, M. A. Markov, and V. F. Mukhanov, “Black Holes as Possible Sources of Closed and Semiclosed Worlds,” Phys. Rev. D 41 (1990) 383.
  * [32] V. F. Mukhanov and R. H. Brandenberger, “A Nonsingular universe,” Phys. Rev. Lett. 68 (1992) 1969–1972.
  * [33] R. H. Brandenberger, V. F. Mukhanov, and A. Sornborger, “A Cosmological theory without singularities,” Phys. Rev. D 48 (1993) 1629–1642, arXiv:gr-qc/9303001.
  * [34] S. Ansoldi, “Spherical black holes with regular center: A Review of existing models including a recent realization with Gaussian sources,” in Conference on Black Holes and Naked Singularities.  2, 2008.  arXiv:0802.0330 [gr-qc].
  * [35] E. B. Gliner, “Algebraic Properties of the Energy-momentum Tensor and Vacuum-like States o+ Matter,” Soviet Journal of Experimental and Theoretical Physics 22 (Feb., 1966) 378.
  * [36] Z.-Y. Fan and X. Wang, “Construction of Regular Black Holes in General Relativity,” Phys. Rev. D 94 no. 12, (2016) 124027, arXiv:1610.02636 [gr-qc].
  * [37] A. D. Sakharov, “The Initial Stage of an Expanding Universe and the Appearance of a Nonuniform Distribution of Matter,” Soviet Journal of Experimental and Theoretical Physics 22 (Jan., 1966) 241.
  * [38] A. Burinskii and S. R. Hildebrandt, “New type of regular black holes and particle - like solutions from NED,” Phys. Rev. D 65 (2002) 104017, arXiv:hep-th/0202066.
  * [39] W. Berej, J. Matyjasek, D. Tryniecki, and M. Woronowicz, “Regular black holes in quadratic gravity,” Gen. Rel. Grav. 38 (2006) 885–906, arXiv:hep-th/0606185.
  * [40] L. Balart and E. C. Vagenas, “Regular black holes with a nonlinear electrodynamics source,” Phys. Rev. D 90 no. 12, (2014) 124045, arXiv:1408.0306 [gr-qc].
  * [41] E. L. B. Junior, M. E. Rodrigues, and M. J. S. Houndjo, “Regular black holes in $f(T)$ Gravity through a nonlinear electrodynamics source,” JCAP 10 (2015) 060, arXiv:1503.07857 [gr-qc].
  * [42] S. N. Sajadi and N. Riazi, “Nonlinear electrodynamics and regular black holes,” Gen. Rel. Grav. 49 no. 3, (2017) 45. [Erratum: Gen.Rel.Grav. 52, 18 (2020)].
  * [43] I. Dymnikova, “Vacuum nonsingular black hole,” Gen. Rel. Grav. 24 (1992) 235–242.
  * [44] K. A. Bronnikov, “Comment on “Construction of regular black holes in general relativity”,” Phys. Rev. D 96 no. 12, (2017) 128501, arXiv:1712.04342 [gr-qc].
  * [45] K. A. Bronnikov and J. C. Fabris, “Regular phantom black holes,” Phys. Rev. Lett. 96 (2006) 251101, arXiv:gr-qc/0511109.
  * [46] R. Kumar and S. G. Ghosh, “Photon ring structure of rotating regular black holes and no-horizon spacetimes,” Class. Quant. Grav. 38 no. 8, (2021) 8, arXiv:2004.07501 [gr-qc].
  * [47] S. G. Ghosh, D. V. Singh, and S. D. Maharaj, “Regular black holes in Einstein-Gauss-Bonnet gravity,” Phys. Rev. D 97 no. 10, (2018) 104050.
  * [48] B. Toshmatov, Z. Stuchlík, and B. Ahmedov, “Comment on “Construction of regular black holes in general relativity”,” Phys. Rev. D 98 no. 2, (2018) 028501, arXiv:1807.09502 [gr-qc].
  * [49] E. T. Newman and A. I. Janis, “Note on the Kerr spinning particle metric,” J. Math. Phys. 6 (1965) 915–917.
  * [50] M. Azreg-Aïnou, “Generating rotating regular black hole solutions without complexification,” Phys. Rev. D 90 no. 6, (2014) 064041, arXiv:1405.2569 [gr-qc].
  * [51] M. Azreg-Aïnou, “From static to rotating to conformal static solutions: Rotating imperfect fluid wormholes with(out) electric or magnetic field,” Eur. Phys. J. C 74 no. 5, (2014) 2865, arXiv:1401.4292 [gr-qc].
  * [52] M. Azreg-Ainou, “Regular and conformal regular cores for static and rotating solutions,” Phys. Lett. B 730 (2014) 95–98, arXiv:1401.0787 [gr-qc].
  * [53] R. Kumar, A. Kumar, and S. G. Ghosh, “Testing Rotating Regular Metrics as Candidates for Astrophysical Black Holes,” Astrophys. J. 896 no. 1, (2020) 89, arXiv:2006.09869 [gr-qc].
  * [54] A. Allahyari, M. Khodadi, S. Vagnozzi, and D. F. Mota, “Magnetically charged black holes from non-linear electrodynamics and the Event Horizon Telescope,” JCAP 02 (2020) 003, arXiv:1912.08231 [gr-qc].
  * [55] S. Vagnozzi, R. Roy, Y.-D. Tsai, and L. Visinelli, “Horizon-scale tests of gravity theories and fundamental physics from the Event Horizon Telescope image of Sagittarius A∗,” arXiv:2205.07787 [gr-qc].
  * [56] A. Uniyal, R. C. Pantig, and A. Övgün, “Probing a nonlinear electrodynamics black hole with thin accretion disk, shadow and deflection angle with M87* and Sgr A* from EHT,” arXiv:2205.11072 [gr-qc].
  * [57] Z. Stuchlík and J. Schee, “Circular geodesic of Bardeen and Ayon–Beato–Garcia regular black-hole and no-horizon spacetimes,” Int. J. Mod. Phys. D 24 no. 02, (2014) 1550020, arXiv:1501.00015 [astro-ph.HE].
  * [58] J. Schee and Z. Stuchlik, “Gravitational lensing and ghost images in the regular Bardeen no-horizon spacetimes,” JCAP 06 (2015) 048, arXiv:1501.00835 [astro-ph.HE].
  * [59] J. Schee and Z. Stuchlík, “Profiled spectral lines generated by Keplerian discs orbiting in the Bardeen and Ayòn-Beato–Garcìa spacetimes,” Class. Quant. Grav. 33 no. 8, (2016) 085004, arXiv:1604.00632 [gr-qc].
  * [60] Z. Stuchlík and J. Schee, “Shadow of the regular Bardeen black holes and comparison of the motion of photons and neutrinos,” Eur. Phys. J. C 79 no. 1, (2019) 44.
  * [61] J. Schee and Z. Stuchlik, “Profiled spectral lines of Keplerian rings orbiting in the regular Bardeen black hole spacetimes,” Eur. Phys. J. C 79 no. 12, (2019) 988, arXiv:1908.07197 [gr-qc].
  * [62] I. Banerjee, “Deciphering signatures of Bardeen black holes from the observed quasi-periodic oscillations,” JCAP 05 no. 05, (2022) 020, arXiv:2201.00679 [gr-qc].
  * [63] I. Banerjee, V. S. Chawan, B. Mandal, S. K. Sahoo, and S. SenGupta, “Quasar continuum spectrum disfavors black holes with a magnetic monopole charge,” Phys. Rev. D 105 no. 6, (2022) 064073, arXiv:2112.05385 [gr-qc].
  * [64] I. Banerjee, “Testing black holes in non-linear electrodynamics from the observed quasi-periodic oscillations,” arXiv:2203.10890 [gr-qc].
  * [65] P. V. P. Cunha and C. A. R. Herdeiro, “Shadows and strong gravitational lensing: a brief review,” Gen. Rel. Grav. 50 no. 4, (2018) 42, arXiv:1801.00860 [gr-qc].
  * [66] A. de Vries, “The apparent shape of a rotating charged black hole, closed photon orbits and the bifurcation setA4,” Classical and Quantum Gravity 17 no. 1, (Dec, 1999) 123–144. https://doi.org/10.1088%2F0264-9381%2F17%2F1%2F309.
  * [67] S. E. Gralla, D. E. Holz, and R. M. Wald, “Black Hole Shadows, Photon Rings, and Lensing Rings,” Phys. Rev. D100 no. 2, (2019) 024018, arXiv:1906.00873 [astro-ph.HE].
  * [68] A. A. Abdujabbarov, L. Rezzolla, and B. J. Ahmedov, “A coordinate-independent characterization of a black hole shadow,” Mon. Not. Roy. Astron. Soc. 454 no. 3, (2015) 2423–2435, arXiv:1503.09054 [gr-qc].
  * [69] A. Abdujabbarov, M. Amir, B. Ahmedov, and S. G. Ghosh, “Shadow of rotating regular black holes,” Phys. Rev. D93 no. 10, (2016) 104004, arXiv:1604.03809 [gr-qc].
  * [70] C. Bambi, K. Freese, S. Vagnozzi, and L. Visinelli, “Testing the rotational nature of the supermassive object M87* from the circularity and size of its first image,” Phys. Rev. D100 no. 4, (2019) 044057, arXiv:1904.12983 [gr-qc].
  * [71] K. Hioki and K.-i. Maeda, “Measurement of the Kerr Spin Parameter by Observation of a Compact Object’s Shadow,” Phys. Rev. D80 (2009) 024042, arXiv:0904.3575 [astro-ph.HE].
  * [72] S. Vagnozzi and L. Visinelli, “Hunting for extra dimensions in the shadow of M87*,” Phys. Rev. D100 no. 2, (2019) 024020, arXiv:1905.12421 [gr-qc].
  * [73] I. Banerjee, S. Chakraborty, and S. SenGupta, “Silhouette of M87*: A New Window to Peek into the World of Hidden Dimensions,” Phys. Rev. D 101 no. 4, (2020) 041301, arXiv:1909.09385 [gr-qc].
  * [74] I. Banerjee, S. Sau, and S. SenGupta, “Signatures of regular black holes from the shadow of Sgr A* and M87*,” arXiv:2206.12125 [gr-qc].
  * [75] I. H. Salazar, A. Garcia, and J. Plebanski, “Duality Rotations and Type $D$ Solutions to Einstein Equations With Nonlinear Electromagnetic Sources,” J. Math. Phys. 28 (1987) 2171–2181.
  * [76] R. Kumar and S. G. Ghosh, “Black hole parameters estimation from its shadow,” arXiv:1811.01260 [gr-qc].
  * [77] S. E. Vazquez and E. P. Esteban, “Strong field gravitational lensing by a Kerr black hole,” Nuovo Cim. B 119 (2004) 489–519, arXiv:gr-qc/0308023.
  * [78] J. L. Walsh, A. J. Barth, L. C. Ho, and M. Sarzi, “The M87 Black Hole Mass from Gas-dynamical Models of Space Telescope Imaging Spectrograph Observations,” Astrophys. J. 770 (2013) 86, arXiv:1304.7273 [astro-ph.CO].
  * [79] K. Gebhardt, J. Adams, D. Richstone, T. R. Lauer, S. M. Faber, K. Gultekin, J. Murphy, and S. Tremaine, “The Black-Hole Mass in M87 from Gemini/NIFS Adaptive Optics Observations,” Astrophys. J. 729 (2011) 119, arXiv:1101.1954 [astro-ph.CO].
  * [80] J. P. Blakeslee, A. Jordan, S. Mei, P. Cote, L. Ferrarese, L. Infante, E. W. Peng, J. L. Tonry, and M. J. West, “The ACS Fornax Cluster Survey. V. Measurement and Recalibration of Surface Brightness Fluctuations and a Precise Value of the Fornax–Virgo Relative Distance,” Astrophys. J. 694 (2009) 556–572, arXiv:0901.1138 [astro-ph.CO].
  * [81] S. Bird, W. E. Harris, J. P. Blakeslee, and C. Flynn, “The Inner Halo of M87: A First Direct View of the Red-Giant Population,” Astron. Astrophys. 524 (2010) A71, arXiv:1009.3202 [astro-ph.GA].
  * [82] M. Cantiello et al., “A Precise Distance to the Host Galaxy of the Binary Neutron Star Merger GW170817 Using Surface Brightness Fluctuations,” Astrophys. J. 854 no. 2, (2018) L31, arXiv:1801.06080 [astro-ph.GA].
  * [83] Event Horizon Telescope Collaboration, K. Akiyama et al., “First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way,” Astrophys. J. Lett. 930 no. 2, (2022) L12.
  * [84] Event Horizon Telescope Collaboration, K. Akiyama et al., “First Sagittarius A* Event Horizon Telescope Results. II. EHT and Multiwavelength Observations, Data Processing, and Calibration,” Astrophys. J. Lett. 930 no. 2, (2022) L13.
  * [85] Event Horizon Telescope Collaboration, K. Akiyama et al., “First Sagittarius A* Event Horizon Telescope Results. III. Imaging of the Galactic Center Supermassive Black Hole,” Astrophys. J. Lett. 930 no. 2, (2022) L14.
  * [86] Event Horizon Telescope Collaboration, K. Akiyama et al., “First Sagittarius A* Event Horizon Telescope Results. IV. Variability, Morphology, and Black Hole Mass,” Astrophys. J. Lett. 930 no. 2, (2022) L15.
  * [87] Event Horizon Telescope Collaboration, K. Akiyama et al., “First Sagittarius A* Event Horizon Telescope Results. V. Testing Astrophysical Models of the Galactic Center Black Hole,” Astrophys. J. Lett. 930 no. 2, (2022) L16.
  * [88] Event Horizon Telescope Collaboration, K. Akiyama et al., “First Sagittarius A* Event Horizon Telescope Results. VI. Testing the Black Hole Metric,” Astrophys. J. Lett. 930 no. 2, (2022) L17.
  * [89] T. Do et al., “Relativistic redshift of the star S0-2 orbiting the Galactic center supermassive black hole,” Science 365 no. 6454, (2019) 664–668, arXiv:1907.10731 [astro-ph.GA].
  * [90] GRAVITY Collaboration, R. Abuter et al., “Mass distribution in the Galactic Center based on interferometric astrometry of multiple stellar orbits,” Astron. Astrophys. 657 (2022) L12, arXiv:2112.07478 [astro-ph.GA].
  * [91] GRAVITY Collaboration, R. Abuter et al., “Detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole,” Astron. Astrophys. 636 (2020) L5, arXiv:2004.07187 [astro-ph.GA].
  * [92] Gravity Collaboration, R. Abuter, A. Amorim, M. Bauböck, J. P. Berger, H. Bonnet, W. Brandner, Y. Clénet, V. Coudé Du Foresto, P. T. de Zeeuw, J. Dexter, G. Duvert, A. Eckart, F. Eisenhauer, N. M. Förster Schreiber, P. Garcia, F. Gao, E. Gendron, R. Genzel, O. Gerhard, S. Gillessen, M. Habibi, X. Haubois, T. Henning, S. Hippler, M. Horrobin, A. Jiménez-Rosales, L. Jocou, P. Kervella, S. Lacour, V. Lapeyrère, J. B. Le Bouquin, P. Léna, T. Ott, T. Paumard, K. Perraut, G. Perrin, O. Pfuhl, S. Rabien, G. Rodriguez Coira, G. Rousset, S. Scheithauer, A. Sternberg, O. Straub, C. Straubmeier, E. Sturm, L. J. Tacconi, F. Vincent, S. von Fellenberg, I. Waisberg, F. Widmann, E. Wieprecht, E. Wiezorrek, J. Woillez, and S. Yazici, “A geometric distance measurement to the Galactic center black hole with 0.3% uncertainty,” aap 625 (May, 2019) L10, arXiv:1904.05721 [astro-ph.GA].
  * [93] C. S. Reynolds, “The Spin of Supermassive Black Holes,” Class. Quant. Grav. 30 (2013) 244004, arXiv:1307.3246 [astro-ph.HE].
  * [94] M. Moscibrodzka, C. F. Gammie, J. C. Dolence, H. Shiokawa, and P. K. Leung, “Radiative Models of Sgr A* from GRMHD Simulations,” Astrophys. J. 706 (2009) 497–507, arXiv:0909.5431 [astro-ph.HE].
  * [95] R. V. Shcherbakov, R. F. Penna, and J. C. McKinney, “Sagittarius A* Accretion Flow and Black Hole Parameters from General Relativistic Dynamical and Polarized Radiative Modeling,” Astrophys. J. 755 (2012) 133, arXiv:1007.4832 [astro-ph.HE].