UNIFORM MODEL OF GEOMETRIC SPACES

Alexandru Popa

Abstract. Full classification of geometric spaces was proposed by Isaak Yaglom
in [2]. By defining the elliptic, parabolic (or linear) and hyperbolic kinds
of measure and applying them to distance, plane and dihedral angle of
different dimensions we get $3^{n}$ geometric spaces of dimension $n$. In his
work [3] Yaglom says that ”finding a general description of all geometric
systems [was] considered by mathematicians the central question of the day.”
A. B. Khachaturean resumed Yaglom’s work in [8].

Author developed a uniform model for all these spaces where distance and angle
measure kinds are parameters. This model is calculus centric, but can also be
used in theoretical research. It is useful in the following domains:

  * •

deduction of uniform equations among geometric spaces;

  * •

uniform model applied to any space, which provides an easy way to calculate
distances, plane and dihedral angles of any dimension, areas and volumes as
well as parallel (where applied) and orthogonal property detection;

  * •

study of not yet described spaces and more.

2000 Mathematics Subject Classification: 51N25, 51N15.

1\. Definitions

As was shown by Yaglom in [2], some $n$-dimensional geometric space can be
defined specifying its $n$ characteristics, or measure kinds. We will use
numbers 1 for elliptic characteristic, 0 for parabolic (or linear) one and
$-1$ for hyperbolic one. So, full space specification is a set of $n$
characteristics $k_{1},...,k_{n}\in\\{-1,0,1\\}$, which can be detected by a
simple algorithm.

Define

$K_{i}=\prod_{j=1}^{i}k_{j},\,\forall i=\overline{0,n}.$ (1)

For two vectors $x,y\in\mathbb{PR}^{n}$, $x=\left<x_{0}:...:x_{n}\right>$,
$y=\left<y_{0}:...:y_{n}\right>$ define a dot product in respect of
characteristics $k_{1}...k_{n}$ as

$x\odot y=\sum_{i=0}^{n}K_{i}x_{i}y_{i}.$ (2)

and cross product in respect of $k_{1}...k_{n}$ so that

$(x\odot y)^{2}+k_{1}(x\otimes y)^{2}=(x\odot x)(y\odot y),\,\forall
x,y\in\mathbb{PR}^{n}.$

It can be checked that111Here and further we will consider for simplicity that
$k^{0}=1$ for $k=0$ too. We will say $x$ divide $k^{i}$, $k=0$ if in
expression $x/k^{i}$ the exponent of $k$ in numerator is greater then or
equals to $i$.

$x\otimes
y=\sqrt{\frac{1}{k_{1}}\sum_{i<j=0}^{n}K_{i}K_{j}(x_{i}y_{j}-x_{j}y_{i})^{2}}.$
(3)

These products were considered by Klein in [1] for elliptic and hyperbolic
spaces.

A $(n+1)\times(n+1)$ matrix is generalized orthogonal in respect of
$k_{1}...k_{n}$ if for all columns $c_{i},c_{j}$ ($i,j=\overline{0,n}$)

$\frac{1}{K_{min(i,j)}}c_{i}\odot c_{j}=\begin{cases}1,i=j,\\\ 0,i\neq
j.\end{cases}$ (4)

Having characteristics $k\in\\{-1,0,1\\}$ consider functions
$C,S,T:\mathbb{R}\to\mathbb{R}$:

$\displaystyle C(x)=C(k,x)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}(-k)^{i}\frac{x^{2i}}{(2i)!},$ (5)
$\displaystyle S(x)=S(k,x)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}(-k)^{i}\frac{x^{2i+1}}{(2i+1)!},$ (6)
$\displaystyle T(x)=T(k,x)$ $\displaystyle=$
$\displaystyle\frac{S(k,x)}{C(k,x)}.$ (7)

It’s easy to see, that

$\displaystyle C(x)=\begin{cases}\cos x,&k=1,\\\ 1,&k=0,\\\ \cosh
x,&k=-1\end{cases}$ $\displaystyle S(s)=\begin{cases}\sin x,&k=1,\\\
x,&k=0,\\\ \sinh x,&k=-1\end{cases}$ $\displaystyle T(x)=\begin{cases}\tan
x,&k=1,\\\ x,&k=0,\\\ \tanh x,&k=-1\end{cases}$

Define a geometric space with characteristics $k_{1}...k_{n}$ as ”unit ball”
in projective space: $\mathbb{B}^{n}=\\{x\in\mathbb{PR}^{n}\,|\,x\odot
x=1\\}$. Consider ”points” $X\in\mathbb{B}^{n}$ corresponding vectors
$x\in\mathbb{PR}^{n}$. Consider ”space transformation” all linear mappings of
$\mathbb{PR}^{n}$ whose matrices are generalized orthogonal. They are also
transformations of $\mathbb{B}^{n}$ as they preserve it. Consider
$m$-dimensional planes images of $\mathbb{B}^{m}\subset\mathbb{B}^{n}$ on some
transformation. All $m$-dimensional planes are (restricted to
$\mathbb{B}^{n}$) linear combination of first $m+1$ columns of some
generalized orthogonal matrix. So, we can identify $m$-dimensional planes,
$m<n$ with such $(n+1)\times(m+1)$ matrices.

For two $m$-dimensional planes $X,Y$ define dot product in respect of
$k_{1}...k_{n}$ as

$X\odot
Y=\sum_{i_{0}<...<i_{m}=0}^{n}X_{i_{0}...i_{m}}Y_{i_{0}...i_{m}}\prod_{p=1}^{m}\frac{K_{i_{p}}}{K_{p}},$
(8)

where

$M_{l_{0}...l_{m}}=\begin{vmatrix}m_{l_{0}0}&\ldots&m_{l_{0}m}\\\
\vdots&\ddots&\vdots\\\ m_{l_{m}0}&\ldots&m_{l_{m}m}\\\ \end{vmatrix}$

and cross product so that

$(X\odot Y)^{2}+k_{m+1}(X\otimes Y)^{2}=(X\odot X)(Y\odot Y)$

It can be checked that

$X\otimes
Y=\sqrt{\frac{1}{k_{m+1}}\sum_{\begin{subarray}{c}i_{0}<...<i_{m}=0\\\
j_{0}<...<j_{m}=0\\\
i_{0}...i_{m}<j_{0}...j_{m}\end{subarray}}^{n}(X_{i_{0}...i_{m}}Y_{j_{0}...j_{m}}-X_{j_{0}...j_{m}}Y_{i_{0}...i_{m}})^{2}\prod_{p=1}^{m}\frac{K_{i_{p}}K_{j_{p}}}{K^{2}_{p}}}.$
(9)

This model generalizes spherical model of elliptic space, hyperboloid model of
hyperbolic space [6], projective euclidean space model [7] and describes many
new spaces.

2\. Calculus in uniform model

Author shows that dot and cross products of points and planes is invariant in
respect of space transformation. Moreover, it can be used for distance and
angle calculus based on equalities ($m<n$).

$\displaystyle X\odot Y$ $\displaystyle=$ $\displaystyle C_{m+1}(\phi),$ (10)
$\displaystyle X\otimes Y$ $\displaystyle=$ $\displaystyle S_{m+1}(\phi),$
(11)

where $X$ and $Y$ are two points (if $m=0$) and $\phi$ is distance between
them or $X$ and $Y$ are $m$-dimensional planes (if $m>0$) and $\phi$ is angle
between them and functions $C_{m+1}(x)=C(k_{m+1},x),S_{m+1}(x)=S(k_{m+1},x)$.

For some figure $F\subset\mathbb{B}^{n}$ volume can be calculated using the
following equation

$V_{\mathbb{R}}(F)=\frac{1}{n+1}V_{\mathbb{B}}(C_{F})$ (12)

where $C_{F}\subset\mathbb{R}^{n+1}$ is cone having origin
$O=\\{0,...,0\\}\notin\mathbb{B}^{n}$ as vertex and figure $F$ as base,
$V_{\mathbb{B}}$ is native volume in $\mathbb{B}^{n}$ and $V_{\mathbb{R}}$ is
volume in sense of $\mathbb{R}^{n+1}$. The advantage of this approach is the
fact $V_{\mathbb{R}}$ is volume in a linear vector space which is usually
easily to find.

Based on this unified model we can deduce common equation among all spaces.
For example, consider $\mathbb{B}^{2}$ with characteristics $k_{1}$ and
$k_{2}$ and triangle $ABC\in\mathbb{B}^{2}$ with edges $a$, $b$ and $c$,
interior angles $\alpha$, $\gamma$ and exterior angle $\beta^{\prime}$
(interior angle $\beta$ may not exist). Then sine and cosine I and II lows
have identical form in all 9 2-dimensional spaces:

$\frac{S_{1}(a)}{S_{2}(\alpha)}=\frac{S_{1}(b)}{S_{2}(\beta^{\prime})}=\frac{S_{1}(c)}{S_{2}(\gamma)},$
(13)

and

$\displaystyle C_{1}(a)$ $\displaystyle=$ $\displaystyle
C_{1}(b)C_{1}(c)+k_{1}S_{1}(b)S_{1}(c)C_{2}(\alpha),$ (14) $\displaystyle
C_{1}(b)$ $\displaystyle=$ $\displaystyle
C_{1}(a)C_{1}(c)-k_{1}S_{1}(a)S_{1}(c)C_{2}(\beta^{\prime}),$ (15)
$\displaystyle C_{1}(c)$ $\displaystyle=$ $\displaystyle
C_{1}(a)C_{1}(b)+k_{1}S_{1}(a)S_{1}(b)C_{2}(\gamma),$ (16) $\displaystyle
C_{2}(\alpha)$ $\displaystyle=$ $\displaystyle
C_{2}(\beta^{\prime})C_{2}(\gamma)+k_{2}S_{2}(\beta^{\prime})S_{2}(\gamma)C_{1}(a),$
(17) $\displaystyle C_{2}(\beta^{\prime})$ $\displaystyle=$ $\displaystyle
C_{2}(\alpha)C_{2}(\gamma)-k_{2}S_{2}(\alpha)S_{2}(\gamma)C_{1}(b),$ (18)
$\displaystyle C_{2}(\gamma)$ $\displaystyle=$ $\displaystyle
C_{2}(\alpha)C_{2}(\beta^{\prime})+k_{2}S_{2}(\alpha)S_{2}(\beta^{\prime})C_{1}(a),$
(19)

or

$\displaystyle
T_{1}^{2}(a)=\frac{T_{1}^{2}(b)+T_{1}^{2}(c)-2T_{1}(b)T_{1}(c)C_{2}(\alpha)+k_{1}k_{2}T_{1}^{2}(b)T_{1}^{2}(c)S_{1}^{2}(\alpha)}{(1+k_{1}T_{1}(b)T_{1}(c)C_{2}(\alpha))^{2}},$
(20) $\displaystyle
T_{1}^{2}(b)=\frac{T_{1}^{2}(a)+T_{1}^{2}(c)+2T_{1}(a)T_{1}(c)C_{2}(\beta^{\prime})+k_{1}k_{2}T_{1}^{2}(a)T_{1}^{2}(c)S_{1}^{2}(\beta^{\prime})}{(1-k_{1}T_{1}(a)T_{1}(c)C_{2}(\beta^{\prime}))^{2}},$
(21) $\displaystyle
T_{1}^{2}(c)=\frac{T_{1}^{2}(a)+T_{1}^{2}(b)-2T_{1}(a)T_{1}(b)C_{2}(\gamma)+k_{1}k_{2}T_{1}^{2}(a)T_{1}^{2}(b)S_{1}^{2}(\gamma)}{(1+k_{1}T_{1}(a)T_{1}(b)C_{2}(\gamma))^{2}},$
(22) $\displaystyle
T_{2}^{2}(\alpha)=\frac{T_{2}^{2}(\beta^{\prime})+T_{2}^{2}(\gamma)-2T_{2}(\beta^{\prime})T_{2}(\gamma)C_{1}(a)+k_{1}k_{2}T_{2}^{2}(\beta^{\prime})T_{2}^{2}(\gamma)S_{1}^{2}(a)}{(1+k_{2}T_{2}(\beta^{\prime})T_{2}(\gamma)C_{1}(a))^{2}},$
(23) $\displaystyle
T_{2}^{2}(\beta^{\prime})=\frac{T_{2}^{2}(\alpha)+T_{2}^{2}(\gamma)+2T_{2}(\alpha)T_{2}(\gamma)C_{1}(b)+k_{1}k_{2}T_{2}^{2}(\alpha)T_{2}^{2}(\gamma)S_{1}^{2}(b)}{(1-k_{2}T_{2}(\alpha)T_{2}(\gamma)C_{1}(b))^{2}},$
(24) $\displaystyle
T_{2}^{2}(\gamma)=\frac{T_{2}^{2}(\alpha)+T_{2}^{2}(\beta^{\prime})-2T_{2}(\alpha)T_{2}(\beta^{\prime})C_{1}(c)+k_{1}k_{2}T_{2}^{2}(\alpha)T_{2}^{2}(\beta^{\prime})S_{1}^{2}(c)}{(1+k_{2}T_{2}(\alpha)T_{2}(\beta^{\prime})C_{1}(c))^{2}}.$
(25)

As another example, consider $\mathbb{B}^{2}$ with characteristics
$k_{1},k_{2}=1$ and $ABC\in\mathbb{B}^{2}$ right triangle with catheti $a,b$,
hypotenuse $c$ and angles $\alpha$ and $\beta$. Equations of $ABC$ have the
same form for elliptic, euclidean and hyperbolic planes.

$\displaystyle T_{1}^{2}(c)$ $\displaystyle=$ $\displaystyle
T^{2}_{1}(a)+T_{1}^{2}(b)+k_{1}T^{2}_{1}(a)T_{1}^{2}(b),$ (26) $\displaystyle
T_{1}(b)$ $\displaystyle=$ $\displaystyle T_{1}(c)\cos\alpha,$ (27)
$\displaystyle T_{1}(a)$ $\displaystyle=$ $\displaystyle T_{1}(c)\cos\beta,$
(28) $\displaystyle S_{1}(a)$ $\displaystyle=$ $\displaystyle
S_{1}(c)\sin\alpha,$ (29) $\displaystyle S_{1}(b)$ $\displaystyle=$
$\displaystyle S_{1}(c)\sin\beta,$ (30) $\displaystyle T_{1}(a)$
$\displaystyle=$ $\displaystyle S_{1}(b)\tan\alpha,$ (31) $\displaystyle
T_{1}(b)$ $\displaystyle=$ $\displaystyle S_{1}(a)\tan\beta,$ (32)
$\displaystyle\cos\alpha$ $\displaystyle=$ $\displaystyle C_{1}(a)\sin\beta,$
(33) $\displaystyle\cos\beta$ $\displaystyle=$ $\displaystyle
C_{1}(b)\sin\alpha,$ (34) $\displaystyle C_{1}(c)$ $\displaystyle=$
$\displaystyle\cot\alpha\cot\beta.$ (35)

References

[1] Felix Klein, Vorlesungen Nicht-Euklidische Geometrie, B.G.Teubner, Leipzig
1890.

[2] Isaak Yaglom, A simple non-euclidean geometry and its physical basis,
Springer, New York 1979.

[3] Isaak Yaglom, Felix Klein and Sophus Lie, Birkhauser, 1988.

[4] Fenchel, Werner, Elementary geometry in hyperbolic space, De Gruyter
Studies in mathematics. 11. Berlin-New York: Walter de Gruyter & Co 1989.

[5] Naber, Gregory L., The Geometry of Minkowski Spacetime. New York,
Springer-Verlag 1992, ISBN 0387978488.

[6] Reynolds, William F, Hyperbolic Geometry on a Hyperboloid, American
Mathematical Monthly 1993, 100:442-455.

[7] Coxeter H. S. M., The Real Projective Plane, 3rd ed, Springer Verlag 1995.

[8] A. B. Khachaturean, Galilean geometry, MCNMO, Moskow, 2005.

[9] James W. Anderson, Hyperbolic Geometry, Springer 2005, ISBN 1852339349

Alexandru Popa
Department of Computer Sciences
Vest University of Timisoara
Address: Blvd. V. Parvan 4, Timisoara 300223, Timis, Romania
<EMAIL_ADDRESS>


#  Thin accretion discs around spherically symmetric configurations with
nonlinear scalar fields

O. S. Stashko  Taras Shevchenko National University of Kyiv, Ukraine  V. I.
Zhdanov  Taras Shevchenko National University of Kyiv, Ukraine  A. N.
Alexandrov  Taras Shevchenko National University of Kyiv, Ukraine

###### Abstract

We study stable circular orbits (SCO) around static spherically symmetric
configuration of General Relativity with a non-linear scalar field (SF). The
configurations are described by solutions of the Einstein-SF equations with
monomial SF potential $V(\phi)=|\phi|^{2n}$, $n>2$, under the conditions of
the asymptotic flatness and behavior of SF $\phi\sim 1/r$ at spatial infinity.
We proved that under these conditions the solution exists and is uniquely
defined by the configuration mass $M>0$ and scalar "charge" $Q$. The solutions
and the space-time geodesics have been investigated numerically in the range
$n\leq 40$, $|Q|\leq 60$, $M\leq 60$. We focus on how nonlinearity of the
field affects properties of SCO distributions (SCOD), which in turn affect
topological form of the thin accretion disk around the configuration. Maps are
presented showing the location of possible SCOD types for different $M,Q,n$.
We found many differences from the Fisher-Janis-Newman-Winicour metric (FJNW)
dealing with the linear SF, though basic qualitative properties of the
configurations have much in common with the FJNW case. For some values of $n$,
a topologically new SCOD type was discovered that is not available for the
FJNW metric. All images of accretion disks have a dark spot in the center
(mimicking an ordinary black hole), either because there is no SCO near the
center or because of the strong deflection of photon trajectories near the
singularity.

naked singularities, test particle motion, accretion disks

###### pacs:

11

##  I Introduction

Scalar field (SF) configurations in General Relativity and its modifications
are interesting for several reasons. Various SF models are extensively used in
cosmology Copeland  _et al._ (2006); Bamba  _et al._ (2012); Novosyadlyj  _et
al._ (2015); Linde (2014). Some propositions to relax the well-known "Hubble
tension" involve SF in diverse approaches to dynamical dark energy (see
Valentino  _et al._ (2021) for a review). It is currently unknown, whether the
cosmological fields of the late epoch (if any) are the same as the fields that
caused the early inflation, or they are of a completely different nature. In
any case, the question arises about possible manifestations of SF in
relativistic astrophysical objects. Interest in alternative models of these
objects has increased significantly after the image of the accretion disk in
the core of M87 had been obtained with the Event Horizon Telescope (EHT)
Collaboration (2019), demonstrating future prospects to distinguish the black
holes from their exotic mimickers.

The first example of the static spherically symmetric solution of Einstein
equations with linear massless SF has been found by Fisher Fisher (1948) and
later, in the other form, by Janis, Newman & Winicour Janis  _et al._ (1968)
(hereinafter FJNW solution; see also Wyman (1981); Virbhadra (1997)). The FJNW
metric does not describe a black hole (BH), it has naked singularity (NS) in
the center not hidded by an event horison. This is a typical feature of static
solutions with SF describing compact objects due to the Bekenstein theorems
Bekenstein (1972a, b), see also a generalisation in case of multiple SFs
Doneva and Yazadjiev (2020). The occurrence of NS in the real Universe is
forbidden by the Penrose Cosmic Censorship hypothesis Penrose (1965, 2002);
though, the question on the validity of this hypothesis remains open
Christodoulou (1984); Ori and Piran (1987); Joshi and Dwivedi (1993). Current
discussions on this topic have shifted to issues of stability and realistic
choices of initial data for the gravitational collapse that may or may not
lead to NS (see, e.g., Ong (2020) for a review).

Anyway, the final answer regarding the role of SF in astrophysics must be
based on observations. It should be noted that there are "exotic" structures
Olivares  _et al._ (2020); Aelst  _et al._ (2021); Herdeiro  _et al._ (2021);
Abdikamalov  _et al._ (2019); Banerjee  _et al._ (2020a, b); Sau  _et al._
(2020); Vincent  _et al._ (2021), which mimic the BHs yielding image similar
to that observed by EHT Collaboration (2019). New theoretical efforts as well
as observations with better resolution are mandatory. Given the progress in
astronomical technology, it is important to study in detail the properties of
relativistic astrophysical objects, which can help to select the appropriate
options from a variety of theoretical models. The main source of observational
information from these objects is associated with the distribution of
surrounding radiating matter (accretion disks, jets etc) and images of this
matter seen by a distant observer. The very first step is to study stable
circular orbits (SCO) of test bodies and their distribution in gravitational
field of the configuration. There are a number of papers on this subject
including those, which use FJNW solution dealing with the linear SF Bambhaniya
_et al._ (2019); Sau  _et al._ (2020); Shaikh and Joshi (2019); Gyulchev  _et
al._ (2019, 2020); Chowdhury  _et al._ (2012); Zhou  _et al._ (2015) and it
would be interesting to study effects of nonlinearity. Several examples
Pugliese  _et al._ (2011, 2013); Slaný and Stuchlík (2020); Stuchlík and Schee
(2015); Dymnikova and Poszwa (2019); Vieira  _et al._ (2014); Stashko and
Zhdanov (2018); Meliani  _et al._ (2015); Chowdhury  _et al._ (2012)
demonstrate occurrence of circular orbit distributions with several non-
connected rings of SCO. This may be of particular interest as observational
signs of differences from ordinary black holes, as well as the unusual form of
the accretion disk images and/or their radiation properties Stuchlík and Schee
(2014); Schee and Stuchlík (2016); Stuchlík  _et al._ (2019); Paul  _et al._
(2019); Shahidi  _et al._ (2020); Abdikamalov  _et al._ (2019); Collodel  _et
al._ (2021); Shaikh and Joshi (2019); Gyulchev  _et al._ (2019, 2020).

In the present paper, we will look for the effects of nonlinear fields on the
SCO distribution (SCOD) around the center. For this purpose, we consider SFs
determined by a sequence of monomial potentials $U(\phi)=\phi^{2n}$, $n>2$,
which have a simple asymptotic behavior $\phi(r)\sim 1/r$ at large distances
(analogous to FJNW)111The cases with $n\leq 2$ lead to asymptotics at infinity
different from $\sim 1/r$.. The reason for such choice is that this is the
simplest nonlinear generalization of the FJNW problem. On the other hand, the
monomial potentials are often used in various cosmological problems (see,
e.g., Smith  _et al._ (2020); Antusch  _et al._ (2020); Ballardini  _et al._
(2019)) We numerically obtain static solutions of the Einstein-SF equations in
the case of spherical symmetry and use these results to study SCO with focus
on the qualitative features of SCOD, as well as on images of these
distributions that can be observed from infinity. Namely, we systematically
analyse the permitted SCO regions.

The paper is organised as follows. In Section II we write down the basic
equations and integrate them numerically. The use of numerical methods
presupposes that the problem is well posed. In this regard, we rely on the
results of Zhdanov and Stashko (2020), which guarantee that our solutions are
regular and they have no singularities outside the center (in contrast, e.g.,
to some special relativistic cases Zhdanov and Stashko (2020)). Also, in
Appendix A we prove that there is a unique solution defined by the boundary
conditions at infinity. In Section III we proceed to test particle motion in
the gravitational field of the configuration. the "equatorial" plane and
present possible SCOD. Four qualitatively different SCOD types are introduced,
differing in the number of individual SCO rings (subsection III.2). Here we
demonstrate how the topology of the SCO rings change with $n$ and present maps
that define types of SCOD for given configuration parameters. The next
subsection III.3 discusses photon trajectories that are used to build the
images of different SCOD. The concluding remarks are summarized in Section IV
where we discuss observational signatures of SCOD, which can be used to
distinguish them.

##  II Spherically symmetric static solutions of Einstein relations with
scalar field

The general metric of a static spherically symmetric space-time in the
"curvature" coordinates (Schwarzschild-like) is

$ds^{2}=e^{\alpha(r)}dt^{2}-e^{\beta(r)}dr^{2}-r^{2}dO^{2},$ (1)

where $dO^{2}=d\theta^{2}+(\sin\theta)^{2}d\varphi^{2}$; radial variable
$r>0$.

We consider one minimally coupled real SF $\phi$ with Lagrangian density

$L=\frac{1}{2}\phi,_{\mu}\phi^{,\mu}-V(\phi),$ (2)

with

$V(\phi)=|\phi|^{2n}\,,\,\,n>2.$ (3)

where $n$ is not necessarily an integer. More general power law potential can
be reduced to (3) by rescaling of the variables. Note that for $V(\phi)=0$ we
have the well known FJNW solution. Some results concerning cases $1\leq n<2$
can be found in Stephenson (1962); Asanov (1974); Stashko and Zhdanov (2019a);
Zhdanov and Stashko (2020).

The Einstein-SF equations are reduced to the following system:

$\frac{d}{dr}\left[r\left(e^{-\beta}-1\right)\right]=-8\pi
r^{2}[e^{-\beta}\phi^{\prime 2}/2+V(\phi)]\,,$ (4)

where $\phi\equiv\phi(r)$,

$re^{-\beta}\frac{d\alpha}{dr}+e^{-\beta}-1=8\pi r^{2}[e^{-\beta}\phi^{\prime
2}/2-V(\phi)],$ (5)

and

$\frac{d}{dr}\left[r^{2}e^{\frac{\alpha-\beta}{2}}\frac{d\phi}{dr}\right]=r^{2}e^{\frac{\alpha+\beta}{2}}V^{\prime}(\phi)\,.$
(6)

Here $\alpha(r),\beta(r)$ are assumed to be $C^{1}$ functions and $\phi(r)$ is
a $C^{2}$ function for $r>0$.

We deal with isolated configurations; correspondingly, we impose the
asymptotic flatness conditions as follows

$\lim\limits_{r\to\infty}\left[r(e^{\alpha}-1)\right]=\lim\limits_{r\to\infty}\left[r(e^{-\beta}-1)\right]=-r_{g},$
(7)

where $r_{g}=2M$ and $M>0$ is the configuration mass; also $\phi(r)\to 0$ for
${r\to\infty}$ and

$\lim\limits_{r\to\infty}r^{2}\frac{d\phi}{dr}=-Q\,,$ (8)

where parameter $Q$ defines the strength of the scalar field at spatial
infinity; we will call it "scalar charge". Relation (8) yields

$\lim\limits_{r\to\infty}r\phi(r)=Q\,.$

The global behavior of the solutions satisfying (7,8) has been studied in a
more general case of multiple scalar fields Zhdanov and Stashko (2020), where
a proof of regularity of solutions on open interval $(0,\infty)$ is given.
Asymptotic properties for $r\to\infty$ have been derived in Zhdanov and
Stashko (2020) for a particular case, under the assumption that the solutions
can be expanded in powers of $1/r$. In Appendix A of the present paper we
provide a more rigorous analysis by means of an iteration procedure. We prove
that there is a solution of equations (4, 5, 6) for $r\geq r_{\rm in}$, where
$r_{\rm in}$ is large enough, with the conditions (8); this solution is
uniquely defined by parameters $M,Q$. The first iterations of this procedure
yield asymptotic relations for large $r$ as follows:

$\phi(r)=\frac{Q}{r}\left[1+\frac{r_{g}}{2r}+\frac{n|Q|^{2n-2}}{(n-2)(2n-3)r^{2n-4}}\right]+O\left[\frac{\mu(r)}{r^{2}}\right]\,,\quad(n>2),$
(9)
$e^{\alpha}=\left(1-\frac{r_{g}}{r}\right)\left[1+O\left(\frac{\mu(r)}{r^{2}}\right)\right],\leavevmode\nobreak\
e^{-\beta}=\left(1-\frac{r_{g}}{r}\right)\left[1+\frac{4\pi
Q^{2}}{r^{2}}+O\left(\frac{\mu(r)}{r^{2}}\right)\right]\,,$ (10)

where $\mu(r)=1/r$ for $n\geq 3$ and $\mu(r)=1/r^{2n-4}$ for $2<n<3$. Note
that in general case (9,10) can contain non-integer powers of $1/r$.

Asymptotics of the metric near the center can be found in Zhdanov and Stashko
(2020): $\alpha(r)\sim{(\eta-1)}{\ln r},\quad\beta(r)\sim{(\eta+1)}{\ln r}$.
Parameter $\eta$ characterizes the strength of the singularity. There is a
critical point $\eta=3$ that separates two types of the singularity with
different behavior of the null geodesics (see below).

We performed a detailed numerical investigation of the problem (4–8) for
$n\leq 40$, $M\leq 60$, $|Q|\leq 60$. To find the solutions for $r\leq r_{\rm
in}$, we proceed numerically starting at sufficiently large initial radius
$r_{\rm in}$ (up to $10^{5}$) to use initial conditions in accordance with
asymptotic relations (9, 10). We integrate backwards from higher to lower
values of $r\in(0,r_{\rm in}]$. The qualitative properties of the metric
coefficients and scalar field are rather similar for different values of
parameters $(Q,M,n)$: ${\alpha(r)}$ is monotonically increasing function
bounded from above by 1 for $r\rightarrow{\infty}$ and ${\beta(r)}$ has a
maximum at some point $r=r_{\rm max}(Q)$. If $Q$ increases, then $r_{\rm
max}(Q)$ is shifted to larger values and the maximum becomes less pronounced.
The scalar field is always a monotonically decreasing function and
$\phi\rightarrow{0}$ as $r\rightarrow{\infty}$. For large $n$ and fixed $M,Q$,
the solutions approach the FJNW curves, except for a small region near the
singularity. An important point is the dependence of $\eta$ upon the
parameters of the configuration at spatial infinity. Qualitatively,
dependencies $\eta(M,Q)$ for different $n$ are rather similar; Fig. 1 shows
the examples for $n=3$ and $n=14$.

Figure 1: Contours of the equal $\eta$ as function of $M,Q$ for $n=3$ (left
panel) and $n=14$ (right panel). Solid white line corresponds to critical
value $\eta=3.$ If $Q\rightarrow{0}$, then $\eta\rightarrow{\infty}$.

##  III Test particle motion

###  III.1 General relations

This Section deals with trajectories of the test particles in the space-time
corresponding to the solutions of the problem (4–8). The equations of the test
particle motion follow from formal Lagrangian

$L=g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau},$ (11)

where $\tau$ is a canonical parameter. Standard procedure involves the first
integrals for trajectories in the equatorial plane ($\theta=\pi/2$):

$e^{\alpha}\left(\frac{dt}{d\tau}\right)^{2}-e^{\beta}\left(\frac{dr}{d\tau}\right)^{2}-r^{2}\left(\frac{d\varphi}{d\tau}\right)^{2}=S\,,$
(12) $e^{\alpha}\left(\frac{dt}{d\tau}\right)=E,\quad
r^{2}\left(\frac{d\varphi}{d\tau}\right)=L\,,$ (13)

where $S=0$ in case of null trajectories and $S=1$ for the test particles with
the non-zero mass; $L,\,E$ are the integrals of motion. This yields

$e^{\alpha+\beta}\left(\frac{dr}{d\tau}\right)^{2}=E^{2}-U_{\rm
eff}(r,L,S)\,,$ (14)

where $U_{\rm eff}(r,L,S)=e^{\alpha}\left(S+{L^{2}}/{r^{2}}\right)$.

Thus, we the problem is reduced to investigation of the one-dimensional
particle motion in the field of effective potential $U_{\rm eff}$. The
asymptotic behavior of the potential depends on asymptotics (10) at infinity
and those near singularity (see Zhdanov and Stashko (2020)). For $L\neq 0$ we
have

$U_{\rm eff}(r\rightarrow{0})\sim r^{\eta-3}\,,\quad U_{\rm
eff}(r\rightarrow{\infty})\rightarrow{S}\,.$ (15)

Thus for $\eta<3$ we have $U_{\rm eff}\rightarrow{\infty}$ at
$r\rightarrow{0}$, i.e. there is an infinite potential barrier that will
reflect falling particles. For $\eta>3$, $U_{\rm eff}\rightarrow{0}$ at
$r\rightarrow{0}$, the particles can approach the center and there is a
maximum at some $r_{u}>0$, which is the radius of an unstable circular orbit.

###  III.2 Circular orbits

We consider the Page-Thorne model of the geometrically thin accretion disk
(AD) Page and Thorne (1974), where the averaged motion of accretion matter is
approximated by means of SCO around gravitating center. A circular orbit is
stable if the right hand side of (14) is zero and $dU^{\prime}_{eff}/dr$
changes its sign at this point. For the congruence of circular orbits with
different radii in equatorial plane we get dependencies of the specific energy
and the specific angular momentum, and the angular velocity
$\Omega=d\varphi/dt$ upon radius $r$ as follows

$\tilde{E}^{2}(r)=\frac{2e^{\alpha(r)}}{2-r\alpha^{\prime}(r)},\leavevmode\nobreak\
\leavevmode\nobreak\
\tilde{L}^{2}(r)=\frac{r^{3}\alpha^{\prime}(r)}{2-r\alpha^{\prime}(r)},\leavevmode\nobreak\
\leavevmode\nobreak\ \Omega^{2}(r)=\frac{\alpha^{\prime}(r)e^{\alpha(r)}}{2r}$
(16)

We use a semi-analytical method of our works Stashko and Zhdanov (2018, 2019b)
to study bifurcations associated with the appearance and disappearance of the
minima of $U_{\rm eff}$. Essentially this is connected with investigation of
joint conditions $U^{\prime}_{\rm eff}=0$ and $U^{\prime\prime}_{\rm eff}=0$,
which allow us to exclude $L$; this leads to a necessary condition $F(r)=0$,
where

$F(r)=\tilde{L}^{2}(r)[r^{2}\alpha^{\prime\prime}(r)-2r\alpha^{\prime}(r)+6]+r^{4}\alpha^{\prime\prime}(r).$
(17)

Using equation (17), we get the bifurcation values
$r_{b},\,L^{2}_{b}=\tilde{L}^{2}(r_{b})$ and $E^{2}_{b}=\tilde{E}^{2}(r_{b})$
under conditions that $\tilde{E}^{2}(r_{b})>0$ and $\tilde{L}^{2}(r_{b})>0$.
We tested our results by considering the explicit forms of $U_{\rm eff}$ near
the bifurcations as in the example in Fig. 2. The right panel of Fig. 2
illustrates occurrence of three minima (which is not observed in case of FJNW
solution); it shows how the shape of the potential changes with increasing the
angular momentum, and as a result, the third minimum appears in the vicinity
of the singularity. Note that the bifurcation radius (boundary radius),
signalizing emergence or disappearance of some minimum of $U_{\rm eff}$,
indicates the boundary of some SCO ring for a given configuration.

Figure 2: Typical dependencies $\tilde{L}(r)$ and $U_{\rm eff}(r)$ in case of
S3 type; $\eta<3$ and $U_{\rm eff}(r)$ is infinite for $r\to 0$. Left panel
shows that there can exist several circular orbits with the same
$\tilde{L}(r)=L$. Roots of this equation correspond to SCO if they belong to
the intervals where $\tilde{L}^{\prime}(r)>0$. Right panel shows effective
potential with three minima ($S3$ type). Three different SCO rings can exist
near $L\sim 6.3$ and about fixed $M,Q,n$ indicated in the figure.

The above reasoning allowed us to obtain the boundaries of rings containing
SCO, which are separated by regions where circular orbits are either unstable
or do not exist. Figs. 3, 4, 5 below present examples of bifurcation curves,
which define these boundaries.

Here we describe main results of our numerical investigation of the
bifurcation radii $r_{b}\equiv r_{i(T)}$, where index $T$ denotes various SCOD
type explained below. We show that at least four SCOD types are possible in
our problem.

The next 3 types deal with the case of $\eta<3$ (effective potentials are
unbounded)

  * •

S1: $\tilde{L}(r)$ is monotonically increasing function. As a result we have
one connected domain of SCO with radii $r\in(0,\infty)$. SCO fill all the
space.

  * •

S2: The effective potentials $U_{\rm eff}$ have two minima, which correspond
to the SCO radii. As a result we have inner disk with SCO radii
$r\in(0,r_{1(2)})$ and outer disk with $r\in(r_{2(2)},\infty)$. Hereinafter,
the index in parenthesis denotes the type.

Note that types U1, S1, S2 exist in the FJNW case.

  * •

S3: We found a new type of the SCO distribution that does not exist in the
FJNW case. The effective potential $U_{\rm eff}$ can have three possible
minima and two maxima corresponding to three SCO regions $r\in(0,r_{1(3)})$,
$(r_{2(3)},r_{3(3)})$,$(r_{4(3)},\infty)$. These regions are separated by two
rings of the unstable orbits with $r\in(r_{1(3)},r_{2(3)})$,
$(r_{3(3)},r_{4(3)})$. Fig. 2 shows typical dependencies of $U_{\rm eff}(r,L)$
and $\tilde{L}(r)$. The emergence of SCO with small radii $r<r_{1(3)})$ occurs
in a neighborhood of a change in the shape of $U_{\rm eff}$ at $\eta$ values
close to 3, where the type of the singularity changes.

If $\eta\geq 3$, then effective potential has a maximum $U_{\rm eff}(r_{\rm
max},L)$ for any $L$. There is a region where either there are no circular
orbits at all, or they are unstable.

Here we have only one type:

  * •

U1 (Schwarzshild-like SCO distribution): effective potential $U_{\rm eff}$ is
bounded from above; $U_{\rm eff}\rightarrow{0}$ for $r\rightarrow{0}$ and only
one minimum can exist. We have one unstable and one stable regions of SCO; the
latter for $r\in(r_{1(U)},\infty)$ with lower boundary $r_{1(U)}$.

Fig. 3 show two illustrative examples of bifurcation radii $r_{b}(Q)\equiv
r_{i(T)}(Q)$, $T$ means type of SCOD, as a functions of $Q$, which realize
different SCOD types. Fig. 4 shows essentially the same, but with more
complete picture of the bifurcation radii, which illustrates how the curves
transform for larger $n$ values. Analogous dependencies are shown on Fig. 5 as
functions on $M$. Figs. 4 and 5 show how the shape of the curves changes when
$n$ passes some critical values.

Figure 3: Boundary radii of SCO regions as a function of $Q$ for some fixed
$M,n$. Vertical dashed lines separate areas of different SCOD types. In the
left panel, we have the S1 area between two branches of the bifurcation curve;
here arbitrary SCO radii are allowed. The other allowed region of radii are
according to definition of U1 and S2. The right panel shows the example with
larger $M,Q,n$, when another type is present, namely S3; corresponding
parameters fall into the yellow region on Figs. 10,11

For every $M$ there are two branches of the curve $r_{b}(Q)$ (unlike the FJNW
case, where there is a single branch exists) and there is a sequence of values
$n^{*}_{1}<n^{*}_{2}<n^{*}_{3}<n^{*}_{4}<n^{*}_{5}$, which has the following
properties.

  * •

The first and the last values denote critical numbers $n^{*}_{1}$,
$n^{*}_{5}$, which correspond to a reshaping (reconnection) of two branches.

  * •

For $2<n<n^{*}_{1}$, both branches are unbounded (like two solid curves on the
left panel of Fig. 4). After the reshaping, for $n\in(n^{*}_{1},n^{*}_{4})$
with some $n^{*}_{4}>n^{*}_{1}$, the left branch comes close to the FJNW curve
(practically merges), whereas the right one is unbounded (see dashed and
dotted curves on the left panel of Fig. 4).

  * •

For $n^{*}_{1}<n<n^{*}_{2}$ the right branch moves away to the right and then,
for $n>n^{*}_{2}$, it starts moving to the left. It is important to note that
for moderate $n\lesssim 7$, the sizes of unstable regions can be noticeably
larger than in the FJNW case.

  * •

At $n=n^{*}_{3}$, an additional wedge-shaped feature in the left branch
appears. This is difficult to show in the right panel of Fig. 4 for $M=1$, but
this is well seen for larger $M,Q$ in the right panel of Fig. 3: the
corresponding feature is formed by sections $r_{1(3)}$ and $r_{2(3)}$ within
the area of $S3$ type. This corresponds to "Pinocchio’s nose" in the right
panel of Fig. 5.

  * •

For $n\in(n^{*}_{4},n^{*}_{5})$ the right branch returns closer enough to the
left one and a new $S3$ area appears. See the right panel of Fig. 3, where
there is an additional small $S3$ area due to the sharp wedge for $r_{b}\sim
10,\,Q\gtrsim 7$.

  * •

At $n=n^{*}_{5}$ the tips of the two wedges (see the right panel of Fig. 3)
touch each other. The new reshaping of the bifurcation curve occurs, after
which the branches reconnect forming a structure represented by the solid
curve on the right panel of Fig.4. For large enough $n$, the lower branch
tends to the abscissa axis.

Figure 4: Boundary radii $r_{b}$ of SCO regions as a functions of $Q$ for
$M=1$ and several fixed values of $n$. The continuous blue line on the left
panel represents the FJNW solution and the orange straight line corresponds to
the Schwarzschild BH case ($r_{\rm ISCO}=6M$). The first reshaping occures at
$n^{*}_{1}\approx 2.38$; then the right branch moves to the right and then
starts backtracking to the left branch when $n>n^{*}_{2}\approx 7.53$. We
found $n^{*}_{3}\approx 13.12$ and $n^{*}_{4}\approx 13.13$. For ($n\geq
n^{*}_{5}\approx 13.15$, the form of the branches is represented by solid
($n=15)$ and dotted ($n=23)$ lines on the right panel; here we also have the
$S3$ type (four-valued function $r_{b}(Q)$ on some interval of $Q$).

Figure 5: Boundary radii $r_{b}$ of SCO regions as a functions of $M$ for
$Q=1$ and several fixed values of $n$. The first reconnection of two branches
occurs at $n\approx 2.62$; dash-dotted ($n=2.4$) and dotted ($n=2.5$) lines
show typical form of branches before the reshaping (left panel), the lines
with $n=2.625$ and $n=3$ show the typical forms after the reshaping. The right
panel demonstrates the branches around the second reconnection near $n\approx
13.86$.

We have determined the number of rings with stable and unstable circular
orbits and built maps of possible SCODs in the $M-Q$ plane for different $n$.
Each point on every map (see Figs. 6 – 7) shows the type of SCOD.

For all $M$ and $n$ there are $S2$ and $U1$ types with appropriate $Q$. We
note that the "thickness" of the ring in the $S2$ case can be considerably
larger than in the FJNW case. The size of $S1$ area changes non-monotonically:
it increases up to $n\sim 7$ and then decreases.

We discovered that the $S3$ region emerges for $n\sim 4.320$ (Fig. 6, right);
for larger $n$ this region becomes to appear for smaller $M$ (Fig. 7, right).

Figure 6: Domains of parameters on $(M,Q)$ plane for different values of $n.$
Here and on figures below white colour denotes the $S2$ type, yellow-$S3$,
light gray-$U1$, dark gray $S1$. For values of $n\sim 7$ the $S1$ area grows.

Figure 7: For larger $n\gtrsim 7$ the black region ($S1$) shrinks to the
origin and $S2$ dominates.

Figure 8: Domains of parameters on the $(n,M)$ plane for different values of
$Q$.

Figure 9: The same as on the previous figure for larger $Q$.

Figure 10: Domains of parameters on $(n,Q)$ plane for different values of $M$.

Figure 11: The same as on the previous figure for larger $M$. For arbitrary
$M$ there is a domain on $(n,Q)$-plane where $S3$ type exist.

###  III.3 Photon trajectories and accretion disk images

In case of null geodesics ($S=0$), the effective potential $U_{\rm
eff}(r,L,0)=e^{\alpha}{L^{2}}/{r^{2}}$ has simple properties. Its asymptotics
are due to (15). The sign of $U_{\rm eff}/dr$ is defined by function

$f(r)=r\alpha^{\prime}(r)-2$ (18)

We have verified numerically that $f(r)$ is a monotonically decreasing
function; evidently $f(\infty)=-2$. For $r\rightarrow{0}$ we have
$f(r)\gtrless 0$ if $\eta\gtrless 3$. Therefore, the root of this function can
exist only if $\eta>3$; then the point of maximum $r_{\rm ph}$ is a single
root of $f(r)$, $r_{\rm ph}$ being the radius of the photon sphere. Fig. 12
shows typical dependencies of the photon orbit‘s radii on the configuration
parameters. One can see that $r_{\rm ph}$ is always less than the
corresponding radius in Schwarzshild/FJNW cases.

Figure 12: The photon orbits radii as a function of scalar "charge" $Q$ (left)
and configuration mass $M$ (right) for different values of $n$. Blue and
orange curves correspond to the cases of FJNW and Schwarzschild metrics.
Curves $r_{\rm ph}$ go below FJNW and tend to them with increasing $n$.

The next step in the study of the configurations in question concerns the
images of accretion discs represented by different types of SCOD. In order to
build direct accretion disk images, we use the ray-tracing algorithm described
in Psaltis and Johannsen (2011); Johannsen and Psaltis (2010); Bambi (2012).
The complete consideration of this problem requires knowledge of the
brightness distribution of radiating matter over the disk. We have estimated
the surface brightness within the Page-Thorne model Page and Thorne (1974)
showing the there is a great enhancement of the radiation flux from the
innermost region with small radii normalized to the mass accretion rate
$\dot{M}_{0}$ (analogously to the results of Chowdhury  _et al._ (2012);
Gyulchev  _et al._ (2019) on FJNW solution). However, the (unknown) value of
$\dot{M}_{0}$ may be very different for outer and inner SCO rings and one can
expect that the input of the inner ring will be much smaller due to a
scattering of the accreting material. This requires a significant modification
of the AD model, presumably within the framework of full-scale hydrodynamic
modeling, which is beyond the scope of this article. In this view, we limited
ourselves to show the observed contours of the SCO regions for accretion disks
and the frequency distributions over these disks due to the gravitational
redshift and the Doppler effects.

To plot the SCOD images as seen by a distant observer, we need the
trajectories of photons falling from infinity. Their properties depend on the
sign of $\eta-3$. For $\eta>3$, when there is maximum of $U_{\rm eff}$, then
the incoming photons with impact parameter $\lambda<[b_{max}]^{-1/2}$, where
$b_{max}=\exp{[\alpha(r_{\rm ph})]}/r^{2}_{\rm ph}$, will reach singularity at
the origin (see the left panel of Fig. 13). For $\eta<3$ the photons with a
nonzero angular momentum will be reflected from the potential (right panel of
Fig. 13). Due to the strong bending of the rays, a scattered photon can hit a
point on the AD plane far enough from the center, where another photon with a
different trajectory also hits. Each such point has two images. On the other
hand, there are no photons falling into the AD region near the singularity;
this area is not visible to a distant observer.

Figure 13: In both panels $n=3$, $M=1$, the red line indicates the equatorial
plane. Left: trajectories of photons incident from infinity to the attracting
singularity; $Q=0.3$, $\eta=4.89$. Right: the same with the repulsing
singularity $Q=2.2$, $\eta=2.33$; in this case, the points near the
singularity form a dark spot around the center, imitating a black hole.

The photon trajectories needed in the ray-tracing method were obtained
numerically, similar to those shown in Fig. 13. We fix a sufficiently large
distance to the static observer, where the geometry can be considered flat,
and we track the photons coming from the observer to the AD plane, where we
take into account only those photons that hit the SCO regions. We do not take
into account the input from the reverse side of AD. The frequency ratio $g$
between the point $(e)$ at the AD surface and static remote observer $(o)$ for
metric (1) is

$g=\frac{k_{\mu}u^{\mu}|_{o}}{k_{\mu}u^{\mu}|_{e}}=\frac{\sqrt{e^{\alpha(r)}-r^{2}\Omega^{2}}}{1+\lambda\Omega}.$
(19)

This formula takes into account both gravitational and Doppler effects. We use
the normalized redshift factor

$\tilde{g}=\frac{g-g_{\rm min}}{g_{\rm max}-g_{\rm min}},$ (20)

where $g_{\rm min}$ and $g_{\rm max}$ is the minimal and maximal frequency
ratios on disk, respectively.

Figs. 14–21 show the SCOD contours and the distribution of $\tilde{g}$ over
the image, visible from infinity. The common feature of all the images is the
existence of dark spot in the center like the ordinary black hole. This is
either due to the properties of the photon trajectories falling from infinity
($\eta<3$, when these photons cannot reach the region near the center), or
simply because of absence of SCO in this region.

Figure 14: AD images for the $U1$ type of SCOD ($M=1$, $Q=0.3$, $n=3$): in
full face and for inclination $i=30^{o}$. White contour corresponds to the
photon orbit at $r_{\rm ph}\approx 2.58$. The ISCO placed at $r_{1(U)}\approx
5.56$ and the outer disk edge was fixed at $r=24$.

Figure 15: The same as on Fig. 14 with inclinations $30^{o}$ and $60^{o}$.

Figure 16: AD images for the $S1$ type ($M=1$, $Q=0.8$, $n=3$); in full face
and for inclination $i=30^{o}$. The outer disk edge radius $r=24$. There is a
dark spot in the center due to the repulsive character of the naked
singularity.

Figure 17: The same as on Fig. 16 with inclinations $60^{o}$ and $80^{o}$.

Figure 18: AD images for the $S2$ type ($M=1$, $Q=2.2$, $n=3$) in full face
and for inclination $i=30^{o}$. The outer disk edge radius is $r=24$. SCO
radii are in intervals $r\in(0,6.5)$ and $r\in(14.5,\infty)$. The inner SCO
region in the first figure cannot be observed due to the repulsive nature of
the naked singularity.

Figure 19: The same as on Fig. 18 with inclinations $60^{o}$ and $80^{o}$. The
outer disk edge is $r=24$. White dashed line here shows the outline of the
shadow of the naked singularity when it is illuminated by background light.

Figure 20: AD images for the $S3$ type ($M=2$, $Q=0.99$, $n=14$, $\eta\approx
2.9998$) in full face and for inclination $i=30^{o}$. The outer disk edge
radius is $r=24$. SCO radii intervals are $(0,0.22)$ (inner SCO region),
$(0.65,2.14)$ (second SCO ring), $(9,\infty)$ (outer unbounded SCO ring). Due
to weak repulsion of the singularity ($\eta\approx 2.9998$) the second SCO
rings is observed (red circles) and the inner SCO region is partially
invisible.

Figure 21: The same as on Fig. 20 with inclinations $60^{o}$ and $80^{o}$.

##  IV Discussion

We have studied isolated static spherically symmetric configurations of
General Relativity with minimally coupled nonlinear SF. The nonlinearity is
introduced due to the SF potential $U(\phi)=\phi^{2n}$. For fixed $n>2$, we
have shown that the solution of the corresponding Einstein-SF system of
equations exists and is unique under the appropriate conditions at spatial
infinity describing an isolated object. This means that the configuration with
scalar field is uniquely defined by two parameters: the configuration mass $M$
and the "scalar charge" $Q$ defined from the SF asymptotics $\phi(r)\approx
Q/r$ for $r\to\infty$.

At the center we have a naked singularity with the asymptotic behaviour that
involves parameter $\eta$, which describes the strength of the singularity
Zhdanov and Stashko (2020)). There is a critical value $\eta=3$ that separates
two types of singularity – attractive and repulsive – with different behaviour
of null geodesics near the center.

The solutions of the Einstein-SF system have been investigated numerically up
to $n\sim 40$ for sufficiently large $M\sim 60$ and $Q\sim 60$. There are a
lot of new elements in comparison with FJNW, which arise in the dependencies
of SCOD characteristics on configuration parameters. The most important
difference from FJNW is associated with the emergence of the $S3$-type of SCOD
with two rings of unstable circular orbits for $n\gtrsim 4.320$. Indeed,
though stable circular geodesics represent the simplified model of an
accretion disk Page and Thorne (1974), the above stability properties can be
important for real AD.

In general, there are 4 possible SCOD types, the first three being similar to
FJNW case. The $U1$ type is similar to the SCO distribution in the case of the
Schwarzschild metric: there is an inner region around the center where
circular orbits are either do not exist or are unstable, and there is an outer
region of SCO that extends to infinity. For $S1$ type, SCO fill all the space
starting from the center. For $S2$ type the stable orbits near the center are
separated by a ring of unstable orbits from the outer SCO region that extends
to infinity. Correspondingly, the images of the thin accretion disks are
qualitatively similar to the FJNW case. And, at last, there is a new type
($S3$) with an additional SCO region and with two rings of the unstable
circular orbits that separate SCO rings. Possible cases of SCOD are presented
on Figs. 6, 7 for different domains on the plane of the configuration
parameters $M,Q$. One can infer that the $S3$ type is less probable, moreover
the innermost SCO rings typically have rather small radii for moderate $M,Q$.

We plotted the observable contours of the accretion disks of the same radius
and images of SCO regions using the ray-tracing algorithm Psaltis and
Johannsen (2011); Johannsen and Psaltis (2010); Bambi (2012). Figs. 14–21 also
show the redshift distribution over AD image, which may be useful to study
deformation of the relativistic lines (e.g., Fe K$\alpha$) in the X-ray
spectra of compact objects. A common feature of all the images is the dark
spot in the center. This is either due to the absence of SCO near the center,
or because strong bending of the photon trajectories near the naked
singularity. In case of $S2$ and $S3$ types, Figs. 16–21 demonstrate features
around the center that are not observed for M87* shadow Collaboration (2019).
Apparently, these types should be ruled out in case of this object, though
observations with better resolution are desirable to have a final answer.

To sum up, we note that basic qualitative properties of static spherically
symmetric solutions of the Einstein-SF equations with monomial potential (3)
have much in common with the FJNW case Fisher (1948); Janis  _et al._ (1968).
However, there are subtle details in the distribution of matter around the
configuration, which distinguish the nonlinear SF.

###### Acknowledgements.

O.S.S. and V.I.Z. acknowledge the support from National Research Foundation of
Ukraine (project No. 2020.02/0073). The work of A.N.A. has been supported by a
scientific program “Astronomy and space physics” of Taras Shevchenko National
University of Kyiv (Project No. 19BF023-01).

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##  Appendix A Iteration method for solutions at large distances

A consideration of existence and uniqueness for an isolated configuration in
the asymptotically flat space time has been carried out in our paper
Alexandrov  _et al._ (2019), where we have used the so called "quasi-global"
coordinates. Here we present a direct proof in the coordinate system defined
by the metric representation (1).

We introduce variables

$X(r)=r\left(e^{-\beta}-1\right),\quad
Y(r)=r^{2}e^{\frac{\alpha-\beta}{2}}\frac{d\phi}{dr}$

The asymptotic flatness conditions (7,8) can be rewritten in terms of
$X,Y,\alpha,\phi$ as

$\lim\limits_{r\to\infty}X(r)=-r_{g},\quad\lim\limits_{r\to\infty}Y(r)=-Q,\quad\lim\limits_{r\to\infty}\left[r\alpha(r)\right]=-r_{g}\,.$
(21)

From (8) we also have

$\lim\limits_{r\to\infty}[r\phi(r)-Q]=-\lim\limits_{r\to\infty}r\int\limits_{r}^{\infty}\frac{x^{2}d\phi/dx+Q}{x^{2}}=0,$

then

$\lim\limits_{r\to\infty}[r\phi(r)]=Q.$ (22)

Einstein equations (4,5) and SF equation (6) can be rewritten in terms of
$Z\equiv\\{X,Y,\alpha,\phi\\}$ as the first order system.

Equations (4, 6) yield

$\frac{dX}{dr}=-8\pi\left[e^{-\alpha}\frac{Y^{2}}{2r^{2}}+r^{2}|\phi|^{2n}\right]\,,$
(23)
$\frac{dY}{dr}=2nr^{2}e^{{\alpha}/{2}}\frac{\phi|\phi|^{2n-2}}{\sqrt{1+X/r}}\,.$
(24)

In equation (5) that takes the form

$\frac{d\alpha}{dr}=\frac{1}{1+X/r}\left\\{-\frac{X(r)}{r^{2}}+8\pi
r\left[e^{-\alpha}\frac{Y^{2}}{2r^{4}}-|\phi|^{2n}\right]\right\\}$

we separate out the dominating term for $r\to\infty$:

$\frac{d\alpha}{dr}=\frac{r_{g}}{r^{2}(1-r_{g}/r)}+D(Z,r)\,,$ (25)

where

$D(Z,r)=\frac{1}{1+X/r}\left\\{-\frac{X+r_{g}}{r^{2}(1-r_{g}/r)}+8\pi
r\left[e^{-\alpha}\frac{Y^{2}}{2r^{4}}-|\phi|^{2n}\right]\right\\}\,.$ (26)

In the equation for $d\phi/dr$, which is expressed by means of $Y$,

$\frac{d\phi}{dr}=e^{-\alpha/{2}}\frac{Y}{r^{2}\sqrt{1+X/r}}\,,$

we also separate out the dominating term:

$\frac{d\phi}{dr}=-\frac{Q}{r^{2}\sqrt{1-r_{g}/r}}+E(Z,r)\,,$ (27)

where

$\displaystyle
E(Z,r)=e^{-\alpha/{2}}\frac{Y}{r^{2}\sqrt{1+X/r}}+\frac{Q}{r^{2}\sqrt{1-r_{g}/r}}=$
$\displaystyle=\frac{Y\left(e^{-\alpha/{2}}-1\right)}{r^{2}\sqrt{1+X/r}}+\frac{Q(X+r_{g})}{r^{3}(\sqrt{1-r_{g}/r}+\sqrt{1+X/r})\sqrt{1-r_{g}/r}\sqrt{1+X/r}}+\frac{Y+Q}{r^{2}\sqrt{1+X/r}}.$

Consider set ${\bf S}$ of continuous vector-functions
$Z(r)=\\{X(r),Y(r),\alpha(r),\phi(r)\\}$ satisfying

$|X(r)|\leq 2r_{g},\quad|\alpha(r)|\leq 2r_{g}/r,\quad|Y(r)|\leq
2|Q|,\quad|\phi(r)|\leq 2|Q|/r\,,\quad r\in[r_{\rm in},\infty),$ (28)

where $r_{\rm in}>0$ will be further assumed to be sufficiently large.

We shall construct a system of integral equations for solutions from ${\bf
S}$, which is equivalent to equations (eqs. 23, 25, 24 and 27) with conditions
(21). For the estimates below, it is essential that $n>2$; the case $n=2$
needs a separate consideration222This can be done in a similar way, but more
cumbersome. not presented here.

Given the conditions (21,22), we get following system.

Equations (23,24) yield

$X(r)=-r_{g}+A_{1}(Z,r),\quad A_{1}(Z,r)\equiv
8\pi\int\limits_{r}^{\infty}ds\,\left[\frac{Y^{2}(s)}{2s^{2}}e^{-\alpha(s)}+s^{2}|\phi(s)|^{2n}\right],$
(29)

where we take into account $n>2$;

$Y(r)=-Q+A_{2}(Z,r),\quad
A_{2}(Z,r)\equiv-2n\int\limits_{r}^{\infty}ds\,\exp\left[\alpha(s)/{2}\right]\frac{s^{2}\phi(s)|\phi(s)|^{2n-2}}{\sqrt{1+X(s)/s}}\,.$
(30)

Equations (25,27) yield

$\alpha(r)=\alpha_{0}(r)+A_{3}(Z,r)\,,\quad\alpha_{0}(r)=\ln(1-r_{g}/r)\,,\quad
A_{3}(Z,r)\equiv-\int\limits_{r}^{\infty}D(Z,s)ds\,;$ (31)
$\phi(r)=\phi_{0}(r)+A_{4}(Z,r)\,,\quad\phi_{0}(r)=\frac{2Q}{r_{g}}\left[1-\sqrt{1-r_{g}/r}\right]\,,\quad
A_{4}(Z,r)\equiv-\int\limits_{r}^{\infty}E(Z,s)ds\,.$ (32)

Here integral operators $A_{i},i=1,...,4,$ are defined on ${\mathbf{S}}$.

Let $Z(r)\equiv\\{X(r),Y(r),\alpha(r),\phi(r)\\}\in{\bf S}$. Denote

$\tilde{X}(r)=-r_{g}+A_{1}(Z,r),\quad\tilde{Y}(r)=-Q+A_{2}(Z,r).$ (33)

For sufficiently large $r$, simple estimates on account of (28) and $n>2$
yield

$|\tilde{X}(r)+r_{g}|=\left|A_{1}(Z,r)\right|\leq\frac{C_{1}}{r},\quad|\tilde{Y}(r)+Q|=\left|A_{2}(Z,r)\right|\leq\frac{C_{3}}{r^{2n-4}},$
(34)

Here and below $C_{i}=C_{i}(M,Q),\,i=1,2,...$ are finite positive constants.
For sufficiently large $r$ inequalities (28,34) yield $|\tilde{X}(r)|\leq
2r_{g},\quad|\tilde{Y}(r)|\leq 2|Q|$; whence
$Z^{\prime}\equiv\\{\tilde{X}(r),\alpha(r),\tilde{Y}(r),\phi(r)\\}\in{\mathbf{S}}$.

Now we denote

$\tilde{\alpha}(r)=\alpha_{0}(r)+A_{3}(Z^{\prime},r)\,,\quad\tilde{\phi}(r)=\phi_{0}(r)+A_{4}(Z^{\prime},r).$
(35)

Note that here we use $Z^{\prime}$ instead of $Z$ and correspondingly
$\tilde{X},\tilde{Y}$ from (33) instead of $\ X,Y$, which modifies the
iteration scheme below. This provides some technical convenience in view of
the specific form of the equations involved and allows us to avoid more
stringent assumptions on ${\mathbf{S}}$. Using (28,34), for $n>2$ we have

$|r\tilde{\alpha}(r)+r_{g}|=\left|rA_{3}(Z^{\prime},r)\right|\leq\frac{C_{4}}{r}.$
(36)

and

$|r\tilde{\phi}(r)+Q|=\left|rA_{4}(Z^{\prime},r)\right|\leq C_{5}\mu(r),$ (37)

where $\mu(r)$ is defined immediately after formula (10).

Whence, for a sufficiently large $r$, $|\tilde{\alpha}(r)|<2r_{g}/r$ and
$|\phi(r)|<2|Q|/r$ that is
$\tilde{Z}\equiv\\{\tilde{X}(r),\tilde{Y}(r),\tilde{\alpha}(r),\tilde{\phi}(r)\\}\in{\bf
S}$. Therefore, we have mapping ${\mathbf{R}}:\,Z\to\tilde{Z}={\mathbf{R}(Z)}$
defined by (33,35), which transforms vector-function $Z$

$Z\to
Z^{\prime}\to\tilde{Z}=\\{\tilde{X}(r),\tilde{Y}(r),\tilde{\alpha}(r),\tilde{\phi}(r)\\}\in{\bf
S},$

i.e. (for sufficiently large $r_{\rm in}$), operator ${\mathbf{R}}$ is
correctly defined and maps functional class ${\mathbf{S}}$ into itself. Thus,
initial equations are reduced to operator equation $Z={\mathbf{R}}(Z)$.

Now we proceed to estimate contraction mapping properties of ${\mathbf{R}}$.
Let $Z_{1}=\\{X_{1},Y_{1},\alpha_{1},\phi_{1}\\}\in{\mathbf{S}}$,
$Z_{2}=\\{X_{2},Y_{2},\alpha_{2},\phi_{2}\\}\in{\mathbf{S}}$;
$\tilde{Z}_{1}={\mathbf{R}}(Z_{1}),\quad\tilde{Z}_{2}={\mathbf{R}}(Z_{2})$;
$\delta Z\equiv Z_{1}-Z_{2}$.

Denote

$\|Z\|\equiv\sup\\{|X(r)|+r|\alpha(r)|+|Y(r)|+r|\phi(r)|,\quad r\in[r_{\rm
in},\infty).$ (38)

Equations (33) on account of (28) yield

$|\delta\tilde{X}(r)|=|A_{1}(Z_{1},r)-A_{1}(Z_{2},r)|\leq\frac{C_{7}}{r}\|\delta
Z\|,$ (39)
$|\delta\tilde{Y}(r)|=|A_{2}(Z_{1},r)-A_{2}(Z_{2},r)|\leq\frac{C_{8}}{r^{2n-4}}\|\delta
Z\|.$ (40)

Here and below we systematically use the Lagrange finite-increments formula.
Using explicit form (26) of $D(X,r)$ we have

$\displaystyle|D(Z_{1},s)-D(Z_{2},s)|\leq$
$\displaystyle\leq\frac{C_{9}}{s^{2}}|\delta X(s)|+\frac{C_{10}}{s^{3}}|\delta
Y(s)|+\frac{C_{11}}{s^{4}}\left(s|\delta\alpha(s)|\right)+\frac{C_{12}}{s^{2n-1}}\left(s|\delta\phi(s)|\right).$

After substitution $Z_{i}\to\tilde{Z}_{i},\,i=1,2$ on account of (39,40) we
have

$|D(Z^{\prime}_{1},s)-D(Z^{\prime}_{2},s)|\leq\left\\{\frac{C_{9}C_{7}}{s^{3}}+\frac{C_{11}}{s^{4}}+\frac{C_{10}C_{8}+C_{12}}{s^{2n-1}}\right\\}\|\delta
Z\|,$

whence

$\displaystyle|\delta\tilde{\alpha}(r)|=|A_{3}(Z^{\prime}_{1},r)-A_{3}(Z^{\prime}_{2},r)|\leq\frac{C_{13}}{r^{2}}\|\delta
Z\|\,.$ (41)

Analogously

$|E(Z_{1},s)-E(Z_{2},s)|\leq\frac{C_{14}}{s^{2}}|\delta
Y(s)|\left(1+O(1/s)\right)+\frac{C_{15}}{s^{4}}|\delta
X(s)|+\frac{C_{16}}{s^{2}}|\delta\alpha(s)|,$

then

$|E(Z^{\prime}_{1},s)-E(Z^{\prime}_{2},s)|\leq\frac{C_{14}C_{8}}{s^{2n-2}}\|\delta
Z\|\left(1+O(1/s)\right)+\frac{C_{15}}{s^{4}}|\delta
X(s)|+\frac{C_{16}}{s^{3}}|s\delta\alpha(s)|),$

$|\delta\tilde{\phi}(r)|=|A_{4}(Z^{\prime}_{1},r)-A_{4}(Z^{\prime}_{2},r)|\leq\frac{C_{17}}{r}\mu(r)\|\delta
Z\|.$ (42)

At last

$\|\mathbf{R}(Z^{\prime})-\mathbf{R}(Z)\|\leq\max\left[\frac{C_{16}}{r_{\rm
in}},\frac{C_{17}}{r_{\rm in}^{2n-4}}\right]\|\delta Z\|$

and we see that, for a sufficiently large $r_{\rm in}$, $\mathbf{R}$ is
contraction mapping. The solution can be obtained by successive approximations
$Z_{(n+1)}={\mathbf{R}}(Z_{(n)})$ with zeroth iteration
$Z_{(0)}=\\{-r_{g},-Q,\alpha_{0},\phi_{0}\\}$. Leaving the main terms for
large $r$, from the first iteration we have

$X_{(1)}(r)=-r_{g}+\frac{4\pi Q^{2}}{r}\left\\{1+O[\mu(r)]\right\\};$ (43)
$Y_{(1)}(r)=-Q-\frac{nQ|Q|^{2n-2}}{(n-2)r^{2n-4}}\left\\{1+O[\mu(r)]\right\\};$
(44) $\alpha_{(1)}(r)=\alpha_{{0}}+O\left[\frac{\mu(r)}{r^{2}}\right]\,;$ (45)
$\phi_{(1)}(r)=\frac{Q}{r}\left\\{1+\frac{r_{g}}{2r}+\frac{n|Q|^{2n-2}}{(n-2)(2n-3)r^{2n-4}}\right\\}+O\left[\frac{\mu(r)}{r^{2}}\right]\,.$
(46)

Here we have left those orders in $r^{-1}$ that will not change in subsequent
iterations. These equations yield asymptotic relations (9,10).