# Singularities in static spherically symmetric configurations of General
Relativity with strongly nonlinear scalar fields

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

###### Abstract

There are a number of publications on relativistic objects dealing either with
black holes or naked singularities in the center. Here we show that there
exist static spherically symmetric solutions of Einstein equations with a
strongly nonlinear scalar field with potential
$V(\varphi)\sim\sinh(\varphi^{2n})$, which allow the appearance of
singularities of a new type (“spherical singularities”) outside the center of
isolated configuration. The space-time is assumed to be asymptotically flat.
Depending on the configuration parameters, we show that the distribution of
the stable circular orbits of test bodies around the configuration is either
similar to that in the case of the Schwarzschild solution (thus mimicking an
ordinary black hole), or it contains additional rings of unstable orbits.

naked singularities, test particle motion, accretion disks

###### pacs:

11

##  I Introduction

Nowadays, the concept of a black hole has become a common element of
astrophysical research Novikov and Thorne (1973); Antonucci (1993); Bianchi
_et al._ (2012). However, in order to be fully confident in the theoretical
idea, it must be compared with alternative models. This is one of the
motivations for many theoretical works devoted to non-canonical and even
exotic solutions of General Relativity and its modifications, which could have
similar astrophysical consequences. The non-exhaustive list below includes
papers on various aspects of compact objects with naked singularities Stuchlík
and Schee (2014); Shao  _et al._ (2021); Chakraborty and Bhattacharyya (2019);
Bhattacharya  _et al._ (2020); Pugliese  _et al._ (2011, 2013); Joshi  _et
al._ (2013); Shahidi  _et al._ (2020); Boshkayev  _et al._ (2016); Stashko and
Zhdanov (2018); Bambhaniya  _et al._ (2019), with wormholes Karimov  _et al._
(2019); Paul  _et al._ (2019); Narzilloev  _et al._ (2021); Abdujabbarov and
Ahmedov (2009); Li and Bambi (2014), boson stars Vincent  _et al._ (2016);
Grandclément  _et al._ (2014); Liebling and Palenzuela (2017); Lamy  _et al._
(2018); Herdeiro  _et al._ (2021), and the other non-singular objects
mimicking the black holes Dymnikova and Poszwa (2019); Stuchlík and Schee
(2015); Schönenbach  _et al._ (2014). The question arises as to how the
aforementioned exotic configurations were formed; probably, not all
corresponding models can reflect real astrophysical situations. However, final
answers about the reality of these models must be based on observations. This
explains why the interest in the “black hole mimickers” greatly increased
after the Event Horizon Telescope observed the image of the accretion disk
around the supermassive black hole in the center of M87 Collaboration (2019).
Correspondingly, much attention is paid to the observational manifestations of
different models dealing with motion of test particles and photons in the
gravitational field of the compact object Chowdhury  _et al._ (2012); Stuchlík
and Schee (2014, 2015); Zhou  _et al._ (2015); Chakraborty and Bhattacharyya
(2019); Dymnikova and Poszwa (2019); an important point concerns the images of
accretion disks, radiation fluxes, forms of the relativistic Fe $K\alpha$ line
Shaikh and Joshi (2019); Gyulchev  _et al._ (2019, 2020, 2021); Collodel  _et
al._ (2021); Sau  _et al._ (2020); Schönenbach  _et al._ (2014); Cao  _et al._
(2016); Yang  _et al._ (2018); Shen  _et al._ (2017); Liu  _et al._ (2018),
etc.

A number of models are based on static solutions of the Einstein equations
with scalar field (SF) Chowdhury  _et al._ (2012); Zhou  _et al._ (2015);
Shaikh and Joshi (2019); Gyulchev  _et al._ (2019, 2020); Martínez  _et al._
(2004); Sau  _et al._ (2020). These papers use analytically defined metrics,
in particular the Fisher–Yanis–Newman–Winicour solution Fisher (1948); Janis
_et al._ (1968); Wyman (1981); Virbhadra (1997). Papers Sau  _et al._ (2020);
Shaikh and Joshi (2019); Gyulchev  _et al._ (2019, 2020); Chowdhury  _et al._
(2012); Zhou  _et al._ (2015) deal with singularities in the center, the
solutions used do not have other “physical” singularities outside the center.
It was proved Zhdanov and Stashko (2020) that this is a fairly general
situation, at least when considering an isolated spherically symmetric
configuration of General Relativity with a scalar field. Here the
gravitational field suppresses the appearance of “spherical singularities” and
only a naked singularity in the very center is possible. The proof uses an
assumption that the SF self-interaction potential is exponentially bounded
Zhdanov and Stashko (2020). Thus, the singularity for any non-zero value of
the radial variable in the curvature (Schwarzschild) coordinates is prohibited
and the question arises whether this result will be preserved in the case of a
sharper dependence of the SF potential for the large field values? This is a
key point of the present paper.

In this article, our goal is to provide examples showing that in the case of a
sufficiently strong nonlinear behavior of the potential (with a sufficiently
fast growth rate), spherical singularities (SS) can arise. To relax the
restrictions of Zhdanov and Stashko (2020), we choose
$V(\phi)=\sinh(\phi^{2n})$, $n>2$ as a representative of such potentials. This
is a technically convenient choice: for large $\phi$, this potential grows
faster than any exponentially bounded function; on the other hand, for small
$\phi$ it behaves as a monomial potential that allows us to use earlier
results concerning asymptotic behavior $\phi(r)\sim r^{-1}$ for $r\to\infty$.

The paper is organized as follows. In Section II we present initial relations.
In Sections III and IV we show the possibility of “spherical singularities”
and present numerical solutions for the Einstein equations with SF in the case
of spherical symmetry. In Section V we study distributions of the stable
circular orbits (SCO), which is important to study a thin accretion disk
around the configuration. We also estimate the radiation flux from the
accretion disk within the Page–Thorne model Page and Thorne (1974).

##  II Initial Relations

We consider a static spherically symmetric space-time metric in the curvature
coordinates, that is

$ds^{2}=e^{\alpha}dt^{2}-e^{\beta}dr^{2}-r^{2}\left[d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right];$ (1)

this fixes the radial variable $r>0$.

The gravitational field interacts with real SF $\phi(r)$ described by
Lagrangian density

$L=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi);$ (2)

the SF potential is

$V(\phi)=\sinh(\phi^{2n}).$ (3)

This potential is strongly nonlinear for $\phi\to\infty$ and for $n\geq 1$ it
grows faster than $|\phi|^{a}\exp(b\phi)$ for any $a,b$.

The SF equation following from (2) is

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

The independent from (4) Einstein equations are reduced to the form (see,
e.g., Zhdanov and Stashko (2020))

$\alpha^{\prime}+\beta^{\prime}=8\pi r\phi^{\prime 2},$ (5)
$\beta^{\prime}-\alpha^{\prime}=\frac{2}{r}\left(1-e^{\beta}\right)+16\pi
re^{\beta}V(\phi),$ (6)

We focus on isolated systems with mass $M$ in the asymptotically flat space-
time assuming for $r\to\infty$

$\mathop{\lim}\limits_{r\to\infty}\left[r{\alpha(r)}\right]=-\mathop{\lim}\limits_{r\to\infty}\left[r\beta(r)\right]=-r_{g},\leavevmode\nobreak\
\leavevmode\nobreak\ r_{g}=2M>0.$ (7)

As for SF, we assume $\phi(\infty)=0$; then for (3) we have
$V(\phi)\approx\phi^{2n}$ for $r\to\infty$ and therefore we can use some of
the results on monomial potentials from Zhdanov and Stashko (2020); Stashko
_et al._ (2021), concerning asymptotic behavior for large $r$:

$\phi(r)\sim\exp(-\mu r)/r^{1+\mu M}$

for $n=1$ (linear massive scalar field with mass $\mu$ Stashko and Zhdanov
(2019));

$\phi(r)\sim\left\\{\frac{1}{r^{1/(n-1)}},\,\,1<n<2;\quad\frac{1}{r\sqrt{|\ln
r|}},\,\,n=2;\quad\frac{1}{r},\,\,n>2;\right\\}$ (8)

(see Appendix A of Zhdanov and Stashko (2020)); asymptotic relations for
$\phi^{\prime}(r)$ can be obtained by formal differentiation of (8).

##  III  Asymptotic Behavior Near Singularity

First of all we note that for regular solutions of (4)–(6) within some
interval $(r_{0},\infty),$ $\,r_{0}>0$, satisfying conditions (8) with non-
trivial $\phi(r)$, we can show that functions $\phi(r)$ and $\phi^{\prime}(r)$
preserve their signs. Indeed, for potential (3) inequality $\phi
V^{\prime}(\phi)>0$ is valid for $\phi\neq 0$, and using Equation (4) we get

$\frac{d}{dr}\left[r^{2}e^{\frac{\alpha-\beta}{2}}\phi\,\frac{d\phi}{dr}\right]=r^{2}e^{\frac{\alpha+\beta}{2}}\phi\,V^{\prime}(\phi)+r^{2}e^{\frac{\alpha-\beta}{2}}\left[\frac{d\phi}{dr}\right]^{2}>0.$
(9)

Therefore, function $r^{2}e^{\frac{\alpha-\beta}{2}}\phi\,\phi^{\prime}$ is
monotonically increasing. On account of conditions (7) and (8), it is strictly
negative for large $r$; therefore it cannot be equal to zero and so is
$\phi(r)\phi^{\prime}(r)<0$. Whence we infer that functions
$\phi(r),\,\phi^{\prime}(r)$ do not change their signs.

Further, for definiteness, we assume $\phi(r)>0$, $\phi^{\prime}(r)<0$.

Now we turn to the singularities at some $r_{s}>0$ in case of potential (3).
We are looking for solutions on $(r_{s},\infty)$ such that

$\phi^{\prime}(r)\to-\infty,\leavevmode\nobreak\ r\to r_{s}+0,$ (10)

for some $r_{s}>0$. Our aim is to estimate asymptotic properties of these
solutions.

Equations (5) and (6) yield

$\beta^{\prime}=4\pi r\phi^{\prime 2}+\frac{1}{r}\left(1-e^{\beta}\right)+8\pi
re^{\beta}V(\phi).$ (11)

Numerical simulations near singularity suggests that $\alpha(r)$ is a slowly
varying function. So, as a first approximation, we are neglecting this
function compared to $\beta(r)$. Under this assumption we get for the leading
terms of Equation (4)

$e^{-\beta/2}\frac{d}{dr}\left[e^{-\beta/2}\phi^{\prime}\right]\simeq
V^{\prime}(\phi).$ (12)

This is a rough approximation that is valid in very small interval near the
singularity. We have then $e^{-\beta}\phi^{\prime 2}\simeq 2V(\phi)+const$,
where the constant will be neglected in comparison with $V(\phi)$ for $r\to
r_{s}+0$, so that

$e^{-\beta}\phi^{\prime 2}\simeq 2V(\phi).$ (13)

In view of (10), $e^{\beta}V(\phi)\simeq\phi^{\prime 2}\to\infty$ for $r\to
r_{s}+0$ and then it is easy to see that the principal terms on the right hand
sides of (5) and (6) are asymptotically the same. This justifies our
assumption about $\alpha(r)$.

Substitution into (11) yields

$\beta^{\prime}\simeq 16\pi re^{\beta}V(\phi),$ (14)

where we discarded the lower order terms. This allows us to reduce the problem
to the system of two equations .

Then $\beta(r)$ is monotonically increasing (for $r\in(r_{s},r_{1}]$, where
$(r_{1}-r_{s})/r_{s}\ll 1$). For $r\to r_{s}+0$ Equation (14) yields

$\frac{d}{dr}\left[e^{-\beta/2}\right]\approx-8\pi r_{s}e^{\beta/2}V(\phi).$
(15)

From Equation (13) we have

$\phi^{\prime}\approx-\sqrt{2}e^{\beta/2}\sqrt{V(\phi)},$ (16)

where we take into account that $\phi(r)$ is decreasing.

Dividing (15) by (16) we have equation

$\frac{d}{d\phi}\left[e^{-\beta/2}\right]=\frac{8\pi
r_{s}}{\sqrt{2}}\sqrt{V(\phi)}$

that can be solved in quadratures. In the leading terms for $r\to r_{s}+0$

$e^{-\beta(r)/2}\simeq\frac{8\pi
r_{s}}{\sqrt{2}}\Phi(\phi)+e^{-\beta(r_{1})/2}\sim\frac{8\pi
r_{s}}{\sqrt{2}}\Phi(\phi)$ (17)

where we denote

$\Phi(\phi)=\int_{\phi(r_{1})}^{\phi}\sqrt{V(x)}dx=\int_{\phi(r_{1})}^{\phi}\exp\left(\frac{1}{2}x^{2n}\right)dx.$

This can be expressed by the incomplete gamma-function, leading to the
asymptotic formula:

$\Phi(\phi)=\frac{1}{n\phi^{2n-1}}\sqrt{V(\phi)}\left[1+O\left(\frac{1}{\phi}\right)\right],\,\,r\to
r_{s}.$ (18)

Then we use (17), (18) to get from (16)

$\frac{d\phi}{dr}=-\frac{\sqrt{V(\phi)}}{4\pi
r_{s}\Phi(\phi)}\simeq-\frac{n}{4\pi r_{s}}\phi^{2n-1}.$

The solution is

$\phi(r)=\left\\{\frac{n(n-1)}{2\pi}\frac{r-r_{1}}{r_{s}}+\frac{1}{\phi_{1}^{2(n-1)}}\right\\}^{-\frac{1}{2(n-1)}},\quad\phi_{1}=\phi(r_{1}).$

The limit (10) occurs if

$\frac{n(n-1)}{2\pi}\cdot\frac{r_{1}-r_{s}}{r_{s}}=\frac{1}{\phi_{1}^{2(n-1)}}$
(19)

Then

$\phi(r)\sim\left\\{\frac{2\pi
r_{s}}{n(n-1)(r-r_{s})}\right\\}^{\frac{1}{2(n-1)}}.$ (20)

Using (17) we have

$\beta(r)\sim-2\ln\Phi(\phi)\sim-\left[\frac{2\pi
r_{s}}{n(n-1)(r-r_{s})}\right]^{n/(n-1)}.$ (21)

The Kretschmann invariant near $r_{s}$ has the form

$R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}\sim\frac{e^{-2\beta(r)}}{\left(r-r_{s}\right)^{\frac{2(2n-1)}{n-1}}}.$
(22)

The main outcome of these considerations is that there exist singularities of
solutions to systems (4)–(6) for some non-zero value of the radial variable in
curvature coordinates, that is, SS are indeed possible. This does not mean
that all the solutions have such singularities. It easy to see that for $n>1$
to have the singularity at some $r=r_{s}>0$, the condition (19) must be
fulfilled (i.e., $\phi_{1}$ must be sufficiently large). The latter depends on
the initial conditions at infinity and this must be derived numerically. This
is a subject of the next section.

##  IV Numerical Solutions

Here we restrict ourselves to the case of a long-range field $n>2$.
Correspondingly, taking into account (8), we assume the conditions for the
field

$\leavevmode\nobreak\ \leavevmode\nobreak\
\mathop{\lim}\limits_{r\to\infty}\left[r^{2}\frac{d\phi}{dr}\right]=-Q$ (23)

yielding

$\mathop{\lim}\limits_{r\to\infty}\left[r\phi(r)\right]=Q.$ (24)

It was shown in Stashko  _et al._ (2021) that there is a unique solution of
the problem for sufficiently large $r$ satisfying (4)–(6) and (24). The
iteration procedure yielding the solution is described in Stashko  _et al._
(2021) for a monomial potential and it can be applied in case of (3) for
sufficiently large $r$ (small $\phi$). This yields initial conditions for the
ordinary differential system (4)–(6) at some (large) $r_{init}$. Instead, one
can directly use asymptotic expansions to derive the solution for large
$r_{init}$. We obtained the solution numerically by integrating the equations
backward from $r_{init}$ to smaller values of $r$, either to a spherical
singularity or to a point-like naked singularity at the origin. The occurrence
of singularity can be checked by means of relation (19); this can be used in
order to define more precisely the singularity radius.

Figures 1 and 2 show typical behavior of solutions in the case of SS; we see
that $e^{\alpha}$ is monotonically increasing, whereas $e^{\beta}$ reaches a
maximum and then decreases to one for $r\to\infty$. Additionally, we note that
near the singularity $e^{\beta-\alpha}\to 0$ for $r\to r_{s}+0$ in accordance
with Section III. Figures 3–5 show a non-trivial dependence of the singularity
radii $r_{s}$ upon parameters $n,M,Q$.

Figure 1: Behavior of metric functions $e^{\alpha}$ (left) and $e^{\beta}$
(right) for $M=1$ and different $Q$.

Figure 2: Behavior of SF (left) and Kretschmann invariant
$R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}$ (right) for the same
parameters.

Figure 3: Radii of SS for different $M,Q$; $n=6$ (left), $n=18$ (right).

Figure 4: Radii of SS for different $n,M$; $Q=0.5$ (left), $Q=1$ (right).

Figure 5:  Radii of SS for different $n,Q$; $M=1$ (left) and $M=5$ (right).

##  V Test Particle Motion

In this section, we consider the massive test particles motion around static
spherically symmetric configurations with SF to describe in order to describe
characteristics of accretion onto a singularity. We use the Page–Thorne model
Page and Thorne (1974) of a thin accretion disk (AD). In this model, the
averaged motion of the accreting matter is essentially described by SCOD in
the equatorial plane.

In the case of spherically symmetric space-time with metric (1), the standard
procedure yields the first integrals for test particle trajectories in the
equatorial plane $\theta=\pi/2$ ($\tau$ is a canonical parameter.):

$e^{\alpha}\left(\frac{dt}{d\tau}\right)^{2}-e^{\beta}\left(\frac{dr}{d\tau}\right)^{2}-r^{2}\left(\frac{d\phi}{d\tau}\right)^{2}=S\,,$
(25) $e^{\alpha}\left(\frac{dt}{d\tau}\right)=E,\quad
r^{2}\left(\frac{d\phi}{d\tau}\right)=L\,,$ (26)

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

$e^{\alpha+\beta}\left(\frac{dr}{d\tau}\right)^{2}=E^{2}-U_{\rm
eff}(r,L,S)\,,$ (27)

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

The form of $U_{\rm eff}(r)$, in particular, the disposition of its minima and
maxima, defines the distribution of stable circular orbits (SCO) and unstable
ones. The SCO distribution (SCOD) is the most important because it forms the
basis for evaluating the properties of an accretion disc around the
configuration described by Equations (4)–(6). Possible types of SCODs are
listed in Table 1 and described schematically on Figure 6.

Type | $r_{\rm stable}$ | $r_{\rm unstable}$ | Photon sphere
---|---|---|---
$U_{1}^{(-)}$ | $(r_{1},\infty)$ | $(r_{s},r_{1})$ | $-$
$U_{1}^{(+)}$ | $(r_{1},\infty)$ | $(r_{s},r_{1})$ | $+$
$U_{2}$ | $(r_{1},r_{2})\cup(r_{3},\infty)$ | $(r_{s},r_{1})\cup(r_{2},r_{3})$ | $-$
Table 1: Possible types of SCOD

We use a method of our works Stashko and Zhdanov (2018, 2019) to study
bifurcations associated with the appearance and disappearance of the minima of
$U_{\rm eff}$. Figure 7 illustrates how the shape of the effective potential
changes with an increase in $L$ in case of the $U_{2}$ SCOD type. 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

$F(r)=r\alpha^{\prime\prime}(r)-r\alpha^{\prime}(r)^{2}+3\alpha^{\prime}(r).$
(28)

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

$\tilde{E}^{2}(r)=\frac{2e^{\alpha(r)}}{2-r\alpha^{\prime}(r)},\leavevmode\nobreak\
\leavevmode\nobreak\
\tilde{L}^{2}(r)=\frac{r^{3}\alpha^{\prime}(r)}{2-r\alpha^{\prime}(r)},\leavevmode\nobreak\
\leavevmode\nobreak\ \Omega^{2}(r)=\frac{\alpha^{\prime}(r)e^{\alpha(r)}}{2r}$
(29)

Using Equation (28), we numerically 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 $\tilde{L}^{2}(r_{b})>0$.
This allows us to determine SCOD types for given configuration parameters.
Figure 8 shows the corresponding results in the $M-Q$ plane. Also, Figure 9
illustrates the dependence of the boundary radii (i.e., radii of the blue
circles in Figure 6) upon $M$ and $Q$.

Figure 6: The schematic examples of possible SCODs in the equatorial plane. In
both cases, black spots in the center represent SS at finite values of $r$.
Dark and light gray rings correspond to the sequences of SCO and unstable
circular orbits correspondingly. Blue circles show the boundary radii of SCOD
as described in Table 1. Figure 7: Typical examples of $U_{\rm eff}$ for
configuration with $U_{2}$ SCOD type. The blue points show the corresponding
extrema. There is only one minimum for large $L$.  Figure 8: Domains of
parameters on $(M,Q)$ plane for $n=3$. Every point of this plane corresponds
to a solution of Equations (4)–(6) with conditions (7), (8). The SCOD type was
determined for every such solution; gray, dark gray and white colors
correspond to the $U_{1}^{(+)}$, $U_{1}^{(-)}$ and $U_{2}$ type, respectively.

Figure 9: Boundary radii $r_{b}/M$ of SCO regions as functions of $Q$ (left)
and $M$ (right) for several values of $M$ and $Q$. The corresponding points
where curves qualitatively change their behavior by reconnection are
$(M,Q)\simeq(2.11,1.39)$ and $(M,Q)\simeq(1.2,1.67)$ for left and right
pictures, respectively and shown by blue squares.

Now we consider radiation from the stationary thin AD described by Page–Thorne
model Page and Thorne (1974). The time averaged radiation flux $F(r)$ from the
surface of the accretion with inner edge located at the boundary SCO $r=r_{\rm
b}$ is

$F(r)=-\frac{\dot{M_{0}}}{4\pi\sqrt{|{}^{(3)}g|}}\frac{\Omega_{,r}}{\left(E-\Omega
L\right)^{2}}\int\limits_{r_{\rm b}}^{r}\left(E-\Omega L\right)L_{,x}\,dx,$
(30)

where $\sqrt{|{}^{(3)}g|}=re^{(\alpha+\beta)/2}$ is the metric’s determinant
in the equatorial plane and $\dot{M_{0}}$ is the mass accretion rate that is
assumed to be constant. Specific particle energy, momentum, and angular
velocity are defined from (29). We obtained the radiation flux for several
values $Q$ in presence/absence of inner stable ring of AD. The corresponding
fluxes are shown in Figure 10. To compare different types, we normalize them
to the maximum flux in the case of a Schwarzschild black hole $F_{\rm
Schw}^{\rm(max)}\simeq 0.0001719\dot{M_{0}}/4\pi M^{2}$. We see that the
maximum value of different types can be less or more then $1$ as distinct from
the case of the massless scalar field Chowdhury  _et al._ (2012); Gyulchev
_et al._ (2020), where $F/F_{\rm Schw}^{\rm(max)}\geq 1$ due to $r_{b}\leq
6M$.

Figure 10: The normalized flux ($M=4$) from inner (right) and outer (left)
parts of AD.

##  VI Discussion

We have shown that General Relativity allows the existence of SS in static
spherically symmetric configurations in the case of the SF potential
$V(\phi)=\sinh(\phi^{2n})$, which represents potentials with fast growth rate.
This statement follows from the analytical reasoning of Section III and is
confirmed by numerical simulations of Section IV. These SS are “physical”
singularities that cannot be removed by a coordinate transformation: this can
be seen from the behavior of the Kretschmann invariant. These are naked
singularities, because they can be observed by a distant observer; it is easy
to verify this by considering the radial motion of photons, taking into
account that near the singularity $\exp(\beta-\alpha)\to 0$ for $r\to
r_{s}+0$.

We note that the appearance of singularities in solutions of nonlinear
equations is a fairly typical case. However, the situation with SS is
different from the case of monomial or the other exponentially bounded
potentials, when SS are suppressed by the gravitational field and we only have
a naked singularity at the center Zhdanov and Stashko (2020). In the case of
$V(\phi)=\sinh(\phi^{2n})$, the sharp growth of the field near the singularity
overcomes this suppression. This suggests that SS can occur for more general
potentials with fairly rapid growth.

Having the numerical solutions, we investigated the location of stable and
unstable circular orbits. There are two main types of SCOD in our case: (i)
$U_{1}$ is similar to SCOD in the case of the Schwarzschild metric, when SCO
radii must be larger than some boundary value; (ii) $U_{2}$ type is formed by
a non-connected distribution with two regions of SCO separated by a ring of
unstable circular orbits. These distributions of the stable orbits are
directly related to the structure of the thin accretion disk in the
Page–Thorne model Page and Thorne (1974). Perhaps this result can also be
important for more complex AD models, if, of course, the above scalar fields
do exist. Note that these types of SCODs also arise in the case of solutions
with monomial potentials Stashko  _et al._ (2021). Additionally, it should be
noted that $U_{2}$ type is not observed for M87* shadow Collaboration (2019).
However, for definite answer observations with better resolution are
mandatory.

We have no answer to the question of whether strongly nonlinear fields really
exist. Moreover, the question remains, how the spherical singularity can form.
This is a question of the same order as the origin of point naked
singularities, as well as other exotic structures such as bosonic stars,
wormholes, etc. Stuchlík and Schee (2014); Shao  _et al._ (2021); Chakraborty
and Bhattacharyya (2019); Bhattacharya  _et al._ (2020); Pugliese  _et al._
(2011, 2013); Joshi  _et al._ (2013); Shahidi  _et al._ (2020); Boshkayev  _et
al._ (2016); Stashko and Zhdanov (2018); Karimov  _et al._ (2019); Paul  _et
al._ (2019); Narzilloev  _et al._ (2021); Abdujabbarov and Ahmedov (2009); Li
and Bambi (2014); Vincent  _et al._ (2016); Grandclément  _et al._ (2014);
Liebling and Palenzuela (2017); Lamy  _et al._ (2018); Dymnikova and Poszwa
(2019); Herdeiro  _et al._ (2021); Stuchlík and Schee (2015). We note,
however, that for some sets of configuration parameters, it could be difficult
to distinguish SS from ordinary black holes relying upon the accretion disk
structures. On the other hand, for other sets of parameters, we may have an
unconnected SCO region, which would be different from the point of view of a
distant observer. Additional information on the existence/non-existence of the
strongly non-linear scalar fields may come from considerations of early
cosmological processes. So the hypothesis of the existence of the spherical
singularities seems to be testable.

###### Acknowledgements.

This work is supported by National Research Foundation of Ukraine (project No.
2020.02/0073).

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