# Nonsingular, lump-like, scalar compact objects in $(2+1)$-dimensional
Einstein gravity

Roberto V. Maluf<EMAIL_ADDRESS>Universidade Federal do Ceará (UFC),
Departamento de Física,
Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil.  Departamento
de Física Teórica and IFIC, Centro Mixto Universidad de Valencia - CSIC.
Universidad de Valencia, Burjassot-46100, Valencia, Spain.  Gerardo Mora-Pérez
<EMAIL_ADDRESS>Departamento de Física Teórica and IFIC, Centro Mixto
Universidad de Valencia - CSIC. Universidad de Valencia, Burjassot-46100,
Valencia, Spain.  Gonzalo J. Olmo<EMAIL_ADDRESS>Departamento de Física
Teórica and IFIC, Centro Mixto Universidad de Valencia - CSIC. Universidad de
Valencia, Burjassot-46100, Valencia, Spain.  Universidade Federal do Ceará
(UFC), Departamento de Física,
Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil.  Diego Rubiera-
Garcia<EMAIL_ADDRESS>Departamento de Física Téorica and IPARCOS,
Universidad Complutense de Madrid, E-28040 Madrid, Spain.

###### Abstract

We study the space-time geometry generated by coupling a free scalar field
with a non-canonical kinetic term to General Relativity in $(2+1)$ dimensions.
After identifying a family of scalar Lagrangians that yield exact analytical
solutions in static and circularly symmetric scenarios, we classify the
various types of solutions and focus on a branch that yields asymptotically
flat geometries. We show that the solutions within such a branch can be
divided in two types, namely, naked singularities and nonsingular objects
without a center. In the latter, the energy density is localized around a
maximum and vanishes only at infinity and at an inner boundary. This boundary
has vanishing curvatures and cannot be reached by any time-like or null
geodesic in finite affine time. This allows us to consistently interpret such
solutions as nonsingular, lump-like, static compact scalar objects, whose
eventual extension to the $(3+1)$-dimensional context could provide structures
of astrophysical interest.

Einstein gravity, Compact objects, Nonlinear scalar field

##  I Introduction

The search for exact analytical solutions to Einstein equations in the
presence of reasonable matter sources and couplings is generally a challenging
problem, particularly if one works in a $(3+1)$ dimensional space-time. Among
the known exact solutions of General Relativity (GR), only a handful are
imbued with relevant physical meaning [1]. Stationary solutions with spherical
and axial symmetries are notable within this group. Indeed, the uniqueness
theorems [2, 3] assure us that the most general possible asymptotically-flat
solution of an electromagnetic nature in vacuum is given by the Kerr-Newman
one, which can be interpreted as the gravitational field external to a body
described solely by its mass, electric charge, and angular momentum [4, 5].
Since charge is typically neglected in astrophysical scenarios, the above
solution reduces to the Kerr one.

The need to confront observations with the Kerr hypothesis, namely, that all
rotating, fully-collapsed, objects in the Universe belong to the Kerr family
of solutions, has sparked much interest in recent years to obtain new
solutions that describe alternative compact astrophysical objects. Solutions
of that type are not interesting only for their observational characteristics,
such as shadows [6, 7] or gravitational wave emission [8], but also from a
theoretical perspective. In fact, given that classical GR is expected to break
down at high enough energies in order to get rid of the various types of
singularities the theory harbours [9], particularly for black holes, new
regular solutions could help us better understand the possibilities beyond
this pessimistic scenario of ill-defined geometries that give support to our
current interpretation of the universe. The theoretical exploration of
nonsingular solutions is thus an important topic on its own and has motivated
many works within and beyond GR, as well as in higher and lower dimensions,
see e.g. [10] for a report on the current observational status of many such
proposals. In this sense, the search for new solutions in $(2+1)$ dimensions
has yielded highly valuable results in a wide range of gravity-matter
scenarios, providing novel perspectives on fundamental questions in classical
and quantum gravity [11]. Among this class of solutions, those obtained by
Bañados, Teitelboim, and Zanelli (BTZ) [12, 13, 14] are of particular interest
and have been extensively investigated in various contexts (see, for instance,
[15, 16, 17]). More recently, studies in $(2+1)$ dimensions coupled to various
types of fields have managed to find regular black holes [18, 19, 20, 21], and
wormhole solutions [22, 23].

In this work we consider $(2+1)$-dimensional scalar fields borrowing
inspiration from Wheeler’s notion of geon [24, 25], in the sense of self-
gravitating free (scalar) fields. We thus focus on a scalar field with no
potential but with a non-canonical kinetic term, bringing in that way some
extra freedom to the problem. Fields of this type are known in the literature
as “k-fields”, and were originally introduced in the context of cosmology [26,
27]. Though the study of geons typically involves some kind of time
dependence, here we consider static scenarios to investigate whether any
interesting structures can be found. As we will see, nontrivial solutions do
emerge.

It is well-known that in $(3+1)$ dimensions, a static, spherically symmetric
free scalar field coupled to Einstein’s gravity may lead to asymptotically
flat, localized solutions that, nonetheless, represent naked singularities. To
obtain regular solutions one must consider oscillating fields, leading to what
are known as boson stars [28], a field of great activity in the last few years
[29, 30, 31, 31]. Exploration of non-canonical scalar fields in $(2+1)$
dimensions can help us shed some light on whether static, nonsingular exotic
solutions could be possible in the more difficult $(3+1)$ case. By considering
a modified ansatz for the line element that is inspired by the $(3+1)$ static,
spherical case, in this work we manage to obtain exact analytical solutions
that represent asymptotically flat self-gravitating scalar objects with a
clearly localized circular structure. The compactness of these structures
depends on the model parameters in a quite transparent functional form, which
facilitates their analysis. Moreover, we show that the geodesic structure of
the family of solutions considered is complete, confirming in that way that
such objects are nonsingular.

The paper is organized as follows. In Sec. II we define our general setting
and look for solutions of the nonlinear scalar matter source. This severely
constrains the form of the scalar field Lagrangian but allows us to find exact
analytical solutions. We then proceed to classify the solutions and analyse
their physical properties in Sec. III, calculating the line element and some
curvature scalars. To better understand the properties of the most physically
appealing solutions, we study their time-like and null geodesic structure in
Sec. IV.1 and their energy distribution in Sec. V. We conclude with a summary
and discussion of the results in Sec. VI.

##  II $(2+1)$-Einstein theory with nonlinear scalar field

Let us start by defining the action for $(2+1)$-Einstein gravity coupled to a
scalar field as

$\mathcal{S}=\int
d^{3}x\sqrt{-g}\left(\frac{1}{2\kappa}\mathcal{R}-\frac{1}{2}L(Y)\right),$ (1)

where $\mathcal{R}\equiv g^{\mu\nu}R_{\mu\nu}$ is the usual curvature scalar
of a space-time metric $g_{\mu\nu}$ and Ricci tensor $R_{\mu\nu}$, while
$L(Y)$ is an arbitrary function of the scalar field invariant $Y\equiv
g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$, and the constant
$\kappa=8\pi\tilde{G}/c^{4}$ where $\tilde{G}$ stands for the Newtonian
gravitational constant in two spatial dimensions, which carries units of
length $[\tilde{G}]=\ell$ in natural units.

The corresponding Einstein field equations are derived by varying the action
(1) with respect to the metric tensor, leading to

$G_{\mu\nu}=\kappa T_{\mu\nu},$ (2)

with the energy-momentum tensor given by

$T_{\mu\nu}=\left(L_{Y}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}L\right),$
(3)

where $L_{Y}\equiv\frac{\partial L}{\partial Y}$. On the other hand, variation
with respect to the scalar field leads to

$\frac{1}{\sqrt{-g}}\nabla_{\mu}\left(\sqrt{-g}L_{Y}g^{\mu\nu}\nabla_{\nu}\phi\right)=0.$
(4)

Taking the trace of Eq.(2) and plugging the result back, one can rewrite this
equation in the more convenient form

$R_{\mu\nu}=\kappa\left(T_{\mu\nu}-g_{\mu\nu}T\right),$ (5)

where $T\equiv g^{\mu\nu}T_{\mu\nu}=(YL_{Y}-\frac{3}{2}L)$ is the trace of the
energy-momentum tensor of the scalar field.

###  II.1 Static and circularly symmetric solutions

For static and circularly symmetric scenarios, one assumes that the scalar
field profile only depends on the radial coordinate, $\phi\equiv\phi(x)$, in a
coordinate system represented by $\\{t,x,\theta\\}$. The equation of motion
(4) can thus be expressed as

$\partial_{x}\left(\sqrt{-g}g^{xx}L_{Y}\phi_{x}\right)=0,$ (6)

where $\phi_{x}\equiv d\phi/dx$. In order to deal with this equation, we adopt
an ansatz for the line element inspired in the choice made by Wyman [32] in
the $(3+1)$-dimensional case but with a modification that is crucial to bring
into the $(2+1)$ scenario the philosophy behind the original choice, namely,

$ds^{2}=-e^{A(x)}dt^{2}+e^{A(x)}\frac{L_{Y}^{2}}{W^{2}(x)}dx^{2}+\frac{1}{W^{2}(x)}d\theta^{2},$
(7)

where $A$ and $W$ are arbitrary functions of the radial coordinate $x$. This
unconventional form of the line element is justified by the fact that it leads
to an almost trivial scalar field equation: $\phi_{xx}=0$. Without loss of
generality, this allows us to take $\phi(x)\equiv x$. As a result, the kinetic
term $Y$ takes the form

$Y=e^{-A}\frac{W^{2}}{L_{Y}^{2}}.$ (8)

which can only be explicitly solved once a concrete function $L(Y)$ is
specified. The metric field equations (5) with this choice take the form

$\displaystyle R_{\ t}^{t}$ $\displaystyle=$ $\displaystyle
Y\left(\frac{A_{x}L_{Y_{x}}}{2L_{Y}}-\frac{A_{xx}}{2}\right)=\kappa\left(L-YL_{Y}\right),$
(9) $\displaystyle R_{\ x}^{x}$ $\displaystyle=$ $\displaystyle
Y\Bigg{(}\frac{A_{x}L_{Y_{x}}}{2L_{Y}}-\frac{A_{x}W_{x}}{W}-\frac{L_{Y_{x}}W_{x}}{L_{Y}W}$
(10) $\displaystyle-$
$\displaystyle\frac{W_{x}^{2}}{W^{2}}-\frac{A_{xx}}{2}+\frac{W_{xx}}{W}\Bigg{)}=\kappa
L,$ $\displaystyle R_{\ \theta}^{\theta}$ $\displaystyle=$ $\displaystyle
Y\left(-\frac{L_{Y_{x}}W_{x}}{L_{Y}W}-\frac{W_{x}^{2}}{W^{2}}+\frac{W_{xx}}{W}\right)$
(11) $\displaystyle=$ $\displaystyle\kappa\left(L-YL_{Y}\right),$

where we used the relation (8) and denoted $L_{Y_{x}}\equiv\frac{\partial
L_{Y}}{\partial x}$.

Once a scalar Lagrangian $L(Y)$ is specified, the above equations represent a
nonlinear coupled system for the variables $A(x)$, $W(x)$, and $L(x)$. A
useful relation can be obtained by evaluating
$R_{x}^{x}-R_{t}^{t}-R_{\theta}^{\theta}$, which leads to a first integral of
the above system in the form

$\frac{A_{x}W_{x}}{W}=\kappa\frac{\left(L-2YL_{Y}\right)}{Y}\ .$ (12)

Another useful expression follows by rewriting Eq.(9) in the form

$A_{xx}-A_{x}\frac{L_{Y_{x}}}{L_{Y}}=-2\kappa\frac{\left(L-YL_{Y}\right)}{Y}\
,$ (13)

which can be seen as a first-order linear ordinary differential equation for
the variable $A_{x}$ that admits the formal solution

$A_{x}=L_{Y}\left(c_{1}-2\kappa\int dx\frac{(L-YL_{Y})}{YL_{Y}}\right)\ .$
(14)

where $c_{1}$ is an integration constant. For arbitrary $L_{Y}$, this is an
integro-differential equation for $A(x)$ coupled to $W(x)$. However, for the
specific choice

$L(Y)=\lambda Y^{\alpha},$ (15)

with $\lambda$ a constant with suitable dimensions and $\alpha$ a
dimensionless parameter111For the static scenarios we are considering, the
kinetic term $Y$ is always positive. In more general settings, and in order to
prevent problems if $Y$ becomes negative, one could consider a redefinition of
$\alpha$ as $\alpha\to 2\tilde{\alpha}$ to force that the Lagrangian is indeed
a real quantity., the integrand of the second term in Eq.(14) becomes a
constant, allowing us to obtain an explicit solution of the form

$A_{x}=-\lambda\alpha Y^{\alpha-1}\Delta_{1}\ ,$ (16)

where we have defined the quantity

$\Delta_{1}\equiv\frac{2\kappa(1-\alpha)}{\alpha}(x-x_{1})\ ,$ (17)

and $x_{1}\equiv c_{1}\alpha/(2\kappa(1-\alpha))$ is an integration constant.
Note that since the theory is invariant under constant shifts of the scalar
field, $\phi(x)\to\phi(x)+\phi_{1}$, and we have taken a coordinate system in
which $\phi(x)=x$, the constant $x_{1}$ does not play any physical role and
can be set to zero for simplicity. Note also that the above expressions lead
to singular results in the cases of $\alpha=1,1/2$, or $0$, so we shall first
elaborate on the general case before discussing these singular ones.

Combining Eq.(16) with Eq.(12), we also find an important simplification,
namely,

$\frac{W_{x}}{W}=\frac{\kappa(2\alpha-1)}{\alpha\Delta_{1}}=\frac{(2\alpha-1)}{2(1-\alpha)}\frac{1}{(x-x_{1})}\
,$ (18)

whose solution can be written as

$W(x)=\frac{1}{r_{0}}\left(\frac{x-x_{1}}{x_{0}}\right)^{\gamma/2}$ (19)

where we have included an integration constant $r_{0}x_{0}^{\gamma/2}$ for
dimensional consistency, and have defined the parameter

$\gamma\equiv\frac{(2\alpha-1)}{(1-\alpha)}\ ,$ (20)

which will play a relevant role in the classification of solutions. The above
expression is valid as long as $\alpha\neq 0,1/2$ and $1$, which belongs to
the singular cases. Using the above results in Eq.(11), a bit of algebra leads
to the following equation for the function $Y$:

$Y_{x}-\frac{2\kappa}{\alpha\Delta_{1}}Y=\frac{\lambda\alpha\Delta_{1}}{(2\alpha-1)}Y^{\alpha}\
,$ (21)

which turns out to be a nonlinear first-order equation of the Bernouilli type.
Such equations can be linearized by the change of variable $\nu=Y^{1-\alpha}$,
leading to

$\nu(x)=(x-x_{1})\Delta_{2}\ ,$ (22)

where we have defined

$\Delta_{2}\equiv c_{2}+\frac{2\kappa\lambda(1-\alpha)}{\gamma}x$ (23)

One can thus put $Y^{\alpha-1}=1/\nu(x)$ into Eq.(16) and integrate to obtain

$e^{A(x)}=\Delta_{2}^{-\gamma}$ (24)

With all the above results, the line element for generic $\alpha\neq 0,1/2,1$
can be written as

$ds^{2}=-\Delta_{2}^{-\gamma}dt^{2}+\frac{(\alpha\lambda)^{2}r_{0}^{2}x_{0}^{\gamma}}{(x\Delta_{2})^{(2+\gamma)}}dx^{2}+r_{0}^{2}\left(\frac{x_{0}}{x}\right)^{\gamma}d\theta^{2}\
,$ (25)

where we have set $x_{1}=0$ for simplicity. We can rewrite this line element
in terms of the usual radial coordinate $r$ by identifying $r^{2}=1/W^{2}$.
Using the relation (19) with $x_{1}=0$ for simplicity, we have that
$r^{2}=r_{0}^{2}x_{0}^{\gamma}/x^{\gamma}$ leads to

$\left(\frac{dr}{r}\right)^{2}=\left(\frac{\gamma}{2}\right)\left(\frac{dx}{x}\right)^{2}\
,$ (26)

and inserting this result in (25), we get

$ds^{2}=-\Delta_{2}^{-\gamma}dt^{2}+{\sigma_{0}^{2}}\Delta_{2}^{-(2+\gamma)}dr^{2}+r^{2}d\theta^{2}\
,$ (27)

where $\sigma_{0}^{2}\equiv(2\alpha\lambda/\gamma)^{2}$ is just a constant.
This completes our construction of the line element of this scalar theory.

##  III Families of solutions

We will now proceed to classify the solutions of our model in terms of the
parameter $\alpha$ that characterizes the scalar field Lagrangian or,
equivalently, in terms of the exponent $\gamma$ defined in Eq.(19). The
relation between these two parameters is as follows:

  * •

$\frac{1}{2}<\alpha<1$ corresponds to the interval $\gamma>0$, with $\gamma=0$
identified with $\alpha=1/2$ and $\gamma\to+\infty$ with $\alpha\to 1^{-}$.

  * •

$\alpha>1$ leads to $-\infty<\gamma<-2$, with $\alpha\to 1^{+}$ corresponding
to $\gamma\to-\infty$ and $\alpha\to\infty$ leading to $\gamma\to-2$.

  * •

$\alpha<0$ is mapped into $-2<\gamma<-1$, with $\alpha=0$ leading to
$\gamma=-1$ and $\alpha\to-\infty$ to $\gamma=-2$.

  * •

$0<\alpha<\frac{1}{2}$ is mapped into $-1<\gamma<0$.

As is evident from this classification, there are three values of the
parameter $\alpha$, corresponding to $\\{0,1/2,1\\}$, that require a separate
discussion and will be considered later. We will address the features of the
general case first. For this purpose, we focus on the radial dependence of the
function $\Delta_{2}$ defined in Eq.(23). Since (19) allows us to write

$x=x_{0}\left(\frac{r_{0}}{r}\right)^{2/\gamma}\ ,$ (28)

we can rewrite $\Delta_{2}$ as

$\Delta_{2}\equiv c_{2}\left[1+\frac{2\lambda\kappa(1-\alpha)x_{0}}{\gamma
c_{2}}\left(\frac{r_{0}}{r}\right)^{2/\gamma}\right]\ .$ (29)

Note that the factor $c_{2}$ in front of this expression can be absorbed into
a redefinition of the time coordinate, in a rescaling of $\sigma_{0}$, and in
a rescaling of $\lambda\kappa x_{0}$ (as long as $c_{2}>0$, which we will
assume from now on). Thus, without loss of generality, we can set $c_{2}=1$ in
$\Delta_{2}$.

Let us now consider the radial dependence of (29). We see that only when
$\gamma>0$ will we have asymptotically flat solutions, which happens in the
interval $1/2<\alpha<1$. These are the solutions we are mostly interested in.
On the other hand, we see that $\lambda$ determines the sign of the second
term in the square bracket. Considering the far limit, $r\to\infty$, of the
$g_{tt}$ component of the metric in its representation (27), we see that

$g_{tt}\approx-\left(1-2\lambda\kappa(1-\alpha)x_{0}\left(\frac{r_{0}}{r}\right)^{2/\gamma}+\ldots\right)\
.$ (30)

This expansion shows that the sign of $\lambda$ in the second term determines
if the source is attractive ($\lambda>0$) or repulsive ($\lambda<0$). We can
thus write $\Delta_{2}$ as

$\Delta_{2}\equiv\left[1\pm\left(\frac{R_{0}}{r}\right)^{2/\gamma}\right]\ ,$
(31)

which will simplify our discussion of these asymptotically flat solutions. In
this last expression we have just defined $R_{0}$ as

$R_{0}^{2/\gamma}\equiv\frac{2|\lambda|\kappa(1-\alpha)x_{0}r_{0}^{2/\gamma}}{\gamma
c_{2}}\ ,$ (32)

and the positive sign in the bracket represents an attractive source with
$\lambda>0$.

###  III.1 Cases $\alpha=0,1/2$, and $1$

Let us consider first the case $\alpha=1$, which represents a canonical
massless scalar field. Going back to Eqs.(16) and (17), we see that
$A_{x}=\lambda c_{1}$ can be trivially integrated to obtain
$A(x)=A_{0}+\lambda c_{1}x$. It is also easy to find that
$W(x)=1/(r_{0}e^{\frac{\kappa x}{c_{1}}})$, which allows us to define
$r(x)=r_{0}e^{\frac{\kappa x}{c_{1}}}$. Combining these results in Eq.(8) we
get that $Y=e^{-\left(\lambda c_{1}+\frac{2\kappa}{c_{1}}\right)x}/(\lambda
r_{0})^{2}$. In terms of the radial coordinate $r$, the line element can thus
be written as

$ds^{2}=-\left(\frac{r}{r_{0}}\right)^{\lambda
c_{1}^{2}/\kappa}dt^{2}+\left(\frac{\lambda
c_{1}}{\kappa}\right)^{2}\left(\frac{r}{r_{0}}\right)^{\lambda
c_{1}^{2}/\kappa}dr^{2}+r^{2}d\theta^{2}\ ,$ (33)

This line element represents the exact solution for a free massless scalar
field in $(2+1)$ dimensions. In $(3+1)$ dimensions, the solution for a free,
massless scalar field is well known in exact form [32], and one can find an
asymptotically flat solution for some choice of parameters. In that case, the
solution approaches the Schwarzschild geometry far from the peak of the matter
distribution, while the solution in the high-density region is formally
similar to the expression found above. In fact, using the notation of [33]
(see Sec.V.A. in that paper), the internal geometry of the $(3+1)$ solution
can be approximated as

$ds^{2}_{3+1}=-\left(\frac{r}{2M}\right)^{\frac{8M^{2}}{\mu^{2}}}dt^{2}+\left(\frac{r}{2M}\right)^{\frac{8M^{2}}{\mu^{2}}}\left(\frac{4Mr}{\mu^{2}}\right)^{2}dr^{2}+r^{2}d\Omega^{2}\
,$ (34)

where $M$ represents the asymptotic mass of the object and $\mu\ll M$ is a
small mass scale. This comparison allows us to see that both cases represent
naked singularities with a strong curvature and energy density divergence at
$r\to 0$ (for instance, the Ricci scalar goes as $\sim r^{-(2+\lambda
c_{1}^{2}/\kappa)}$). This suggests that the $(2+1)$ solutions may be seen as
a rough description of the innermost regions of the $(3+1)$ configurations (at
least qualitatively).

On the other hand, the case $\alpha=1/2$ is very peculiar because Eq.(18)
implies that either $W_{x}$ or $A_{x}$ must vanish. In both cases such a fact
leads to $Y=0$, which generates inconsistencies in the equations. We will thus
not explore this case in any further detail.

Finally, when $\alpha=0$, the original matter action reduces to a cosmological
constant-type term, resulting in the well-known BTZ black hole solution [12,
13].

##  IV Asymptotically flat solutions

Let us now focus on the line element (27) with $\Delta_{2}$ defined as in
(31), and with $\gamma>0$. For positive $\lambda$, the line element becomes

$ds^{2}=-\frac{1}{\left(1+\left(\frac{R_{0}}{r}\right)^{\frac{2}{\gamma}}\right)^{\gamma}}dt^{2}+\frac{\sigma_{0}^{2}dr^{2}}{\left(1+\left(\frac{R_{0}}{r}\right)^{\frac{2}{\gamma}}\right)^{2+\gamma}}+r^{2}d\theta^{2}\
,$ (35)

and represents a horizonless space-time with a delicate point at $r\to 0$. For
negative $\lambda$, the line element is

$ds^{2}=-\frac{1}{\left(1-\left(\frac{R_{0}}{r}\right)^{\frac{2}{\gamma}}\right)^{\gamma}}dt^{2}+\frac{\sigma_{0}^{2}dr^{2}}{\left(1-\left(\frac{R_{0}}{r}\right)^{\frac{2}{\gamma}}\right)^{2+\gamma}}+r^{2}d\theta^{2}\
,$ (36)

there are no horizons either, but the delicate point arises as $r\to R_{0}>0$.
In both cases, the problems cannot be avoided by a redefinition of the radial
coordinate because they affect also the $g_{tt}$ component. A look at the
Ricci and Kretschmann curvature scalars leads to

$\displaystyle R$ $\displaystyle=$
$\displaystyle\pm\frac{2(\gamma+4)\left(1\pm\left(\frac{R_{0}}{r}\right)^{\frac{2}{\gamma}}\right)^{\gamma+1}\left(\frac{R_{0}}{r}\right)^{\frac{2}{\gamma}}}{\gamma\sigma_{0}^{2}r^{2}}$
(37) $\displaystyle K$ $\displaystyle=$
$\displaystyle\frac{4\left(3\gamma^{2}+8\gamma+8\right)\left(1\pm\left(\frac{R_{0}}{r}\right)^{\frac{2}{\gamma}}\right)^{2(\gamma+1)}\left(\frac{R_{0}}{r}\right)^{\frac{4}{\gamma}}}{\gamma^{2}\sigma_{0}^{4}r^{4}}\
,$

where the $\pm$ sign corresponds to the sign of $\lambda$. When $\lambda>0$,
curvature scalars diverge as $r\to 0$, pointing towards a curvature
singularity caused by the concentration of energy at the center (recall that
$\lambda>0$ represents an attractive field). For negative $\lambda$, instead,
the curvature vanishes as $r\to R_{0}$ and we gain no new information about
what may be happening in that region to generate a divergence in the metric.
To deepen into this aspect and try to unveil what is really going on in that
region, we must study the behavior of geodesics.

###  IV.1 Geodesics

We will now explore whether the solutions found above represent singular or
nonsingular spacetimes from the perspective of their geodesic structure. For
this purpose, we must determine if the affine parameter is defined over the
whole real line (complete geodesics) or it in can only cover a portion of it
(incomplete geodesics). A space-time with any nonzero number of incomplete
geodesics is regarded as singular. This is so because incomplete null
geodesics imply that information (light rays) can be created and/or destroyed,
while incomplete time-like geodesics imply that observers can be created
and/or destroyed, which is physically unacceptable. This is the key notion
behind the theorems proving the existence of space-time singularities within
GR, see e.g. [9] for a discussion of this topic.

The Lagrangian from which the geodesic equations can be obtained can be
written as

$L=\frac{1}{2}\left(\frac{ds}{d\tau}\right)^{2}=\frac{1}{2}\left[-\left(1\pm\left(\frac{R_{0}}{r}\right)^{\frac{2}{\gamma}}\right)^{-\gamma}\dot{t}^{2}+\sigma_{0}^{2}\left(1\pm\left(\frac{R_{0}}{r}\right)^{\frac{2}{\gamma}}\right)^{-2-\gamma}\dot{r}^{2}+r^{2}\dot{\theta}^{2}\right],$
(38)

where $\tau$ is an affine parameter (the proper time for time-like observers),
and the overdot denotes differentiation with respect to it. Taking into
account the symmetries of the Lagrangian, i.e., its static and invariant
nature under rotations, we find the presence of two conserved quantities, as
given by

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\partial
L}{\partial\dot{t}}=\left(1\pm\left(\frac{R_{0}}{r}\right)^{\frac{2}{\gamma}}\right)^{-\gamma}\dot{t},$
(39) $\displaystyle J$ $\displaystyle=$ $\displaystyle\frac{\partial
L}{\partial\dot{\theta}}=r^{2}\dot{\theta},$ (40)

where $E$ and $J$ denote the energy and angular momentum per unit mass of the
particle, respectively. As usual, we can normalize the 4-velocity $U^{\mu}$ to
one, so that

$U^{\mu}U_{\mu}=\left(\frac{ds}{d\tau}\right)^{2}\equiv\epsilon=\pm 1,0,$ (41)

where the parameter $\epsilon$ characterizes the type of geodesics we are
dealing with: time-like ($\epsilon=-1$), space-like ($\epsilon=1$), or null
($\epsilon=0$).

Combining Eqs.(39), (40), and (41), we can conveniently write the geodesic
equation as

$\frac{\sigma_{0}^{2}\left(\frac{dr}{d\tau}\right)^{2}}{\left(1\pm\left(\frac{R_{0}}{r}\right)^{2/\gamma}\right)^{2+2\gamma}}=E^{2}-\frac{1}{\left(1\pm\left(\frac{R_{0}}{r}\right)^{2/\gamma}\right)^{\gamma}}\left(\frac{J^{2}}{r^{2}}-\epsilon\right)\
.$ (42)

Since the left-hand side of this equation must be positive by construction, if
$\epsilon=-1$ or $J\neq 0$ the domain of $r(\tau)$ must be restricted to the
region

$E^{2}\geq\frac{1}{\left(1\pm\left(\frac{R_{0}}{r}\right)^{2/\gamma}\right)^{\gamma}}\left(\frac{J^{2}}{r^{2}}-\epsilon\right)$
(43)

This means that any massive particle ($\epsilon=-1$) or massless particle with
angular momentum ($J\neq 0$) that moves inwards in the radial direction, will
eventually reach a minimum, $r_{m}\geq 0$ if $\lambda>0$ and $r_{m}\geq R_{0}$
if $\lambda<0$, at which the equality above is satisfied. The motion then must
continue towards increasing values of $r$ (the particle moves away after
reaching the closest radial distance), thus guaranteeing the completeness of
all such geodesics.

Considering now radial null geodesics, i.e., those with $\epsilon=0$ and
$J=0$, then Eq.(42) can be written in the simpler form

$\frac{d\hat{r}}{\left(1\pm\left(\frac{1}{\hat{r}}\right)^{2/\gamma}\right)^{1+\gamma}}=\pm
d\hat{\tau}\ ,$ (44)

where we have defined the dimensionless variables $\hat{r}\equiv r/R_{0}$ and
also $\hat{\tau}={E\tau}/{R_{0}\sigma_{0}}$. This equation can be formally
integrated for arbitrary $\gamma$, yielding the result

$\,_{2}F_{1}\left(-\frac{\gamma}{2},\gamma+1;1-\frac{\gamma}{2};\mp\hat{r}^{-2/\gamma}\right)\hat{r}=\pm\hat{\tau}+\beta\
,$ (45)

where $\beta$ is an integration constant and ${}_{2}F_{1}$ a hypergeometric
function. Note that the $\pm$ sign in ${}_{2}F_{1}$ is associated with the
sign of $\lambda$, while on the right-hand side it represents if the geodesic
is outgoing ($+$) or ingoing ($-$). In the far limit, where
$\hat{r}\to\infty$, we can approximate this hypergeometric function by $\
{}_{2}F_{1}\left(-\frac{\gamma}{2},\gamma+1;1-\frac{\gamma}{2};0\right)=1$,
which leads to $\hat{r}\mp\hat{\tau}=\alpha$ and represents the usual straight
lines of light rays in asymptotically flat geometries. In the opposite limit,
we need to split the discussion because for $\lambda>0$, the limit corresponds
to $\hat{r}\to 0$, while for $\lambda<0$, we have $\hat{r}\to 1$.

The expansion around $\hat{r}\to 0$ can be easily derived from (44) by
approximating the left-hand side as $\hat{r}^{2+2/\gamma}d\hat{r}$. By direct
integration we find that radial null geodesics in this region behave as

$\frac{\hat{r}^{3+2/\gamma}}{3+2/\gamma}\approx\pm\hat{\tau}+\tilde{\beta}\ ,$
(46)

which implies that they reach $\hat{r}\to 0$ in finite affine time, confirming
that this space-time is singular, as we had guessed from the curvature
scalars.

Let us now focus on the case with $\lambda<0$. When $\hat{r}\to 1$, one can
show that the dominant term of the solution takes the form

$-\frac{1}{2}\left(\frac{\gamma/2}{\hat{r}-1}\right)^{\gamma}\approx\pm\hat{\tau}+\beta$
(47)

and diverges as $\hat{r}\to 1$. This means that the affine parameter always
diverges as the minimal circumference $r=R_{0}$ is approached, implying that
all these geodesics are complete. Thus, the circumference $r=R_{0}$ represents
a boundary of the manifold and cannot be reached in finite affine time.
Together with the completeness of the other geodesics discussed above, this
results into a nonsingular space-time despite the divergence of the metric
functions in that region. Note that this situation has been found before in
the literature within other gravitational settings, see e.g. [34].

The left-hand side of Eq.(45) admits a representation in terms of elementary
functions for some values of the parameter $\gamma$. Some examples for
$\lambda<0$ are as follows:

  * •

$\gamma=1\Rightarrow\ \ \
\frac{\hat{r}}{2-2\hat{r}^{2}}+\hat{r}+\frac{3}{4}\log\frac{(\hat{r}-1)}{(\hat{r}+1)}$

  * •

$\gamma=2\Rightarrow\ \ \
\hat{r}+\frac{5-6\hat{r}}{2(\hat{r}-1)^{2}}+3\log(\hat{r}-1)$

  * •

$\gamma=3\Rightarrow\ \ \
\frac{1}{16}\left(\frac{-693\hat{r}^{5/3}+144\hat{r}^{7/3}+16\hat{r}^{3}-315\sqrt[3]{\hat{r}}+840\hat{r}}{\left(\hat{r}^{2/3}-1\right)^{3}}\right.$
$\ \ \ \ \ \ \ \ \
\left.-315\left[\tanh^{-1}\left(\sqrt[3]{\hat{r}}\right)+\frac{i\pi}{2}\right]\right)$

The representation of radial null geodesics for these and other values of the
parameter $\gamma$ appears in Fig. 1, where their completeness, as given by
the affine parameter $\tau$ going to $\pm\infty$ at both ends of the
coordinate $\hat{r}$, i.e., $\hat{r}\to\infty$ and $\hat{r}\to 1$, is evident.

Figure 1: Representation of the (normalized) affine parameter
$\frac{E}{R_{0}\sigma_{0}^{1/2}}\tau$ as a function of the radial coordinate
$\hat{r}$ for ingoing (blue) and outgoing (red) geodesics. The dashed lines
depict the trajectories $\pm\hat{r}$ that represent the Minkowskian geodesics
(for illustration). Note that for $0<\gamma<1$ the convergence to the
Minkowskian value is very fast, being fastest in the limit $\gamma\to 0$. The
divergence of the affine parameter as $\hat{r}\to 1$ shows that this region is
a boundary of the manifold that cannot be reached by any observer or light
signal.

##  V Energy density distribution

Let us now focus on how the energy density is distributed in the solutions
studied above. From Eq.(8) and a little algebra using the line element (36),
we see that the kinetic term $Y$ can be written as

$Y=Y_{0}\frac{\left(1\pm\hat{r}^{-2/\gamma}\right)^{2+\gamma}}{\hat{r}^{\frac{2(2+\gamma)}{\gamma}}}\
,$ (48)

where $Y_{0}\equiv\left(\lambda\alpha
R_{0}\right)^{-\frac{2(2+\gamma)}{\gamma}}$ is an irrelevant constant factor.

For the singular solutions corresponding to $\lambda>0$, it is easy to see
that this kinetic energy density diverges when $\hat{r}\to 0$ as
$Y/Y_{0}\approx 1/r^{4(2+\gamma)/\gamma}$, which provides further evidence
about its pathological nature.

On the contrary, for $\lambda<0$ the energy density goes to zero both at
infinity and at the minimal circumference $\hat{r}=1$, both of which represent
boundaries of the manifold. One thus expects the existence of a maximum
located somewhere in between these two asymptotic regions. An elementary
calculation indicates that $Y_{\hat{r}}$ vanishes at $\hat{r}=1$, at infinity,
and at $\hat{r}_{M}=2^{\gamma/2}$, where the kinetic term takes the maximum
value $Y=Y_{0}2^{-2(2+\gamma)}$. Therefore, our nonsingular solutions
represent localized energy distributions with a maximum around the
circumference of radius $r=R_{0}2^{\gamma/2}$. A representation of the
amplitude of the kinetic term $Y$ on the plane is provided in Figs. 2 and 3.
One can verify by direct calculation that the total energy of the system is
finite (integrating the scalar action from $\hat{r}=1$ to infinity) for all
$\gamma>0$.

Figure 2:  Representation of the kinetic term
$Y=g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ for the cases
$\gamma=1/2,1,$ and $2$ (recall that $\alpha=(1+\gamma)/(2+\gamma)$) when
$\lambda<0$. Vertical dashed lines indicate the location of the maximum. The
localized nature of these solutions is evident. Note that the smaller the
value of $\gamma$, the higher the peak and the more compact the structure.

Figure 3: Left: three-dimensional representation of the kinetic term
$Y=g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ when $\lambda<0$ for the
case $\gamma=1$. Right: same representation for $\gamma=1/2$ (blue) and
$\gamma=2$ (orange). Note how the more compact solution $\gamma=1/2$ is always
hidden by the $\gamma=2$ one except at the innermost region. The different
amplitudes of the maxima are also evident in this plot.

##  VI Summary and conclusion

In this work we have studied $(2+1)$-dimensional Einstein gravity coupled to a
static, nonlinear scalar field with a purely kinetic term and circular
symmetry. The search for analytical solutions led us to consider a family of
power-law models with a Lagrangian density of the form given in Eq.(15). After
classifying the various branches of solutions, we focused on the case
$\gamma>0$ (equivalently $1/2<\alpha<1$) and showed that the resulting
geometries are determined by the line element (36), which represents
asymptotically flat spaces. We showed that when the parameter $\lambda$ that
sets the amplitude of the scalar Lagrangian is positive, we have an attractive
source, whereas for negative $\lambda$ we have a repulsive source. All
solutions with $\lambda>0$ represent naked singularities (divergent curvatures
and energy density, and incomplete geodesics), whereas for $\lambda<0$ all
solutions are regular and nonsingular.

Though in the $\lambda<0$ case the $g_{tt}$ and $g_{rr}$ components of the
metric diverge at $r=R_{0}>0$, we found that curvature invariants vanish at
that location. Furthermore, we showed that the circumference $r=R_{0}$
represents a boundary of the manifold, as all radial null geodesics take an
infinite affine time to reach there. Time-like geodesics and null rays with
nonzero angular momentum never reach this boundary and have an $r>R_{0}$ as
their minimal radial coordinate. The analysis of the kinetic term of the
scalar field shows that these geometries are generated by localized volcano-
like lumps of energy with maximum amplitude at $r=2^{\gamma/2}R_{0}$,
remaining positive everywhere and vanishing only at $r=R_{0}$ and at infinity
(see Figs. 2 and 3).

In our view, the most relevant result of this paper is the discovery of exact
analytical solutions that represent nonsingular static compact scalar objects
in an asymptotically flat geometry. These localized structures are possible
thanks to the exotic (non-canonical) dynamics of the scalar field, and the
fact that they generate an inner boundary of radius $R_{0}$ is a surprise that
could have not been anticipated a priori. In practical terms, this boundary
and its neighborhood act like a region of repulsive forces (because geodesics
bounce) that prevent the collapse of the energy distribution and regularise
its maximum amplitude. Even though this kind of exotic matter source has
repulsive gravitational properties, it is worth exploring its stability and
potential interactions with other sources to better understand alternative
singularity avoidance mechanisms.

If analogous structures could be found in $(3+1)$-dimensional extensions of
this model, there could be important implications for the astrophysics of
compact objects and dark matter/energy models. In particular, boson stars are
regarded as spherical distributions of scalar matter with peak density at the
center. Our analysis puts forward that nonsingular compact objects without a
center do exist within GR. This means that contrary to the standard approach,
one should look for new solutions of self-gravitating scalar fields with
boundary conditions which are not defined at a center, since the later may not
exist. These and other related questions are currently under study.

## Acknowledgments

The authors thank the Fundação Cearense de Apoio ao Desenvolvimento Científico
e Tecnológico (FUNCAP), the Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior (CAPES), and the Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), Grant no. 200879/2022-7 (RVM). This work is also supported
by the Spanish Agencia Estatal de Investigación (grants PID2020-116567GB-C21
and PID2022-138607NB-I00, funded by MCIN/AEI/10.13039/501100011033, FEDER, UE,
and ERDF A way of making Europe) and by the project PROMETEO/2020/079
(Generalitat Valenciana). R. V. Maluf thanks the Department of Theoretical
Physics & IFIC of the University of Valencia - CSIC, and the Department of
Theoretical Physics and IPARCOS of the Complutense University of Madrid, for
the kind hospitality during the elaboration of this work. This article is
based upon work from COST Action CA21136, supported by COST (European
Cooperation in Science and Technology).

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