# Shadows of rotating hairy Kerr black holes coupled to time periodic scalar
fields with non-flat target space

Galin N. Gyulchev<EMAIL_ADDRESS>Department of Theoretical
Physics, Faculty of Physics, Sofia University, Sofia 1164, Bulgaria  Ayush Roy
<EMAIL_ADDRESS>School of Mathematical Sciences and STAG Research Centre,
University of Southampton, Southampton, SO17 1BJ, United Kingdom  Lucas G.
Collodel<EMAIL_ADDRESS>Theoretical Astrophysics,
Eberhard Karls University of Tübingen, Tübingen 72076, Germany  Petya G.
Nedkova<EMAIL_ADDRESS>Department of Theoretical Physics, Faculty
of Physics, Sofia University, Sofia 1164, Bulgaria  Stoytcho S. Yazadjiev
<EMAIL_ADDRESS>Department of Theoretical Physics, Faculty of Physics,
Sofia University, Sofia 1164, Bulgaria  Institute of Mathematics and
Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev St. 8, Sofia
1113, Bulgaria  Daniela D. Doneva<EMAIL_ADDRESS>Theoretical
Astrophysics, Eberhard Karls University of Tübingen, Tübingen 72076, Germany

###### Abstract

We study the shadows cast by rotating hairy black holes with two non-trivial
time-periodic scalar fields having a non-flat Gaussian curvature of the target
space spanned by the scalar fields. Such black holes are a viable alternative
to the Kerr black hole, having a much more complicated geodesic structure and
resulting shadows. We investigate how a nontrivial Gauss curvature alters the
pictures for different amounts of scalar hair around the black holes,
quantified by a normalized charge. Our results show that for high values of
this charge, close to a boson star limit, chaotic shadows are observed with
multiple small disconnected components for all considered Gaussian curvatures.
For moderately large amounts of scalar hair and corresponding normalized
charge, although the shadows still exhibit chaotic behavior, a dominant shadow
component emerges, the size and shape of which are substantially influenced by
the Gaussian curvature. For instance, highly chaotic shadows for flat target
space, start developing a large central shadow region with the increase of the
Gauss curvature even for black holes with substantially heavy scalar hair. For
lower values of the normalized charge, the shadows resemble qualitatively the
Kerr black hole while the Gaussian curvature has a small impact on their
properties.

##  I Introduction

Through the years, general relativity (GR) has always emerged victorious when
subjected to various tests [42, 8, 1]. From accurately predicting the right
amount of Mercury’s perihelion precession, to the direct detection of
gravitational waves from merging compact objects, Einstein’s gravity continues
to perform extremely well. Yet, most of these experiments are either in the
weak field regime or have limited accuracy that hinders us from testing strong
gravity in fine detail. That is why many well-motivated generalizations of
Einstein’s theory still remain viable.

Very recently the Event Horizon Telescope (EHT) Collaboration has opened up
the gate to new tests of the strong field regime of gravity through
observations of black hole shadows. In 2019, the EHT Collaboration captured
the image of the central supermassive black hole (SMBH) in the M87 galaxy [9].
Three years later in 2022, the EHT Collaboration also produced an image of
${\rm Sgr}A^{*}$, the central supermassive black hole in the Milky Way [18].
This success motivated the proposition of a Next Generation EHT that will be
able to take black hole snapshots with a much higher accuracy. The advance in
observations pushed the theoretical development with the idea of challenging
the Kerr hypothesis. Building upon the black hole images obtained by the EHT,
numerous alternative models to the Kerr black hole have been proposed and
examined. These include horizonless compact objects such as naked
singularities and wormholes [39, 37, 39, 20, 32, 31], as well as beyond-Kerr
black holes arising from modified theories of gravity [6, 4, 5, 13, 41, 45,
30]. While a diverse range of Kerr black hole mimickers exists, certain
configurations exhibit distinct features in their shadows, leading to multi-
connected or highly distorted images. Examples of such scenarios comprise
black holes with scalar or Proca hair [23, 22, 38], black holes in binary
systems [44, 35, 40], and black holes interacting with external matter
distributions [2, 3, 19]. In the present paper, we will consider further
examples of scalarized black holes interacting with multiple scalar fields.
These solutions can be classified according to the Gaussian curvature of the
target space spanned by the scalar fields and may manifest diverse behaviors
depending on its value.

Stationary asymptotically flat black hole solutions predicted by vacuum GR are
hairless and completely determined by their mass and angular momentum. This
also applies to the case when the Einstein equations are coupled to a single
real scalar field – it is a classical result that the black hole can not
support a single scalar field hair [36, 25, 24]. However, the scenario changes
when multiple scalar fields are considered. In the case of a complex time-
dependent scalar field, or equivalently, two real scalar fields forming a flat
2-dimensional manifold (target space), it was initially discovered that black
holes with hair can emerge within the perturbative regime [26, 27, 7, 28].
Shortly after that, the self-consistent non-linear solutions were generated
numerically [23, 22]. The scalar fields in this scenario do not inherit the
stationary and axisymmetric properties of the spacetime but instead have a
harmonic dependence on the $t$ and $\phi$ coordinates. Even though the scalar
field is time-dependent, the generalized Einstein equations, as well as the
spacetime metric, are stationary. The regularity at the black hole horizon
requires, though, synchronizing the angular velocity at the black hole horizon
with the scalar field oscillation frequency. That is why these solutions were
dubbed black holes with synchronized scalar hair.

The black hole solutions discussed in [23, 22] indeed involve two scalar
fields whose target space is flat, as previously mentioned. However, this
represents just the simplest choice, and the geometry of the target space can
be substantially more intricate [16, 29, 43]. In fact, nonlinear rotating
black hole solutions with synchronized hair, comprising two real scalar fields
forming a non-flat manifold (non-flat target space) that is maximally
symmetric, were constructed in [10]. It was demonstrated that the curvature of
the target space can significantly alter the domain of existence and the
properties of the hairy black holes.

Kerr black holes with synchronized scalar hair are ones of the very few
astrophisically relevant candidates that can produce deviations from GR not
only for stellar black holes but also for supermassive ones (for another
interesting very recent example see [17]). This motivated the study of their
shadows that was performed in the case of a flat target space [14, 15]. The
purpose of the present paper is to extend these studies to a non-flat target
space, more precisely for a maximally symmetric space with curvature $\kappa$.
We will systematically examine the influence of nonzero $\kappa$ on the black
hole shadow and eventually explore the inverse problem – what the back hole
shadow can tell us about the target space formed by the scalar fields.

The paper is organized as follows. In the next section, we present the
scalarized black holes which we consider, describing their basic properties.
In section 3 we outline the theoretical background necessary for obtaining the
shadow images. Then, in section 4 we construct explicitly the shadows cast by
selected solutions with positive, negative, and zero curvature of the target
space. The shadow images are compared and analyzed according to the influence
of the normalized charge of the solutions and the Gaussian curvature of the
target space. In the last section, we present our conclusions.

##  II Kerr black holes with synchronized scalar hair – non-flat target space
geometry

In the present paper, we focus on multiple dynamical scalar fields
$\varphi^{a}$ minimally coupled to the Einstein gravity. The scalar fields
$\varphi^{a}$ can be considered as generalized coordinates on an abstract
$N$-dimensional Riemmanian space $({\cal E}_{N},\gamma_{ab}(\varphi))$, the
so-called target space. The target space metric $\gamma_{ab}(\varphi)$ should
be positively defined on ${\cal E}_{N}$. The general action of the theory is
then given by

$S=\frac{1}{4\pi
G}\int\sqrt{-g}\left(\frac{R}{4}-\frac{1}{2}g^{\mu\nu}\gamma_{ab}(\varphi)\partial_{\mu}\varphi^{a}\partial_{\nu}\varphi^{b}-V(\varphi)\right)\,d^{4}x,$
(1)

where $V(\varphi)$ is the scalar field potential. This action can be also
interpreted as the vacuum action of the tensor-multi-scalar-theories of
gravity ([16, 29]). Varying it with respect to the spacetime metric and the
scalar fields, we get the following field equations

$\displaystyle R_{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\gamma_{ab}(\varphi)\partial_{\mu}\varphi^{a}\partial_{\nu}\varphi^{b}+2V(\varphi)g_{\mu\nu},$
(2) $\displaystyle\square\varphi^{a}$ $\displaystyle=$
$\displaystyle-\gamma^{a}_{bc}(\varphi)g^{\mu\nu}\partial_{\mu}\varphi^{b}\partial_{\nu}\varphi^{c}+\gamma^{ab}(\varphi)\frac{\partial
V(\varphi)}{\partial\varphi^{b}},$ (3)

where $\square$ is the d’Alembert operator associated with the spacetime
metric and $\gamma^{a}_{bc}(\varphi)$ are the Christoffel symbols with respect
to the target space metric.

From now on, we shall focus on two scalar fields (effectively modeling the
complex scalar field originally considered in [23]), possessing maximally
symmetric target spaces $({\cal E}_{2},\gamma_{ab}(\varphi))$. In this case we
have globally defined coordinates, the so-called isothermal coordinates, in
which the target space metric can be written in the conformally flat form

$\gamma_{ab}(\varphi)=\Omega^{2}(\varphi)\delta_{ab}.$ (4)

Here $\delta_{ab}$ is the usual Kronecker Delta and the conformal factor
$\Omega(\varphi)$ is given by

$\Omega^{2}(\varphi)=\frac{1}{\left(1+\frac{\kappa}{4}\psi^{2}\right)^{2}},$
(5)

with $\psi^{2}=\delta_{ab}\varphi^{a}\varphi^{b}$ and $\kappa$ being the
Gaussian curvature of the target space.

For the scalar fields potential $V(\varphi)$ we assume the simplest standard
massive potential given by

$V(\psi)=\frac{1}{2}\mu^{2}\psi^{2}$ (6)

where $\mu$ is the scalar field mass.

Since we are interested in rotating black holes, the ansatz for the stationary
and axisymmetric line element is chosen to be

$ds^{2}=-\mathcal{N}e^{2F_{0}}dt^{2}+e^{2F_{1}}\left(\frac{dr^{2}}{\mathcal{N}}+r^{2}d\theta^{2}\right)+e^{2F_{2}}r^{2}\sin^{2}\theta\left(d\phi-\frac{\omega}{r}dt\right)^{2},$
(7)

where $\mathcal{N}=1-\frac{r_{H}}{r}$ ($r_{H}$ is the location of the horizon
in these coordinates), while $F_{0},F_{1},F_{2}$, and $\omega$ are functions
of $r$ and $\theta$ only. The interested reader can consult the Appendix of
[22] for an isometry between the line element in eq. (7) and the Kerr line
element in Boyer-Lindquist coordinates.

In the present paper, we will be interested in scalar field endowed black
holes. In order to violate the no-scalar-hair theorems it is not enough to
consider multiple scalar fields. Additionally, one should let the fields be
time-dependent [24]. Thus, an ansatz for these fields, which is also
consistent with the circularity of the metric (7), is the following

$\varphi^{1}=\psi(r,\theta)\cos(\omega_{s}t+m\phi),\quad\varphi^{2}=\psi(r,\theta)\sin(\omega_{s}t+m\phi),$
(8)

where $\omega_{s}$ is a real parameter and $m$ is an integer. One can easily
check that although $\varphi^{1}$ and $\varphi^{2}$ are time-dependent, the
resulting field equations (2), (3) are stationary. More details can be found
in [10].

Assuming these forms of the scalar field and metric ansatze, plus the
appropriate boundary conditions that are regularity at the event horizon and
the axes, and asymptotic flatness at infinity, one is able to find stationary
black hole solutions with nontrivial scalar hair. It is interesting to note
that the regularity at the horizon leads to the condition

$\omega|_{r=r_{H}}=-\frac{r_{H}\omega_{s}}{m},$ (9)

which also ensures that there is no scalar flux into the black hole.
Therefore, the angular velocity at the black hole horizon
$\Omega_{H}=\omega_{H}/r_{H}$ should be equal to $\omega_{s}$. Thus the scalar
field is synchronized with the black hole rotation. Additionally, at infinity

$\lim_{r\to\infty}\psi\propto\frac{1}{r}exp\left(-\sqrt{\mu^{2}-\omega_{s}^{2}}r\right),$
(10)

thereby bounding $\omega_{s}$ via $\omega_{s}^{2}\leq\mu^{2}$.

Figure 1: Tracks for fixed $r_{H}$ in the $M-\omega_{s}$ plane, taken from
[10] for $\kappa=\\{0,-5,5\\}$. The solutions are obtained by solving a
coupled set of partial differential equations found by varying the Einstein-
Hilbert action coupled to two scalar fields. The black line and below holds
Kerr solutions, whereas new exotic solutions are held in the yellow bubble. It
is bounded by three classes of solutions, solitonic, extremal hairy black
hole, and cloud solutions, depending on the value of $r_{H}$ and normalized
charge $q=\frac{mQ}{J}$, as explained in the text. The positions of selected
solutions are indicated by red and blue dots, which, for a given $r_{H}$ and
various $\kappa$, share the same $q$. Each configuration is identified by
X${}^{\,u}_{\,v}$, where the symbol X represents the configuration number,
superscript $u$ represents $\kappa$, and the subscript $v$ denotes $r_{H}$.
The physical characteristics of these selected solutions are given in Table 1
of the Appendix.

The ADM mass and angular momentum can be calculated from by the metric
asymptotic

$M=\frac{1}{2}\lim_{r\to\infty}r^{2}\partial_{r}F_{0},\quad
J=\frac{1}{2}\lim_{r\to\infty}r^{2}\omega.$ (11)

They have varying contributions from the black hole bare mass and the hair. To
specify the hairiness, the normalized charge $q$ is introduced, defined as
$q=\frac{mQ}{J}$, where $Q$ is the conserved Noether charge. Interestingly,
the angular momentum of the scalar field is quantized in terms of the Noether
Charge $Q$, i.e, $J_{\psi}=mQ$, so we simply have $q=\frac{J_{\psi}}{J}$. This
quantization is similar to the rotating boson stars [33]. The $q\approx 0$
solutions are the limit of scalar clouds non-backreacting the metric, whereas
$q\approx 1$ solutions are the boson star limit (potentially with non-flat
target space metric [43, 11]).

In the present paper we will utilize the numerical solutions obtained in [10]
for Gaussian curvatures $\kappa\in\\{-5,-1,0,1,5\\}$ and fixed $m=1$. The
domains of existence of hairy black holes are presented in a $M-\omega_{s}$
plane in Fig. 1 for different $\kappa$. The black line marks the extremal Kerr
limit, with $a=M$, and thus Kerr black holes exist only below it in the grey-
shaded region. The yellow region is the domain of existence of hairy Kerr
black hole solutions having the following boundaries – the red line is the
solitonic (boson star) limit with $q=1$ and $r_{H}=0$, extremal hairy black
holes are denoted with a green line (again $r_{H}=0$ but $q\neq 0$), while
blue lines are used to depict the cloud solutions ($q=0$) 111The fact that
some of these lines look incomplete is due to numerical difficulties in
constructing the solutions that in general become worse for more negative
$\kappa$.. A general trend is that with the decrease of $\kappa$ the domain of
existence of hairy black holes becomes more deformed. Dashed lines mark
sequences of solutions with constant horizon radii while red dots on these
lines are the particular black hole solutions we used for building shadows.

From the point of view of $r_{H}$, black holes with smaller values of $r_{H}$
have a larger domain of existence in $\omega_{s}$: They move closer to the
pure solitonic ($q=1$) and extermal black hole limit, both of which have a
vanishing horizon radius. Tracks denoted with an extremely small value of
$r_{H}$ start and stop at the cloud line. More generally, however, each fixed
track of $r_{H}$ starts at the Minkowski limit with $\omega_{s}/\mu=1$ and
stops at the cloud line.

##  III Calculating the black hole shadow

###  III.1 Photon Equations of Motion and Initial Conditions

To generate the black hole’s shadow through backward ray tracing, we
numerically integrate photon geodesics until they either fall into the black
hole or intersect the celestial sphere. The paths of these geodesics are
governed by the geodesic equations, which are obtained from Hamilton’s
equations:

$\dot{x}^{\mu}=\frac{\partial\mathcal{H}}{\partial p_{\mu}}\
,\qquad\dot{p}_{\mu}=-\frac{\partial\mathcal{H}}{\partial x^{\mu}}\ ,$ (12)

where the overdot implies differentiation with respect to an affine parameter.
The Hamiltonian is defined in terms of the contravariant components of the
metric tensor, $g^{\mu\nu}$. Assuming minimal coupling between the null
particles with 4–momentum $p_{\mu}$ components and the spacetime geometry, it
satisfies that

$\mathcal{H}\equiv\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu}=0.$ (13)

Due to stationarity and axisymmetry, $\mathcal{H}$ does not depend on $t$ and
$\phi$, and both $p_{t}$ and $p_{\phi}$ are conserved quantities of the
geodesic motion. Assuming the asymptotically flatness of the spacetime, we can
then define the integrals of motion $E$ and $L$, which are interpreted as the
energy and azimuthal angular momentum of the photon as measured by an
asymptotic static observer:

$E=-p_{t}=-g_{tt}\dot{t}-g_{t\phi}\dot{\phi},\quad\quad
L=p_{\phi}=g_{t\phi}\dot{t}+g_{\phi\phi}\dot{\phi}.$ (14)

Therefore, by decoupling the variables, we obtain

$\dot{t}=\frac{g_{\phi\phi}E+g_{t\phi}L}{g_{t\phi}^{2}-g_{tt}\,g_{\phi\phi}},\quad\quad\dot{\phi}=-\frac{g_{t\phi}E+g_{tt}L}{g_{t\phi}^{2}-g_{tt}\,g_{\phi\phi}}.$
(15)

Taking into account that for the photon motion $\mathcal{H}=0$ is a conserved
quantity along the trajectory, we can represent the Hamiltonian in the form:

$\mathcal{H}=p_{r}^{\,2}g^{rr}+p_{\theta}^{\,2}g^{\theta\theta}+V_{\rm
eff}(r,\theta)=0,$ (16)

where the quantity $T\equiv
p_{r}^{\,2}g^{rr}+p_{\theta}^{\,2}g^{\theta\theta}\geq 0$ related to the
kinetic energy is positive definite. Here, the effective potential $V_{\rm
eff}(r,\theta)\leq 0$ is negative definite and reveals the allowed region
$(r,\theta)$ of the photon’s motion. It is given by

$V_{\rm
eff}(r,\theta)=\frac{E^{2}g_{\phi\phi}+2ELg_{t\phi}+L^{2}g_{tt}}{g_{tt}g_{\phi\phi}-g_{t\phi}^{2}}=\frac{E^{2}g_{tt}}{(g_{tt}g_{\phi\phi}-g_{t\phi}^{2})}(\eta-
h_{+})(\eta-h_{-}),$ (17)

where $\eta=L/E$ is the impact parameter. The functions $h_{\pm}(r,\theta)$
are the photon’s effective potentials defined as:

$h_{\pm}=\frac{-g_{t\phi}\pm\sqrt{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}}{g_{tt}}.$
(18)

The discriminant $\mathcal{D}=g_{t\phi}^{2}-g_{tt}g_{\phi\phi}$ is positive
outside the black hole horizon, while $g_{tt}$ is positive inside (negative
outside) the ergoregion.

To begin ray tracing, we require information about a photon’s initial
conditions, namely $t,r,\theta,\phi,E,L,p_{r}$ and $p_{\theta}$. These are
obtained by constructing a local orthonormal basis at the position of the
observer, generally placed at $(\tilde{r}_{obs}=15M$ and
$\theta_{obs}=\frac{\pi}{2})$ throughout this work. Here $\tilde{r}$ is the
circumferential radius of a point at the equatorial plane and is given by:

$\tilde{r}=\frac{1}{2\pi}\int_{0}^{2\pi}d\phi\,\sqrt{g_{\phi\phi}}=r\,e^{F_{2}(r)},$
(19)

where $r$ is the coordinate radius. To establish the initial conditions for
the photon’s 4-momentum, we adhere to the formalism discussed in [15, 34]. For
convenience, we introduce two angles, $\alpha$ and $\beta$, which parameterize
the impact parameters of a photon positioned at the observer’s location, as
follows:

$x=-\tilde{r}_{obs}\tan\beta,\quad y=\tilde{r}_{obs}\sin\alpha.$ (20)

At a chosen observer’s position $r_{obs}$, first, we calculate the
circumferential distance $\tilde{r}_{obs}(r_{obs})$, according to Eq. (19).
Positioned at the point $(\tilde{r}_{obs},\theta_{obs})$, the observer views a
two-dimensional flat screen, $(x,y)$, passing through the center of the black
hole and perpendicular to the axis between the observer and the origin.
Additionally, the magnitude of the 4–momentum of the photon in the observer’s
frame of reference satisfies the Hamiltonian constraint

$\mathcal{P}^{2}=-(\mathcal{P}^{t})^{2}+(\mathcal{P}^{r})^{2}+(\mathcal{P}^{\theta})^{2}+(\mathcal{P}^{\phi})^{2}=0.$
(21)

Then, the components of the spatial momentum $\mathcal{\vec{P}}$ in terms of
the angles $(\alpha,\beta)$ are given by,

$\mathcal{P}^{r}=|\mathcal{\vec{P}}|\cos\alpha\cos\beta,\quad\mathcal{P}^{\theta}=|\mathcal{\vec{P}}|\sin\alpha,\quad\mathcal{P}^{\phi}=|\mathcal{\vec{P}}|\cos\alpha\sin\beta,$
(22)

and the photon has $\mathcal{P}^{t}=|\mathcal{\vec{P}}|$. Since
$|\mathcal{\vec{P}}|$ only determines the photon’s frequency and does not
alter the trajectory, it is set to unity, implying that $\mathcal{P}^{t}=1$ as
well.

We must now relate the components of the 4-momentum in generally curved
coordinates to the quantities
$\mathcal{P}^{t},\mathcal{P}^{r},\mathcal{P}^{\theta},\mathcal{P}^{\phi}$.
This requires an understanding of how the basis vectors appear in the local
orthonormal basis. The chosen basis takes the following form,

$\hat{e}_{\theta}=A^{\theta}\partial_{\theta},\quad\hat{e}_{\phi}=A^{\phi}\partial_{\phi},\quad\hat{e}_{t}=\zeta\partial_{t}+\gamma\partial_{\phi},\quad\hat{e}_{r}=A^{r}\partial_{r}.$
(23)

Now imposing the conditions
$\hat{e}_{r}\cdot\hat{e}_{r}=1,\hat{e}_{\theta}\cdot\hat{e}_{\theta}=1,\hat{e}_{\phi}\cdot\hat{e}_{\phi}=1$,
$\hat{e}_{t}\cdot\hat{e}_{t}=-1$ and $\hat{e}_{t}\cdot\hat{e}_{\phi}=0$, we
arrive at

$A^{r}=\frac{1}{\sqrt{g_{rr}}},\quad
A^{\theta}=\frac{1}{\sqrt{g_{\theta\theta}}},\quad
A^{\phi}=\frac{1}{\sqrt{g_{\phi\phi}}},\quad\gamma=-\zeta\frac{g_{t\phi}}{g_{\phi\phi}},\quad\zeta=\sqrt{\frac{g_{\phi\phi}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}}.$
(24)

The final form of the four basis vectors describing a Minkowski frame is as
follows,

$\hat{e}_{t}=\zeta\left(\partial_{t}-\frac{g_{t\phi}}{g_{\phi\phi}}\partial_{\phi}\right),\quad\hat{e}_{r}=\frac{1}{\sqrt{g_{rr}}}\partial_{r},\quad\hat{e}_{\theta}=\frac{1}{\sqrt{g_{\theta\theta}}}\partial_{\theta},\quad\hat{e}_{\phi}=\frac{1}{\sqrt{g_{\phi\phi}}}\partial_{\phi}.$
(25)

Using this information, we project the components of the 4-momentum into this
basis as follows,

$\displaystyle\mathcal{P}^{t}=-\hat{e}^{\mu}_{t}p_{\mu}=-\left(\zeta
p_{t}+\gamma
p_{\phi}\right)=\zeta\left(E+\frac{g_{t\phi}}{g_{\phi\phi}}L\right),$ (26)
$\displaystyle\mathcal{P}^{r}=\hat{e}^{\mu}_{r}p_{\mu}=\frac{1}{\sqrt{g_{rr}}}p_{r},$
(27)
$\displaystyle\mathcal{P}^{\theta}=\hat{e}^{\mu}_{\theta}p_{\mu}=\frac{1}{\sqrt{g_{\theta\theta}}}p_{\theta},$
(28)
$\displaystyle\mathcal{P}^{\phi}=\hat{e}^{\mu}_{\phi}p_{\mu}=\frac{1}{\sqrt{g_{\phi\phi}}}L,$
(29)

where the identification of the conserved quantities $E=-p_{t}$ and
$L=p_{\phi}$ was used wherever they appear. Given the earlier presentation of
the equations of motion, it is more desirable to obtain starting conditions
for $E,L,p_{r}$ and $p_{\theta}$. Explicitly, inverting the above expressions
yields,

$\displaystyle
E=\frac{1+\gamma\sqrt{g_{\phi\phi}}\sin\beta\cos{\alpha}}{\zeta},\quad
p_{r}=\sqrt{g_{rr}}\cos\beta\cos\alpha,$ (30) $\displaystyle
L=\sqrt{g_{\phi\phi}}\sin\beta\cos{\alpha},\qquad\quad\;\;p_{\theta}=\sqrt{g_{\theta\theta}}\sin\alpha.$
(31)

After positioning the observer on the equatorial plane $(\theta=\pi/2)$ at a
circumferential radius $\tilde{r}_{obs}=15M$, the angles $\alpha$ and $\beta$
determine the impact parameters $(x,y)$ and initial momenta using the local
orthonormal basis. With these initial conditions, photons evolve backward in
time until they encounter the celestial sphere, situated at a circumferential
distance $\tilde{r}_{cel}=30M$ or they hit the black hole horizon. To
interpret the resulting gravitational lensing patterns, we divide the
celestial sphere into four quadrants, each assigned a distinct color based on
the angular distribution between the polar angle $\theta_{cel}$ and the
azimuthal angle $\phi_{cel}$ as follows. In the top hemisphere
($0<\theta_{cel}<\pi/2$): the green quadrant if $0<\phi_{cel}<\pi$, and the
red quadrant if $\pi<\theta_{cel}<2\pi$. In the bottom hemisphere
($\pi/2<\theta_{cel}<\pi$): the blue quadrant if $0<\phi_{cel}<\pi$, and the
yellow quadrant if $\pi<\phi_{cel}<2\pi$. We introduce a grid of thin black
meridian and parallel lines spaced $10^{\circ}$ apart in each quadrant.

###  III.2 Light Rings and Ergoregions

In this subsection, we will consider the properties of the light rings and
ergoregions for the chosen configurations shown in Fig. 1. As we will see,
some of the studied configurations have stable light rings that, placed in
their ergoregions, become light beam attractors, which can lead to the
appearance of chaotic patterns in the black hole shadows.

The light rings are circular null geodesics considered in the equatorial plane
of symmetry, $\theta=\pi/2$, that satisfy the conditions $p_{\theta}=0$ and
$p_{r}=\dot{p_{r}}=0$, from which it follows that

$V_{\rm eff}=0,\quad\quad\partial_{r}V_{\rm eff}=0,$ (32)

at the location of the orbit. Following Eq. (17), these conditions can be
reduced to the next system of two algebraic equations that must be satisfied
simultaneously

$\displaystyle g_{\phi\phi}+2\eta g_{t\phi}+g_{tt}\eta^{2}=0,$ (33)
$\displaystyle\partial_{r}g_{\phi\phi}+2\eta\partial_{r}g_{t\phi}+\eta^{2}\partial_{r}g_{tt}=0.$
(34)

Solving the first of these equations for the impact parameter $\eta=L/E$ and
substituting in the second, we obtain the light ring equation, which predicts
the existence of a photon circular orbit with a radius $r_{LR}$

$\partial_{r}g_{\phi\phi}+2h_{\pm}\partial_{r}g_{t\phi}+h_{\pm}^{2}\partial_{r}g_{tt}=0,$
(35)

where the functions $h_{\pm}$ are defined via Eq. (18). Moreover, the radial
condition for the existence of stable (unstable) light rings imposes
$\partial^{2}V_{\rm eff}>0$, ($\partial^{2}V_{\rm eff}<0$), which is reduced
to the restriction

$\partial^{2}_{r}g_{\phi\phi}+2h_{\pm}\partial^{2}_{r}g_{t\phi}+h_{\pm}^{2}\partial^{2}_{r}g_{tt}\begin{cases}<0,&\text{if
light ring is stable.}\\\ >0,&\text{otherwise.}\end{cases}$ (36)

Besides, taking into account that $\eta=h_{\pm}$ on the circular orbit and
$h_{+}\neq h_{-}$ outside the horizon, after straightforward calculations, one
can show that the conditions for the effective potential (32) are reduced to
an analogical light ring equation

$\partial_{r}h_{\pm}=0.$ (37)

The solutions to that equation predict stable (unstable) light rings if the
radial condition $\pm\partial^{2}_{r}h_{\pm}>0$,
($\pm\partial^{2}_{r}h_{\pm}<0$) is satisfied. Moreover, the normalized
timelike Killing vector field at infinity, denoted as $\partial_{t}$, becomes
null over the surface $g_{tt}=0$, defining the ergoregions [21]. In that
special case, in the limit $g_{tt}\rightarrow 0$, one of the functions
$h_{\pm}$ diverges, and the other converges to $-g_{\phi\phi}/2g_{t\phi}$.
Generally, outside the ergoregions $g_{tt}<0$, while passing within the
ergoregions $g_{tt}>0$.

Light rings can also be classified in terms of their direction of rotation.
Taking into account Eqs. (15), as well as that Eq. (33) is satisfied on the
light ring, one can show that the angular velocity, $\Omega=d\phi/dt$, of the
photons moving on circular orbits is connected to the impact parameter via the
relation

$\Omega=\frac{1}{\eta}.$ (38)

Hence, it follows that the light ring’s rotational direction is given by the
sign of the impact parameter $\eta$ for a static observer at spatial infinity.
In general, the orbital angular frequency of rotating photons at the light
rings is given by

$\Omega_{\pm}=\frac{-\partial_{r}g_{t\phi}\pm\sqrt{\partial_{r}g_{t\phi}^{2}-\partial_{r}g_{tt}\partial_{r}g_{\phi\phi}}}{\partial_{r}g_{\phi\phi}},$
(39)

where the above expressions are evaluated at the location of the light ring.
In the above equation $\Omega_{+}$ is the angular frequency of co-rotating
photons, and $\Omega_{-}$ is the angular frequency of the counter-rotating
photons.

###  III.3 Numerical Ray Tracing Methods

To compute the cast shadows of rotating tensor-multiscalar black holes with
scalar hair, a program code was developed using the Wolfram Language
Mathematica. The code uses built-in interpolation and differential solver
routines to integrate the Hamiltonian equations of motion numerically. For the
numerical implementation of the equations, the metric functions $F_{0}$,
$F_{1}$, $F_{2}$, and $\omega$ were interpolated using a two-dimensional cubic
spline interpolation concerning the variables $r$ and $\theta$, which was set
as the default interpolation procedure in the programming language. In the
Hamiltonian equations, the interpolated functions were substituted into the
analytical expressions for the metric functions and their derivatives
concerning $r$ and $\theta$. For the numerical solution of the system of
differential equations, we utilize time integration methods inherent to
Mathematica, such as the Adams/BDF multi-step method with automatic step size
control.

Additional information is required to generate the shadows’ image on the
observer’s screen. This includes the observer’s position, the two-dimensional
width of the field of view, the image resolution, and the radius of the
celestial sphere where the rays, scattered by the black hole, will complete
their trajectory.

In this context, without loss of generality, we choose the observer to be
located on the equatorial plane at a specific circumferential distance from
the black hole, such that
$(t_{obs},r_{obs},\theta_{obs},\phi_{obs})=(0,r_{obs}(\tilde{r}_{obs}),\pi/2,0)$.
By selecting the circumferential distance as $\tilde{r}_{obs}=15M$, we use Eq.
(19) to numerically determine the radial coordinate of the observer,
$r_{obs}$. Simultaneously, we set the circumferential distance
$\tilde{r}_{cel}=30M$ as the second boundary condition for any geodesic
reaching the celestial sphere. After performing the ray tracing procedure, we
apply an equiangular projection that directly maps the angles $\theta_{cel}$
and $\phi_{cel}$, labeling the photon scattered by the black hole onto the
celestial sphere with the observed angles $(\beta,\alpha)$ onto the coordinate
axes $(x,y)$. To implement the projection, we establish the desired field of
view, which specifies the angles between the optical axis and the image
boundaries. Subsequently, by determining the desired image resolution,
corresponding to $1024\times 1024$ photon trajectories, we found the required
step size for uniform grid formation across the entire field of view.

Increasing the geodesic flux density is crucial to generating a high-
resolution image of the shadow, revealing more precisely the complexity of the
chaotic patterns. To significantly diminish the integration time for the
entire field of view, we first take advantage of the reflection symmetry
inherent in the shadow image as perceived by an equatorial observer. This
involves integrating only half of the shadow–specifically, the geodesics
either above or below the equatorial plane of symmetry. Subsequently,
exploiting the reflection symmetry of the examined solution enables us to
reconstruct the complete shadow image, filled with its intrinsic colours
denoting the geodesics scattered to different parts of the celestial sphere.

On the other hand, since modern workstations contain multiple computing cores,
we utilize the built-in FinestGrained method to break down the overall
computation into the minor possible subunits, whose evaluations take different
amounts of time. Employing this approach facilitates the optimization of the
integration time, a notably time-consuming process for scalarized solutions
with stable light rings. The geometry of these solutions contributes to the
chaotic behavior of geodesics near the event horizon. This is illustrated by
the shadow image and the heat map of the time delay function in Fig. 3. This
function is defined as the variation of the time required for a photon
geodesic to travel from a particular pixel on the observer’s screen to a
corresponding point on the celestial sphere, measured in units of $\mu^{-1}$.
The heat map provides valuable information about the distribution of time-
delay function values on the observer’s screen. It is particularly useful in
diagnosing the light ring system since photon trajectories that come close to
the light rings take a significant amount of time to return to spatial
infinity. For a more detailed discussion of the time delay heat map in various
scenarios, such as rotating boson stars and hairy black hole solutions, please
refer to [12].

##  IV Results

In this section, we will examine the shadow images for seven groups of
selected configurations, I–VII, all highlighted in Fig. 1. Each configuration
is identified by X${}^{\,u}_{\,v}$, where the symbol X represents the
configuration number, superscript $u$ represents the value of the Gaussian
curvature $\kappa$, and the subscript $v$ denotes the value of the black hole
horizon $r_{H}$. Each group contains three configurations for a unique
normalized charge, certain black hole horizon, and various Gaussian curvatures
$\kappa\in\\{-5,0,5\\}$. Generally, the higher values of $\kappa$ reduce the
chaotic patterns in the shadows of black holes with different horizons,
affecting the radii and rotational direction of the light rings (LR) and the
equatorial radii of the ergoregions (ER). Henceforth, in the paper, we will
utilize the dimensionless compactified radial coordinate $R\in[0,\,1]$ defined
as follows:

$R=\frac{\tilde{R}}{1+\tilde{R}},\quad\mbox{with}\quad\tilde{R}=\sqrt{r^{2}-r_{H}^{2}}.$
(40)

In the following subsection, we explore a particular configuration,
VII${}^{\,-5}_{\,0.05}$, which allows us to observe not only a complex
structure of ergoregions and a system of multiple light rings but also a
variety of non-simply connected shadow images, accompanied by the chaotic
behavior of scattered photons.

###  IV.1 Photon Potential, Ergoregions and Light Rings

In this section, we will focus on a specific configuration distinguished by
one of the most intricate shadows. Beyond the appearance of numerous non-
simply connected shadows with various shapes and regions highly saturated with
chaotically scattered orbits, there lies a complex structure of light rings
and multiple ergoregions. A notable aspect of this particular solution is the
presence of a stable light ring characterized by torus topology situated
within one of the ergoregions. To explore the mechanisms behind the formation
of these unusual shadows, in this subsection, we will present the contour
plots of the $h_{+}$ and $h_{-}$ photon potentials, the shadow image, and the
photon’s time delay heat map for the configuration VII${}^{\,-5}_{\,0.05}$,
with Gaussian curvature $\kappa=-5$, a black hole horizon of $r_{H}=0.05$ and
a normalized charge $q\simeq 0.996$.

Figure 2: Contour plots of the two effective potentials $h_{+}$ (left panel)
and $h_{-}$ (right panel) for configurations VII${}^{\,-5}_{0.01}$ at Gaussian
curvature $\kappa=-5$. The configuration contains two disconnected ergoregions
pointed out with red contours on the upper panel, the first among which is
located near the horizon. Each of the potentials $h_{+}$ and $h_{+}$ possess a
saddle point between both ergoregions, allowing two unstable, closely spaced
light rings to form. In the center of the second ergoregion, the potential
$h_{+}$ possesses a local minimum, corresponding to a third stable light ring.
Beyond the second ergoregion, the function $h_{+}$ has a second saddle point,
which causes the formation of a fourth outermost unstable light ring. The
orange dots indicate the locations of the four light rings.

Fig. 2 exhibits the effective potentials $h_{+}$ and $h_{-}$ for the spacetime
configuration VII${}^{\,-5}_{\,0.05}$, and Table 1 in the Appendix presents
the corresponding physical quantities of the selected solution. The contour
lines of the function $h_{+}$ reveal a singular behavior of the potential at
the boundary of two detached ergoregions, the first located near the black
hole’s event horizon ($R_{H}=0$) in the equatorial domain
$R_{ER}\in[0,\,0.036]$. The second ergoregion extending in the interval
$R_{ER}\in[0.154,\,0.484]$, possesses toroidal topology and contains a global
minimum corresponding to the existence of an equatorial stable light ring for
$R_{LR}\simeq 0.307$, rotating in the same direction as the black hole. The
presence of two saddle points reveals the existence of two equatorial unstable
light rings, the first located between the two ergoregions for $R_{LR}\simeq
0.070$ and the second formed beyond the ergotorus for $R_{LR}\simeq 0.733$.
Both rings correspond to photons, circling the black hole in the opposite
spinning direction as the black hole.

Fig. 2, bottom panel, exhibits $h_{-}$ contour lines, which demonstrate the
existence of a saddle point, corresponding to an equatorial unstable light
ring situated at $R_{LR}\simeq 0.041$ outside the inner ergoregion. The sign
of $\eta$ indicates that the photons in the light ring are circling the black
hole in the same spinning direction as the black hole.

The sign of $\eta$, $d\phi/dt$ and $g_{tt}$ as well as the corresponding
values of the dimensionless coordinate $R$, pointing the positions of the
light rings and the equatorial domain of existence of the ergoregions are
organized in the following table.

Configuration | Fig. | Ergoregions | $R_{ER}$ | LR | $R_{LR}$ | stability | $\eta$ | $g_{tt}$ | $d\varphi/dt$ | Chaos
---|---|---|---|---|---|---|---|---|---|---
VII${}^{\,-5}_{\,0.05}$ | 3 |  |  | $h_{-}$ | 0.041 | unstable | $+$ | $-$ | $+$ | Yes
1 ER | $[0,\;0.036]$ | $h_{+}$ | 0.070 | unstable | $-$ | $-$ | $-$
2 ER | $[0.154,\;0.484]$ | $h_{+}$ | 0.307 | stable | $+$ | $+$ | $+$
|  | $h_{+}$ | 0.733 | unstable | $-$ | $-$ | $-$

Figure 3: Zoom time delay heat map associated with the scattering orbits
(right panel) and lensed image (left panel) for configuration
VII${}^{\,-5}_{\,0.05}$, with $\kappa=-5$, $r_{H}=0.05$,
$\omega_{s}/\mu=0.648538$, $M\mu=0.915671$ and $q=0.996185$. The regions
corresponding to shadow points are shown in black. The physical quantities of
the solution are presented in Table 1 in Appendix A.

A correlation has been observed between the chaotic patterns found in the
shadow’s image and the characteristics of the corresponding geodesic motion
[12]. In Fig. 3, on the right, you can see the black hole shadow alongside the
time delay heat function, which is depicted in units of the reciprocal values
of the scalar field mass, $\mu^{-1}$. The photon’s variation in the coordinate
time, $t$, illustrates the expected sensitivity in mapping between the
photon’s coordinates $(x,y)$ in the image plane and the corresponding arrival
point on the celestial sphere or the black hole horizon. The heat map in this
case is very informative, as it reveals a strong correlation between the shape
of the regions with bright pixels with a significant time delay and the
chaotic patterns created by the scattered photons on the shadow image. This
direct relationship indicates that some photon bundle are sensitive to the
subset of their impact parameters, in which photons are propagated near the
light rings.

This sensitivity arises due to the effective potential $h_{+}$, which permits
the existence of quasi-bound orbits for specific impact parameters $\eta$.
Some of these orbits are situated in a pocket with a narrow throat containing
an unstable light ring ($LR_{4}$), enabling photons to enter the pocket.
Subsequently, the photons traverse the ergotorus, housing the stable light
ring ($LR_{3}$), near which the orbits undergo multiple radial turning points
before leaving the pocket and reaching the celestial sphere. Since the number
of radial turning points depends on the frequency of changes in the sign of
$\dot{p}_{r}$ during the photon’s motion along the light ray’s trajectory,
this can be interpreted as a deviation from Kerr spacetime, where null
geodesics possess at most one turning point. All trajectories of this class,
semi-trapped in a pocket, take significantly longer to escape, resulting in a
more significant time delay. These light beams are depicted as bright dots on
the time delay heat map shown in Fig. 3 on the right.

On the contrary, for specific values of the impact parameter $\eta$, the
analysis of the effective potential $h_{+}$ reveals the possible appearance of
a second inner throat in the pocket. This configuration allows photons to
approach the unstable light ring $LR_{2}$. Due to the radial instability of
this light ring, for some impact parameters, photons may either retrace their
path back into the pocket or proceed towards the fourth unstable light ring,
$LR_{1}$, located near the inner ergoregion, as the potential $h_{-}$ shows.
Ultimately, these photons may fall into the black hole, especially with small
perturbations in the impact parameter. The configuration of two throats in the
effective potential may give rise to the formation of multiple disconnected
shadows resembling black hole shadows reported in [14].

###  IV.2 Black hole shadows and influence of the Gaussian curvature

To illustrate the impact of different Gaussian curvatures on the light ring’s
formation, the ergoregions of the rotating solutions, and the visual features
of their shadows in the following figures, we expose the shadows of the
selected configurations, emphasizing the influence of this parameter on the
observed phenomena. We have limited ourselves to the solutions already
generated in [10], which constitute sequences of black holes with fixed
horizon radii. That is why the solutions are grouped with respect to $r_{H}$
while the normalized charge $q$ is also kept as similar in value as possible
in each group.

Given the significant variation in the shape, size, and characteristic pattern
of the shadow within the domain of existence of hairy Kerr black holes, as
depicted in Fig. 1, we will perform a detailed survey of the 18 most
interesting cases. The physical characteristics of these cases are listed in
detail in Table I of the Appendix. Our focus will predominantly be on models
close to the limiting red curve in that figure, marking the boson star limit.
These cases exhibit the most peculiar characteristics and the appearance of
chaotic regions in the shadow image. Below, the discussion is grouped
according to the horizon radius and normalized charge of the explored
solutions, which, roughly speaking, indicates how far away the black hole
solutions are from the boson star limit.

####  IV.2.1 Model I, $r_{H}=0.01$, $q\simeq 0.999$: Black hole image domited
by chaotic regions.

To study the impact of the normalized charge, $q$, on shadow formation,
ergoregions, and the system of light rings, we focus on configurations
I${}^{\,\kappa}_{\,0.01}$, with $\kappa\in\\{-5,0,5\\}$ (highlighted in Fig.
1). All these configurations exhibit relatively higher values of the
normalized charge, approximately $q\simeq 0.999$. The corresponding physical
parameters for these configurations are detailed in Table 1 in Appendix A.

Similar to the solution VII${}^{\,-5}_{\,0.05}$, the selected configurations
feature a system of four light rings, two ergoregions, and distinct shadow
images saturated with multiple chaotic patterns. Among these light rings,
there is always one that remains stable, contributing to the observation of
numerous chaotic regions in shadow images. The peculiarity of these
configurations lies in their ability to generate multiple miniature, highly
elongated, single-connected shadows. Furthermore, even though shadows exist
for negative $\kappa$, the size of the shadow is marginal, while this size
increases with increasing $\kappa$.

The distinctive features of the ergoregions and light rings for the analyzed
configurations are presented in the following table.

Configuration | Fig. | Ergoregions | $R_{ER}$ | LR | $R_{LR}$ | stability | $\eta$ | $g_{tt}$ | $d\varphi/dt$ | Chaos
---|---|---|---|---|---|---|---|---|---|---
I${}^{\,-5}_{\,0.01}$ | 4 |  |  | $h_{-}$ | 0.010 | unstable | $+$ | $-$ | $+$ | Yes
1 ER | $[0,\;0.004]$ | $h_{+}$ | 0.012 | unstable | $-$ | $-$ | $-$
2 ER | $[0.045,\;0.474]$ | $h_{+}$ | 0.186 | stable | $+$ | $+$ | $+$
|  | $h_{+}$ | 0.727 | unstable | $-$ | $-$ | $-$
I${}^{\,0}_{\,0.01}$ | 4 |  |  | $h_{-}$ | 0.010 | unstable | $+$ | $-$ | $+$ | Yes
1 ER | $[0,\;0.003]$ | $h_{+}$ | 0.012 | unstable | $-$ | $-$ | $-$
2 ER | $[0.063,\;0.462]$ | $h_{+}$ | 0.196 | stable | $+$ | $+$ | $+$
|  | $h_{+}$ | 0.722 | unstable | $-$ | $-$ | $-$
I${}^{\,5}_{\,0.01}$ | 4 |  |  | $h_{-}$ | 0.009 | unstable | $+$ | $-$ | $+$ | Yes
1 ER | $[0,\;0.005]$ | $h_{+}$ | 0.013 | unstable | $-$ | $-$ | $-$
2 ER | $[0.043,\;0.445]$ | $h_{+}$ | 0.155 | stable | $+$ | $+$ | $+$
|  | $h_{+}$ | 0.718 | unstable | $-$ | $-$ | $-$

Figure 4: Examples of shadows which illustrate the transition between shadows
for different Gaussian curvatures $\kappa\in\\{-5,0,5\\}$ and black hole
horizon $r_{H}=0.01$. The left panel corresponds to configuration
I${}^{\,-5}_{\,0.01}$ with $\kappa=-5$, $\omega_{s}/\mu=0.607387$,
$M\mu=0.890489$ and $q=0.999866$, the center panel corresponds to
configuration I${}^{\,0}_{\,0.01}$ with $\kappa=0$, $\omega_{s}/\mu=0.679241$,
$M\mu=0.881991$ and $q=0.999875$, while the right panel corresponds to
configuration I${}^{\,5}_{\,0.01}$ with $\kappa=5$, $\omega_{s}/\mu=0.731639$,
$M\mu=0.883209$ and $q=0.999733$. The physical quantities of the solutions are
presented in Table 1 in Appendix A.

In all three geometries, there exists an inner ergoregion extending from the
horizon ($R_{H}=0$) to a distance $R_{ER}$, which slightly depends on the
Gaussian curvature. A notable distinction is observed when the curvature of
the target space is zero; the equatorial extension of the inner ergoregion is
minimal, $R_{ER}\in[0,\,0.003]$, in contrast to the extensions,
$R_{ER}\in[0,\,0.004]$ and $R_{ER}\in[0,\,0.005]$, for negative and positive
curvatures, respectively. No light ring is placed in any of these inner
ergoregions.

All three configurations feature an ergotorus, corresponding to zero Gaussian
curvature with the minor equatorial section, $R_{LR}\in[0.063,\,0.462]$. In
the other two cases, the ergotorus possesses a slightly larger width:
$R_{ER}\in[0.045,\,0.474]$ for negative curvature and
$R_{ER}\in[0.043,\,0.445]$ for positive curvature. Each ergotorus houses a
stable light ring that co-rotates with the black hole, irrespective of the
Gaussian curvature of the target space.

Regardless of the magnitude of the Gaussian curvature, each configuration has
two inner unstable light rings located near each other, positioned between the
two ergoregions. The innermost of these rings co-rotates with the black hole,
while the second larger ring rotates in the opposite direction. A distinctive
feature is observed in the ergotorus of the solution with zero Gaussian
curvature, where the stable light ring has the largest radius, $R_{LR}\simeq
0.196$, in contrast to the smaller-radius stable rings, $R_{LR}\simeq 0.155$,
for positive curvature, and $R_{LR}\simeq 0.186$, for negative curvature.
Outside the ergotorus is positioned the outermost fourth unstable light ring,
which rotates in the opposite direction to the black hole.

The shadows of the considered solutions are depicted in Fig. 4. Under negative
Gaussian curvature, the shadow is densely filled with chaotic patterns that
exhibit slight changes in both structure and location at zero and positive
curvature. A significant feature at this nearly extreme value of the normal
charge in scalar hair black hole configurations is the emergence of multiple
highly elongated shadows. Specifically, under negative Gaussian curvature, the
shadow displays 11 distinguishable single-connected miniature dark regions.
This number decreases to 7 shadow regions at zero and positive curvature.
During this transition, four dark regions merge into two slightly more
extensive regions, while two other dark regions disappear. This change results
in a noticeable, albeit small, reduction in chaotic patterns. Furthermore, the
increase in curvature convincingly demonstrated an enlargement in the width
and overall size of the multiple dark areas formed, contributing to the entire
shadow image.

####  IV.2.2 Model II, $r_{H}=0.01$, $q\simeq 0.997$: Shadows with large
chaotic regions.

Analyzing the influence of the normalized charge, $q$, on shadow formation,
ergoregions, and the system of light rings, we move through the $M-\omega_{s}$
space, shown in Fig. 1, to the right while maintaining a constant horizon
radius, $r_{H}=0.01$. We select configurations II${}^{\,\kappa}_{\,0.01}$,
with $\kappa\in\\{-5,0,5\\}$, all with slightly smaller values of the
normalized charge, roughly $q\simeq 0.997$. The corresponding physical
parameters for these configurations are provided in Table 1 in Appendix A.

Each configuration with a different Gaussian curvature has one ergoregion and
four light rings. Among these light rings, there is always one that remains
stable, contributing to the observation of chaotic regions in shadow images.
The peculiarity of these configurations is their ability to form multiple non-
simply connected images, which shows significant chaotic behaviour in
trajectories of light orbits near the black hole horizon.

The ergoregion of the II${}^{\,-5}_{\,0.01}$ configuration has the most
spacious equatorial cross-section, extending from the black hole’s event
horizon ($R_{H}=0$) to $R_{ER}\simeq 0.410$. The outermost light ring at
$R_{LR}\simeq 0.674$, is unstable and rotates opposite to the black hole.
Moving towards the event horizon, we pass through the ergosurface ($g_{tt}=0$)
and discover a system of three equatorial light rings relatively close to the
horizon. The outermost of these rings, at $R_{LR}\simeq 0.071$, is stable and
rotates in the direction of the black hole’s rotation. The next inner ring, at
$R_{LR}\simeq 0.014$, is unstable and rotates oppositely to the black hole’s
rotation. The innermost light ring, closest to the horizon, is also unstable,
rotates in the direction of the black hole’s rotation, and has a radius
corresponding to coordinate $R_{LR}\simeq 0.009$. The positions of the light
rings and the equatorial domain of ergoregion existence, expressed in terms of
the dimensionless parameter $R$, are summarized in the following table.

Configuration | Fig. | Ergoregions | $R_{ER}$ | LR | $R_{LR}$ | stability | $\eta$ | $g_{tt}$ | $d\varphi/dt$ | Chaos
---|---|---|---|---|---|---|---|---|---|---
II${}^{\,-5}_{\,0.01}$ | 5 | 1 ER | $[0,\;0.410]$ | $h_{-}$ | 0.009 | unstable | $+$ | $+$ | $+$ | Yes
$h_{+}$ | 0.014 | unstable | $-$ | $+$ | $-$
$h_{+}$ | 0.071 | stable | $+$ | $+$ | $+$
$h_{+}$ | 0.674 | unstable | $-$ | $-$ | $-$
II${}^{\,0}_{\,0.01}$ | 5 | 1 ER | $[0,\;0.387]$ | $h_{-}$ | 0.008 | unstable | $+$ | $+$ | $+$ | Yes
$h_{+}$ | 0.016 | unstable | $-$ | $+$ | $-$
$h_{+}$ | 0.069 | stable | $+$ | $+$ | $+$
$h_{+}$ | 0.657 | unstable | $-$ | $-$ | $-$
II${}^{\,5}_{\,0.01}$ | 5 | 1 ER | $[0,\;0.397]$ | $h_{-}$ | 0.007 | unstable | $+$ | $+$ | $+$ | Yes
$h_{+}$ | 0.020 | unstable | $-$ | $+$ | $-$
$h_{+}$ | 0.063 | stable | $+$ | $+$ | $+$
$h_{+}$ | 0.680 | unstable | $-$ | $-$ | $-$

Figure 5: Examples of shadows which illustrate the transition between shadows
for different Gaussian curvatures $\kappa\in\\{-5,0,5\\}$ and black hole
horizon $r_{H}=0.01$. The left panel corresponds to configuration
II${}^{\,-5}_{\,0.01}$ with $\kappa=-5$, $\omega_{s}/\mu=0.739809$,
$M\mu=0.690217$ and $q=0.997229$, the center panel corresponds to
configuration II${}^{\,0}_{\,0.01}$ with $\kappa=0$,
$\omega_{s}/\mu=0.835272$, $M\mu=0.648229$ and $q=0.997293$, while the right
panel corresponds to configuration II${}^{\,5}_{\,0.01}$ with $\kappa=5$,
$\omega_{s}/\mu=0.821927$, $M\mu=0.742514$ and $q=0.997069$. The physical
quantities of the solutions are presented in Table 1 in Appendix A.

Referring to Fig. 5, we observe that an increase in Gaussian curvature results
in a noticeable expansion of the visible area of black hole shadows,
maintaining the same normal charge. For the most substantial negative value of
the Gaussian curvature, $k=-5$, multiple nested ovals and non-simply connected
images are observed, indicating a significant chaotic behavior in the
trajectories of light orbits. As the Gaussian curvature increases to $k=0$,
chaotic patterns diminish in size, forming distinguishable multiple-shadow
images of the black hole. In the case of the minor event horizon,
$r_{H}=0.01$, one reason for the reduction in chaotic patterns within the
shadow images is the shrinking of the radius of stable light rings with
increasing Gaussian curvature. Simultaneously, in the considered
configurations, the outermost unstable light rings move away from the event
horizon, and their radius further expands with the increasing curvature of the
target space. This phenomenon is an essential condition for increasing the
capture cross-section of the black holes, ultimately resulting in the
formation of a larger shadow.

####  IV.2.3 Model III, $r_{H}=0.05$, $q\simeq 0.994$: Larger size black hole
shadows harbouring chaotic regions.

Increasing the size of the event horizon to $r_{H}=0.05$, we consider
configurations III${}^{\,\kappa}_{\,0.05}$, with $\kappa\in\\{-5,0,5\\}$,
highlighted in Fig. 1. In this scenario, the selected configurations feature a
normalized charge of $q\simeq 0.994$. The physical quantities of these
solutions are detailed in Table 1 in Appendix A, showcasing variations for
different Gaussian curvatures of the target space. As in configurations I and
VII, in configuration III, we observe the appearance of the same structure of
four light rings and two ergoregions. The table below presents the technical
characteristics of the geometric structures under consideration.

Configuration | Fig. | Ergoregions | $R_{ER}$ | LR | $R_{LR}$ | stability | $\eta$ | $g_{tt}$ | $d\varphi/dt$ | Chaos
---|---|---|---|---|---|---|---|---|---|---
III${}^{\,-5}_{\,0.05}$ | 6 |  |  | $h_{-}$ | 0.040 | unstable | $+$ | $+$ | $+$ | Yes
1 ER | $[0,\;0.057]$ | $h_{+}$ | 0.073 | unstable | $-$ | $-$ | $-$
2 ER | $[0.095,\;0.476]$ | $h_{+}$ | 0.274 | stable | $+$ | $+$ | $+$
|  | $h_{+}$ | 0.727 | unstable | $-$ | $-$ | $-$
III${}^{\,0}_{\,0.05}$ | 6 |  |  | $h_{-}$ | 0.037 | unstable | $+$ | $+$ | $+$ | Yes
1 ER | $[0,\;0.047]$ | $h_{+}$ | 0.080 | unstable | $-$ | $-$ | $-$
2 ER | $[0.157,\;0.471]$ | $h_{+}$ | 0.297 | stable | $+$ | $+$ | $+$
|  | $h_{+}$ | 0.739 | unstable | $-$ | $-$ | $-$
III${}^{\,5}_{\,0.05}$ | 6 |  |  | $h_{-}$ | 0.036 | unstable | $+$ | $+$ | $+$ | Yes
1 ER | $[0,\;0.045]$ | $h_{+}$ | 0.087 | unstable | $-$ | $-$ | $-$
2 ER | $[0.238,\;0.444]$ | $h_{+}$ | 0.333 | stable | $+$ | $+$ | $+$
|  | $h_{+}$ | 0.757 | unstable | $-$ | $-$ | $-$

In configuration III, similar to configurations I and VII, we note the
presence of the same structure, featuring four light rings and two
ergoregions. However, in contrast to configurations I and VII, where both four
light rings and two ergoregions coexist, configuration III exhibits a
different arrangement – the innermost unstable ring is located within the
inner ergoregion. Transitioning into the zone between the two ergoregions, a
single unstable light ring is found. Similar to solutions I and VII, the
ergoregion also hosts a stable light ring. Extending beyond the ergoregion, we
establish the existence of the fourth outermost unstable light ring.

Analyzing the dynamics of the light rings, we observe that, moving from the
innermost to the outermost ring, the first co-rotates with the black hole,
while subsequent rings alternate in spin direction. As a result, the fourth
and final light ring counter-rotates with the black hole. When increasing the
Gaussian curvature of the target space, we note a corresponding growth in the
radius of light rings of the same type. Consequently, the system of light
rings for positive Gaussian curvature exhibits the most extensive spatial
distribution. At the same time, the equatorial region of both inner
ergoregions and ergotoruses decreases with an increase in Gaussian curvature
across different configurations.

At these values of the horizon and the normal charge, we observe a significant
change in the shape and size of the shadow when transitioning from negative to
positive Gaussian curvature. At these values of the horizon and the normal
charge, we observe a significant change in the shape and size of the shadow
when transitioning from negative to positive Gaussian curvature. Black hole
shadows for configurations III${}^{\,\kappa}_{\,0.05}$, with
$\kappa\in\\{-5,0,5\\}$ are exposed in Fig. 6. In instances of negative
curvature, configuration III${}^{\,-5}_{\,0.05}$ not only features a series of
single-connected dark regions but simultaneously exhibits chaotic patterns. At
zero curvature of the target space, the individual single-connected regions
merge into a larger shadow that continues to show signs of randomness. In
contrast to these cases, with positive curvature, the image of the shadow is
fully connected, relatively more compact, and exhibits chaoticity of the
scattered photon orbits outside the dark region of the shadow.

Figure 6: Examples of shadows which illustrate the transition between shadows
for different Gaussian curvatures $\kappa\in\\{-5,0,5\\}$ and black hole
horizon $r_{H}=0.05$. The left panel corresponds to configuration
III${}^{\,-5}_{\,0.05}$ with $\kappa=-5$, $\omega_{s}/\mu=0.653743$,
$M\mu=0.886503$ and $q=0.994449$, the center panel corresponds to
configuration III${}^{\,0}_{\,0.05}$ with $\kappa=0$,
$\omega_{s}/\mu=0.706437$, $M\mu=0.908153$ and $q=0.994490$, while the right
panel corresponds to configuration III${}^{\,5}_{\,0.05}$ with $\kappa=5$,
$\omega_{s}/\mu=0.702254$, $M\mu=1.100262$ and $q=0.994709$. The physical
quantities of the solutions are presented in Table 1 in Appendix A.

####  IV.2.4 Model IV, $r_{H}=0.1$, $q\simeq 0.96$: Black hole shadows with
decreasing chaotic regions.

Let’s consider the configurations IV${}^{\,\kappa}_{\,0.1}$, with
$\kappa\in\\{-5,0,5\\}$, highlighted in Fig. 1, with an event horizon set at
$r_{H}=0.1$. In these instances, the normal charge of the selected solutions
varies around $q\simeq 0.96$. A notable feature of these configurations is
that, for every value of the Gaussian curvature, a system of four light rings
and one ergoregion exists. The positions of the light rings, the equatorial
domain of the existence of the ergoregions, and additional information about
their characteristics are presented in the following table.

The equatorial domain of ergoregion existence is defined from the event
horizon ($R_{H}=0$) to a distance $R_{ER}$, which decreases significantly as
the Gaussian curvature of the target space increases. Therefore, at negative
$\kappa=-5$, the relatively large width of the ergoregion,
$R_{ER}\in[0,\,0.439]$, encloses three of the inner light rings with radii
$R_{LR}=0.061$, $R_{LR}=0.199$, and $R_{LR}=0.331$. The innermost and middle
of these rings are unstable, while the third is stable, and all rotate in the
direction of the black hole’s rotation. In contrast, the outermost fourth
light ring is located outside the ergoregion at $R_{LR}=0.724$. Moreover, it
is unstable and rotates contrary to the black hole. In the absence of Gaussian
curvature, the equatorial domain of the ergoregion is bounded by
$R_{ER}\in[0,\,0.147]$, signifying a shift of the outer boundary towards the
event horizon. Additionally, the ergoregion hosts only one unstable light ring
at $R_{LR}=0.057$, rotating in the direction of the black hole. Consequently,
the other three light rings, situated at $R_{LR}=0.238$, $R_{LR}=0.395$, and
$R_{LR}=0.742$, respectively, remain external to the ergoregion, each
exhibiting retrograde rotation concerning the black hole. This pattern
persists even in the presence of positive Gaussian curvature. The ergoregion
contracts, narrowing to $R_{ER}\in[0,\,0.133]$, housing only one unstable
light ring at $R_{LR}=0.055$, which rotates prograde to the black hole. All
remaining light rings are located outside the ergoregion, with radii
corresponding to $R_{LR}=0.251$, $R_{LR}=0.501$, and $R_{LR}=0.759$,
respectively, and exhibit retrograde rotation about the black hole. Generally,
the third light ring from the inside out is consistently stable for any
Gaussian curvature. Furthermore, as the Gaussian curvature increases, the
innermost ring moves closer to the black hole, while the other three move
away.

Figure 7: Examples of shadows which illustrate the transition between shadows
for different Gaussian curvatures $\kappa\in\\{-5,0,5\\}$ and black hole
horizon $r_{H}=0.1$. The left panel corresponds to configuration
IV${}^{\,-5}_{\,0.1}$ with $\kappa=-5$, $\omega_{s}/\mu=0.713790$,
$M\mu=0.873847$ and $q=0.965279$, the center panel corresponds to
configuration IV${}^{\,0}_{\,0.1}$ with $\kappa=0$, $\omega_{s}/\mu=0.738499$,
$M\mu=1.00043$ and $q=0.964477$, while the right panel corresponds to
configuration IV${}^{\,5}_{\,0.1}$ with $\kappa=5$, $\omega_{s}/\mu=0.748666$,
$M\mu=1.16102$ and $q=0.968930$. The physical quantities of the solutions are
presented in Table 1.

Configuration | Fig. | Ergoregions | $R_{ER}$ | LR | $R_{LR}$ | stability | $\eta$ | $g_{tt}$ | $d\varphi/dt$ | Chaos
---|---|---|---|---|---|---|---|---|---|---
IV${}^{\,-5}_{\,0.1}$ | 7 | 1 ER | $[0,\;0.439]$ | $h_{-}$ | 0.061 | unstable | $+$ | $+$ | $+$ | Yes
$h_{+}$ | 0.199 | unstable | $+$ | $+$ | $+$
$h_{+}$ | 0.331 | stable | $+$ | $+$ | $+$
$h_{+}$ | 0.724 | unstable | $-$ | $-$ | $-$
IV${}^{\,0}_{\,0.1}$ | 7 | 1 ER | $[0,\;0.147]$ | $h_{-}$ | 0.057 | unstable | $+$ | $+$ | $+$ | Yes
$h_{+}$ | 0.238 | unstable | $-$ | $-$ | $-$
$h_{+}$ | 0.395 | stable | $-$ | $-$ | $-$
$h_{+}$ | 0.742 | unstable | $-$ | $-$ | $-$
IV${}^{\,5}_{\,0.1}$ | 7 | 1 ER | $[0,\;0.133]$ | $h_{-}$ | 0.055 | unstable | $+$ | $+$ | $+$ | Yes
$h_{+}$ | 0.251 | unstable | $-$ | $-$ | $-$
$h_{+}$ | 0.501 | stable | $-$ | $-$ | $-$
$h_{+}$ | 0.759 | unstable | $-$ | $-$ | $-$

While stable light rings persist in the geometry of the considered
configurations IV${}^{\,\kappa}_{\,0.1}$, with $\kappa\in\\{-5,0,5\\}$, for
the given value of the normal charge, chaotic patterns in the shadow images
(as shown in Fig. 7) are notably lacking. One distinctive feature in the
formed shadows is the emergence of tooth-like regions around the polar parts
of the shadow contour. With an increase in Gaussian curvature, these regions
gradually diminish, causing the images of the chaotically scattered orbits to
shift from the inner part of the shadow towards its boundary. Accordingly, the
chaotic patterns manifest at positive curvature in a thin, crescent-shaped
region outside the shadow boundary. It is worth noting that the situation with
configuration III${}^{\,5}_{\,0.05}$ is analogous. This observation emphasizes
that solutions with positive Gaussian curvatures result in shadows with
considerably smaller areas and fewer chaotic patterns compared to those with
zero or negative curvature of the target space.

####  IV.2.5 Model V, $r_{H}=0.2$, $q\simeq 0.85$: Single ergoregion models
with deformed quasi-circular shadows.

Last but not least, the table below provides data on the characteristics of
the ergoregions and the light ring system for configurations
V${}^{\,\kappa}_{\,0.2}$, where $\kappa\in\\{-5,0,5\\}$ (as indicated in Fig.
1). These configurations correspond to an event horizon of $r_{H}=0.2$, and a
normal charge $q\simeq 0.845$. Corresponding shadow images are depicted in
Fig. 8.

Configuration | Fig. | Ergoregions | $R_{ER}$ | LR | $R_{LR}$ | stability | $\eta$ | $g_{tt}$ | $d\varphi/dt$ | Chaos
---|---|---|---|---|---|---|---|---|---|---
V${}^{\,-5}_{\,0.2}$ | 8 | 1 ER | $[0,\;0.248]$ | $h_{-}$ | 0.095 | unstable | $+$ | $+$ | $+$ | No
$h_{+}$ | 0.471 | unstable | $-$ | $-$ | $-$
V${}^{\,0}_{\,0.2}$ | 8 | 1 ER | $[0,\;0.252]$ | $h_{-}$ | 0.094 | unstable | $+$ | $+$ | $+$ | No
$h_{+}$ | 0.478 | unstable | $-$ | $-$ | $-$
V${}^{\,5}_{\,0.2}$ | 8 | 1 ER | $[0,\;0.258]$ | $h_{-}$ | 0.093 | unstable | $+$ | $+$ | $+$ | No
$h_{+}$ | 0.488 | unstable | $-$ | $-$ | $-$

The considered configurations stand out from those previously studied in this
paper due to the presence of a single ergoregion. Interestingly, these
configurations exhibit a unique feature – a system with two unstable light
rings for all Gaussian curvatures of the target space. This distinctive
characteristic draws comparisons with Kerr’s rotating black hole, known for
having one ergoregion and a system of two counter-rotating unstable light
rings.

A notable feature of the examined configurations is that, as the Gaussian
curvature increases, the equatorial domain of the ergoregions negligibly
expands from $R_{ER}\in[0,\,0.248]$ for negative $\kappa$, through
$R_{ER}\in[0,\,0.252]$ for zero $\kappa$, to $R_{ER}\in[0,\,0.258]$ for
positive $\kappa$. Each region hosts only one unstable light ring, prograde
rotating regarding the black hole’s rotation. Beyond the ergoregions, each
configuration allows for a second, outer unstable light ring retrograde
rotating concerning the black hole.

Interestingly, with an increase in Gaussian curvature, the inner rings move
slightly closer to the horizon, acquiring radii $R_{LR}=0.095$,
$R_{LR}=0.094$, and $R_{LR}=0.093$ for negative, zero, and positive $\kappa$,
respectively. Simultaneously, the outer rings move away from the black hole as
photons adopt orbits with radii $R_{LR}=0.471$, $R_{LR}=0.478$, and
$R_{LR}=0.488$, correspondingly for negative, zero, and positive $\kappa$.

The distinctive behaviour of unstable prograde and retrograde light rings
results in a shift in the equatorial portions of the shadows from west to east
with increasing Gaussian curvature. Notably, this effect occurs when adjusting
the curvature of the target space while keeping a constant the normal charge
$q$ for each solution. On the contrary, the phenomenon intensifies with an
increase in the frequency $\omega_{s}/\mu$, or equivalently, with a rise in
the angular velocity of the horizon as the scalar field is synchronized with
the rotation of the black hole. Furthermore, as depicted in Fig. 8, negative
Gaussian curvature of the target space leads to forming a shadow with a larger
area than the case without curvature. Conversely, with positive Gaussian
curvature, the shadow area decreases. The reduction in shadow size appears as
an outcome of the proportionally smaller fraction of the horizon mass to the
ADM mass. Nevertheless, despite the varied Gaussian curvature, the shadows
maintain a D-like shape, a distinctive feature resembling a rotating Kerr
black hole as observed from the equatorial plane.

Figure 8: Examples of shadows which illustrate the transition between shadows
for different Gaussian curvatures $\kappa\in\\{-5,0,5\\}$ and black hole
horizon $r_{H}=0.2$. The left panel corresponds to configuration
V${}^{\,-5}_{\,0.2}$ with $\kappa=-5$, $\omega_{s}/\mu=0.883526$,
$M\mu=0.847374$ and $q=0.848888$, the center panel corresponds to
configuration V${}^{\,0}_{\,0.2}$ with $\kappa=0$, $\omega_{s}/\mu=0.895538$,
$M\mu=0.878726$ and $q=0.850778$, while the right panel corresponds to
configuration V${}^{\,5}_{\,0.2}$ with $\kappa=5$, $\omega_{s}/\mu=0.907762$,
$M\mu=0.884665$ and $q=0.843196$. The physical quantities of the solutions are
presented in Table 1 in Appendix A.

####  IV.2.6 Model VI, $r_{H}=0.3$, $q\simeq 0.65$: Single ergoregion models
with quasi-circular shadows.

Finally, to study the structure of the system of light rings and ergoregions
for an event horizon $r_{H}=0.3$, we choose configurations
VI${}^{\,\kappa}_{\,0.3}$, where $\kappa\in\\{-5,0,5\\}$ (as indicated in Fig.
1). These configurations are characterized by a normal charge $q\simeq 0.647$.
The corresponding characteristics of the light rings are presented in the
table below, and shadow images are exposed in Fig. 9.

Configuration | Fig. | Ergoregions | $R_{ER}$ | LR | $R_{LR}$ | stability | $\eta$ | $g_{tt}$ | $d\varphi/dt$ | Chaos
---|---|---|---|---|---|---|---|---|---|---
$-5.6$ | 9 | 1 ER | $[0,\;0.1968]$ | $h_{-}$ | 0.1599 | unstable | $+$ | $+$ | $+$ | No
$h_{+}$ | 0.4306 | unstable | $-$ | $-$ | $-$
$0.6$ | 9 | 1 ER | $[0,\;0.1972]$ | $h_{-}$ | 0.1598 | unstable | $+$ | $+$ | $+$ | No
$h_{+}$ | 0.4310 | unstable | $-$ | $-$ | $-$
$5.6$ | 9 | 1 ER | $[0,\;0.1976]$ | $h_{-}$ | 0.1597 | unstable | $+$ | $+$ | $+$ | No
$h_{+}$ | 0.4314 | unstable | $-$ | $-$ | $-$

Similar to solutions V with an event horizon of $r_{H}=0.2$, configurations VI
also possess a system of two unstable light rings and one ergoregion for each
of the Gaussian curvatures of the target space $\kappa$. The main
distinguishing feature here is that the Gaussian curvature has a negligible
effect on the radii of the two light rings and the equatorial domain of the
ergoregion. Specifically, the ergoregions extend from the event horizon and
reach distances, $R_{ER}\in[0,\,0.1968]$, $R_{ER}\in[0,\,0.1972]$, and
$R_{ER}\in[0,\,0.1976]$, corresponding to negative, zero, and positive
$\kappa$, respectively. Additionally, within the ergoregions, a prograde
rotating light ring exists. With an increase in the curvature of the target
space, the radii of those light rings decrease insignificantly, estimated as
$R_{LR}=0.1599$, $R_{LR}=0.1598$, and $R_{EL}=0.1597$, for negative, zero, and
positive $\kappa$, respectively. Moreover, the outer light rings, situated
beyond the ergoregions, rotate retrograde relative to the black hole rotation
and expand with increasing Gaussian curvature: $R_{LR}=0.4306$,
$R_{LR}=0.4310$, and $R_{LR}=0.4314$, for negative, zero, and positive
$\kappa$, respectively. Concurrently, as exposed in Fig. 9, despite the
different Gaussian curvature, the shadows exhibit an O-like shape, and due to
the almost constant fraction of the horizon mass to the ADM mass, their area
appears the same to an observer from the equatorial plane.

Figure 9: Examples of shadows which illustrate the transition between shadows
for different Gaussian curvatures $\kappa\in\\{-5,0,5\\}$ and black hole
horizon $r_{H}=0.3$. The left panel corresponds to configuration
VI${}^{\,-5}_{\,0.3}$ with $\kappa=-5$, $\omega_{s}/\mu=0.988000$,
$M\mu=0.319008$ and $q=0.646883$, the center panel corresponds to
configuration VI${}^{\,0}_{\,0.3}$ with $\kappa=0$, $\omega_{s}/\mu=0.988000$,
$M\mu=0.320010$ and $q=0.647791$, while the right panel corresponds to
configuration VI${}^{\,5}_{\,0.3}$ with $\kappa=5$, $\omega_{s}/\mu=0.988000$,
$M\mu=0.321013$ and $q=0.648605$. The physical quantities of the solutions are
presented in Table 1 in Appendix A.

##  V Conclusion

In this work we consider the shadow cast by rotating hairy black holes coupled
to two non-trivial time-periodic scalar fields. The scalar fields may be
viewed as coordinates in a nonphysical Riemannian space possessing a constant
curvature. Thus, the black hole solutions can be divided into three classes
according to the sign of its Gaussian curvature. Each class encompasses a wide
range of solutions with different phenomenology depending on the amount of
scalar hair, which can be measured by a normalized charge $q\in(1,0)$,
representing the ratio between the scalar field Noether charge and the black
hole spin. Solutions with high values of the normalized scalar charge
approaching $q=1$ possess similar properties as the soliton-like solutions of
the Einstein-scalar field equations representing boson stars. On the other
hand, for small $q\approx 0$ the back-reaction of the scalar fields on the
spacetime geometry vanishes and the solutions reduce to scalar clouds. Near
this limit, the hairy black holes interact weakly with the scalar fields and
resemble in their properties the Kerr black hole.

In our analysis, we examine the influence of the Gaussian curvature and the
normalized charge $q$ on the properties of the hairy black hole shadow. For
that purpose, we consider configurations with several representative values of
$q$ and study the corresponding solutions for positive, negative and zero
curvature of the target space. The shadows of the selected solutions are
constructed numerically using a ray-tracing procedure. We observe the
following systematical behavior. The shadows for extremely high values of the
normalized charge ($q>0.997$ for the presented models) exhibit large regions
of chaotically scattering geodesics for all considered Gaussian curvatures.
This leads to the formation of multiple disconnected shadow images. Increasing
the Gaussian curvature, the chaotic behavior becomes milder producing more
coherent shadow images. The chaotic behavior further reduces if we consider
lower normalized charges. The shadows for charges in the range $q\sim
0.96-0.994$ become more compact possessing fewer disconnected components and a
clearly defined dominant component. Positive Gaussian curvatures lead again to
more regular shadow images compared to negative ones, as well as a larger size
of the central dark region. Conversely, for negative Gaussian curvatures the
shadow size decreases substantially and the chaotic region dominates. Finally,
for values of the normalized charge around $q=0.85$ the chaotic behavior of
the scattering geodesics becomes negligible for all the Gaussian curvatures.
For lower $q$ the shadow of the hairy black holes is represented by a single
oval component and resembles qualitatively the Kerr black hole while its size
is influenced relatively weakly by the Gaussian curvature.

The presence of chaotic regions and the shadow shape significantly influence
the optical appearance of accretion disks around hairy black holes. Exploring
this problem is ongoing work, but we can offer some qualitative insights. What
matters most when calculating the actual image produced by a given accretion
disk around the black hole is the presence of a central compact shadow. Having
extensive chaotic regions, like those observed very close to the boson star
limit with $q$ approaching $1$, would likely not produce significant or any
dark region of the image, which practically contradicts present observations.
Thus, models with very large $q$, or models with hefty scalar hair, can be
excluded by the Event Horizon Telescope observations. Large positive Gauss
curvature can potentially relieve this problem because it increases the size
of the central dark shadow and decreases the chaotic regions close to the
boson star limit. In the present paper, we have considered only moderate high
curvatures since the models calculated in [10] were taken as a background. It
will be interesting to investigate whether this pattern remains with a further
increase in Gauss curvature and whether one can extend the dark region of the
shadow, as supported by current observations, even close to the boson star
limit ($q=1$).

## Acknowledgements

This study is financed by the European Union-NextGenerationEU, through the
National Recovery and Resilience Plan of the Republic of Bulgaria, project No.
BG-RRP-2.004-0008-C01. DD acknowledges financial support via an Emmy Noether
Research Group funded by the German Research Foundation (DFG) under grant no.
DO 1771/1-1.

##  Appendix A Physical quantities of selected solutions

Below we present in detail the characteristics of the solutions used to
produce the black hole images. They are taken from [10] where sequences of
constant horizon radii were constructed for various Gauss curvatures $\kappa$.

Label | $\omega_{s}/\mu$ | $M\mu$ | $J\mu^{2}$ | $M_{BH}\mu$ | $J_{BH}\mu^{2}$ | $M_{\psi}\mu$ | $J_{\psi}\mu^{2}$ | $J_{\psi}/J$ | $J/M^{2}$ | $J_{BH}/M_{BH}^{2}$
---|---|---|---|---|---|---|---|---|---|---
I${}^{\,-5}_{\,0.01}$ | 0.7398 | 0.6902 | 0.4734 | 0.0056 | 0.0013 | 0.6778 | 0.4721 | 0.9972 | 0.9938 | 42.544
I${}^{\,0}_{\,0.01}$ | 0.8353 | 0.6482 | 0.4068 | 0.0049 | 0.0011 | 0.6434 | 0.4057 | 0.9973 | 0.9680 | 46.444
I${}^{\,5}_{\,0.01}$ | 0.8219 | 0.7425 | 0.5041 | 0.0049 | 0.0015 | 0.7469 | 0.5026 | 0.9971 | 0.9144 | 60.307
II${}^{\,-5}_{\,0.01}$ | 0.6074 | 0.8905 | 0.7813 | 0.0039 | 0.0001 | 0.8576 | 0.7812 | 0.9999 | 0.9853 | 6.7133
II${}^{\,0}_{\,0.01}$ | 0.6792 | 0.8820 | 0.7259 | 0.0035 | 0.0001 | 0.8785 | 0.7258 | 0.9999 | 0.9331 | 5.8772
II${}^{\,5}_{\,0.01}$ | 0.7316 | 0.8832 | 0.6890 | 0.0031 | 0.0002 | 0.8984 | 0.6888 | 0.9997 | 0.8833 | 11.727
III${}^{\,-5}_{\,0.05}$ | 0.6537 | 0.8865 | 0.7567 | 0.0256 | 0.0042 | 0.8244 | 0.7525 | 0.9944 | 0.9629 | 6.3547
III${}^{\,0}_{\,0.05}$ | 0.7064 | 0.9082 | 0.8329 | 0.0252 | 0.0045 | 0.9415 | 0.8283 | 0.9945 | 0.8915 | 7.1310
III${}^{\,5}_{\,0.05}$ | 0.7023 | 1.1003 | 0.9812 | 0.0254 | 0.0052 | 1.1078 | 0.9760 | 0.9947 | 0.8105 | 8.0147
IV${}^{\,-5}_{\,0.1}$ | 0.7138 | 0.8738 | 0.7122 | 0.0780 | 0.0248 | 0.7639 | 0.6875 | 0.9653 | 0.9327 | 4.0651
IV${}^{\,0}_{\,0.1}$ | 0.7385 | 1.0004 | 0.8535 | 0.0870 | 0.0302 | 0.9134 | 0.8232 | 0.9645 | 0.8528 | 3.9982
IV${}^{\,5}_{\,0.1}$ | 0.7487 | 1.1610 | 1.0394 | 0.0904 | 0.0322 | 1.1001 | 1.0071 | 0.9689 | 0.7711 | 3.9335
V${}^{\,-5}_{\,0.2}$ | 0.8835 | 0.8474 | 0.6478 | 0.2706 | 0.0976 | 0.5709 | 0.5499 | 0.8489 | 0.9022 | 1.3339
V${}^{\,0}_{\,0.2}$ | 0.8955 | 0.8787 | 0.6790 | 0.2796 | 0.1012 | 0.5990 | 0.5776 | 0.8508 | 0.8793 | 1.2946
V${}^{\,5}_{\,0.2}$ | 0.9078 | 0.8847 | 0.6814 | 0.2921 | 0.1065 | 0.5951 | 0.5746 | 0.8432 | 0.8707 | 1.2486
VI${}^{\,-5}_{\,0.3}$ | 0.9880 | 0.3190 | 0.1252 | 0.2373 | 0.0442 | 0.0817 | 0.0810 | 0.6469 | 1.2300 | 0.7848
VI${}^{\,0}_{\,0.3}$ | 0.9880 | 0.3200 | 0.1260 | 0.2377 | 0.0444 | 0.0823 | 0.0816 | 0.6478 | 1.2304 | 0.7856
VI${}^{\,5}_{\,0.3}$ | 0.9880 | 0.3210 | 0.1268 | 0.2380 | 0.0446 | 0.0830 | 0.0823 | 0.6486 | 1.2306 | 0.7865
VII${}^{\,-5}_{\,0.05}$ | 0.6485 | 0.9157 | 0.8048 | 0.0243 | 0.0030 | 0.8493 | 0.8017 | 0.9962 | 0.9598 | 5.0973
Table 1: Physical quantities of hairy black hole solutions as depicted in Fig.
1, corresponding to different Gaussian curvatures of the target space,
$\kappa=\\{-5,\,0,\,5\\}$. Each configuration is identified by
X${}^{\,u}_{\,v}$, where the symbol X represents the configuration number,
superscript $u$ represents the value of the Gaussian curvature $\kappa$, and
the subscript $v$ denotes the value of the black hole horizon $r_{H}$. In a
single group of solutions the horizon radii $r_{H}$ is constant and the
normalized charge $q=\frac{J_{\psi}}{J}$ (represented in the 9th column) is
adjusted to be nearly constant.

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