# Testing Rotating Regular Metrics with EHT Results of Sgr A*

Rahul Kumar Walia Astrophysics Research Centre, School of Mathematics,
Statistics and Computer Science, University of KwaZulu-Natal, Private Bag
54001, Durban 4000, South Africa  Sushant G. Ghosh Centre for Theoretical
Physics, Jamia Millia Islamia, New Delhi 110025, India  Astrophysics Research
Centre, School of Mathematics, Statistics and Computer Science, University of
KwaZulu-Natal, Private Bag 54001, Durban 4000, South Africa  Sunil D. Maharaj
Astrophysics Research Centre, School of Mathematics, Statistics and Computer
Science, University of KwaZulu-Natal, Private Bag 54001, Durban 4000, South
Africa

###### Abstract

The Event Horizon Telescope (EHT) observation unveiled the first image of
supermassive black hole Sgr A* showing a shadow of diameter
$\theta_{sh}=48.7\pm 7\,\mu$as with fractional deviation from the
Schwarzschild black hole shadow diameter
$\delta=-0.08^{+0.09}_{-0.09}\leavevmode\nobreak\
\text{(VLTI)},-0.04^{+0.09}_{-0.10}\leavevmode\nobreak\ \text{(Keck)}$. The
Sgr A* shadow size is within $\leavevmode\nobreak\ 10\%$ of the Kerr
predictions, providing us another tool to investigate the nature of strong-
field gravity. We use the Sgr A* shadow observables to constrain metrics of
four independent and well-motivated, parametrically different from Kerr
spacetime, rotating regular spacetimes and the corresponding no-horizon
spacetimes. We present constraints on the deviation parameter $g$ of rotating
regular black holes. The shadow angular diameter $\theta_{sh}$ within the
$1\sigma$ region places bounds on the parameters $a$ and $g$. Together with
EHT bounds on the $\theta_{sh}$ and $\delta$ of Sgr A*, our analysis concludes
that the three rotating regular black holes, i.e., Bardeen Hayward, and
Simpson-Visser black holes, and corresponding no-horizon spacetimes agree with
the EHT results of Sgr A*. Thus, these three rotating regular spacetimes and
Kerr black holes are indiscernible in some parameter space, and one cannot
rule out the possibility of the former being strong candidates for the
astrophysical black holes.

Galaxy: center– gravitation – black hole physics -black hole shadow-
gravitational lensing: strong

##  1 Introduction

Black holes are one of the most profound predictions of the theory of general
relativity (GR) (Einstein, 1915), which were initially thought to be a
mathematical consequence of GR rather than physically relevant objects
(Schwarzschild, 1916). However, later it was found that they are a generic and
mostly inevitable outcome of gravitational collapse (Oppenheimer & Snyder,
1939; Penrose, 1965). The no-hair theorem epitomizes the remarkable features
of the GR black hole, stating that the Kerr (1963) and Kerr-Newman (Newman et
al., 1965) are the only stationary, axially symmetric, and asymptotically
flat, respectively, vacuum and electro-vacuum solutions to the Einstein
equations (Israel, 1967, 1968; Carter, 1971; Hawking, 1972; Robinson, 1975).
Black holes embedded in the optically thin accreting region are expected to
reveal a dark “shadow” encircled by a bright ring (Bardeen, 1973; Cunningham &
Bardeen, 1973). The shadow’s boundary is defined by the photon ring, which
explicitly depends on the black hole parameters (Johannsen & Psaltis, 2010).
Therefore, the photon ring structure is an essential tool to test the theories
of gravity through the astrophysical black holes’ observational predictions.
Applications of shadow in understanding the near-horizon geometry have aroused
a flurry of activities analyzing, both analytically and numerically, shadows
for black holes in GR (Falcke et al., 2000; de Vries, 2000; Shen et al., 2005;
Yumoto et al., 2012; Atamurotov et al., 2013; Abdujabbarov et al., 2015; Cunha
& Herdeiro, 2018; Kumar & Ghosh, 2020a; Afrin & Ghosh, 2022; Hioki & Maeda,
2009) and modified theories of gravity (MTGs) (Amarilla et al., 2010; Amarilla
& Eiroa, 2012, 2013; Amir et al., 2018; Singh & Ghosh, 2018; Mizuno et al.,
2018; Allahyari et al., 2020; Papnoi et al., 2014; Kumar et al., 2020a; Kumar
& Ghosh, 2020b; Ghosh et al., 2021; Afrin & Ghosh, 2021; Vagnozzi et al.,
2022; Vagnozzi & Visinelli, 2019; Afrin et al., 2021; Afrin & Ghosh, 2021),
and found the shadows of black holes in MTGs to be smaller and more distorted
when compared with the Kerr black hole shadow. Also, the shadow of rotating
regular spacetimes, which may be realized when GR is coupled to nonlinear
electrodynamics, has been widely investigated to extract limits on the black
hole magnetic charge (Allahyari et al., 2020; Kumar et al., 2020b; Kumar &
Ghosh, 2021, 2020a; Amir & Ghosh, 2016; Kumar et al., 2019). In addition, by
observing the size and deformation of the shadow, the spin, mass parameter,
and possibly other global charges of the black holes can be estimated (Hioki &
Maeda, 2009; Tsupko, 2017; Cunha et al., 2019a, b; Kumar & Ghosh, 2020a;
Khodadi et al., 2020; Ghosh et al., 2021). Besides, the black hole shadow is
also valuable for testing theories of gravity.

The event horizon is accessible only for tests with strong field phenomena
near black holes, e.g., with the black hole shadow and gravitational waves. In
2019, the Event Horizon Telescope (EHT) collaboration released the first
horizon-scale image of the M87* supermassive black holes (Akiyama et al.,
2019a, b, c) which made black holes a physical reality. Using the M87*
distance from the Earth $D=16.8$ Mpc and estimated mass $M=(6.5\pm 0.7)\times
10^{9}M_{\odot}$, bounds could be placed on the compact emission region size
with angular diameter $\theta_{d}=42\pm 3\,\mu$as along with the central flux
depression with a factor of $\gtrsim 10$, and circularity deviation $\Delta
C\leq 0.10$. Recently, the EHT published the Sgr A* black hole shadow results,
showing shadow angular diameter $\theta_{sh}=48.7\pm 7\,\mu$as with
surrounding bright and thick emission ring of diameter $\theta_{d}=51.8\pm
2.3\mu$as (Akiyama et al., 2022a, b, c, d, e, f). Considering a black hole of
mass $M=4.0^{+1.1}_{-0.6}\times 10^{6}M_{\odot}$ and distance $D=8$kpc from
Earth, the EHT demonstrate that the Sgr A* images are consistent with the
expected appearance of a Kerr black hole (Akiyama et al., 2022a, f).
Furthermore, when compared with the EHT results for M87*, it exhibits
consistency with the predictions of GR stretching across three orders of
magnitude in central mass (Akiyama et al., 2022e).

Nevertheless, the current uncertainty in the measurement of spin and the
relative deviation of quadrupole moments do not eliminate black holes arising
in MTG (Akiyama et al., 2019a, b, c; Cardoso & Pani, 2019; Kocherlakota et
al., 2021). Sgr A* black hole shadow observational results have already been
used to test the viabilities of various non-Kerr black holes (Vagnozzi et al.,
2022; Khodadi & Lambiase, 2022; Kuang et al., 2022; Pantig & Övgün, 2022; He
et al., 2022; Uniyal et al., 2022; Ghosh & Afrin, 2022; Chen, 2022; Khodadi &
Lambiase, 2022; Uniyal et al., 2022). Photons that originate in the deep
gravitational fields of black holes form the images, and thereby carry
imprints of the spacetime properties in the strong-field regime (Jaroszynski &
Kurpiewski, 1997; Falcke et al., 2000). Therefore, the black hole shadows can
be used to perform the strong-field gravitational tests and eventually to test
the no-hair theorem (Johannsen & Psaltis, 2010; Cunha & Herdeiro, 2018; Baker
et al., 2015). The horizon-scale images of supermassive black holes provide a
theoretically new route for testing the theory of GR. Interestingly, the
rotating regular black hole shadow’s radius decreases and distortion in shadow
increases monotonically with the parameters $g$ apart from spin $a$, and also
a violation of the no-hair theorem significantly alters the shadow’s shape and
size (Bambi & Freese, 2009; Johannsen & Psaltis, 2010; Falcke & Markoff, 2013;
Johannsen, 2013a, 2016).

Until recently, however, precision gravity tests with black holes were
impossible. Regardless, the recent observations of black holes have become a
reality, e.g., the detection of gravitational waves by LIGO/Virgo (Abbott et
al., 2016a, b, 2019), the detection of relativistic effects in the orbits of
stars around Sgr A* (Abuter et al., 2018, 2020) and the EHT observations of
the supermassive black holes M87* and Sgr A* (Akiyama et al., 2019a). The
horizon-scale image of Sgr A* can help test the black hole physics and GR
predictions more satisfactorily than the image of M87*. The detection of
relativistic effects precisely realized the mass and distance of Sgr A* in
stellar orbits (Do et al., 2019). One can implement strong constraints on
spacetime properties. Therefore, it would be interesting to use shadow
observations of Sgr A* to distinguish rotating regular black holes/no-horizon
spacetimes from the Kerr black hole/naked singularity in order to place
constraints on the deviation parameters. There is extensive evidence that Sgr
A* contains a large amount of mass confined within a minimal volume. Whether
it is an actual black hole with an event horizon remains unresolved. Sgr A*
shadow observational results are consistent with the presence of an event
horizon, proving that it ruled out all alternatives, e.g., naked
singularities, and horizonless objects. Here, we discuss what EHT observations
of Sgr A* can add to this question. If Sgr A* does not have an event horizon,
it will probably have some surface (Akiyama et al., 2022e, f). Alternatively,
the object might be a boson star, naked singularity, or no-horizon spacetimes
(Cardoso & Pani, 2019). Here, we test both rotating regular black holes and
no-horizon spacetimes using Sgr A* observational data, and then the case for
Sgr A* having an event horizon or being a no-horizon spacetime becomes
significantly more robust.

The paper is organized as follows: In Section 2, we discuss the null geodesic
equations for most general stationary, axially symmetric regular black hole
spacetimes as well as the mathematical technique for investigating the black
hole shadow. Section 3 presents the shadows of four well-motivated rotating
regular black holes, including those of their corresponding no-horizon
spacetimes. Prior constraints on $(a,\,g)$ are also discussed. In Section 4 we
numerically compute the shadow observables in order to compare the simulated
shadows with the observed Sgr A* shadow. We derive constraints on the rotating
regular black hole parameters with the aid of the Sgr A* black hole shadow.
Finally, in Section 5 we summarize the obtained results and discuss the
possibility of the candidature of rotating regular black holes for the
astrophysical black holes.

##  2 Analytic expressions for black hole shadows: General Rotating spacetimes

A black hole, placed in a bright background, casts a shadow that appears as a
dark region in the observer’s plane, edged by a characteristic sharp and
bright ring, known as a photon ring, whose structure solely depends upon the
spacetime geometry (Cunningham & Bardeen, 1973). The black hole shadow results
from the strong-field gravity near the event horizon, and it can in principle
be used to determine the properties of the black hole spacetime, such as its
spin (Kumar & Ghosh, 2020a). Hence, we investigate the photon geodesics around
the rotating regular black hole spacetime, whose line element in
Boyer$-$Lindquist coordinates ($t,r,\theta,\phi$) reads (Bambi & Modesto,
2013; Kumar & Ghosh, 2020a, 2021)

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

where $\Sigma=r^{2}+a^{2}\cos^{2}\theta,\;\;\;\;\;\Delta=r^{2}+a^{2}-2m(r)r.$,
$m(r)$ is the mass function with $\lim_{r\to\infty}m(r)=M$ and $a$ is the spin
parameter defined as $a=J/M$; $J$ and $M$ are, respectively, the angular
momentum and ADM mass of rotating black hole. The metric (1) goes over to Kerr
(1963) and Kerr$-$Newman (Newman et al., 1965) spacetimes, respectively, when
$m(r)=M$ and $m(r)=M-Q^{2}/2r$. In the metric (1), one can suitably replace
$m(r)$ to get a wide range of rotating regular black hole spacetime (Ghosh,
2015; Abdujabbarov et al., 2016; Amir & Ghosh, 2016; Kumar & Ghosh, 2020a,
2021; Islam et al., 2021; Ghosh et al., 2021).

The Hamilton$-$Jacobi equation determines the photon motion in the spacetime
(1) (Carter, 1968)

$\displaystyle\frac{\partial
S}{\partial\tau}=-\frac{1}{2}g^{\alpha\beta}\frac{\partial S}{\partial
x^{\alpha}}\frac{\partial S}{\partial x^{\beta}},$ (2)

The corresponding geodesics equations can be obtained from Eq. (2). Here,
$\tau$ is the affine parameter along the geodesics, and $S$ is the Jacobi
action, given by

$\displaystyle S=-{\cal E}t+{\cal L}\phi+S_{r}(r)+S_{\theta}(\theta),$ (3)

$S_{r}(r)$ and $S_{\theta}(\theta)$, respectively, are functions only of the
$r$ and $\theta$ coordinates. The metric (1) is time translational and
rotational symmetry, which leads to two conserved quantities, namely energy
$\mathcal{E}=-p_{t}$ and axial angular momentum $\mathcal{L}=p_{\phi}$, where
$p_{\mu}$ is the photon’s four-momentum. The Petrov-type $D$ character of
metric (1) guarantees the separable constant $\mathcal{K}$ (Carter, 1968), and
then we get a set of null geodesics equations in the first-order differential
form (Carter, 1968; Chandrasekhar, 1985):

$\displaystyle\Sigma\frac{dt}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{r^{2}+a^{2}}{\Delta}\left({\cal E}(r^{2}+a^{2})-a{\cal
L}\right)-a(a{\cal E}\sin^{2}\theta-{\mathcal{L}})\ ,$ (4)
$\displaystyle\Sigma\frac{dr}{d\tau}$ $\displaystyle=$
$\displaystyle\pm\sqrt{\mathcal{V}_{r}(r)}\ ,$ (5)
$\displaystyle\Sigma\frac{d\theta}{d\tau}$ $\displaystyle=$
$\displaystyle\pm\sqrt{\mathcal{V}_{\theta}(\theta)}\ ,$ (6)
$\displaystyle\Sigma\frac{d\phi}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{a}{\Delta}\left({\cal E}(r^{2}+a^{2})-a{\cal
L}\right)-\left(a{\cal E}-\frac{{\cal L}}{\sin^{2}\theta}\right)\ ,$ (7)

where $\mathcal{V}_{r}(r)$ and $\mathcal{V}_{\theta}(\theta)$, respectively,
are related to the following effective potentials for radial and polar motion:

$\displaystyle\mathcal{V}_{r}(r)$ $\displaystyle=$
$\displaystyle\left[(r^{2}+a^{2}){\cal E}-a{\cal L}\right]^{2}-\Delta[{\cal
K}+(a{\cal E}-{\cal L})^{2}],\quad$ (8)
$\displaystyle\mathcal{V}_{\theta}(\theta)$ $\displaystyle=$
$\displaystyle{\cal K}-\left[\frac{{\cal L}^{2}}{\sin^{2}\theta}-a^{2}{\cal
E}^{2}\right]\cos^{2}\theta.$ (9)

The $\mathcal{K}$ is related to the Carter constant
$\mathcal{Q}=\mathcal{K}+(a\mathcal{E}-\mathcal{L})^{2}$ (1968), which is
essentially a manifestation of the isometry of metric (1) along the second-
order Killing tensor field. To proceed further, we introduce two dimensionless
impact parameters (Chandrasekhar, 1985)

$\xi\equiv\mathcal{L}/\mathcal{E},\qquad\eta\equiv\mathcal{K}/\mathcal{E}^{2},$
(10)

to reduce the degrees of freedom of the photon geodesics from three to two.
Depending on the critical parameters’ values, the photon may either get
captured, scatter to infinity, or form bound orbits, which depend on the
radial effective potential $\mathcal{V}_{r}(r)$ and the constants of motion.
It turns out that the unstable photon orbits outline the black hole shadow,
which can be determined by

$\left.\mathcal{V}_{r}\right|_{(r=r_{p})}=\left.\frac{\partial\mathcal{V}_{r}}{\partial
r}\right|_{(r=r_{p})}=0,\,\,\text{and}\,\,\left.\frac{\partial^{2}\mathcal{V}_{r}}{\partial
r^{2}}\right|_{(r=r_{p})}>0,$ (11)

where $r_{p}$ is the radius of an unstable photon orbit. Solving Eq. (11) for
Eq. (8) results in the critical impact parameters (Abdujabbarov et al., 2016;
Kumar & Ghosh, 2020a)

$\displaystyle\xi_{c}=$
$\displaystyle\frac{[a^{2}-3r_{p}^{2}]m(r_{p})+r_{p}[a^{2}+r_{p}^{2}][1+m^{\prime}(r_{p})]}{a[m(r_{p})+r_{p}[-1+m^{\prime}(r_{p})]]},$
$\displaystyle\eta_{c}=$
$\displaystyle-\frac{r_{p}^{3}}{a^{2}[m(r_{p})+r_{p}[-1+m^{\prime}(r_{p})]]^{2}}\Big{[}r_{p}^{3}+9r_{p}m(r_{p})^{2}$
$\displaystyle+r_{p}m^{\prime}(r_{p})[4a^{2}+2r_{p}^{2}+r_{p}^{2}m^{\prime}(r_{p})]$
$\displaystyle-2m(r_{p})[2a^{2}+3r_{p}^{2}+3r_{p}^{2}m^{\prime}(r_{p})]\Big{]},$
(12)

where ′ stands for the derivative with respect to the radial coordinate.
Equation (12) reduces to that of the Kerr black hole when $m(r)=M$. In
particular, photons with $\mathcal{\eta}_{c}=0$ form planar circular orbits
confined only to the equatorial plane, whereas $\mathcal{\eta}_{c}>0$ leads to
three-dimensional spherical orbits (Chandrasekhar, 1985). In rotating
spacetimes, photons can either have prograde motion or retrograde motion,
whose respective radii at the equatorial plane, $r_{p}^{-}$ and $r_{p}^{+}$,
can be identified as the real positive roots of $Y=0$ for $r_{p}\geq r_{+}$,
and all other spherical photon orbits have radii of
$r_{p}^{-}<r_{p}<r_{p}^{+}$. Further, the maximum latitude of spherical orbits
depends on the angular momentum of photons, i.e., the smaller the angular
momentum, the larger the orbit latitude. Photons must have zero angular
momentum to reach the polar plane of the black hole, whose orbit radius
$r_{p}^{0}$, such that $r_{p}^{-}\leq r_{p}^{0}\leq r_{p}^{+}$, can be
determined by the zeros of $\xi_{c}=0$.

The prograde and retrograde circular photon orbits radii, respectively, in the
equatorial plane read (Teo, 2003)

$\displaystyle r_{p}^{-}$ $\displaystyle=$ $\displaystyle
2M\left[1+\cos\left(\frac{2}{3}\cos^{-1}\left[-\frac{|a|}{M}\right]\right)\right],$
$\displaystyle r_{p}^{+}$ $\displaystyle=$ $\displaystyle
2M\left[1+\cos\left(\frac{2}{3}\cos^{-1}\left[\frac{|a|}{M}\right]\right)\right],$
(13)

and they are, respectively, in the ranges $M\leq r_{p}^{-}\leq 3M$ and $3M\leq
r_{p}^{+}\leq 4M$. Indeed, for the Schwarzschild black hole ($a=0$), these two
circular orbits coincide into a single orbit, and the photon region in Eq. (2)
degenerates into a photon sphere of constant radius $r_{p}=3M$. The
$r_{p}^{-}\leq r_{p}^{+}$ can be attributed to the Lense-Thirring effect
(Johannsen & Psaltis, 2010), whereas the prograde orbits and event horizon
radii coincide with $r_{p}^{-}=r_{E}=M$ for the extremal Kerr black hole
($a=M$).

Next, we derive the relation between two constants, $\xi_{c}$ and $\eta_{c}$,
and the observer’s image plane coordinates, $X$ and $Y$. For an observer at
the position ($r_{o},\theta_{o}$) and using the tetrad components of the four-
momentum $p^{(\mu)}$ and geodesic Eqs. (4), (6), and (7), we obtain:

$\displaystyle
X=-r_{o}\frac{p^{(\phi)}}{p^{(t)}}=-\left.r_{o}\frac{\xi_{c}}{\sqrt{g_{\phi\phi}}(\zeta-\gamma\xi_{c})}\right|_{(r_{o},\theta_{o})},$
$\displaystyle
Y=r_{o}\frac{p^{(\theta)}}{p^{(t)}}=\pm\left.r_{o}\frac{\sqrt{\mathcal{V}_{\theta}(\theta)}}{\sqrt{g_{\theta\theta}}(\zeta-\gamma\xi_{c})}\right|_{(r_{o},\theta_{o})},\leavevmode\nobreak\
$ (14)

where

$\displaystyle\zeta=\sqrt{\frac{g_{\phi\phi}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}},\qquad\gamma=-\frac{g_{t\phi}}{g_{\phi\phi}}\zeta.$
(15)

where the coordinates $X$ and $Y$ in Eq. (14), respectively, denote the
apparent displacement along the perpendicular and parallel axes to the
projected axis of the black hole symmetry. For an observer in an
asymptotically flat region ($r_{o}\to\infty$), the celestial coordinate Eq.
(14), yields (Bardeen, 1973)

$X=-\xi_{c}\csc\theta_{o},\qquad
Y=\pm\sqrt{\eta_{c}+a^{2}\cos^{2}\theta_{o}-\xi_{c}^{2}\cot^{2}\theta_{o}}.$
(16)

The parametric plots of Eqs. (14) or (16) in the ($X,Y$) plane can cast a
variety of black hole shadows for different choices of black hole mass
function $m(r)$ (Abdujabbarov et al., 2016; Amir & Ghosh, 2016; Kumar et al.,
2019; Kumar & Ghosh, 2020a, 2021), and when $m(r)=M$, we get the shadows of
the Kerr black holes (Bardeen, 1973; Kumar & Ghosh, 2020a).

|
---|---
Figure 1: Shadow silhouette for a Kerr black hole (left) and the Kerr naked
singularity (right) for an inclination angle $\theta_{0}=50^{o}$. Kerr naked
singularity ($a>M$) always cast open arc shadow.

The apparent shape of a photon ring is perfectly circular for all values of
$a$ and inclination angle $\theta_{o}=0,\pi$. Indeed, the Kerr black holes
($a\leq M$) always admit both prograde and retrograde unstable photon orbits
and form a closed photon ring or shadow silhouette (Kumar et al., 2020b; Kumar
& Ghosh, 2021). However, for a Kerr naked singularity $a>M$, prograde photons
on the equatorial plane and a few neighboring spherical photon orbits spiral
in and end up in the central singularity. The retrograde photon orbits cause
an open arc ring with a tiny segment missing near the left endpoint (Charbulák
& Stuchlík, 2018; Hioki & Maeda, 2009). Figure 1 shows that the photon ring
structure for the Kerr naked singularity is markedly different from that of a
Kerr black hole. For a Kerr black hole, the shape of this photon ring is
nearly circular unless the black hole is rapidly spinning. According to the
cosmic censorship conjecture (CCC), the event horizon hides the black hole
singularity, forbidding naked singularities (Penrose, 1969). The proof of
CCC’s validity is still an open problem in relativity, and CCC remains
unproven to date. Indeed, the closed photon ring in the Kerr spacetime is in
accordance with the CCC; the photon ring in Kerr spacetime is closed as long
as the horizon is present. The naked singularity of spherically symmetric
Janis-Newman-Winicour (Shaikh & Joshi, 2019) and Joshi-Malafarina-Narayan
(Shaikh et al., 2019) spacetimes, for some values of parameters, possess
photon spheres and cast shadows similar to the Schwarzschild black hole
shadow. However, in the absence of photon spheres, the images of naked
singularities significantly differ from those of black holes (Shaikh & Joshi,
2019). Therefore, it would be interesting to use shadow observations of Sgr A*
to distinguish rotating regular black holes/no-horizon spacetimes from the
Kerr black hole/naked singularity in order to place constraints on the
deviation parameters. In the next section, we consider models involving four
well-motivated rotating regular spacetimes, as well as other exotic
possibilities, e.g., corresponding rotating regular no-horizon spacetimes.
Hereafter, we consider the inclination angle $\theta_{0}=50^{o}$, which is
relevant for the Sgr A* black hole (Akiyama et al., 2022a).

Rotating Regular Spacetimes | $g_{E}/M$ | $g_{p}/M$
---|---|---
Bardeen | 0.763332 | 0.82472
Hayward | 1.05231 | 1.17566
Ghosh-Culetu | 1.202 | 1.2291
Simpson-Visser | 1.9950 | 2.9083
Table 1: The values of $g_{E}$ and $g_{p}$ for the rotating regular spacetimes
($a=0.1M$) and $\theta_{0}=50^{o}$. While the horizon disappears for
$g>g_{E}$, the closed photon ring exists even for no-horizon spacetime with
$g_{E}<g\leq g_{p}$.

##  3 Regular black holes and no-horizon spacetimes

|
---|---
|
Figure 2: Parameter space ($a,g$) corresponding to a black hole and no-horizon
spacetime. Parameters in the grey regions define a black hole such that
regions outside the red line represent no-horizon spacetime. No-horizon
spacetime with parameters only in the black shaded region leads to the closed
photon ring structure

Here, we examine four well-known rotating regular black holes, namely Bardeen
(Bambi & Modesto, 2013; Kumar et al., 2020b; Kumar & Ghosh, 2021), Hayward
(Bambi & Modesto, 2013; Kumar et al., 2020b; Kumar & Ghosh, 2021), Ghosh-
Culetu (Ghosh, 2015) and Simpson-Visser black holes (Mazza et al., 2021; Islam
et al., 2021). The regularity of these black holes spacetimes are investigated
in terms of the curvature scalars, which are finite everywhere including at
the center, as shown in Appendix A. It is important to note that the Hawking-
Penrose singularity theorem talks about the spacetime singularities in terms
of the geodesics incompleteness rather than the singularity of curvature
invariant (Hawking, 1972). Thus, the ultimate proof of regularity is the
geodesic completeness. Nevertheless, for $r\geq 0$, these regular black holes
are geodesically complete (Zhou & Modesto, 2022). The shadows of these regular
models have received significant attention and their shape and size are
considerably different from those of the Kerr black hole shadows (Abdujabbarov
et al., 2016; Amir & Ghosh, 2016; Kumar et al., 2019; Kumar & Ghosh, 2020a,
2021). The rotating regular spacetimes in question belong to a family of
prototype non-Kerr black hole metrics with an additional deviation parameter
of $g$ related to the nonlinear electrodynamics charge besides the $a$ and $M$
of the Kerr black hole, which is included as a special case of vanishing
deviation parameter $g=0$. Interestingly, for a given $a$, there is a critical
value of $g$, $g_{E}$ such that $\Delta=0$ has no zeros for $g>g_{E}$ and one
double zero at $r=r_{E}$ for $g=g_{E}$, respectively corresponding to a no-
horizon regular spacetime and an extremal black hole with degenerate horizons.
We show that, unlike the Kerr naked singularity, no-horizon regular spacetimes
can possess a closed photon ring when $g_{E}<g\leq g_{p}$ (cf. Fig. 2 and
Table 1) (Kumar & Ghosh, 2021). Interestingly, the value of $g_{p}$ depends on
the inclination angle $\theta_{0}$.

The EHT collaboration released the first image of the supermassive black hole
M87* in 2019, delivering an emission ring of diameter $\theta_{d}=42\pm
3\,\mu$as (Akiyama et al., 2019a, c, b). In Ref. (Kumar et al., 2020b), we
have shown that the shadows of Bardeen black holes ($g\lesssim 0.26M$),
Hayward black holes ($g\lesssim 0.65M$), Ghosh-Culetu black holes ($g\lesssim
0.25M$) and Simpson-Visser (independent of $g$) are indistinguishable from
Kerr black hole shadows within the current observational uncertainties. Thus,
we can consider these black holes as powerful, viable candidates for
astrophysical black holes. Furthermore, Bardeen black holes ( $g\leq
0.30182M$), Hayward black holes ($g\leq 0.73627M$), and Ghosh-Culetu black
holes ($g\leq 0.30461M$), within the $1\sigma$ region for
$\theta_{d}=39\,\mu$as, are consistent with the observed angular diameter of
M87* (Kumar et al., 2020b). In addition, unlike the Kerr naked singularity,
no-horizon regular spacetimes can possess a closed photon ring when
$g_{E}<g\leq g_{p}$, e.g., for $a=0.10M$, Bardeen ($g_{E}=0.763332M<g\leq
g_{p}=0.816792M$), Hayward ($g_{E}=1.05297M<g\leq g_{p}=1.164846M$) and Ghosh-
Culetu ($g_{E}=1.2020M<g\leq g_{p}=1.222461M$) no-horizon spacetimes have
closed photon ring (cf. Fig. 2 and Table 1) (Kumar & Ghosh, 2021).

|
---|---
|
Figure 3: Regular black hole shadows silhouette with varying charge $g$.  |
---|---
|
Figure 4: Figure showing a comparison of the silhouette of regular black holes
and no-horizon spacetime shadows with varying charge $g$. The black curve
represents the black hole case $g\leq g_{E}$, whereas the red and green curves
are for the no-horizon spacetime $g\geq g_{E}$. Interestingly, no-horizon
spacetime with $g_{E}\leq g\leq g_{p}$ casts a closed shadow boundary, as
shown with the red curve.

###  3.1 Bardeen Spacetime

An idea first put forth by Sakharov (Sakharov, 1966) suggests that matter
could avoid singularities, i.e., with a de Sitter core having an equation of
state $P=-\rho$. Based on this idea, Bardeen (1968) proposed the first regular
black hole solution, which has horizons but no curvature singularity and is
asymptotically ($r\to\infty$) flat. The rotating regular Bardeen black hole
(Bambi & Modesto, 2013; Kumar & Ghosh, 2021, 2020a; Kumar et al., 2020b; Ghosh
& Amir, 2015) is described by metric (1) with the mass function (Bardeen,
1968)

$m(r)=M\left(\frac{r^{2}}{r^{2}+g^{2}}\right)^{3/2},$ (17)

where the deviation parameter $g$ can be identified as the magnetic monopole
charge (Ayon-Beato & Garcia, 2000) and at the limit $g\to 0$, one gets a Kerr
black hole. First, the rotating Bardeen black hole shadows ($g\neq 0$) become
smaller and more distorted with increasing $g$ when compared with the Kerr
black hole shadows (cf. Fig. 3 ) (Abdujabbarov et al., 2016; Kumar et al.,
2020b; Kumar & Ghosh, 2021; Tsukamoto et al., 2014). Utilizing X-ray data from
the disk surrounding the black hole candidate in Cygnus X-1, the rotating
Bardeen black hole metric has been put to the test, such that the $3\sigma$
bounds $a_{K}>0.95M$ (Gou et al., 2011) and $a_{K}>0.983M$ (Gou et al., 2014)
for the Kerr metric infer respective bounds on Bardeen black hole parameters,
respectively, $a>0.78M$ and $g<0.41M$, and $a>0.89M$ and $g<0.28M$ (Bambi,
2014). Although for $a=0$, the black hole horizons disappear for
$g>g_{E}=0.7698M$, the prograde and retrograde photon orbits exist for $g\leq
0.85865M$, resulting in a closed photon ring structure. In Figs. 3 and 4, we
have shown the black holes and corresponding no-horizon shadow variation with
varying charge $g$. Black hole shadow size decreases with both $a$ and $g$.
For the Sgr A* model, the shadow angular diameter falls in
$(43.7792,50.2831)\mu$as, whereas the no-horizon spacetime shadow diameter is
$\theta_{sh}\geq 40.7098\mu$as (cf. Fig. 6).

###  3.2 Hayward Spacetime

Another thoroughly studied regular black hole model was proposed by Hayward
(2006). Besides the mass $M$ it has one additional parameter $\ell$, which
determines the length associated with the region concentrating the central
energy density, such that modifications in the spacetime metric appear when
the curvature scalar becomes comparable with $\ell^{-2}$. The spherically
symmetric Hayward black hole model is identified as an exact solution of the
general relativity minimally coupled to NED with magnetic charge $g$, where
$g$ is related to $\ell$ via $g^{3}=2M\ell^{2}$ (Fan & Wang, 2016). The
rotating Hayward black hole is also described by the metric (1) with the mass
function (Hayward, 2006; Kumar & Ghosh, 2020a)

$m(r)=\frac{Mr^{3}}{r^{3}+g^{3}}.$ (18)

|
---|---

Figure 5: (Left:) Bardeen black holes’ shadow angular diameter $\theta_{sh}$,
in units of $\mu$as, as a function of ($a,g$). Black lines are for $46.9\mu$as
and $50\mu$as corresponding to the Sgr A* black hole shadow bounds. Bardeen
black holes with parameter region between these two black lines cast shadows
that are consistent with the Sgr A* shadow size. (Right:) Bardeen no-horizon
spacetime shadow angular diameter $\theta_{sh}$, in units of $\mu$as, as a
function of ($a,g$). The black line is for $41.7\mu$as corresponding to the
Sgr A* black hole shadow $1\sigma$ bounds, such that the region under the
black line satisfies the Sgr A* shadow $1\sigma$ bound.

|
---|---

Figure 6: Bardeen black holes (left) and no-horizons spacetime (right) shadows angular diameter deviation from that of a Schwarzschild black hole as a function of ($a,g$). The black and green lines are, respectively, for the $\delta=-0.04$ (Keck) and $\delta=-0.08$ (VLTI) corresponding to the Sgr A* black hole shadow bounds.  |
---|---
Figure 7: Bardeen black holes (left) and no-horizons spacetime (right) shadows
circularity deviation observable $\Delta C$ as a function of ($a,g$).

The photon region and shadows of rotating Hayward black holes have been
extensively discussed (Abdujabbarov et al., 2016; Kumar et al., 2019; Liu et
al., 2019). It turns out that the bound $\Delta C\leq 0.1$ for the measured
asymmetry of M87* shadow constrains $g\leq 1.0582M$ for the Hayward black
holes (Kumar et al., 2020b). The angular diameter
$\theta_{sh}=39.6192\;\mu$as, which falls within the $1\sigma$ confidence
region with the observed angular diameter of the EHT observation of M87* black
hole, strongly constrains the parameter $g$, i.e., $g\leq 0.73627M$ for the
Hayward black holes (Kumar et al., 2020b). Although for $a=0$, the black hole
horizons disappear for $g>g_{E}=1.05827M$, the prograde and retrograde photon
orbits exist for $g\leq 1.2183407M$ and lead to a closed photon ring
structure. For $a=0.1M$, $g_{E}=1.05231M$ and $g_{p}=1.17566M$ (cf. Fig. 4).
The sizes of the black hole shadows decrease with both $a$ and $g$, and
constraints on the parameters are derived from the Sgr A* black hole shadow
observation from the EHT.

###  3.3 Ghosh-Culetu Spacetime

Bardeen and Hayward regular black holes have an asymptotically de-Sitter core.
The next regular model (Ghosh, 2015; Culetu, 2015; Singh et al., 2022), which
we hereafter refer to as Ghosh-Culetu black holes, is a novel class of regular
black hole with an asymptotic Minkowski core (Simpson & Visser, 2020). While
these regular models share many features with Bardeen and Hayward black holes,
there are also notable differences, especially at the deep core (Simpson &
Visser, 2020). This solution was found by Ghosh (Ghosh, 2015) generalizing the
spherically symmetric regular (i.e., singularity-free) black hole solution
(Culetu, 2015) to the rotating case. The mass function of rotating non-
singular black hole reads (Ghosh, 2015)

$\displaystyle m(r)=Me^{-g^{2}/2Mr},$ (19)

where $g$ is the NED charge. The Ghosh-Culetu black hole is a generalization
of the Kerr black hole that goes over asymptotically as the Kerr-Newman black
hole and, in the limit, $g\to 0$ as the Kerr black hole. The size of a shadow
reduces as the parameter $g$ increases, and the shadow becomes more distorted
as we increase the parameter $g$. Again, the bound $\Delta C\leq 0.1$ merely
constrains the black hole parameter space, and theoretically allowed values of
$g$, that are consistent with the observed asymmetry of M87*, is $g\leq
1.2130M$. On the other hand, the observed angular diameter $\theta_{sh}$ of
the M87* black hole, within the $1\sigma$ confidence level, requires $g\leq
0.30461M$ for the Ghosh-Culetu black holes (Kumar et al., 2020b).

Although, for a fixed value of $a$, the black hole horizons disappear for
$g>g_{E}$, the prograde and retrograde photon orbits exist for $g\leq g_{p}$
and lead to a closed photon ring structure. The values of $g_{E}$ and $g_{p}$
are summarized in Table 1. For $a=0$, the $g_{E}=1.21306M$ and
$g_{p}=1.26453M$, whereas for $a=0.1M$, $g_{E}=1.202M$ and $g_{p}=1.2291M$.
Black hole shadow size decreases with both $a$ and $g$, and constraints on
parameters are derived from the Sgr A* black hole shadow observation from the
EHT (cf. Fig. 3 ).

###  3.4 Simpson-Visser Spacetime

Simpson and Visser proposed an interesting spherically symmetric regular black
hole spacetime (Simpson et al., 2019) with an additional parameter $g$ that
characterize the least modification of the Schwarzschild black hole spacetime.
Simpson-Visser spacetime interpolates between the Schwarzschild black hole and
the Morris–Thorne traversable wormhole and is regular everywhere. It has been
shown that the Simpson-Visser metric solves the Einstein field equations
sourced by a phantom scalar field with minimally coupled to the nonlinear
electrodynamics (Bronnikov & Walia, 2022).

|
---|---

Figure 8: (Left:) Hayward black holes’ shadow angular diameter $\theta_{sh}$,
in $\mu$as, as a function of ($a,g$). The black line is for
$\theta_{sh}=50\mu$as corresponding to the Sgr A* black hole shadow. Hayward
black holes that have parameters outside of the black line produce a shadow
that is consistent with the Sgr A* shadow size. (Right:) Hayward no-horizon
spacetime shadow angular diameter $\theta_{sh}$, in units of $\mu$as, as a
function of ($a,g$). The black line is for $\theta_{sh}=46.9\mu$as
corresponding to the Sgr A* black hole shadow, such that the region under the
black line satisfies the Sgr A* shadow size.

|
---|---

Figure 9: Hayward black holes (left) and no-horizon spacetime (right) shadow angular diameter deviation from that of a Schwarzschild black hole as a function of ($a,g$). The black and green lines are, respectively, for the $\delta=-0.04$ (Keck) and $\delta=-0.08$ (VLTI) corresponding to the Sgr A* black hole shadow bounds.  |
---|---
Figure 10: Hayward black holes (left) and no-horizon spacetime (right) shadows
circularity deviation observable $\Delta C$ as a function of ($a,g$).

The rotating Simpson-Visser black holes metric is constructed as a
modification from the Kerr black hole metric in Refs. (Shaikh et al., 2021;
Mazza et al., 2021), and is given by the (1) with mass function

$\displaystyle m(r)=M\sqrt{1+\frac{g^{2}}{r^{2}}}.$ (20)

$\Sigma=r^{2}+g^{2}+a^{2}\cos^{2}\theta$ and $\Delta=r^{2}+g^{2}+a^{2}-2m(r)r$
are different from earlier three black holes. It is important to note here
that the radial coordinate $r$ is not the same as the areal radius, which is
$\sqrt{r^{2}+g^{2}}$. The Simpson-Visser spacetime has an interesting feature,
namely that the areal radii of the event horizon, the photon sphere, and the
shadow are completely independent of parameter $g$, but, the proper distance
between these surfaces is $g-$dependent (Lima et al., 2021). Several studies
have been done on the observational features of Simpson-Visser black hole
spacetime, namely gravitational lensing (Tsukamoto, 2021; Ghosh &
Bhattacharyya, 2022), the black hole shadow (Lima et al., 2021; Shaikh et al.,
2021; Jafarzade et al., 2021), the optical appearance of a thin accretion disk
(Bambhaniya et al., 2022; Guerrero et al., 2021), quasi-periodic oscillations
(Jiang et al., 2021). Although, for $a=0$, the black hole horizons disappear
for $g>g_{E}=2.0M$, the prograde and retrograde photon orbits exist for $g\leq
3.0M$ and lead to a closed photon ring structure. For $a=0.1M$, $g_{E}=1.995M$
and $g_{p}=2.9083M$. Interestingly, the black hole shadow size decreases only
with $a$, and it remains completely independent of $g$. We assume that the
rotating regular spacetimes defined by the metric (1) describe supermassive
compact objects Sgr A* focusing on four well-motivated different spacetimes,
and determine whether their images are compatible with those observed with
EHT, and constraints the deviation parameter.

##  4 Observational constraints from the EHT results of Sgr A*

|
---|---

Figure 11: (Left:) Ghosh-Culetu black holes’ shadow angular diameter
$\theta_{sh}$, in units of $\mu$as, as a function of ($a,g$). Black lines are
for $46.9\mu$as and $50\mu$as corresponding to the Sgr A* black hole shadow
bounds. Ghosh-Culetu black holes that have parameters between the two black
lines produce a shadow that is consistent with the magnitude of the Sgr A*
shadow. (Right:) Bardeen no-horizon spacetime shadow angular diameter
$\theta_{sh}$, in units of $\mu$as, as a function of ($a,g$).

|
---|---

Figure 12: Ghosh-Culetu black holes (left) and no-horizon spacetime (right) shadow angular diameter deviation from that of a Schwarzschild black hole as a function of ($a,g$). The black and green lines are, respectively, for the $\delta=-0.04$ (Keck) and $\delta=-0.08$ (VLTI) corresponding to the Sgr A* black hole shadow bounds.  |
---|---
Figure 13: Ghosh-Culetu black holes (left) and no-horizon spacetime (right)
shadows circularity deviation observable $\Delta C$ as a function of ($a,g$).

|
---|---

Figure 14: Simpson-Visser black holes (left) and no-horizon spacetime (right)
shadow angular diameter $\theta_{sh}$, in units of $\mu$as, as a function of
($a,g$). Black line is for $46.9\mu$as corresponding to the Sgr A* black hole
shadow bounds.

|
---|---

Figure 15: Simpson-Visser black holes (left) and no-horizon spacetime (right) shadows angular diameter deviation from that of a Schwarzschild black hole as a function of ($a,g$). The black and green lines are, respectively, for the $\delta=-0.04$ (Keck) and $\delta=-0.08$ (VLTI) corresponding to the Sgr A* black hole shadow bounds.  |
---|---
Figure 16: Simpson-Visser black holes (left) and no-horizon spacetime (right)
shadows circularity deviation observable $\Delta C$ as a function of ($a,g$).

The EHT collaboration has released the shadow results for the Sgr A* black
hole based on the VLBI observational campaign at 1.3mm wavelength conducted in
2017 (Akiyama et al., 2022a, b, c, d, e, f). Even though the mass and distance
of Sgr A* are already well-known from the resolved individual star orbits, the
horizon scaled shadow measurements also provided similar estimates (Do et al.,
2019; Abuter et al., 2019). Various imaging and modeling techniques were used
to construct the shadow images, which are all remarkably consistent in having
a central brightness depression encircled by a bright emission ring (Akiyama
et al., 2022a, f). The emission ring corresponds to the lensed position of the
surrounding accretion disk inner edge and thus different from the
characteristics photon ring (shadow boundary), which preferentially falls
inside the emission ring. The photon ring solely depends on the black hole
spacetime, whereas the emission ring additionally depends on the accretion
model. Thus, the photon ring can be used as a tool to test the black hole
matrices. In contrast to the M87* black hole, for which the EHT only provided
an estimate of the emission ring diameter (Akiyama et al., 2019a), for Sgr A*,
the EHT not only measured the emission ring angular diameter
$\theta_{d}=(51.8\pm 2.3)\mu$as but also estimated the shadow diameter
$\theta_{sh}=(48.7\pm 7)\mu$as with the priors $M=4.0^{+1.1}_{-0.6}\times
10^{6}M_{o}$ and $D_{LS}=8.15\pm 0.15$ kpc (Akiyama et al., 2022f). For
instance, EHT employed three independent imaging algorithms,  eht-imaging,
SIMLI, and DIFMAP, to determine the Sgr A* shadow morphology. The most likely
averaged measured value of the shadow angular diameter from these three
algorithms is in the range $\theta_{sh}\in(46.9-50)\,\mu$as, and the $1\sigma$
credible interval is $41.7-55.6\,\mu$as (Akiyama et al., 2022f). Here, we
consider the Sgr A* black hole as the rotating regular black hole model and
use the Sgr A* shadow results to test the viability of these models to explain
the astrophysical black hole spacetimes. For this purpose, we will use three
observables, the shadow angular diameter $\theta_{sh}$, the deviation of the
shadow size from that of a Schwarzschild black hole $\delta$, and the circular
asymmetry $\Delta C$. The shadow angular diameter, defined in terms of the
shadow cone opening angle, reads as

$\theta_{sh}=\frac{2}{D_{LS}}\sqrt{\frac{A}{\pi}},$ (21)

where $A$ is the black hole shadow area (Kumar & Ghosh, 2020a; Abdujabbarov et
al., 2015)

$A=2\int{Y(r_{p})dX(r_{p})}=2\int_{r_{p}^{-}}^{r_{p}^{+}}\left(Y(r_{p})\frac{dX(r_{p})}{dr_{p}}\right)dr_{p}.$
(22)

Furthermore, EHT used the observable $\delta$ to quantify the deviation
between the inferred shadow diameter and that of a Schwarzschild black hole as
follows (Akiyama et al., 2022b, f)

$\delta=\frac{\theta_{sh}}{\theta_{sh,Sch}}-1.$ (23)

Clearly, $\delta$ takes positive (negative) values if the black hole shadow
size is greater (smaller) than the Schwarzschild black hole of the same mass.
EHT used the two separate priors for the Sgr A* angular size from the Keck and
Very Large Telescope Interferometer (VLTI) observations and the three
independent imaging models to estimate the bounds on the fraction deviation
observable $\delta$ (Akiyama et al., 2022a, f)

$\displaystyle\delta=\begin{dcases*}-0.08^{+0.09}_{-0.09}\;\;\;\;\;&\text{VLTI}\\\
-0.04^{+0.09}_{-0.10}\;\;\;\;\;&\text{Keck}\end{dcases*}$ (24)

In addition, we define the dimensionless circularity deviation observable as a
measure of the distortion of the shadow from a perfect circle described by
polar coordinates ($R(\varphi),\varphi$) (Johannsen & Psaltis, 2010;
Johannsen, 2013b; Kumar et al., 2020b)

$\Delta
C=\frac{1}{\bar{R}}\sqrt{\frac{1}{\pi}\int_{0}^{2\pi}\left(R(\varphi)-\bar{R}\right)^{2}d\varphi},$
(25)

where $\bar{R}$ is the average shadow radius defined as (Johannsen & Psaltis,
2010)

$\bar{R}=\frac{1}{2\pi}\int_{0}^{2\pi}R(\varphi)d\varphi.$ (26)

The deviation of the black hole shadow from a circle strongly depends on the
spin parameter. However, the EHT 2017 observations could not estimate the Sgr
A* shadow circularity deviation nor the Sgr A* black hole spin (Akiyama et
al., 2022f). Therefore, to constrain the rotating regular black hole
parameters ($a,g$), we will use the observed bounds of $\theta_{sh}$ and
$\delta$ for the Sgr A* shadow.

We calculated and depicted the Bardeen black holes and corresponding no-
horizon spacetime shadow angular diameter $\theta_{sh}$ in Fig. 6, the shadow
angular diameter deviation $\delta$ in Fig. 6, and the circularity deviation
$\Delta C$ in Fig. 7. As shown in Fig. 6, for $a=0$, Bardeen black holes with
$0.18M\leq g\leq 0.595M$ cast shadows with an angular diameters
$\theta_{sh}\in(46.9,50)\mu$as corresponding to the observed Sgr A* shadow
angular diameter, whereas no-horizon spacetimes with $0.7698M\leq g\leq
0.83794M$ satisfy the $1\sigma$ bound $41.7-55.6\,\mu$as. Similarly, for
$a=0$, Bardeen black holes with $g=0.47165M$ and $g=0.64025M$, respectively,
lead to $\delta=-0.04$ (consistent with the Keck) and $\delta=-0.08$
(consistent with the VLTI). The constrained parameter space is shown in Figs.
6 and 6. Clearly, rotating Bardeen black holes and no-horizon spacetime,
within a finite parameter space, categorically satisfy the observed Sgr A*
black hole shadow observables. Even though the EHT did not give any bound on
the Sgr A* black hole shadow $\Delta C$, we show the circularity deviation
$\Delta C$ as a function of ($a,g$) in Fig. 7. Evidently, for black hole
shadows, $\Delta C\leq 0.04$ and no-horizon spacetime shadows, $\Delta C\leq
0.017$.

Figures 9, 9, and 10, respectively, show the Hayward black holes along with
the corresponding no-horizon shadow observables $\theta_{sh}$, $\delta$ and
$\Delta C$. We find that, for $a=0$, Hayward black holes with $g\geq 0.5302M$
and no-horizon spacetimes with $g\leq 1.11921M$ cast shadow radii with
$\theta_{sh}\in(46.9,50)\mu$as corresponding to the observed Sgr A* shadow
angular diameter (cf. Fig. 9). Similarly, for $a=0$, Hayward black holes with
$g=0.9768M$ and no-horizon spacetime with $g=1.16356M$, respectively, leads
$\delta=-0.04$ (consistent with the Keck) and $\delta=-0.08$ (consistent with
the VLTI) (cf. Fig. 9). The Hayward no-horizon spacetime shadows lead to the
maximum $\delta=-0.103376$ for $a=0$ and $g=1.2183407M$. It is interesting to
note that both the Hayward black holes and the no-horizon spacetime are
consistent models for the Sgr A* shadow. Moreover, Hayward black hole shadows
are bounded by $\Delta C\leq 0.06$, whereas no-horizon spacetime shadows
$\Delta C\leq 0.0304$.

Figures 12, 12, and 13, respectively, show the Ghosh-Culetu black holes and
corresponding no-horizon shadow observables $\theta_{sh}$, $\delta$ and
$\Delta C$ in the parameter space ($a,g$). It is important to note that the
Ghosh-Culetu black hole shadows cover a large window
$\theta_{sh}\in(33.72,50.28)\mu$as. For $a=0$, Ghosh-Culetu black holes with
$0.18345M\leq g\leq 0.62058M$ cast shadow radius with
$\theta_{sh}\in(46.9,50)\mu$as corresponding to the observed Sgr A* shadow
angular diameter, whereas the corresponding no-horizon spacetimes shadow size
fall outside the $1\sigma$ bound $41.7-55.6\,\mu$as. In particular, the no-
horizon spacetime with $a=0$ and $g=1.26453M$ produced the smallest shadow
with an angular diameter of $\theta_{sh}=30.6156\mu$as. Similarly, for $a=0$,
Ghosh-Culetu black holes with $g=0.483224M$ and $g=0.673512M$, respectively,
lead to $\delta=-0.04$ (Keck) and $\delta=-0.08$ (VLTI). In contrast to
Bardeen and Hayward black holes, the Ghosh-Culetu black holes parameters are
tightly constrained from the Sgr A* shadow (cf. Figs. 12 and 12). As a result,
the Sgr A* black hole shadow is consistent with the Ghosh-Culetu black holes
and not with the corresponding no-horizon spacetimes. Recently, constraints on
the Ghosh-Culetu black hole parameters have been derived from the angular
diameter of the shadow of the Sgr A* black hole (Banerjee et al., 2022).

We calculated and depicted the Simpson-Visser black hole and corresponding no-
horizon spacetime shadow angular diameter $\theta_{sh}$ in Fig. 15, the shadow
angular diameter deviation $\delta$ in Fig. 15, and the circularity deviation
$\Delta C$ in Fig. 16. It is worthwhile to mention that the Simpson-Visser
spacetime shadows are completely independent of the parameter $g$. The
rotating Simpson-Visser black hole and no-horizon spacetimes cast shadows with
an angular diameter $\theta_{sh}\in(47.4074,50.2831)\mu$as. In particular,
Simpson-Visser black holes with $a\geq 0.3722M$ cast shadow radius with
$\theta_{sh}\in(46.9,50)\mu$as corresponding to the observed Sgr A* shadow
angular diameter. Similarly, Simpson-Visser black holes with $a=0.8884M$ cast
a shadow that leads to $\delta=-0.04$ and satisfies the Keck bound for Sgr A*
black holes.

##  5 Conclusion

The supermassive black hole, Sgr A*, at the center of the Milky Way, is an
ideal and realistic laboratory for testing the strong-field predictions of
general relativity. Recently, in a series of papers, the EHT collaboration
released the first horizon-scale images of Sgr A*, acquired at a wavelength of
1.3 mm (Akiyama et al., 2022a, b, c, d, e, f). A bright ring of emission that
enclosed a central brightness depression characterizes the images (Akiyama et
al., 2022d). The EHT collaboration’s predicted size of shadow is based on the
prior information on the mass-to-distance ratio of the Sgr A* black hole. The
EHT results are consistent with the prediction on the Kerr metric, and there
is no evidence for any breaches of the theory of GR. Here, we used the EHT
observation of the black hole shadow in Sgr A* to place constraints on
deviation parameters associated with four well-motivated (non-Kerr) rotating
regular black holes.

The rotating regular spacetimes considered here possess a de Sitter core,
except for the Ghosh–Culetu spacetime, which has a Minkowski core and a
Simpson– Visser spacetime with a bounce geometry. The latter is of particular
interest because, depending on the value of the regularization parameter $g$,
it can describe either a regular black hole or a traversable wormhole. The
shadows of Bardeen, Hayward, and Ghosh-Culetu rotating regular black holes are
smaller and more distorted than those of the Kerr black hole. On the other
hand, the Simpson-Visser black hole shadow is identical to that of a Kerr
black hole. We have found that, for a finite parameter space, these rotating
regular spacetimes, even in the horizon’s absence, lead to a closed photon
ring. The Simpson-Visser no-horizon spacetime with an arbitrarily large spin
can cast a closed photon ring, whereas Bardeen, Hayward, and Ghosh-Culetu no-
horizon spacetimes are restricted to small values of $a$, $a\lesssim 0.5M$. We
have further charcaterized the shadows using the angular diameter
$\theta_{sh}$, the shadow angular diameter deviation from the Schwarzschild
black hole shadow $\delta$, and the circularity deviation $\Delta C$. Using
the bounds on $\theta_{sh}$ and $\delta$ for the Sgr A*, we derive constraints
on the spacetime parameters ($a,g$).

The shadows of Bardeen black holes with $(a=0,\;\;0.18M\leq g\leq 0.595M)$ and
$(a=0.2M,\;\;0.154M\leq g\leq 0.58367M)$ are consistent with the Sgr A* black
hole shadow angular diameter $\theta_{sh}$, whereas, the corresponding no-
horizon spacetime with $g\leq 0.83794M$ also satisfies the $1\sigma$ bound. In
comparison, the Sgr A* shadow angular size placed a relatively weaker
constraint on Hayward spacetime, i.e.., black holes with $g\geq 0.5302M$ and
no-horizon spacetimes with $g\leq 1.11921M$ produced a shadow that is
consistent with the Sgr A* shadow size. On the other hand, the Ghosh-Culetu
spacetime parameters are tightly constrained, e.g., for $a=0$, $0.18345M\leq
g\leq 0.62058M$ and for $a=0.20M$, $0.155M\leq g\leq 0.61116M$. While the
photon rings of Bardeen, Hayward, and Simpson-Visser no-horizon spacetime are
consistent with Sgr A* shadow size, the photon ring of Ghosh-Culetu no-horizon
spacetime is considerably smaller and falls outside the $1\sigma$ bound. For
mathematical calculations, we have considered the inclination angle
$\theta_{o}=50^{o}$, as relevant for the Sgr A* black hole. If we consider
different inclination angle, the Sgr A* shadow observables will lead to
weaker/stronger constraints on black hole parameters.

The Sgr A* shadow fits with the projections of the Kerr metric, which,
according to the GR summarized no-hair theorem, represents any astrophysical
black holes spacetimes. The celebrated theorem may not hold for the modified
theories of gravity that admit non-Kerr black holes. Our study indicates that
rotating regular spacetimes, except possibly Ghosh-Culetu no-horizon
spacetime, agree with the results of SgrA* and they are strong candidates for
the astrophysical black holes.

Given the significance of the accretion onto black holes, relativistic
magnetohydrodynamic simulations of the hot plasma flow with the effects of
magnetic fields in the accretion disk are necessary for a proper analysis of
shadow. One can, however, take the existing results as an encouraging sign to
analyze these effects.

##  Appendix A Regularity of the rotating metric

For the rotating metric (1), the metric function diverges for $\Sigma=0$,
which is $r=0,\theta=\pi/2$. However, for the mass functions discussed in
subsections 3.1–3.4, the rotating metric (1) defines a globally regular
spacetime. This can be seen from the expressions of its curvature invariant,
namely Ricci scalar, Ricci square, and Kretschmann scalar (Bambi & Modesto,
2013). We calculate these curvature scalars for all four of the rotating
regular black holes. For the Bardeen and the Hayward spacetimes, they reduce
to simpler forms at the origin $(r=0,\theta=\pi/2)$

$\displaystyle\lim_{r\to 0}\Big{(}\lim_{\theta\to\pi/2}R\Big{)}$
$\displaystyle=$ $\displaystyle-\frac{24M}{g^{3}},$ $\displaystyle\lim_{r\to
0}\Big{(}\lim_{\theta\to\pi/2}R_{\mu\nu}R^{\mu\nu}\Big{)}$ $\displaystyle=$
$\displaystyle\frac{144M^{2}}{g^{6}},$ $\displaystyle\lim_{r\to
0}\Big{(}\lim_{\theta\to\pi/2}R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}\Big{)}$
$\displaystyle=$ $\displaystyle\frac{96M^{2}}{g^{6}}.$

Clearly, the curvature scalars are finite at the black hole center ring as
long as $g\neq 0$. Whereas for the Ghosh-Culetu spacetime, all the curvature
scalars vanish at $(r=0,\theta=\pi/2)$ (Ghosh, 2015):

$\displaystyle\lim_{r\to 0}\Big{(}\lim_{\theta\to\pi/2}R\Big{)}=\lim_{r\to
0}\Big{(}\lim_{\theta\to\pi/2}R_{\mu\nu}R^{\mu\nu}\Big{)}=\lim_{r\to
0}\Big{(}\lim_{\theta\to\pi/2}R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}\Big{)}=0.$

In contrast, for the Simpson-Visser spacetime the surface $r=0$ is an oblate
spheroid of size $g$. Only when $g=0$, the spheroid collapse to a ring at
$\theta=\pi/2$ and the usual singularity of the Kerr geometry isthen
recovered. When instead $g\neq 0$, the singularity is excised and $r=0$ is a
regular surface of finite size. The curvature scalars read

$\displaystyle\lim_{r\to 0}\Big{(}\lim_{\theta\to\pi/2}R\Big{)}$
$\displaystyle=$
$\displaystyle-\frac{2\left(2a^{4}+a^{2}g(g-6M)+g^{2}(g-2M)^{2}\right)}{a^{2}g^{4}},$
$\displaystyle\lim_{r\to
0}\Big{(}\lim_{\theta\to\pi/2}R_{\mu\nu}R^{\mu\nu}\Big{)}$ $\displaystyle=$
$\displaystyle\frac{2\left(4a^{8}+4a^{6}g(g-4M)+a^{4}g^{2}\left(4g^{2}-19gM+34M^{2}\right)+a^{2}g^{3}(g-2M)^{2}(2g-9M)+g^{4}(g-2M)^{4}\right)}{a^{4}g^{8}},$
$\displaystyle\lim_{r\to
0}\Big{(}\lim_{\theta\to\pi/2}R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}\Big{)}$
$\displaystyle=$
$\displaystyle\frac{4\left(4a^{8}+2a^{6}g(2g-3M)+a^{4}g^{2}\left(3g^{2}-8gM+12M^{2}\right)+2a^{2}g^{3}(g-2M)^{2}(g-3M)+g^{4}(g-2M)^{4}\right)}{a^{4}g^{8}}.$

Here, $R$, $R_{\mu\nu}$ and $R_{\mu\nu\alpha\beta}$ are, respectively, the
Ricci scalar, Ricci tensor, and Riemann tensor.

##  Appendix B Acknowledgement

S.G.G. would like to thank SERB-DST for project No. CRG/2021/005771. R.K.W.
would like to thank the University of KwaZulu-Natal for the continued support
and the NRF for the postdoctoral fellowship. S.D.M acknowledges that this work
is based upon research supported by the South African Research Chair
Initiative of the Department of Science and Technology and the National
Research Foundation.

## References

  * Abbott et al. (2016a) Abbott, B. P., et al. 2016a, Phys. Rev. Lett., 116, 061102, doi: 10.1103/PhysRevLett.116.061102
  * Abbott et al. (2016b) —. 2016b, Phys. Rev. Lett., 116, 221101, doi: 10.1103/PhysRevLett.116.221101
  * Abbott et al. (2019) —. 2019, Phys. Rev. D, 100, 104036, doi: 10.1103/PhysRevD.100.104036
  * Abdujabbarov et al. (2016) Abdujabbarov, A., Amir, M., Ahmedov, B., & Ghosh, S. G. 2016, Phys. Rev., D93, 104004, doi: 10.1103/PhysRevD.93.104004
  * Abdujabbarov et al. (2015) Abdujabbarov, A. A., Rezzolla, L., & Ahmedov, B. J. 2015, Mon. Not. Roy. Astron. Soc., 454, 2423, doi: 10.1093/mnras/stv2079
  * Abuter et al. (2018) Abuter, R., et al. 2018, Astron. Astrophys., 615, L15, doi: 10.1051/0004-6361/201833718
  * Abuter et al. (2019) Abuter, R., Amorim, A., Bauböck, M., et al. 2019, Astronomy & Astrophysics, 625, L10
  * Abuter et al. (2020) Abuter, R., et al. 2020, Astron. Astrophys., 636, L5, doi: 10.1051/0004-6361/202037813
  * Afrin & Ghosh (2021) Afrin, M., & Ghosh, S. G. 2021.  https://arxiv.org/abs/2110.05258
  * Afrin & Ghosh (2022) —. 2022, Universe, 8, 52, doi: 10.3390/universe8010052
  * Afrin et al. (2021) Afrin, M., Kumar, R., & Ghosh, S. G. 2021, Mon. Not. Roy. Astron. Soc., 504, 5927, doi: 10.1093/mnras/stab1260
  * Akiyama et al. (2019a) Akiyama, K., et al. 2019a, Astrophys. J., 875, L1, doi: 10.3847/2041-8213/ab0ec7
  * Akiyama et al. (2019b) —. 2019b, Astrophys. J., 875, L5, doi: 10.3847/2041-8213/ab0f43
  * Akiyama et al. (2019c) —. 2019c, Astrophys. J., 875, L6, doi: 10.3847/2041-8213/ab1141
  * Akiyama et al. (2022a) —. 2022a, Astrophys. J. Lett., 930, L15, doi: 10.3847/2041-8213/ac6736
  * Akiyama et al. (2022b) —. 2022b, Astrophys. J. Lett., 930, L16, doi: 10.3847/2041-8213/ac6672
  * Akiyama et al. (2022c) —. 2022c, Astrophys. J. Lett., 930, L13, doi: 10.3847/2041-8213/ac6675
  * Akiyama et al. (2022d) —. 2022d, Astrophys. J. Lett., 930, L14, doi: 10.3847/2041-8213/ac6429
  * Akiyama et al. (2022e) —. 2022e, Astrophys. J. Lett., 930, L12, doi: 10.3847/2041-8213/ac6674
  * Akiyama et al. (2022f) —. 2022f, Astrophys. J. Lett., 930, L17, doi: 10.3847/2041-8213/ac6756
  * Allahyari et al. (2020) Allahyari, A., Khodadi, M., Vagnozzi, S., & Mota, D. F. 2020, JCAP, 02, 003, doi: 10.1088/1475-7516/2020/02/003
  * Amarilla & Eiroa (2012) Amarilla, L., & Eiroa, E. F. 2012, Phys. Rev., D85, 064019, doi: 10.1103/PhysRevD.85.064019
  * Amarilla & Eiroa (2013) —. 2013, Phys. Rev., D87, 044057, doi: 10.1103/PhysRevD.87.044057
  * Amarilla et al. (2010) Amarilla, L., Eiroa, E. F., & Giribet, G. 2010, Phys. Rev., D81, 124045, doi: 10.1103/PhysRevD.81.124045
  * Amir & Ghosh (2016) Amir, M., & Ghosh, S. G. 2016, Phys. Rev., D94, 024054, doi: 10.1103/PhysRevD.94.024054
  * Amir et al. (2018) Amir, M., Singh, B. P., & Ghosh, S. G. 2018, Eur. Phys. J., C78, 399, doi: 10.1140/epjc/s10052-018-5872-3
  * Atamurotov et al. (2013) Atamurotov, F., Abdujabbarov, A., & Ahmedov, B. 2013, Phys. Rev., D88, 064004, doi: 10.1103/PhysRevD.88.064004
  * Ayon-Beato & Garcia (2000) Ayon-Beato, E., & Garcia, A. 2000, Phys. Lett., B493, 149, doi: 10.1016/S0370-2693(00)01125-4
  * Baker et al. (2015) Baker, T., Psaltis, D., & Skordis, C. 2015, Astrophys. J., 802, 63, doi: 10.1088/0004-637X/802/1/63
  * Bambhaniya et al. (2022) Bambhaniya, P., K, S., Jusufi, K., & Joshi, P. S. 2022, Phys. Rev. D, 105, 023021, doi: 10.1103/PhysRevD.105.023021
  * Bambi (2014) Bambi, C. 2014, Phys. Lett., B730, 59, doi: 10.1016/j.physletb.2014.01.037
  * Bambi & Freese (2009) Bambi, C., & Freese, K. 2009, Phys. Rev., D79, 043002, doi: 10.1103/PhysRevD.79.043002
  * Bambi & Modesto (2013) Bambi, C., & Modesto, L. 2013, Phys. Lett., B721, 329, doi: 10.1016/j.physletb.2013.03.025
  * Banerjee et al. (2022) Banerjee, I., Sau, S., & SenGupta, S. 2022.  https://arxiv.org/abs/2206.12125
  * Bardeen (1968) Bardeen, J. 1968, Tbilisi, USSR
  * Bardeen (1973) —. 1973, Black Holes, edited by C. DeWitt and BS DeWitt, Gordon and Breach, New York
  * Bronnikov & Walia (2022) Bronnikov, K. A., & Walia, R. K. 2022, Phys. Rev. D, 105, 044039, doi: 10.1103/PhysRevD.105.044039
  * Cardoso & Pani (2019) Cardoso, V., & Pani, P. 2019, Living Rev. Rel., 22, 4, doi: 10.1007/s41114-019-0020-4
  * Carter (1968) Carter, B. 1968, Phys. Rev., 174, 1559, doi: 10.1103/PhysRev.174.1559
  * Carter (1971) —. 1971, Phys. Rev. Lett., 26, 331, doi: 10.1103/PhysRevLett.26.331
  * Chandrasekhar (1985) Chandrasekhar, S. 1985, The mathematical theory of black holes (Oxford: Oxford Univ. Press)
  * Charbulák & Stuchlík (2018) Charbulák, D., & Stuchlík, Z. 2018, Eur. Phys. J. C, 78, 879, doi: 10.1140/epjc/s10052-018-6336-5
  * Chen (2022) Chen, C.-Y. 2022.  https://arxiv.org/abs/2205.06962
  * Culetu (2015) Culetu, H. 2015, Int. J. Theor. Phys., 54, 2855, doi: 10.1007/s10773-015-2521-6
  * Cunha & Herdeiro (2018) Cunha, P. V. P., & Herdeiro, C. A. R. 2018, Gen. Rel. Grav., 50, 42, doi: 10.1007/s10714-018-2361-9
  * Cunha et al. (2019a) Cunha, P. V. P., Herdeiro, C. A. R., & Radu, E. 2019a, Phys. Rev. Lett., 123, 011101, doi: 10.1103/PhysRevLett.123.011101
  * Cunha et al. (2019b) —. 2019b, Universe, 5, 220, doi: 10.3390/universe5120220
  * Cunningham & Bardeen (1973) Cunningham, C. T., & Bardeen, J. M. 1973, ApJ, 183, 237, doi: 10.1086/152223
  * de Vries (2000) de Vries, A. 2000, Class. Quantum Gravity, 17, 123, doi: 10.1088/0264-9381/17/1/309
  * Do et al. (2019) Do, T., et al. 2019, Science, 365, 664, doi: 10.1126/science.aav8137
  * Einstein (1915) Einstein, A. 1915, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ), 1915, 844
  * Falcke & Markoff (2013) Falcke, H., & Markoff, S. B. 2013, Class. Quant. Grav., 30, 244003, doi: 10.1088/0264-9381/30/24/244003
  * Falcke et al. (2000) Falcke, H., Melia, F., & Agol, E. 2000, Astrophys. J., 528, L13, doi: 10.1086/312423
  * Fan & Wang (2016) Fan, Z.-Y., & Wang, X. 2016, Phys. Rev., D94, 124027, doi: 10.1103/PhysRevD.94.124027
  * Ghosh & Bhattacharyya (2022) Ghosh, S., & Bhattacharyya, A. 2022.  https://arxiv.org/abs/2206.09954
  * Ghosh (2015) Ghosh, S. G. 2015, Eur. Phys. J., C75, 532, doi: 10.1140/epjc/s10052-015-3740-y
  * Ghosh & Afrin (2022) Ghosh, S. G., & Afrin, M. 2022.  https://arxiv.org/abs/2206.02488
  * Ghosh & Amir (2015) Ghosh, S. G., & Amir, M. 2015, Eur. Phys. J. C, 75, 553, doi: 10.1140/epjc/s10052-015-3786-x
  * Ghosh et al. (2021) Ghosh, S. G., Kumar, R., & Islam, S. U. 2021, JCAP, 03, 056, doi: 10.1088/1475-7516/2021/03/056
  * Gou et al. (2011) Gou, L., McClintock, J. E., Reid, M. J., et al. 2011, Astrophys. J., 742, 85, doi: 10.1088/0004-637X/742/2/85
  * Gou et al. (2014) Gou, L., McClintock, J. E., Remillard, R. A., et al. 2014, Astrophys. J., 790, 29, doi: 10.1088/0004-637X/790/1/29
  * Guerrero et al. (2021) Guerrero, M., Olmo, G. J., Rubiera-Garcia, D., & Gómez, D. S.-C. 2021, JCAP, 08, 036, doi: 10.1088/1475-7516/2021/08/036
  * Hawking (1972) Hawking, S. W. 1972, Commun. Math. Phys., 25, 152, doi: 10.1007/BF01877517
  * Hayward (2006) Hayward, S. A. 2006, Phys. Rev. Lett., 96, 031103, doi: 10.1103/PhysRevLett.96.031103
  * He et al. (2022) He, A., Tao, J., Wang, P., Xue, Y., & Zhang, L. 2022.  https://arxiv.org/abs/2205.12779
  * Hioki & Maeda (2009) Hioki, K., & Maeda, K.-i. 2009, Phys. Rev., D80, 024042, doi: 10.1103/PhysRevD.80.024042
  * Islam et al. (2021) Islam, S. U., Kumar, J., & Ghosh, S. G. 2021, JCAP, 10, 013, doi: 10.1088/1475-7516/2021/10/013
  * Israel (1967) Israel, W. 1967, Phys. Rev., 164, 1776, doi: 10.1103/PhysRev.164.1776
  * Israel (1968) —. 1968, Commun. Math. Phys., 8, 245, doi: 10.1007/BF01645859
  * Jafarzade et al. (2021) Jafarzade, K., Zangeneh, M. K., & Lobo, F. S. N. 2021.  https://arxiv.org/abs/2106.13893
  * Jaroszynski & Kurpiewski (1997) Jaroszynski, M., & Kurpiewski, A. 1997, Astron. Astrophys., 326, 419.  https://arxiv.org/abs/astro-ph/9705044
  * Jiang et al. (2021) Jiang, X., Wang, P., Yang, H., & Wu, H. 2021, Eur. Phys. J. C, 81, 1043, doi: 10.1140/epjc/s10052-021-09816-z
  * Johannsen (2013a) Johannsen, T. 2013a, Phys. Rev., D87, 124017, doi: 10.1103/PhysRevD.87.124017
  * Johannsen (2013b) —. 2013b, Astrophys. J., 777, 170, doi: 10.1088/0004-637X/777/2/170
  * Johannsen (2016) —. 2016, Class. Quant. Grav., 33, 124001, doi: 10.1088/0264-9381/33/12/124001
  * Johannsen & Psaltis (2010) Johannsen, T., & Psaltis, D. 2010, Astrophys. J., 718, 446, doi: 10.1088/0004-637X/718/1/446
  * Kerr (1963) Kerr, R. P. 1963, Phys. Rev. Lett., 11, 237, doi: 10.1103/PhysRevLett.11.237
  * Khodadi et al. (2020) Khodadi, M., Allahyari, A., Vagnozzi, S., & Mota, D. F. 2020, JCAP, 09, 026, doi: 10.1088/1475-7516/2020/09/026
  * Khodadi & Lambiase (2022) Khodadi, M., & Lambiase, G. 2022.  https://arxiv.org/abs/2206.08601
  * Kocherlakota et al. (2021) Kocherlakota, P., et al. 2021, Phys. Rev. D, 103, 104047, doi: 10.1103/PhysRevD.103.104047
  * Kuang et al. (2022) Kuang, X.-M., Tang, Z.-Y., Wang, B., & Wang, A. 2022.  https://arxiv.org/abs/2206.05878
  * Kumar & Ghosh (2020a) Kumar, R., & Ghosh, S. G. 2020a, Astrophys. J., 892, 78, doi: 10.3847/1538-4357/ab77b0
  * Kumar & Ghosh (2020b) —. 2020b, JCAP, 07, 053, doi: 10.1088/1475-7516/2020/07/053
  * Kumar & Ghosh (2021) —. 2021, Class. Quant. Grav., 38, 8, doi: 10.1088/1361-6382/abdd48
  * Kumar et al. (2019) Kumar, R., Ghosh, S. G., & Wang, A. 2019, Phys. Rev., D100, 124024, doi: 10.1103/PhysRevD.100.124024
  * Kumar et al. (2020a) —. 2020a, Phys. Rev. D, 101, 104001, doi: 10.1103/PhysRevD.101.104001
  * Kumar et al. (2020b) Kumar, R., Kumar, A., & Ghosh, S. G. 2020b, Astrophys. J., 896, 89, doi: 10.3847/1538-4357/ab8c4a
  * Lima et al. (2021) Lima, Junior., H. C. D., Crispino, L. C. B., Cunha, P. V. P., & Herdeiro, C. A. R. 2021, Phys. Rev. D, 103, 084040, doi: 10.1103/PhysRevD.103.084040
  * Liu et al. (2019) Liu, C., Ding, C., & Jing, J. 2019, Sci. China Phys. Mech. Astron., 62, 10411, doi: 10.1007/s11433-018-9243-x
  * Mazza et al. (2021) Mazza, J., Franzin, E., & Liberati, S. 2021, JCAP, 04, 082, doi: 10.1088/1475-7516/2021/04/082
  * Mizuno et al. (2018) Mizuno, Y., Younsi, Z., Fromm, C. M., et al. 2018, Nat. Astron., 2, 585, doi: 10.1038/s41550-018-0449-5
  * Newman et al. (1965) Newman, E. T., Couch, R., Chinnapared, K., et al. 1965, J. Math. Phys., 6, 918, doi: 10.1063/1.1704351
  * Oppenheimer & Snyder (1939) Oppenheimer, J. R., & Snyder, H. 1939, Phys. Rev., 56, 455, doi: 10.1103/PhysRev.56.455
  * Pantig & Övgün (2022) Pantig, R. C., & Övgün, A. 2022.  https://arxiv.org/abs/2206.02161
  * Papnoi et al. (2014) Papnoi, U., Atamurotov, F., Ghosh, S. G., & Ahmedov, B. 2014, Phys. Rev., D90, 024073, doi: 10.1103/PhysRevD.90.024073
  * Penrose (1965) Penrose, R. 1965, Phys. Rev. Lett., 14, 57, doi: 10.1103/PhysRevLett.14.57
  * Penrose (1969) —. 1969, Riv. Nuovo Cim., 1, 252, doi: 10.1023/A:1016578408204
  * Robinson (1975) Robinson, D. C. 1975, Phys. Rev. Lett., 34, 905, doi: 10.1103/PhysRevLett.34.905
  * Sakharov (1966) Sakharov, A. D. 1966, Sov. Phys. JETP, 22, 241
  * Schwarzschild (1916) Schwarzschild, K. 1916, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ), 1916, 189.  https://arxiv.org/abs/physics/9905030
  * Shaikh & Joshi (2019) Shaikh, R., & Joshi, P. S. 2019, JCAP, 10, 064, doi: 10.1088/1475-7516/2019/10/064
  * Shaikh et al. (2019) Shaikh, R., Kocherlakota, P., Narayan, R., & Joshi, P. S. 2019, Mon. Not. Roy. Astron. Soc., 482, 52, doi: 10.1093/mnras/sty2624
  * Shaikh et al. (2021) Shaikh, R., Pal, K., Pal, K., & Sarkar, T. 2021, Mon. Not. Roy. Astron. Soc., 506, 1229, doi: 10.1093/mnras/stab1779
  * Shen et al. (2005) Shen, Z.-Q., Lo, K. Y., Liang, M. C., Ho, P. T. P., & Zhao, J. H. 2005, Nature, 438, 62, doi: 10.1038/nature04205
  * Simpson et al. (2019) Simpson, A., Martin-Moruno, P., & Visser, M. 2019, Class. Quant. Grav., 36, 145007, doi: 10.1088/1361-6382/ab28a5
  * Simpson & Visser (2020) Simpson, A., & Visser, M. 2020, Universe, 6, 8
  * Singh & Ghosh (2018) Singh, B. P., & Ghosh, S. G. 2018, Annals Phys., 395, 127, doi: 10.1016/j.aop.2018.05.010
  * Singh et al. (2022) Singh, D. V., Ghosh, S. G., & Maharaj, S. D. 2022, Nucl. Phys. B, 981, 115854, doi: 10.1016/j.nuclphysb.2022.115854
  * Teo (2003) Teo, E. 2003, General Relativity and Gravitation, 35, 1909
  * Tsukamoto (2021) Tsukamoto, N. 2021, Phys. Rev. D, 103, 024033, doi: 10.1103/PhysRevD.103.024033
  * Tsukamoto et al. (2014) Tsukamoto, N., Li, Z., & Bambi, C. 2014, JCAP, 06, 043, doi: 10.1088/1475-7516/2014/06/043
  * Tsupko (2017) Tsupko, O. Yu. 2017, Phys. Rev., D95, 104058, doi: 10.1103/PhysRevD.95.104058
  * Uniyal et al. (2022) Uniyal, A., Pantig, R. C., & Övgün, A. 2022.  https://arxiv.org/abs/2205.11072
  * Vagnozzi et al. (2022) Vagnozzi, S., Roy, R., Tsai, Y.-D., & Visinelli, L. 2022.  https://arxiv.org/abs/2205.07787
  * Vagnozzi & Visinelli (2019) Vagnozzi, S., & Visinelli, L. 2019, Phys. Rev., D100, 024020, doi: 10.1103/PhysRevD.100.024020
  * Yumoto et al. (2012) Yumoto, A., Nitta, D., Chiba, T., & Sugiyama, N. 2012, Phys. Rev., D86, 103001, doi: 10.1103/PhysRevD.86.103001
  * Zhou & Modesto (2022) Zhou, T., & Modesto, L. 2022.  https://arxiv.org/abs/2208.02557