# Shadows in dyonic Kerr-Sen black holes

Soumya Jana<EMAIL_ADDRESS>Department of Physics, Sitananda
College, Nandigram, 721631, India  Department of Physics, Indian Institute of
Technology, Kharagpur - 721302, India  Sayan Kar<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology, Kharagpur - 721302,
India

###### Abstract

Black holes with dyonic charges in Einstein-Maxwell-dilaton-axion supergravity
theory are revisited in the context of black hole shadows. We consider static
as well as rotating (namely the dyonic Kerr-Sen) black holes. The matter
stress-energy tensor components, sourced by the Maxwell, axion and dilaton
fields satisfy the standard energy conditions. The analytical expressions for
the horizon and the shadow radius of the static spacetimes demonstrate their
dependence on $P^{2}+Q^{2}$ ($P$, $Q$ the magnetic and electric charges,
respectively) and the mass parameter $M$. The shadow radius lies in the range
$2M<R_{shadow}<3\sqrt{3}M$ and there is no stable photon orbit outside the
horizon. Further, shadows cast by the rotating dyonic Kerr-Sen black holes are
also studied and compared graphically with their Kerr-Newman and Kerr-Sen
counterparts. Deviation of the shadow boundary is prominent with the variation
of the magnetic charge, for the relatively slowly rotating dyonic Kerr-Sen
spacetimes. We test any possible presence of a magnetic monopole charge in the
backdrop of recent EHT observations for the supermassive black holes M87∗ and
Sgr A∗. Deviation from circularity of the shadow boundary ($\Delta C$) and
deviation of the average shadow radius from the Schwarzschild shadow radius
(quantified as the fractional deviation parameter $\delta$) are the two
observables used here. Observational bound on $\Delta C$ (available only for
M87∗) is satisfied for all theoretically allowed regions of parameter space
and thus cannot constrain the parameters. The observational bound on $\delta$
available for Sgr A∗ translates into an upper limit on any possible magnetic
monopole charge linked to Sgr A∗ and is given as $P\lesssim 0.873\,M$. Such a
constraint on $P$ is however expected to be far more stringent for other
astrophysical tests.

##  I Introduction

The Reissner–Nordstrom (RN) geometry representing the gravitational field due
to a charged massive object is among the earliest known exact solutions in
General Relativity coupled to electromagnetism, i.e. the Einstein–Maxwell
theory. A straightforward generalisation of this solution is the dyonic RN
spacetime Chen and Zhang (2013) which can be written down by just replacing
the $Q^{2}$ in RN spacetime with $P^{2}+Q^{2}$, where $P$ represents the
‘magnetic’ charge and $Q$ is its ‘electric’ counterpart. However, for the
dyonic solution, the definition of the electromagnetic potential $A_{i}$ is a
little tricky –one needs, as expected for a dyon, a two-patch definition – one
for the northern hemisphere and the other for the southern. The standard
electric-magnetic duality which arises when both magnetic and electric charges
are present keeps the solution unchanged. The horizons and other features for
the dyonic RN spacetime resemble those for the usual RN geometry modulo the
presence of the extra magnetic charge.

It is also known that additional matter fields (other than Maxwell) such as
the dilaton and/or the axion appear in the context of supergravity theories or
in low energy effective actions which emerge out of full string theory Gibbons
and Maeda (1988); Garfinkle  _et al._ (1991); Sen (1992). In such scenarios
too one expects dyonic solutions representing gravitational fields of such
objects. Among various known solutions Gibbons and Maeda (1988); Garfinkle
_et al._ (1991); Campbell  _et al._ (1991); Sen (1992); Shapere  _et al._
(1991); Kallosh and Ortín (1993) there are static, spherically symmetric ones
as well stationary spacetimes wherein rotation is present. The purpose of this
article is to revisit such known solutions without and with rotation. Our
primary aim is to learn how the various theory parameters (eg. electric,
magnetic as well as other charges) control the nature and profile of the
shadow/silhouette created by the gravitational field representing such
solutions. We also try and see if any meaningful constraint can be placed on
the various charges, using the known shadow observations for the supermassive
compact object present in M87∗ Collaboration (2019a, b, c) and for Sgr A*
Collaboration (2022a, b). Though, dyonic scenarios presently have little to do
with observations in other contexts, we will see how one may place bounds on
their viability through shadow observations. In other words, we try to show
that what is seen in the images may also be explained using hypothetical
constructs which, by no means can be ruled out altogether, unless other
observations imply mismatches and contradictions.

Shadows in Kerr-Sen Black holes Sen (1992) have already been studied by
several authors Hioki and Miyamoto (2008); Xavier  _et al._ (2020); Narang
_et al._ (2020). The rotating version of the dyonic black holes in Einstein-
Maxwell-dilaton theory Gibbons and Maeda (1988); Cheng  _et al._ (1994) and
its shadows were studied in Shaikh (2019). In Banerjee  _et al._ (2022), the
authors investigated shadows of regular (Bardeen) black holes having magnetic
monopole charge sourced by nonlinear electrodynamics coupled to GR. Presence
of axionic hair or the Kalb-Ramond field and their influence on the shadow of
M87∗ was investigated in Banerjee  _et al._ (2020a). In Hu  _et al._ (2021),
the authors investigated the effect of QED on the shadows of the static black
holes with magnetic monopoles. There are also several other studies on the
shadows of hairy black holes. For example, the authors of Lee  _et al._ (2021)
studied the shadows cast by the rotating black holes with anisotropic matter
fields which could describe an extra $U(1)$ field as well as diverse dark
matters. Studies on the shadows of braneworld blackholes such as in Banerjee
_et al._ (2020b); Hou  _et al._ (2021) are among other examples. For a more
recent study on the shadows of the black holes in the extended or alternative
gravity theories, in the light of the observations of Sgr A∗, see Vagnozzi
_et al._ (2022).

Our article is organised as follows. In Section II we recall the static black
hole solutions and discuss the energy conditions in Einstein-Maxwell–dilaton-
axion (EMDA) supergravity theory. Section III provides a summary of the
corresponding stationary solutions (dyonic) which include rotation. Shadow
calculations and related details are presented in Section IV and
connections/comparisons with observations are outlined in Section V. The final
Section VI is a summary with concluding remarks.

##  II Black holes in Einstein -Maxwell- Dilaton-Axion supergravity theory

The Einstein-Maxwell -Dilaton- Axion (EMDA) supergravity theory is described
by the action Shapere  _et al._ (1991)

$S_{\text{EMDA}}=\int\text{d}^{4}x\sqrt{-g}\left[R-\frac{1}{2}(\partial\Phi)^{2}-\frac{1}{2}e^{2\Phi}(\partial\xi)^{2}-e^{-\Phi}F^{2}+\xi
F_{\mu}\tilde{F}^{\mu\nu}\right]$ (1)

where $\Phi$ and $\xi$ are the dilaton and axion fields, $F_{\mu\nu}$ is the
usual electromagnetic field tensor, $F^{2}=F_{\mu\nu}{F^{\mu\nu}}$, and
$\tilde{F}^{\mu\nu}=\frac{1}{2\sqrt{-g}}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$
is the dual electromagnetic field tensor. The equations of motion of the
dilaton, axion fields are obtained as

$\Box\Phi-e^{2\Phi}(\partial\xi)^{2}+e^{-\Phi}F^{2}=0,$ (2)

and

$\Box\xi+2\nabla^{\mu}\Phi\nabla_{\mu}\xi+e^{-2\Phi}F_{\mu\nu}\tilde{F}^{\mu\nu}=0.$
(3)

The equation of motion for the electromagnetic vector potential $A_{\mu}$ (
where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$) is obtained as

$\nabla_{\mu}\left(-e^{-\Phi}F^{\mu\nu}+\xi\tilde{F}^{\mu\nu}\right)=0,$ (4)

along with the usual Bianchi identity,

$\nabla_{\mu}\tilde{F}^{\mu\nu}=0.$ (5)

The equation of motion for the metric tensor $g_{\mu\nu}$ is obtained by
varying the action (1) with respect to $g_{\mu\nu}$. We get

$R_{\mu\nu}=\frac{1}{2}\nabla_{\mu}\nabla_{\nu}\Phi+\frac{1}{2}e^{2\Phi}\nabla_{\mu}\xi\nabla_{\nu}\xi+2e^{-\Phi}F_{\mu\alpha}F_{\nu}{}^{\alpha}-\frac{1}{2}g_{\mu\nu}e^{-\Phi}F^{2},$
(6)

where $R_{\mu\nu}$ are the Ricci tensor components.

###  II.1 Static Black Hole solution

The static black hole solution in such a system has already been obtained in
Shapere  _et al._ (1991), where the authors used symmetry transformations on
the axion and dilaton fields to obtain its form. In this section, we first
outline the derivation of the same solution by solving directly, the Einstein
field equations. Thereafter, we analyze the structure of the black hole
solution.

We consider the ansatz for the spherically symmetric static line element

$\text{d}s^{2}_{static}=-\Delta^{2}(R)\text{d}t^{2}+\frac{\psi^{2}(R)}{\Delta^{2}(R)}\text{d}R^{2}+R^{2}\left(\text{d}\theta^{2}+\sin^{2}\theta\text{d}\phi^{2}\right).$
(7)

We also assume nonvanishing components of the electromagnetic field tensor

$F_{01}=-F_{10}=\mathcal{E}(R),\,\quad~{}F_{23}=-F_{32}=\mathcal{B}(R)\sin\theta.$
(8)

Then the Eq. (4) becomes

$\left(\frac{e^{-\Phi}\mathcal{E}R^{2}}{\psi}+\xi\mathcal{B}\right)^{\prime}=0,$
(9)

where the prime (′) denotes the derivative with respect to the radial
coordinate $R$. The Bianchi identity is satisfied for $\mathcal{B}(R)=P$ (a
constant) and $P$ is therefore identified as the magnetic charge since

$\tilde{F}^{10}=-\frac{P}{R^{2}\psi}.$ (10)

Integrating Eq. (9), we get

$\mathcal{E}=\psi e^{\Phi}\left(\frac{Q-\xi P}{R^{2}}\right),$ (11)

where the integration constant $Q$ is identified as the electric charge,
since, at large $R$, $\mathcal{E}\sim Q/R^{2}$. The equations of motion
[Eqs.(2)-(3)] for the dilaton and the axion field become

$\left(\frac{\Delta^{2}R^{2}\Phi^{\prime}}{\psi}\right)^{\prime}=2e^{-\Phi}\psi
R^{2}\left(\frac{\mathcal{E}^{2}}{\psi^{2}}-\frac{P^{2}}{R^{4}}\right)+e^{2\Phi}\frac{R^{2}\Delta^{2}}{\psi}\xi^{\prime}{}^{2},$
(12)

and

$\left(\frac{\Delta^{2}R^{2}\xi^{\prime}}{\psi}\right)^{\prime}=-2\frac{\Delta^{2}R^{2}}{\psi}\Phi^{\prime}\xi^{\prime}-4e^{-2\Phi}P\mathcal{E}.$
(13)

Note from Eq. (13) that for $\xi=0$, $P=0$ or $Q=0$. From Eq. (6), we get
three non-vanishing components which are

$\displaystyle R_{00}$ $\displaystyle=$
$\displaystyle\frac{\Delta^{4}}{\psi^{2}}\left[\frac{\Delta^{\prime}{}^{2}}{\Delta^{2}}-\frac{\Delta^{\prime}\psi^{\prime}}{\Delta\psi}+\frac{(\Delta^{\prime}R^{2})^{\prime}}{\Delta
R^{2}}\right]=e^{-\Phi}\Delta^{2}\left(\frac{\mathcal{E}^{2}}{\psi^{2}}+\frac{P^{2}}{R^{4}}\right),$
(14) $\displaystyle R_{11}$ $\displaystyle=$
$\displaystyle\frac{2\psi^{\prime}}{R\psi}-\frac{\Delta^{\prime}{}^{2}}{\Delta^{2}}+\frac{\Delta^{\prime}\psi^{\prime}}{\Delta\psi}-\frac{(\Delta^{\prime}R^{2})^{\prime}}{\Delta
R^{2}}=\frac{1}{2}\Phi^{\prime}{}^{2}+\frac{1}{2}e^{2\Phi}\xi^{\prime}{}^{2}-e^{-\Phi}\frac{\psi^{2}}{\Delta^{2}}\left(\frac{\mathcal{E}^{2}}{\psi^{2}}+\frac{P^{2}}{R^{4}}\right),$
(15) $\displaystyle R_{22}$ $\displaystyle=$ $\displaystyle
1-2R\frac{\Delta\Delta^{\prime}}{\psi^{2}}+\frac{\Delta^{2}R^{2}}{\psi^{3}}\left(\frac{\psi}{R}\right)^{\prime}=R^{2}e^{-\Phi}\left(\frac{\mathcal{E}^{2}}{\psi^{2}}+\frac{P^{2}}{R^{4}}\right).$
(16)

Using Eqs. (14) and (15), we get

$\frac{2}{R}\frac{\psi^{\prime}}{\psi}=\frac{1}{2}\Phi^{\prime}{}^{2}+\frac{1}{2}e^{2\Phi}\xi^{\prime}{}^{2}.$
(17)

Demanding proper asymptotic behaviour, i.e. $\Phi\rightarrow 0$ and
$\xi\rightarrow 0$, $\psi\rightarrow 1$ for $R\rightarrow\infty$, we assume,

$\frac{\psi^{\prime}}{\psi}=\frac{\sigma^{2}}{R\left(R^{2}+\sigma^{2}\right)},\quad~{}\text{or,}\quad~{}\quad~{}\psi^{2}=\frac{R^{2}}{R^{2}+\sigma^{2}},$
(18)

where $\sigma^{2}$ is a constant. From Eqs. (14) and (16), we get

$\left[\frac{1}{2}\frac{\left(\Delta^{2}R^{2}\right)^{\prime}}{\psi}\right]^{\prime}=\psi.$
(19)

Solving this equation with the assumption on $\psi$ given in Eq. (18), we
obtain the solution for $\Delta(r)$ as

$\Delta^{2}(R)=1-\frac{2M\sqrt{R^{2}+\sigma^{2}}}{R^{2}}+\frac{P^{2}+Q^{2}}{R^{2}},$
(20)

where the integration constants are identified as the mass $M$ and the sum of
the square of the charges ($P^{2}+Q^{2}$). The black hole resembles the
Reissner-Nordström black holes asymptotically. Using the Eqs. (18) and (20)
the solution for the equations of motion for the dilaton and axion fields are

$\displaystyle e^{\Phi}$ $\displaystyle=$ $\displaystyle
1+\frac{2d}{R^{2}}\sqrt{R^{2}+k^{2}+d^{2}}+\frac{2(k^{2}+d^{2})}{R^{2}},$ (21)
$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{2k\sqrt{R^{2}+k^{2}+d^{2}}}{R^{2}+2d\sqrt{R^{2}+k^{2}+d^{2}}+2(k^{2}+d^{2})},$
(22)

where, $d=\frac{(P^{2}-Q^{2})}{2M}$ and $k=\frac{PQ}{M}$ are dilaton and axion
charges respectively and $\sigma^{2}=k^{2}+d^{2}$.

In the absence of both electric and magnetic charges (i.e. $P=Q=0$), the
dyonic and axionic charges also vanish, i.e. $k=d=0$, and we recover the
Schwarzschild black hole. For any non-zero $P$ and/or $Q$, the line element
does not resemble the Riessner-Nordström black holes. This signifies that
these black holes are hairy. Another distinguishing feature of these black
holes are that they have single horizons – unlike the Riessner-Nordström.
Using the relation $k^{2}+d^{2}=(P^{2}+Q^{2})^{2}/{4M^{2}}$ in the
$f(R_{hz})=0$, we get the horizon radius as,

$R_{hz}=2M\sqrt{1-\frac{P^{2}+Q^{2}}{2M^{2}}}.$ (23)

This feature is also different from Riessner-Nordström and the dyonic black
holes with dilaton field as the only scalar hair. We notice a double horizon
in general and extremal horizon in a special situation. However, the static
version of the Kerr-Sen black holes share the similar feature of a single
horizon. For black holes,

$M\geq\sqrt{\frac{P^{2}+Q^{2}}{2}},$ (24)

otherwise, we have naked singularities. This is illustrated in Fig. 1.

Figure 1: $\frac{P}{M}$ vs. $\frac{Q}{M}$ parameter space is plotted. The
shaded circular region indicates the allowed parameter space for black holes
and the exterior region in parameter space corresponds to naked singularities.

###  II.2 Energy conditions

By identifying the non-zero components of the stress-energy tensors as
$T^{0}{}_{0}=-\rho$, $T^{1}{}_{1}=\tau$, and $T^{2}{}_{2}=T^{3}{}_{3}=p$,
where $\rho$ (energy density), $\tau$ (radial pressure), and $p$ (tangential
pressure) are defined in the orthonormal frame basis, and using the Einstein
field equations ($T_{\mu\nu}=G_{\mu\nu}/{8\pi G}$), we analyze all the energy
conditions. We find that:

$(a)$ The Null Energy Conditions (NEC), i.e. $\rho+\tau\geq 0$ and $\rho+p\geq
0$ are satisfied when $\Delta^{2}(R)\geq 0$. This implies that the NEC is
satisfied on and outside the horizon of the black hole. For a naked
singularity, the NEC is satisfied for all $R$.

$(b)$ The Weak Energy Conditions (WEC) implies $\rho\geq 0$ in addition to the
NEC. Using the Einstein field equation

$\rho=\frac{1}{8\pi
G}\left[\frac{1}{R^{2}}-\frac{(\Delta^{2})^{\prime}}{R\psi^{2}}-\frac{\Delta^{2}}{R^{2}\psi^{2}}+2\frac{\Delta^{2}\psi^{\prime}}{R\psi^{3}}\right].$
(25)

One can check that $\rho\geq 0$ for $R\geq R_{hz}/\sqrt{3}$ for black holes,
and for all $R$ in case of naked singularities. Thus, the WEC is also
satisfied on and outside the horizon of the black holes.

$(c)$ The Strong Energy Condition (SEC), i.e. $\rho+\tau+2p\geq 0$ is
satisfied for all $R$ irrespective of black holes or naked singularities.

$(d)$ The Dominant Energy Conditions (DEC), i.e. $\rho\geq 0$,
$\rho\geq|\tau|$, and $\rho\geq|p|$ are satisfied on and outside of the black
hole horizon, and for all $R$ in case of naked singularities.

It was conjectured Cvetič  _et al._ (2016), Guo  _et al._ (2022) that “a
violation of either the dominant or the strong energy condition is a necessary
condition for the existence of an anti-photon sphere outside a regular black
hole horizon”. Thus, according to our analysis of the energy conditions, the
static dyonic black holes in the EMDA theory do not consist anti-photon sphere
or stable photon orbits.

##  III Dyonic Kerr-Sen black holes

Dyonic Kerr-Sen black holes are the rotating versions of the static black
holes. The Newman-Janis (NJ) algorithm Newman and Janis (1965); Newman  _et
al._ (1965) can be applied to obtain such rotating black holes. After
introducing a new radial coordinate $r$ such that the squared area radius
$R^{2}=r^{2}-2dr-k^{2}$ or $r=\sqrt{R^{2}+k^{2}+d^{2}}+d$, the static line
element (7,18, 20) becomes

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{g(r)}+h(r)\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (26)

where

$\begin{split}f(r)=g(r)&=1-\frac{2M(r-d)-P^{2}-Q^{2}}{r^{2}-2dr-k^{2}}\\\
&=\left(1-\frac{2(d+M)}{r}+\frac{2P^{2}-k^{2}}{r^{2}}\right)\left(1-\frac{2d}{r}-\frac{k^{2}}{r^{2}}\right)^{-1},\end{split}$
(27)

and

$\begin{split}h(r)=R^{2}(r)&=r^{2}-2dr-k^{2}\\\
&=r^{2}\left(1-\frac{2d}{r}-\frac{k^{2}}{r^{2}}\right).\end{split}$ (28)

In terms of the advanced Eddington-Finkelstein coordinates
($u,r,\theta,\phi$), where $du=dt-dr/f(r)$, Eq. (26) is written as

$ds^{2}=-f(r)du^{2}-2dudr+h(r)\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (29)

The inverse metric components of the line element (29) can be decomposed using
the null tetrad
$Z^{\mu}{}_{\alpha}=\left(l^{\mu},n^{\mu},m^{\mu},\bar{m}^{\mu}\right)$ as

$g^{\mu\nu}=-l^{\mu}n^{\nu}-l^{\nu}n^{\mu}+m^{\mu}\bar{m}^{\nu}+m^{\nu}\bar{m}^{\mu},$
(30)

where

$l^{\mu}=\delta^{\mu}{}_{r},\quad~{}n^{\mu}=\delta^{\mu}_{u}-\frac{f}{2}\delta^{\mu}_{r},\quad~{}m^{\mu}=\frac{1}{\sqrt{2h}}\left(\delta^{\mu}_{\theta}+\frac{i}{\sin\theta}\delta^{\mu}_{\phi}\right).$
(31)

Using the complex transformation

$r\rightarrow r^{\prime}=r+ia\cos\theta,\quad~{}u\rightarrow
u^{\prime}=u-ia\cos\theta,$ (32)

where $a$ is the rotation parameter, and replacing the terms $r^{2}$ by
$\hat{\rho}^{2}=r^{\prime}r^{\prime*}=r^{2}+a^{2}\cos^{2}\theta$ and
$\frac{2}{r}$ by
$(\frac{1}{r^{\prime}}+\frac{1}{r^{\prime*}})=\frac{2r}{\hat{\rho}^{2}}$, we
get the new metric in the Eddington-Finkelstein coordinates Drake and Szekeres
(2000); Shaikh (2019)

$\begin{split}ds^{2}&=-F(r,\theta)du^{2}-2dudr+2a\sin^{2}\theta\left[F(r,\theta)-1\right]dud\phi+2a\sin^{2}\theta
drd\phi\\\
&\quad~{}+H(r,\theta)d\theta^{2}+\sin^{2}\theta\left[H(r,\theta)+a^{2}\sin^{2}\theta(2-F)\right]d\phi^{2},\end{split}$
(33)

where $F(r,\theta)$ and $H(r,\theta)$ are complexified forms of $f(r)$ and
$h(r)$ respectively. In our case using Eqs. (27), (28) we get

$\displaystyle f(r)\rightarrow F(r,\theta)$ $\displaystyle=$
$\displaystyle\left(1-\frac{2(d+M)r}{\hat{\rho}^{2}}+\frac{2P^{2}-k^{2}}{\hat{\rho}^{2}}\right)\left(1-\frac{2dr}{\hat{\rho}^{2}}-\frac{k^{2}}{\hat{\rho}^{2}}\right)^{-1},$
(34) $\displaystyle h(r)\rightarrow H(r,\theta)$ $\displaystyle=$
$\displaystyle\hat{\rho}^{2}\left(1-\frac{2dr}{\hat{\rho}^{2}}-\frac{k^{2}}{\hat{\rho}^{2}}\right).$
(35)

In Boyer-Lindquist coordinates, the new metric Eq. (33) for the rotating black
hole finally takes the form Shaikh (2019)

$\begin{split}ds^{2}&=-Fdt^{2}-2a(1-F)\sin^{2}\theta
dtd\phi+\frac{H}{FH+a^{2}\sin^{2}\theta}dr^{2}+Hd\theta^{2}\\\
&\quad~{}+\sin^{2}\theta\left[H+a^{2}\sin^{2}\theta(2-F)\right]d\phi^{2}.\end{split}$
(36)

After simplification using the explicit forms of $F(r,\theta)$ and
$H(r,\theta)$ in our case, we arrive at the line element for a rotating dyonic
black hole in Boyer-Lindquist coordinates

$\begin{split}\text{d}s^{2}=&-\left(1-\frac{2M(r-d)-P^{2}-Q^{2}}{\hat{\Sigma}}\right)\text{d}t^{2}-\frac{2a\sin^{2}\theta}{\hat{\Sigma}}(2M(r-d)-P^{2}-Q^{2})\text{d}t\text{d}\phi\\\
&+\left(r^{2}-2dr-k^{2}+a^{2}+\frac{a^{2}\sin^{2}\theta}{\hat{\Sigma}}\left(2M(r-d)-P^{2}-Q^{2}\right)\right)\sin^{2}\theta\text{d}\phi^{2}+\frac{\hat{\Sigma}}{\hat{\Delta}}\text{d}r^{2}+\hat{\Sigma}\text{d}\theta^{2}\end{split}$
(37)

where the functions $\hat{\Delta}(r)$ and $\hat{\Sigma}(r,\theta)$ are

$\hat{\Delta}=r^{2}-2dr-2M(r-d)-k^{2}+a^{2}+P^{2}+Q^{2},$ (38)

and

$\hat{\Sigma}=r^{2}-2dr-k^{2}+a^{2}\cos^{2}\theta.$ (39)

This metric was already derived in Gal’tsov and Kechkin (1994), Wu  _et al._
(2021) using a different method. This is known as the dyonic Kerr-Sen black
hole spacetime. Here $M$ and $a$ are the mass and rotation parameters of the
black hole. $Q$ and $P$ are the electric and magnetic charges, respectively.
$d=(P^{2}-Q^{2})/{2M}$ and $k=PQ/M$ are the dyonic charge and axion charge
respectively. If the magnetic charge of the black hole vanishes, i.e. $P=0$
then it reduces to the Kerr-Sen black hole. For the special case, $P=Q$, the
dyonic charge vanishes, i.e. $d=0$, but axion charge $k\neq 0$. This is a
distinguishing feature of dyonic Kerr-Sen black holes.

Figure 2: In (a) the shaded region in the $a/M$\- $Z_{c}$ parameter space
corresponds to black holes. In (b) the plot is extended into the full 3-D
parameter space of $a/M$, $P/M$, and $Q/M$. Note that $Z_{c}^{2}=P^{2}+Q^{2}$.

There is a curvature singularity at $r=0$ covered by the radius

$r_{\pm}=M+\frac{P^{2}-Q^{2}}{2M}\pm\sqrt{\left(M-\frac{P^{2}+Q^{2}}{2M}\right)^{2}-a^{2}}.$
(40)

The corresponding event horizon and Cauchy horizon are given by
$R_{+}=\sqrt{r_{+}^{2}-2dr_{+}-k^{2}}$ and
$R_{-}=\sqrt{r_{-}^{2}-2dr_{-}-k^{2}}$ respectively. The horizon can exist
only for

$\left(1-\frac{Z_{c}^{2}}{2M^{2}}\right)^{2}\geq\frac{a^{2}}{M^{2}},$ (41)

where $Z_{c}^{2}=P^{2}+Q^{2}$. Otherwise, the spacetime describes a naked
singularity. This is illustrated in Fig. 2.

##  IV Black hole shadows

In this section we study the shadow cast by the black holes (both rotating and
non-rotating) on the observer’s sky. In order to separate the radial ($r$) and
angular ($\theta$) equation of motion for photons, we use the Hamilton-Jacobi
method. For the rotating case we use the Boyer-Lindquist coordinates, while
for the non-rotating case Schwarzschild coordinates are used.

###  IV.1 Shadows of (rotating) dyonic Kerr-Sen black holes

The Hamilton-Jacobi (HJ) equation for photon trajectories is given by

$H(x^{\mu},p_{\mu})+\frac{\partial S}{\partial\lambda}=0$ (42)

where $S(x^{\mu},\lambda)$ is the Jacobi action, $\lambda$ is the affine
parameter, and $H(x^{\mu},p_{\mu})$ is the Hamiltonian corresponding to the
Lagrangian null trajectories given by
$\mathcal{L}=\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=0$, and ‘dot’
represents the derivative with respect to the affine parameter $\lambda$. The
conjugate momentum is $p_{\mu}=\frac{\partial S}{\partial
x^{\mu}}=\frac{\partial\mathcal{L}}{\partial\dot{x}^{\mu}}$.

We can write $S$, using separation of variables, as

$S=-Et+L\phi+S^{r}(r)+S^{\theta}(\theta),$ (43)

where $E=-p_{t}$ and $L=p_{\phi}$ are the constants of motion. As $S$ does not
depend explicitly on $\lambda$, the HJ equation becomes
$H=\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu}=0$. Using $p_{r}=\frac{\partial
S}{\partial r}=\frac{\text{d}S^{r}}{\text{d}r}$ and $p_{\theta}=\frac{\partial
S}{\partial\theta}=\frac{\text{d}S^{\theta}}{\text{d}\theta}$ and the inverse
metric components

$\displaystyle g^{tt}$ $\displaystyle=$
$\displaystyle-\frac{\left(r^{2}-2dr-k^{2}+a^{2}\right)^{2}-\hat{\Delta}a^{2}\sin^{2}\theta}{\hat{\Sigma}\hat{\Delta}},$
(44) $\displaystyle g^{t\phi}$ $\displaystyle=$ $\displaystyle g^{\phi
t}=-\frac{a}{\hat{\Sigma}\hat{\Delta}}\left(2M(r-d)-P^{2}-Q^{2}\right),$ (45)
$\displaystyle g^{\phi\phi}$ $\displaystyle=$
$\displaystyle\frac{\hat{\Delta}-a^{2}\sin^{2}\theta}{\hat{\Sigma}\hat{\Delta}\sin^{2}\theta},$
(46) $\displaystyle g^{rr}$ $\displaystyle=$
$\displaystyle\frac{\hat{\Delta}}{\hat{\Sigma}},\quad~{}g^{\theta\theta}=\frac{1}{\hat{\Sigma}},$
(47)

we expand the HJ equation and obtain the separated angular and radial
equations of motion of photons. The angular equation of motion is

$\begin{split}\frac{\text{d}S^{\theta}}{\text{d}\theta}=&E\sqrt{\Theta(\theta)}\\\
=&E\sqrt{\chi-l^{2}\cot^{2}\theta+a^{2}\cos^{2}\theta},\\\ \end{split}$ (48)

where $\chi=C/E^{2}$ ($C$ is the Carter constant) and $l=L/E$. The radial
equation is

$\frac{\text{d}S^{r}}{\text{d}r}=E\sqrt{-V(r)},$ (49)

where the effective potential

$V(r)=\frac{(l-a)^{2}+\chi}{\hat{\Delta}}-\frac{(r^{2}-2dr-k^{2}+a^{2}-al)^{2}}{\hat{\Delta}^{2}}.$
(50)

For unstable photon orbits $V(r_{ph})=V^{\prime}(r_{ph})=0$. After
simplification , we obtain $l(r_{ph})$ and $\chi(r_{ph})$ as

$\displaystyle l(r_{ph})$ $\displaystyle=$
$\displaystyle\frac{1}{a}\left(r^{2}_{ph}+a^{2}-2dr_{ph}-k^{2}-4(r_{ph}-d)\frac{\hat{\Delta}(r_{ph})}{\hat{\Delta}^{\prime}(r_{ph})}\right),$
(51) $\displaystyle\chi(r_{ph})$ $\displaystyle=$
$\displaystyle\frac{16(r_{ph}-d)^{2}\hat{\Delta}(r_{ph})}{\hat{\Delta}^{\prime
2}(r_{ph})}-\frac{1}{a^{2}}\left(r^{2}_{ph}-2dr_{ph}-k^{2}-4(r_{ph}-d)\frac{\hat{\Delta}(r_{ph})}{\hat{\Delta}^{\prime}(r_{ph})}\right)^{2},$

where $\hat{\Delta}(r_{ph})$ and $\hat{\Delta}^{\prime}(r_{ph})$ can be
obtained from Eq. (38) . The celestial coordinates for the observer’s sky are

$\alpha=-\frac{l}{\sin\theta_{0}},\quad~{}\beta=\pm\sqrt{\Theta(\theta_{0})},$
(53)

where $\theta_{0}$ is the inclination angle . The parametric plot
$\alpha(r_{ph})$ versus $\beta(r_{ph})$ using the Eqs. (51),
(LABEL:eq:chi(rph)), and (53) gives the shadow profile.

In Fig. 3, the comparison between the shadow profiles for different values of
$a/M$, $Q/M$, and $P/M$ are shown for the Kerr-Newmann, Kerr-Sen, and dyonic
Kerr-Sen black holes. We note that as we increase the $P/M$ values the
deviations from the Kerr-Sen and the Kerr-Newmann black holes are more
prominent. However, as the $a/M$ value (rotation parameter) is increased the
maximum deviation from the Kerr-Sen black holes is found to decrease. This is
more prominent in Fig. 4, where all the shadow boundaries approach the
outermost black solid curve for the Kerr black holes, as the rotation
parameter value is increased.

(a) $a/M=0.5,\,Q/M=0.85,\,P/M=0.25$

(b) $a/M=0.5,\,Q/M=0.85,\,P/M=0.35$

(c) $a/M=0.5,\,Q/M=0.85,\,P/M=0.45$

(d) $a/M=0.65,\,Q/M=0.75,\,P/M=0.20$

(e) $a/M=0.65,\,Q/M=0.75,\,P/M=0.25$

(f) $a/M=0.65,\,Q/M=0.75,\,P/M=0.35$

Figure 3: Shadow boundaries are plotted in the observer’s sky, i.e.
$\frac{\alpha}{M}$ vs $\frac{\beta}{M}$ space. The black dashed and the solid
blue curves in each figure denote the shadow boundaries of the Kerr-Newmann
and the Kerr-Sen black holes respectively. The solid red curve denotes the
dyonic Kerr-Sen black holes. In the top panel, $a/M=0.5$ and $Q/M=0.85$ for
all three figures ($(a),\,(b),\,(c)$) but $P/M$ increases for the dyonic Kerr-
Sen black holes as we go from $(a)$ to $(c)$. In the bottom panel, the $a/M$
value is increased to $a/M=0.65$ for the figures $(d)$ to $(f)$. The $Q/M$
value is also fixed at $Q/M=0.75$ but $P/M$ values are increased for dyonic
Kerr-Sen black holes as in the top panel figures. The inclination angle
$\theta_{0}=90$ degrees for all the figures.

(a) $a/M=0.75,\,Q/M=0.65,\,P/M=0.27$

(b) $a/M=0.85,\,Q/M=0.52,\,P/M=0.165$

(c) $a/M=0.95,\,Q/M=0.30,\,P/M=0.09$

Figure 4: Shadow boundaries are plotted in the observer’s sky, i.e.
$\frac{\alpha}{M}$ vs $\frac{\beta}{M}$ space. The black dashed and the solid
blue curves in each figure denote the shadow boundaries of the Kerr-Newmann
and the Kerr-Sen black holes respectively. The solid red curve denotes values
for the dyonic Kerr-Sen black holes. The outer black solid curve is for Kerr
black holes with corresponding rotation parameter ($a/M$) value. The
inclination angle $\theta_{0}=90$ degrees for all the figures.

###  IV.2 Shadows of static black holes

Using the line element [Eq. (7)] for static black holes, the equations of
motion for the photon trajectories are

$\displaystyle\frac{\text{d}S^{\theta}}{\text{d}\theta}=E\sqrt{\chi-l^{2}\cot^{2}\theta},$
(54) $\displaystyle\frac{\text{d}S^{R}}{\text{d}R}=E\sqrt{-V(R)},$ (55)

where $\chi=C/E^{2}$, $l=L/E$, $C$ is Carter constant, $L$ is angular
momentum. The effective potential for the radial equation of motion

$V(R)=\frac{\psi^{2}}{\Delta^{2}}\left[\frac{\chi+l^{2}}{R^{2}}-\frac{1}{\Delta^{2}}\right].$
(56)

For the photon sphere radius (corresponding to the unstable orbits),
$V(R_{ph})=V^{\prime}(R_{ph})=0$, which leads to the relation

$R_{ph}=\frac{\Delta(R_{ph})}{\Delta^{\prime}(R_{ph})}.$ (57)

Using the expression of $\Delta(R)$ [Eq. (20)], we obtain a quadratic equation

$\begin{split}x^{2}+bx+c=&0,\\\
\text{where},\quad~{}b=&\frac{\tilde{Z}_{c}^{4}}{4}+4\tilde{Z}_{c}^{2}-9,\\\
\text{and},\quad~{}c=&\tilde{Z}_{c}^{6}-2\tilde{Z}_{c}^{4},\end{split}$ (58)

where $x=R_{ph}^{2}/M^{2}$ and $\tilde{Z}_{c}^{2}=(P^{2}+Q^{2})/M^{2}$. Then
we obtain the photon radius from the root of the quadratic equation, i.e.

$x=\frac{R_{ph}^{2}}{M^{2}}=\frac{-b+\sqrt{b^{2}-4c}}{2}.$ (59)

For the celestial coordinates $\alpha$ and $\beta$ as defined earlier [Eq.
(53)] with $a=0$, we obtain

$\alpha^{2}+\beta^{2}=\chi+l^{2}=\frac{R^{2}_{ph}}{\Delta^{2}(R_{ph})}=\frac{1}{\Delta^{\prime
2}(R_{ph})}.$ (60)

Therefore the shadow radius for static black holes is

$\begin{split}R_{shadow}=&\frac{R_{ph}}{\Delta(R_{ph})}\\\
=&M\left[\frac{x^{2}}{x-2\sqrt{x+\frac{\tilde{Z}_{c}^{4}}{4}}+\tilde{Z}_{c}^{2}}\right]^{1/2},\end{split}$
(61)

where $x$ is given by Eq. (59). For the critical value $\tilde{Z}_{c}^{2}=2$
the photon-sphere radius vanishes ($R_{ph}=0$) but the shadow radius does not
vanish ($R_{shadow}=2M$). Thus, the shadow does not exist for naked
singularities. For, $P=Q=0$, i.e. $\tilde{Z}_{c}=0$, $R_{shadow}=3\sqrt{3}M$,
which is the case for the Schwarzschild black hole.

##  V Observational bound on rotating black holes

We can test the possible existence of rotating dyonic Kerr-Sen black holes
using the observations of black hole shadows of M87∗ and Sgr A∗. To do this,
we may define two observational quantities which are– $(i)$ deviation from
circularity ($\Delta C$) Bambi  _et al._ (2019) and $(ii)$ fractional
deviation parameter ($\delta$) related to the average shadow diameter
Collaboration (2022b); Afrin  _et al._ (2023); Shaikh (2022). These two
quantities are described as follows.

We note that the shadow profile is symmetric about $\beta=0$, i.e. the
$\alpha$ axis. Geometric centre of the shadow image on the $\alpha$\- axis is
obtained by taking its mean. Therefore, the centre of the shadow profile is

$\alpha_{c}=\frac{\int\alpha\,\text{d}A}{\int\,\text{d}A},\quad~{}\beta_{c}=0,$
(62)

where $\text{d}A=2\beta d\alpha$ is the area element on the shadow image. From
the geometric centre of the shadow image, the radial distance $\ell(\phi)$ to
any point on the shadow boundary, making an angle $\phi$ with respect
$\alpha$\- axis, can be expressed as

$\ell(\phi)=\sqrt{\left(\alpha(\phi)-\alpha_{c}\right)^{2}+\beta(\phi)^{2}}.$
(63)

Then the average shadow radius can be defined as the root mean squared radius,
i.e.

$R_{avg}^{2}=\frac{1}{2\pi}\int^{2\pi}_{0}\text{d}\phi\ell^{2}(\phi).$ (64)

Finally, the deviation from circularity is defined as Bambi  _et al._ (2019)

$\Delta
C=\frac{1}{R_{avg}}\sqrt{\frac{1}{2\pi}\int^{2\pi}_{0}\text{d}\phi\left(\ell(\phi)-R_{avg}\right)^{2}}.$
(65)

For the computation, it is more convenient to use $r_{ph}$ as the parameter
instead of $\phi$. Then, we can express $R_{avg}$ and $\Delta C$ as

$\displaystyle R^{2}_{avg}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{r_{ph-}}_{r_{ph+}}\left(\beta^{\prime}(\alpha-\alpha_{c})-\beta\alpha^{\prime}\right)\text{d}r_{ph},$
(66) $\displaystyle\Delta C$ $\displaystyle=$
$\displaystyle\frac{1}{R_{avg}}\sqrt{\frac{1}{\pi}\int^{r_{ph-}}_{r_{ph+}}\left(\beta^{\prime}(\alpha-\alpha_{c})-\beta\alpha^{\prime}\right)\left(1-\frac{R_{avg}}{\ell}\right)^{2}\text{d}r_{ph}},$
(67)

where $\beta^{\prime}=\frac{\text{d}\beta}{\text{d}r_{ph}}$ and
$\alpha^{\prime}=\frac{\text{d}\beta}{\text{d}r_{ph}}$. Here $r_{ph+}$ and
$r_{ph-}$ are obtained from the roots of $\beta(r_{ph})=0$, i.e. the values of
$r_{ph}$ for which the shadow boundary cuts the $\alpha$-axis. In other words,
$\phi(r_{ph+})=0$ and $\phi(r_{ph-})=\pi$. The geometric centre of the shadow
($\alpha_{c}$, $\beta_{c}$) can also be expressed in terms of the parameter
$r_{ph}$ as

$\alpha_{c}=\frac{\int^{r_{ph-}}_{r_{ph+}}\alpha\beta\alpha^{\prime}\text{d}r_{ph}}{\int^{r_{ph-}}_{r_{ph+}}\beta\alpha^{\prime}\text{d}r_{ph}},\quad~{}\beta_{c}=0.$
(68)

Note that $\alpha(r_{ph})$, $\beta(r_{ph})$ are obtained from the Eqs. (51),
(LABEL:eq:chi(rph)), and (53).

From the observation of shadow of M87∗, the EHT collaboration has given a
bound $\Delta C\lesssim 0.1$ for an inclination angle $\theta_{0}=17^{o}$
Collaboration (2019a, b, c). However, the bound on $\Delta C$ from SgrA∗ is
not available. From Fig. 5, we note that for Kerr black holes the maximum
value $\Delta C\lesssim 0.07$ for all inclination angle ($\theta_{0}$).
Considering the orientation of the observed relativistic jets from the M87∗,
the inclination angle is estimated to be $\theta_{0}=17^{o}$ Walker  _et al._
(2018). In Fig. 5, the variation of $\Delta C$ is shown as the function of
$a/M$. Note that the maximum value of $\Delta C\lesssim 0.005$ for the
inclination angle $\theta_{0}=17^{o}$. For the same inclination angle, we
scanned the parameter space $a/M-Z_{c}/M$ for the dyonic Kerr-Sen black holes.
From the Fig. 6 we note that the for $\Delta C\lesssim 0.00534$. As $\Delta C$
increases with the inclination angle $\theta_{0}$, we have also scanned the
parameter space for the inclination angle $\theta_{0}=90^{o}$ in Fig. 6. The
maximum possible deviation is $\Delta C\lesssim 0.072$. Therefore, we conclude
that all black hole parameters are allowed and the present observational bound
on the deviation from circularity cannot constrain the parameter space of the
dyonic Kerr-Sen black holes.

Figure 5:  $(a)$ The contour plots for different values of $\Delta C$ is shown
over $a/M$ vs. $\theta_{0}$ parameter space, for Kerr black holes. The black
dashed line corresponds to $\theta_{0}=17^{o}$. $(b)$ $\Delta C$ is plotted as
a function of $a/M$ for the inclination angle $\theta_{0}=17^{o}$.

Figure 6:  The contour plots for different values of $\Delta C$ is shown over
$a/M$ vs. $Z_{c}/M$ parameter space for dyonic Kerr-Sen black holes. For plot
$(a)$ the inclination angle is $\theta_{0}=17^{o}$ and for plot $(b)$
$\theta_{0}=90^{o}$. In the plots, $Z_{c}=\sqrt{P^{2}+Q^{2}}$. The white
excluded region of the parameter space is for naked singularities.

The recent EHT papers on SgrA∗ observations have used the fractional deviation
parameter $\delta$ to constrain the spacetime geometries different from the
Schwarzschild or Kerr black holes. The definition of $\delta$ is as follows

$\delta=\frac{d_{sh}}{d_{sh,Sch}}-1=\frac{R_{avg}}{3\sqrt{3}M}-1,$ (69)

where the average diameter of the shadow , $d_{sh}=2R_{avg}$. Using the
observations of the shadow of SgrA∗ and two separate set of prior values of
mass and distance of SgrA∗ from the VLTI and Keck observations, the EHT
collaboration provided the bound on $\delta$ Collaboration (2022a, b) as

$\delta=\begin{cases}-0.08^{+0.09}_{-0.09}&\text{(VLTI)}\\\
-0.04^{+0.09}_{-0.10}&\text{(Keck)}\\\ \end{cases}$ (70)

Therefore, combining both these bounds, we have $-0.14<\delta<0.01$.

Figure 7: The contour plots for different values of $\delta$ is shown over
$a/M$ vs. $\theta_{0}$ parameter space, for Kerr black holes. The black dashed
line corrspends to $\theta_{0}=50^{o}$.

(a) $\theta_{0}=50^{o}$

(b) $\theta_{0}=50^{o}$, $a=0$

Figure 8:  $(a)$ The contour plots for different values of $\delta$ is shown
over $a/M$ vs. $Z_{c}/M$ parameter space for dyonic Kerr-Sen black holes. The
inclination angle is $\theta_{0}=50^{o}$. The blue shaded region is
observationally disfavored as there $\delta<-0.14$. $(b)$ The black solid line
is the plot of $\delta$ as the function of $Z_{c}/M$ for static black holes,
i.e. $a=0$. The red dashed line corresponds to $\delta=-0.14$ (the
observational limit). It intersects the black solid curve at $Z_{C}/M=0.873$,
meaning $Z_{C}/M\lesssim 0.873$ to satisfy the observational constraint.

In Fig. 7, we scan the parameter space $a/M-\theta_{0}$ with the contours
labelled by different values of $\delta$ for the Kerr black holes. It is noted
that $-0.0704\lesssim\delta<0$ for all parameter values. Thus, the Kerr black
hole parameters are unconstrained from the observational bound on $\delta$
from SgrA∗. Further, we note that the variation in $\delta$ is less sensitive
to the variation of $\theta_{0}$. Moreover, in the observation of SgrA∗,
inclination angle greater than $50^{o}$ is disfavored.

Therefore, we choose $\theta_{0}=50^{o}$ in the Fig. 8(a), where we scan the
parameter space $a/M-Z_{c}/M$ with contours of different $\delta$ values for
the dyonic Kerr-Sen black holes. In the blue shaded region of the plot
$\delta<-0.14$ which means the parameter values ($a/M,Z_{c}/M$) are not
allowed according to the observations of the SgrA∗. Further, we note that for
a $Z_{c}/M$ parameter value greater than the critical limit $0.873$, any $a/M$
parameter value does not satisfy the observational constraint. This critical
value of $Z_{c}/M$ parameter is independent of the inclination angle
$\theta_{0}$ as the critical limit corresponds to the $a=0$, i.e. the static
black holes. In Fig. 8(b), we show the variation of $\delta$ as the function
of $Z_{c}/M$ for static black holes using the analytical expression for the
shadow radius given in Eq. (61). There, we explicitly show the critical limit
of $Z_{c}/M$. We conclude that $Z_{c}/M\lesssim 0.873$ and for any of allowed
$Z_{c}/M$ parameter value, the allowed range of $a/M$ parameter values must be
in the allowed region of space shown in Fig. 8(a).

##  VI Discussion and conclusions

In this paper we have revisited black holes with dyonic charges in Einstein-
Maxwell-Dilaton-Axion (EMDA) theory in the context of the observations on
shadows of the supermassive black holes M87∗ and SgrA∗.

First we outlined the derivation of the static black hole solution by direct
integration of the field equations of EMDA theory. Further, the rotating
version (the dyonic Kerr-Sen black hole) was obtained from the static solution
by applying the Newman-Janis algorithm. Thereafter, we analyzed the structure
of the static black holes in detail. The differences with the standard
Reissner-Nordström (RN) black holes are parametrized by the quantity
$Z_{c}^{2}=P^{2}+Q^{2}$. These static black holes have a single horizon (the
event horizon) unlike non-extremal RN black holes. For either $P=0$ or $Q=0$
the axion field vanishes. However the dilaton field vanishes only when both
$P=0$ and $Q=0$, which is the case for the Schwarzschild black hole. Thus
there is no solution with only axion and Maxwell’s electromagnetic field but
without dilaton field. The reason lies in the structure of the EMDA action
where the axion field is linearly coupled with $F_{\mu\nu}\tilde{F}^{\mu\nu}$.
The static dyonic black holes satisfy all energy conditions outside the event
horizon. There is also no stable photon orbit outside the event horizon of
these static dyonic black holes.

For the rotating case ($a\neq 0$) the axion field is non-zero even if the
axion charge is zero ($k=PQ/M=0$) which is the case for the Kerr-Sen black
hole ($P=0$, $Q\neq 0$, and $a\neq 0$). In the static limit of the Kerr-Sen
black holes (i.e. setting $a=0$), the axion field vanishes. This is an
interesting difference between the Kerr-Sen black holes and the dyonic Kerr-
Sen black holes, in general.

We study the shadow profiles for both rotating and static black holes. We have
obtained the exact expression for the shadow radius for static black holes.
The parameter $Z_{c}<\sqrt{2}M$ and the shadow radius $R_{shadow}$ obeys
$2M<R_{shadow}<3\sqrt{3}M$. The effect of magnetic charge/ monopole on the
shadow profile of rotating black holes has been investigated graphically by
comparing the dyonic Kerr-Sen black holes with the Kerr-Newman and the Kerr-
Sen black holes (Fig. 3). The deviation increases with the increase in
magnetic charge. However, the deviation is not so prominent for higher
rotation parameter (Fig. 4).

Finally, we test whether the known supermassive black holes at galactic
centres could be modeled as such dyonic Kerr-Sen black holes. In other words,
we look for metric parameter values for which the shadow features match with
those in the observed shadow images of M87∗ and SgrA∗. We have used two
observational quantities related to black hole shadows – $(i)$ the deviation
from circularity of the observed shadow boundary ($\Delta C$) and $(ii)$ the
fractional deviation parameter ($\delta$) representing the deviation of the
observed average shadow diameter from that for a Schwarzschild black hole. The
observational bound on $\Delta C$ for M87∗ is satisfied for all parameter
values for the dyonic black holes. Thus, it cannot be constrained. For SgrA∗,
the EHT collaboration provided a bound on $\delta$ which gives a constraint on
the black hole parameter $Z_{c}=\sqrt{P^{2}+Q^{2}}\lesssim 0.873M$. Thus we
get an upper bound on the magnetic monopole charge (if any) for SgrA∗ as
$P\lesssim 0.873M$ where $M=4.154\times 10^{6}M_{\odot}$ for SgrA∗. In natural
units, a magnetic charge $Q_{m}=P/m_{P}$ where $m_{P}=G^{-1/2}=1.22\times
10^{19}$ $GeV$ is the Planck mass. In these units, the obtained bound on the
magnetic charge of SgrA∗ is $Q_{m}\lesssim 6.955\times 10^{44}$.

Magnetic monopoles are a natural outcome of grand-unified theories (GUTs)
Hooft (1974) and was an original motivation for cosmic inflation Guth (1981);
Preskill (1979). They have been searched in various experiments Ambrosio  _et
al._ (2002); Orito  _et al._ (1991); Balestra  _et al._ (2008). Primordial
black holes with magnetic charges, which reached the extremality condition in
due course of cosmic evolution and do not Hawking radiate anymore, can be a
possible dark matter candidate Diamond and Kaplan (2022). They are termed as
the extremal Magnetic Black Holes (EMBHs). Various astrophysical limits on
such EMBHs was discussed in Diamond and Kaplan (2022). However, the authors
put constraints on the black holes with mass range having the upper limit
$M<10^{33}\,g$. Therefore these black holes cannot be modeled as the
supermassive black holes like M87∗ and SgrA∗ and thus cannot be directly
compared with the bound that we obtained from the observations of shadow
images. However, we can have some idea on what type of physical process can
give bounds on the magnetic charge of the supermassive black holes at the
center of our galaxy. For example, from the observed temperature of clouds of
Warm Ionized Medium (WIM) in the Milky Way Lehner  _et al._ (2004), we can put
some upper bound on the magnetic charge of SgrA*. Following the calculations
of Diamond and Kaplan (2022), we get a rough estimation on the upper bound on
the magnetic charge of SgrA∗ to be $P\lesssim 10^{-13}M$. Another
astrophysical constraint comes from the Parker bound Parker (1970) which is
based upon the survival of today’s galactic magnetic field, as the field
energy is drained out by the magnetic monopoles while moving through the
field. This puts an upper limit on the flux of magnetic monopoles. From the
Parker bound our rough estimate on the magnetic charge of SgrA∗ is $P\lesssim
359\,M_{\odot}$ and thus $P/M\lesssim 10^{-4}$. Clearly, the other
astrophysical constraints are expected to be far more stringent by several
orders of magnitude.

Thus, any possible observational relevance of our results, atleast in the
context of shadows and the viable presence of magnetic charges seems quite a
distant dream. However, the theoretical analysis we have carried out, may be
of use once more imaging observations on shadows of black holes are carried
out and presented in future.

## Acknowledgements

Research of SJ is partially supported by the SERB, DST, Govt. of India,
through a TARE fellowship grant no. TAR/2021/000354, hosted by the department
of Physics, Indian Institute of Technology Kharagpur.

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