# Global Existence and Singularity Formation of Classical Solutions in
Einstein-Skyrme System

Emir Syahreza Fadhilla♯,‡, Ardian Nata Atmaja‡, and Bobby Eka
Gunara♯,‡111Corresponding author

♯ Theoretical High Energy Physics Research Division,
Faculty of Mathematics and Natural Sciences,
Institut Teknologi Bandung
Jl. Ganesha no. 10 Bandung, Indonesia, 40132
and
‡Research Center for Quantum Physics, National Research and Innovation Agency
(BRIN),
Kompleks PUSPIPTEK Serpong, Tangerang 15310, Indonesia

email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(August 27, 2024)

###### Abstract

In this paper, we elucidate the problem of gravitating Skyrmion governed by
field equations of the Einstein-Skyrme system with no potential term in the
Bondi coordinate. The spherical symmetry has to be assumed and both the metric
functions and Skyrme ansatz depend on radial and retarded time coordinates
which implies that the system is dynamic. We show that unique smooth solutions
with arbitrary initial data exist for a certain time interval with the extra
condition that the time interval must be finite in order to have a non-zero
topological charge. Then, the strategy to show that global smooth solutions
exist is to restrict the initial data and proceed to find suitable
configurations for extending the time interval to infinity. We also discuss
the possible configurations within The Einstein-Skyrme System which develop
singularities in coordinate origin.

##  1 Introduction

The Skyrme model was initially proposed as a model for strongly interacting
particles in flat spacetime and is proven to be a good model in low energy
region. It is constructed from the usual sigma model with a $S^{3}$ target
space and an extra quartic term, instead of just the usual quadratic term, in
the Lagrangian density of the model. The quartic term is formulated in such a
way that the resulting term is still proportional to the square of time
derivative of the field. Furthermore, this model possess a conserved current
known as topological current that is proportional to the volume form on the
target space. The corresponding charge is considered as the baryon number and
the model is considered as the unified field theory of mesons and baryons [1,
2]. The classical solution of the dynamical equation in Skyrme model are known
as Skyrmion, which belong to the topological solitons due to the topological
properties that have been mentioned above and it propagates in the form of a
solitary wave that does not obey superposition principles due to its non-
linear nature [3]. The Einstein-Skyrme system is the system of dynamical field
equations arising from the least-action principle of the following action

$\mathcal{S}=\int\sqrt{-\gamma}\frac{R}{16\pi}\leavevmode\nobreak\
d^{4}x+\mathcal{S}_{\text{Skyrme}},$ (1.1)

where $\gamma$ is the determinant of metric tensor, $R$ is the Ricci scalar of
corresponding spacetime manifold and $\mathcal{S}_{\text{Skyrme}}$ is the
action of the Skyrme model in $3+1$ dimensional spacetime [4]. The expression
of the action of the Skyrme model is given by

$\displaystyle\mathcal{S}_{\text{Skyrme}}=-\int\sqrt{-\gamma}\left[C_{1}\phi^{a}_{\kappa}\phi^{a}_{\tau}\gamma^{\kappa\tau}+C_{2}\phi^{a}_{[\kappa}\phi^{b}_{\alpha]}\phi^{a}_{[\tau}\phi^{b}_{\beta]}\gamma^{\kappa\tau}\gamma^{\alpha\beta}\right]d^{4}x$
(1.2)

where $\phi^{a}$’s are the scalar multiplets and
$C_{1},C_{2}\in\mathbb{R}^{+}$ are real coupling constants. As we know, the
Euler-Lagrange equations corresponding to the variation in metric tensor
components are equivalent to Einstein’s field equations with the Skyrme field
acting as its matter field. This system models the interaction between
Skyrmion and gravity that are minimally coupled.

The study of Skyrmions under the effect of gravity is first introduced in [5,
6] where it shows that such a model admits a large family of spherically
symmetric asymptotically flat solutions. The stable spherically symmetric
solutions exist for both with and without horizons provided that the coupling
constant is greater than a certain critical value [7]. This led to an
extensive study on the Skyrmion hairy black hole as a candidate for counter-
example of the no-hair conjecture, for example in [8] where the Skyrme field
is located on a Schwarzschild background due to the assumption that the
gravitational coupling is small [9]. The result proposes that it is possible
for a black hole to have a Skyrmion hair, but this model was, actually,
incomplete because the back-reaction from the Skyrme field to the gravity is
ignored. Later, it is found that stable black holes with back-reacting
massless Skyrmion hair exist for asymptotically AdS four-dimensional
spacetime, reported in [10]. In this setup, the Skyrmion possess a fractional
topological charge, in contrast with the usual Skyrmion in flat spacetime
which has an integer-valued topological charge. This property is quite
peculiar because it implies that the gravitational collapse of a Skyrme field
to a black hole does not conserve the topological charge. This property is
also found in a more general system where higher order terms are included [11,
12] and also in higher dimensional cases such as five-dimensional
asymptotically flat Skyrme black hole [13] and AdS black holes in higher
dimensional Einstein-Skyrme system [4].

It is also possible to construct a star-like solution for a gravitating
Skyrmion. The star solution does not possess a horizon and typically has an
integer-valued topological charge. Some of the known solutions are gravitating
BPS Skyrmions as a neutron star [14, 15, 16], and a more general five-
dimensional Skyrmions and Skyrme stars [13, 17] which includes up to the octic
terms of the Skyrme Lagrangian. All of the solutions from gravitating Skyrmion
systems mentioned above have a similarity, namely, they assume time-
independency by taking the ansatz to be either static or stationary. From the
known examples, we conclude that the time dependency in those models is
introduced perturbatively around the time-independent solutions to study the
stability under small, time-dependent, disturbances. Thus, it is interesting
to generalize the study of the Einstein-Skyrme system by assuming time-
dependency on the solutions. Such solutions could become the model of
Skyrmions dynamics under gravitational effect and show us how the static and
stationary solutions can form in their steady-state limit.

It is known that the time-dependent solution of simpler gravitating scalar
field systems do exist. The study is first proposed in [18, 19] where the
authors shows that unique spherically symmetric global solutions of Einstein-
Klein-Gordon system do exist. Furthermore, they showed that such system could
develop naked singularity in finite time [20] although the singularity turns
out to be unstable, hence it cannot be the counter-example of the cosmic
censorship conjecture [21]. Regardless of the results found in the
singularities in Einstein-Klein-Gordon system, the method demonstrated in
those works shows promising future and can be developed further for a wider
class of gravitating scalar fields. An example of this can be found in [22]
where the author shows that spherically symmetric global solutions of the
Einstein-Klein-Gordon system with a special type of polynomial potential exist
for small initial data. It is also possible to adopt the method for
gravitating scalar multi-fields where the fields possess internal symmetries.

In this work, we propose that a similar method to show the global existence of
the Einstein-Klein-Gordon system can be adopted for a more general Einstein-
Skyrme system. The idea of the proof is as follows: The Skyrme model is
characterized by a charge, known as the topological charge, and it coincides
with the Klein-Gordon model when the topological charge, $B$, is set to be
zero. While the $B=0$ limit of the model (the Klein-Gordon model) is a linear
model, the Skyrme model in general, with $B\geq 0$, is a non-linear scalar
theory which is characterized by the scalar fields equipped with internal
symmetries, that is, the symmetries of the target space (see [3] for further
details on non-linear scalar theory). This raises a challenge since we cannot
solve the components of the Skyrme field multiplet independently due to the
constraint that it must preserve the internal symmetry. A common solution to
circumvent this complexity is the introduction of a coordinate system on the
target space. As such, the coordinates on the target space act as the new
dynamical fields which we can guarantee to obey the symmetries of its internal
structure. Due to the non-linear nature of the theory, we choose to work with
a certain ansatz in order to simplify the problem which serves as the
coordinate system on the target space mentioned before. These ansatzes also
obey the assumed symmetry of the spacetime and in this work, we choose to work
with the dynamical hedgehog ansatz that is suitable for spherically symmetric
models with time dependence. This way, we can reduce the complexity of the
problem of non-linear scalar theory into a system of non-linear wave
equations.

Furthermore, the dynamical equation of the Skyrme model itself is highly non-
linear in the sense that we cannot superpose two independent solutions222This
property is known as the common property of soliton. A solitary wave solution
of non-linear dynamical equations.. This non-linearity does not just come from
the implementation of ansatz, but also from the higher power derivative terms
and potential terms in the action of the Skyrme model which is explicitly
given in (1.2). Within the spherically symmetric hedgehog ansatz, the
dynamical equations for the system are the set of Einstein’s equations
supplied by the conservation of energy-momentum tensor that is equivalent to
the Euler-Lagrange equations of action (1.2) for non-trivial solutions. In
order to use Christodoulou’s method, proposed in [19], to the Einstein-Skyrme
system we need to find well-behaving estimates that are always bigger than the
terms in the dynamical equations but decay fast enough at spatial infinity. We
found that polynomials are able to estimate such non-linear terms in the
Einstein-Skyrme system. This approach transforms our problem from highly non-
linear wave equations into a slightly simpler problem, that is, non-linear
wave equations with only polynomial terms. A similar problem of wave equations
with polynomial terms has been addressed by Chae in [22] with Christodoulou’s
method, as mentioned previously. Thus, the main procedure that is frequently
employed in this work is finding appropriate polynomial estimates for the
terms in the wave equations of the Einstein-Skyrme system.

The idea for the proof of local existence is to find a sufficiently small
finite time interval where we can guarantee that the smooth solutions are
locally Lipschitz. This approach is similar to Christodoulou’s approach to the
Einstein-Klein-Gordon system, except that we employ the estimates and
Lipschitz condition to the ansatz instead of the field, $\phi$. Thus, the
resulting estimates highly depend on the corresponding coordinate system and
its assumed symmetries. The idea for the existence of globally regular
solutions follows similar reasoning, that is, in order to find globally
regular solutions (with semi-infinite time intervals) we constrain the norm of
the initial data to be bounded above. The bound for the intervals of local
solutions and the norm of the initial data for global solutions are deduced
from Lipschitz’s condition that must be satisfied to ensure uniqueness. On the
other hand, the topology of the Skyrmions forbids the maximum values of
initial data to be less than multiples of $\pi$ [2, 3]. Thus, we expect that
the initial data is both bounded from above and below.

It is also known in the soliton theory that such a non-linear system could
develop singularities in finite time under certain conditions [23]. These
singularities in the Einstein-Skyrme system are not directly evident in the
case of travelling wave solutions considered within Christodoulou’s method. To
observe the possible singular solutions we consider the self-similar solutions
of the Einstein-Skyrme system. In order to do this, firstly, we study the
existence and uniqueness of smooth self-similar solutions in a semi-infinite
interval of self-similar coordinates. From this step, we found that unique
self-similar solutions do exist with a remark that in order to include the
origin vertex of the coordinate system ($r=0$) the ansatz of the Skyrme field
must satisfy the boundary condition of topological Skyrmion. The implication
of this boundary condition is that some of the metric tensor components have
to change their sign at a finite distance from the origin, implying the
existence of a horizon. From here, we proceed to show that the self-similar
solutions we found above are, indeed, singular at the origin after a finite
interval of time. This result implies that the singularity is possibly hidden
inside a horizon, in contrast with the result found for the Einstein-Klein-
Gordon system where the singularity is naked [20].

The article is organized as follows: Section 2 is devoted to introduce the
details of the Einstein-Skyrme system as a model of gravitating Skyrmions. In
this section, we describe the characteristics of the Skyrme model and how we
can formulate the system of equations under the assumption of spherical
symmetry. In Section 3 we show the main results for both the local and global
existence of spherically symmetric time-dependent solutions in the Einstein-
Skyrme system. Lastly, In section 4 we discuss some possible configurations
which develop singularities within the Einstein-Skyrme system and we give some
outlooks for further study in the last section, namely, Section 5.

##  2 The Einstein-Skyrme System in Bondi Coordinate

We specifically consider a spherically symmetric spacetime for the model. The
metric of the spacetime is written in Bondi coordinate that is given by [24]

$ds^{2}=\gamma_{\mu\nu}dx^{\mu}dx^{\nu}=-e^{2\rho}du^{2}-2e^{\rho+\sigma}dudr+r^{2}d\Omega^{2},$
(2.1)

where $\rho\equiv\rho(u,r)$ and $\sigma\equiv\sigma(u,r)$, both are functions
of $u$ and $r$ and $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$ is
the metric of 2-sphere, $S^{2}$. Einstein’s summation convention is to be
understood for every repeated indices throughout this paper unless stated
otherwise. The coordinate $r$ is the radial coordinate which is defined
through the area of spatial rotation $SO(3)$ group orbit, $\mathcal{A}$, that
satisfies $\mathcal{A}=4\pi r^{2}$, implying that the central world-line is
the set of points $r=0$. The coordinate $u$ is defined by requiring it to be
constant on the future light cone of each point on the central world-line and
$u$ tends to the proper time on the world line of constant $r$ as $r$ goes to
infinity [19, 18]. In this way, we can consider the coordinate $u$ as the
retarded time. The null tetrad in this coordinate consists of two orthonormal
bases on $S^{2}$ and two vectors

$n=e^{-\rho}\frac{\partial}{\partial
u}-\frac{1}{2}e^{-\sigma}\frac{\partial}{\partial r},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
l=e^{-\sigma}\frac{\partial}{\partial r},$ (2.2)

both $n$ and $r$ are orthogonal to the bases on $S^{2}$ [19, 20].

The Einstein equation of the Einstein-Skyrme system that models the Skyrme
field minimally coupled to gravity is given by

$R_{\mu\nu}=8\pi\left[T_{\mu\nu}-\frac{1}{2}\gamma_{\mu\nu}T\right]=8\pi\mathcal{T}_{\mu\nu},$
(2.3)

with $R_{\mu\nu}$ is the Ricci tensor corresponding to the metric tensor
$\gamma_{\mu\nu}$, $R=R_{\mu\nu}\gamma^{\mu\nu}$ is the Ricci scalar, and
$T_{\mu\nu}$ is the energy-momentum tensor of matter fields, which, in this
case, is in the form of Skyrme field.

In order to find an explicit expression of the energy-momentum tensor of
Skyrme fields, we consider the action of the Skyrme model, that is expressed
in equation (1.2). This model contains a real scalar multiplet,
$\phi^{a}=(\phi^{0},\phi^{1},\phi^{2},\phi^{3})$ which satisfies the $O(4)$
model constraint, $\phi^{a}\phi^{a}=1$. This constraint implies that the
Skyrme field maps the spacetime manifold to a three-dimensional sphere
$S^{3}$. Here, we have changed the representation of the Skyrme field to
$O(4)$ from the usual $SU(2)$ valued field to simplify the calculation,
without any loss of generality, by following the construction of $O(N)$
Skyrme-Sigma Models (see [25] for details). Since the target space is $S^{3}$,
which is the same target space manifold of an $SU(2)$ valued field, then the
model obeys the same symmetry. In fact, the relationship between $\phi$ and
the usual $SU(2)$ valued field is straight forward, namely

$U=\phi^{0}\textbf{1}_{2\times
2}+i(\phi^{1}\sigma^{1}+\phi^{2}\sigma^{2}+\phi^{3}\sigma^{3}),$ (2.4)

where $\textbf{1}_{2\times 2}$ is the $2\times 2$ identity matrix and
$(\sigma^{1},\sigma^{2},\sigma^{3})$ are the Pauli matrices. We can see that
the $O(4)$ model constraints, $\phi^{a}\phi^{a}=1$, is equivalent to
$U^{\dagger}U=\textbf{1}_{2\times 2}$.

With the action of the Skyrme model in hand, we can derive the energy-momentum
tensor corresponding to Skyrme field by employing the following equation

$T_{\mu\nu}=-\frac{2}{\sqrt{-\gamma}}\frac{\delta\mathcal{S}_{\text{Skyrme}}}{\delta\gamma^{\mu\nu}},$
(2.5)

where $\delta$ is the variational operator. This formula will ensure that
Einstein’s equations are equivalent to
$\frac{1}{\sqrt{-\gamma}}\frac{\delta\mathcal{S}}{\delta\gamma^{\mu\nu}}=0$.
As such, the expression of energy-momentum tensor for the Skyrme field is as
follows

$\displaystyle
T_{\mu\nu}=2\left[C_{1}\phi^{a}_{\mu}\phi^{a}_{\nu}+2C_{2}\phi^{a}_{[\mu}\phi^{b}_{\alpha]}\phi^{a}_{[\nu}\phi^{b}_{\beta]}\gamma^{\alpha\beta}\right]-\gamma_{\mu\nu}\left[C_{1}\phi^{a}_{\kappa}\phi^{a}_{\tau}\gamma^{\kappa\tau}+C_{2}\phi^{a}_{[\kappa}\phi^{b}_{\alpha]}\phi^{a}_{[\tau}\phi^{b}_{\beta]}\gamma^{\kappa\tau}\gamma^{\alpha\beta}\right].$
(2.6)

From equation (2.3) we can see that $T_{\mu\nu}$ must have the same dimensions
as $R_{\mu\nu}$. Because scalar multiplets $\phi^{a}$ are dimensionless
quantities (see, [1, 26, 13]), then $C_{1}$ must be a dimensionless constant
and $C_{2}$ must have a dimension of length squared. This analyses of
dimension for coupling constants is useful for constructions of dimensionless
equation of motion we show in later sections. The Einstein-Skyrme system is
supplemented by matter field equation, given by

$\nabla_{\mu}T^{\mu\nu}=0,$ (2.7)

such that we have all dynamical equations for both the metric tensor and
Skyrme (matter) field.

The original Skyrme model, firstly proposed by T. Skyrme in [1, 2], does not
accommodate potential. Such a model has minimal terms to be considered as
stable in the sense that Derrick stability conditions are satisfied and the
lack of potential term implies that the corresponding Skyrmion is massless
[3]. It has been proven that this model satisfies BPS
(Bogomol’nyi–Prasad–Sommerfield) inequality $E\geq B$ where $B$ is a constant
topological charge and it becomes equality if the map $\phi$ is given by
$S^{3}\rightarrow S^{3}$, implying a spherical base manifold [26]. Thus, the
original Skyrme model is topologically supported333In the sense that the
energy cannot go to zero for higher topological degree, $B\geq 1$, hence,
skyrmions cannot disperse into vacuum spontaneously even though it comes from
different mechanism from the BPS Skyrme model and the generalized Skyrme
model.

For the geometric features of the spacetime of our model, we start by showing
the corresponding Ricci tensor. The components of the Ricci tensor of this
Bondi coordinate are given by

$\displaystyle R_{uu}$ $\displaystyle=$
$\displaystyle\frac{e^{\rho-\sigma}}{r}\left(e^{\rho-\sigma}\left(\rho_{,r}\left(r\rho_{,r}-r\sigma_{,r}+2\right)+r\rho_{,rr}\right)-\left(r\left(\rho_{,ur}+\sigma_{,ur}\right)+2\sigma_{,u}\right)\right),$
(2.8) $\displaystyle R_{ur}$ $\displaystyle=$ $\displaystyle
R_{ru}=\frac{e^{\rho-\sigma}}{r}\left(\rho_{,r}\left(r\rho_{,r}-r\sigma_{,r}+2\right)+r\rho_{,rr}\right)-\rho_{,ur}-\sigma_{,ur},$
(2.9) $\displaystyle R_{rr}$ $\displaystyle=$
$\displaystyle\frac{2}{r}\left(\rho_{,r}+\sigma_{,r}\right),$ (2.10)
$\displaystyle R_{\theta\theta}$ $\displaystyle=$ $\displaystyle
1-e^{-2\sigma}\left(r\rho_{,r}-r\sigma_{,r}+1\right),$ (2.11) $\displaystyle
R_{\varphi\varphi}$ $\displaystyle=$
$\displaystyle\sin^{2}\theta\left(1-e^{-2\sigma}\left(r\rho_{,r}-r\sigma_{,r}+1\right)\right).$
(2.12)

In this work, we use spherically symmetric ansatz possessing unit topological
charge, known as the hedgehog ansatz, given by

$\displaystyle\phi^{0}=\cos\xi,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \phi^{1}=\sin\xi\cos(A_{3}\theta),$
$\displaystyle\phi^{2}=\sin\xi\sin(A_{3}\theta)\cos(A_{4}\varphi),\leavevmode\nobreak\
\leavevmode\nobreak\ \phi^{3}=\sin\xi\sin(A_{3}\theta)\sin(A_{4}\varphi),$
(2.13)

with $\xi\equiv\xi(u,r)$ is a real valued function, $A_{3},A_{4}\in\mathbb{Z}$
to ensure the uniqueness of $\phi$ in every point in spacetime, and both
$\theta$ and $\varphi$ take values in the usual range for angular coordinates
on $S^{2}$, namely $\theta\in[0,\pi)$ and $\varphi\in[0,2\pi)$. The boundary
conditions for $\xi$ also come from the requirement that $\phi$ must be
unique, then at $r=0$, $\sin\xi$ must be zero which leads to $\xi(u,0)=n\pi$
where $n\in\mathbb{Z}$. We restrict the solutions, from this point on to the
decaying solutions of $\xi$ with respect to $r$ such that we always assume
$\lim_{r\rightarrow\infty}\xi(u,r)=0$, and then we show the proof that such
smooth decaying solutions exist in the following sections. Now, we proceed by
substituting this ansatz into
$\mathcal{T}_{\mu\nu}=T_{\mu\nu}-\frac{1}{2}\gamma_{\mu\nu}T$ to have the
following results

$\displaystyle\mathcal{T}_{rr}$ $\displaystyle=$
$\displaystyle\left[C_{1}\left(2\xi_{,r}^{2}\right)+C_{2}\left(2\left(A_{3}^{2}+A_{4}^{2}\frac{\sin^{2}(A_{3}\theta)}{\sin^{2}\theta}\right)\frac{\sin^{2}\xi}{r^{2}}\xi_{,r}^{2}\right)\right]\leavevmode\nobreak\
,$ (2.14) $\displaystyle\mathcal{T}_{\theta\theta}$ $\displaystyle=$
$\displaystyle
C_{2}A_{3}^{2}A_{4}^{2}\frac{\sin^{2}(A_{3}\theta)}{\sin^{2}\theta}\frac{\sin^{4}\xi}{r^{2}}+C_{1}\left(2A_{3}^{2}\sin^{2}\xi\right)$
$\displaystyle-
C_{2}\left(A_{3}^{2}-A_{4}^{2}\frac{\sin^{2}(A_{3}\theta)}{\sin^{2}\theta}\right)\xi_{,r}\sin^{2}\xi
e^{-\rho-2\sigma}\left(2\xi_{,u}e^{\sigma}-\xi_{,r}e^{\rho}\right),$
$\displaystyle\mathcal{T}_{\varphi\varphi}$ $\displaystyle=$
$\displaystyle\left[C_{2}A_{3}^{2}A_{4}^{2}\frac{\sin^{2}(A_{3}\theta)}{\sin^{2}\theta}\frac{\sin^{4}\xi}{r^{2}}+C_{1}\left(2A_{4}^{2}\frac{\sin^{2}(A_{3}\theta)}{\sin^{2}\theta}\sin^{2}\xi\right)\right.$
$\displaystyle\left.-C_{2}\left(A_{3}^{2}-A_{4}^{2}\frac{\sin^{2}(A_{3}\theta)}{\sin^{2}\theta}\right)\xi_{,r}\sin^{2}\xi
e^{-\rho-2\sigma}\left(2\xi_{,u}e^{\sigma}-\xi_{,r}e^{\rho}\right)\right]\sin^{2}\theta.$

The angular components are singular at $\theta\rightarrow 0$ and
$\theta\rightarrow\pi$, thus in order to have a well-behaved energy-momentum
tensor and a spherically symmetric $rr$ component, we need to set $|A_{3}|=1$.
This leads to a specific value for $A_{4}$, since in order to make
$\theta\theta$ and $\varphi\varphi$ components of Einstein’s equation
consistent requires $A_{3}^{2}-A_{4}^{2}=0$, then $|A_{4}|=1$. This way,
spherical symmetry is restored because
$\mathcal{T}_{\varphi}^{\varphi}=\mathcal{T}_{\theta}^{\theta}$.

By exploiting the same formulation for $O(d+1)$ models in [25, 13, 17], we
have the following expression of topological charge, given by

$\displaystyle B$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi^{2}}\int\sqrt{-g}A_{3}A_{4}\frac{\sin(A_{3}\theta)}{r^{2}\sin\theta}e^{-(\rho+\sigma)}\sin^{2}\xi\xi^{\prime}\leavevmode\nobreak\
d\varphi d\theta dr$ (2.15) $\displaystyle=$
$\displaystyle\frac{A_{3}A_{4}}{2\pi^{2}}\int_{0}^{n\pi}\int_{0}^{\pi}\int_{0}^{2\pi}\sin^{2}\xi\sin(A_{3}\theta)\leavevmode\nobreak\
d\varphi d\theta d\xi.$

Evaluating the integral expression above and using the argument,
$|A_{3}|=|A_{4}|=1$, in the preceding paragraph gives

$B=\pm n.$ (2.16)

The spherically symmetric minimal energy Skyrmion has $n=1$. From this point
on, we are going to consider only positive $n$ and positive signed $B$ in
order to simplify the analysis. The negative signed $B$ are called the anti-
Skyrmions and they behave similarly to the Skyrmions.

To proceed, we choose to take the ${rr}$ and ${\theta\theta}$ components of
(2.3), namely

$\displaystyle\frac{1}{r}\left(\rho_{,r}+\sigma_{,r}\right)$ $\displaystyle=$
$\displaystyle
8\pi\xi_{,r}^{2}\left[C_{1}+C_{2}\left(2\frac{\sin^{2}\xi}{r^{2}}\right)\right]\leavevmode\nobreak\
,$ (2.17)
$\displaystyle\frac{e^{2\sigma}-1}{r^{2}}-\frac{1}{r}\left(\rho_{,r}-\sigma_{,r}\right)$
$\displaystyle=$ $\displaystyle 8\pi
e^{2\sigma}\left[C_{1}\left(2\frac{\sin^{2}\xi}{r^{2}}\right)+C_{2}\left(\frac{\sin^{4}\xi}{r^{4}}\right)\right],$
(2.18)

and the rest of the equations from (2.3) should be consistent with (2.17) and
(2.18). The ${u}$ and ${r}$ components of matter field equation (2.7) is given
by

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left[\frac{2}{r^{2}}\left(re^{-2\sigma}\left(\xi_{,r}\left(-r\rho_{,r}+r\sigma_{,r}-2\right)+2\left(\xi_{,u}+r\xi_{,ur}\right)e^{\sigma-\rho}-r\xi_{,rr}\right)+\sin(2\xi)\right)\right]$
(2.19)
$\displaystyle+\left[\frac{4\sin\xi}{r^{4}}\left(2r^{2}e^{-\rho-\sigma}\left(\xi_{,ur}\sin\xi+\xi_{,r}\xi_{,u}\cos\xi\right)+\sin^{2}\xi\cos\xi\right.\right.$
$\displaystyle\left.\frac{\left.\right.\left.\right.}{\left.\right.}r^{2}\left.e^{-2\sigma}\left(\sin\xi\left(\xi_{,r}\left(\rho_{,r}-\sigma_{,r}\right)+\xi_{,rr}\right)+\xi_{,r}^{2}\cos\xi\right)\right)\right],$

and the rest of the components of (2.7) are trivial. In equation (2.19) we
have rescaled the coordinates as
$(u,r)\rightarrow\left(\sqrt{\frac{C_{1}}{C_{2}}}u,\sqrt{\frac{C_{1}}{C_{2}}}r\right)$
such that the resulting equation becomes dimensionless. From the set of field
equations above, we can see that the topological charge does not explicitly
appear in the differential equations. Thus, the estimates for upper bound and
lower bounds in our analysis will not depend on the topological charge.
Instead, the dependency on the topological charge appears in the analysis of
suitable initial data where boundary conditions play some important roles,
especially for the lower bounds of the norm of initial data and the size of
the interval of local existence that is going to be demonstrated in details in
the next section.

##  3 Existence of Smooth Solutions

In this section, we show that some classes of smooth solutions exist in this
system by using contraction mapping theorem. To do so, we recast the system of
equations (3.6), (3.7), and (2.19) into a single integro-differential equation
as demonstrated below.

Firstly, it is useful to introduce averaged field notation for any function,
$f$, of $u,r$ on $r$, given by

$\bar{f}=\frac{1}{r}\int_{0}^{r}f(u,s)ds.$ (3.1)

Following the prescription on [19, 22], we introduce a new field
$h=h(u,r)\equiv\frac{\partial}{\partial r}\left(r\xi\right)$ which allows us
to transform $\xi$ and its derivative as

$\xi=\bar{h},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\xi_{,r}=\frac{h-\bar{h}}{r}.$ (3.2)

By defining another functions of $u,r$ for metric fields, namely

$\displaystyle g$ $\displaystyle\equiv$ $\displaystyle e^{\rho+\sigma},$ (3.3)
$\displaystyle\tilde{g}$ $\displaystyle\equiv$ $\displaystyle
e^{\rho-\sigma},$ (3.4)

and introducing a differential operator parallel to $n$

$D=\frac{\partial}{\partial u}-\frac{\tilde{g}}{2}\frac{\partial}{\partial r}$
(3.5)

we rewrite (2.17) and (2.18) as follows

$\displaystyle g$ $\displaystyle=$ $\displaystyle\exp\left[-8\pi
C_{1}\int^{\infty}_{r}\frac{(h-\bar{h})^{2}}{s}\left(1+2\frac{\sin^{2}\bar{h}}{s^{2}}\right)\leavevmode\nobreak\
ds\right],$ (3.6) $\displaystyle\tilde{g}$ $\displaystyle=$
$\displaystyle\bar{g}-\frac{8\pi
C_{1}}{r}\int_{0}^{r}s^{2}g\left(2\frac{\sin^{2}\bar{h}}{s^{2}}+\frac{\sin^{4}\bar{h}}{s^{4}}\right)\leavevmode\nobreak\
ds.$ (3.7)

which are analogous to the relations we have for Einstein-Klein-Gordon system
[22]. Now, with these two relations for $g$ and $\tilde{g}$ in hand, we can
proceed by reducing the system into a single integro-differential equation for
$h$ on a curve $\chi(u^{\prime},r^{\prime})$ as a solution of

$\frac{dr}{du}=-\frac{1}{2}\tilde{g}$ (3.8)

possessing tangential vector field which is parallel to $D$. Thus, the
characteristic curve is located on the null region of the spacetime.

Notice that after these rescaling, $u,r$ become dimensionless parameters. Let
us define a new coupling constant between matter field and gravity, given by

$\alpha=8\pi C_{1}.$ (3.9)

Thus, by substituting (3.6) and (3.7) to (2.19) we arrive at the following
integro-differential equation

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
Dh-\frac{(h-\bar{h})(g-\tilde{g})}{2r}+\frac{\alpha}{2}(h-\bar{h})gr\left(2\frac{\sin^{2}\bar{h}}{r^{2}}+\frac{\sin^{4}\bar{h}}{r^{4}}\right)+\frac{g}{r}\sin\bar{h}\cos\bar{h}$
(3.10)
$\displaystyle+2\frac{\sin^{2}\bar{h}}{r^{2}}Dh+2\frac{\sin\bar{h}}{r}\left(hD\left(\frac{\sin\bar{h}}{r}\right)-D\left(\sin\bar{h}\frac{\bar{h}}{r}\right)-\sin\bar{h}\frac{(h-\bar{h})(g-\tilde{g})}{2r}\right.$
$\displaystyle+\left.\frac{\alpha}{2}\sin\bar{h}(h-\bar{h})g\left(2\frac{\sin^{2}\bar{h}}{r^{2}}+\frac{\sin^{4}\bar{h}}{r^{4}}\right)\right)+\frac{g}{r^{3}}\sin^{3}\bar{h}\cos\bar{h}$

with only one dimensionless free coupling constant, $\alpha$ to be tuned.

Now, consider the solution of (3.10) along the curve $\chi$, then we can
reduce the partial integro-differential equation into an ordinary one,
parameterized by $u^{\prime}$, namely

$\frac{dh}{du^{\prime}}=Ph+Q,$ (3.11)

with

$\displaystyle P$ $\displaystyle\equiv$
$\displaystyle\left[\frac{g-\tilde{g}}{2r}-\frac{\alpha}{2}gr\left(2\frac{\sin^{2}\bar{h}}{r^{2}}+\frac{\sin^{4}\bar{h}}{r^{4}}\right)\right]_{\chi}-\frac{1}{2}\frac{d}{du^{\prime}}\ln\left(1+2\frac{\sin^{2}\bar{h}}{r^{2}}\right)_{\chi}$
(3.12) $\displaystyle Q$ $\displaystyle\equiv$
$\displaystyle-\left[\bar{h}\frac{g-\tilde{g}}{2r}-\frac{\alpha}{2}\bar{h}gr\left(2\frac{\sin^{2}\bar{h}}{r^{2}}+\frac{\sin^{4}\bar{h}}{r^{4}}\right)\right]_{\chi}+\left(\frac{2\frac{\sin^{2}\bar{h}}{r^{2}}}{1+2\frac{\sin^{2}\bar{h}}{r^{2}}}\right)_{\chi}\left.\frac{d}{du^{\prime}}\bar{h}\right|_{\chi}$
(3.13)
$\displaystyle-\left.g\cos\bar{h}\frac{\sin\bar{h}}{r}\right|_{\chi}\left(\frac{1+\frac{\sin^{2}\bar{h}}{r^{2}}}{1+2\frac{\sin^{2}\bar{h}}{r^{2}}}\right)_{\chi}+\frac{\bar{h}}{2}\frac{d}{du^{\prime}}\ln\left(1+2\frac{\sin^{2}\bar{h}}{r^{2}}\right)_{\chi}.$

Thus we can define a map $h\rightarrow\mathcal{F}(h)$ as

$\mathcal{F}(h)=h_{0}\exp\left(\int_{0}^{u^{\prime}}Pdu\right)+\left[\int_{0}^{u^{\prime}}Q\exp\left(\int_{u}^{u^{\prime}}Pd\tilde{u}\right)du\right],$
(3.14)

where $h_{0}\equiv h_{0}(r)=h(0,r)$ is the initial data of $h$. With equation
(3.14) in hand, the proof of local and global existence of the original
Einstein-Skyrme system solution can be provided by showing that $\mathcal{F}$
is a contractive mapping. It is worth noting that $P$ and $Q$ satisfy the
following Inequality

$\displaystyle\left|P\right|$ $\displaystyle\leq$
$\displaystyle\left|\frac{g-\tilde{g}}{2r}\right|+\left|\frac{\alpha}{2}gr\left(2\frac{\bar{h}^{2}}{r^{2}}+\frac{\bar{h}^{4}}{r^{4}}\right)\right|+\frac{d}{du^{\prime}}\left|2\frac{\bar{h}^{2}}{r^{2}}\right|,$
(3.15) $\displaystyle\left|Q\right|$ $\displaystyle\leq$
$\displaystyle\left|\bar{h}\frac{g-\tilde{g}}{2r}\right|+\left|\frac{\alpha}{2}\bar{h}gr\left(2\frac{\bar{h}^{2}}{r^{2}}+\frac{\bar{h}^{4}}{r^{4}}\right)\right|+\left|g\frac{\bar{h}^{3}}{2r}\right|+\frac{d}{du^{\prime}}\left|2\frac{\bar{h}^{3}}{r^{2}}\right|$
(3.16)

because $\sin\bar{h}\leq\bar{h}$ and $\frac{\sin{\bar{h}}}{r}\geq 0$ for every
point on $\chi$. The dynamical equation of first derivative of $h$ can be
found by taking the derivative of (3.10) with respect to $r$ to give

$\frac{d}{du^{\prime}}\left(\frac{\partial h}{\partial
r}\right)=P\left(\frac{\partial h}{\partial r}\right)+U,$ (3.17)

where $U$ satisfies,

$\displaystyle\left|U\right|$ $\displaystyle\leq$
$\displaystyle\left(\left|2\alpha B^{2}\left(7g+r\frac{\partial g}{\partial
r}\right)\left(\frac{\bar{h}^{2}}{r^{2}}+\frac{\bar{h}^{4}}{r^{4}}\right)\right|+\left|\frac{1}{2r}\frac{\partial
g}{\partial
r}\right|+\frac{d}{du^{\prime}}\left|10B\frac{\bar{h}^{2}}{r^{3}}\right|\right)\left(h+\bar{h}\right)$
(3.18)
$\displaystyle+\left(\left|\frac{g-\tilde{g}}{2r^{2}}\right|+\left|\frac{4B}{r^{2}}\bar{h}^{2}\left(g+r\frac{\partial
g}{\partial r}\right)\right|\right)\left|h-\bar{h}\right|.$

Thus, by similar approach with estimation of $h$ using $\mathcal{F}$ [22], we
define a map $\mathcal{G}:\frac{\partial h}{\partial
r}\rightarrow\mathcal{G}\left(\frac{\partial h}{\partial r}\right)$ as follows

$\mathcal{G}\left(\frac{\partial h}{\partial r}\right)=\frac{\partial
h_{0}}{\partial r}\exp\left(\int_{0}^{u^{\prime}}P\leavevmode\nobreak\
du\right)+\left[\int_{0}^{u^{\prime}}\exp\left(\int_{u}^{u^{\prime}}P\leavevmode\nobreak\
d\tilde{u}\right)U\leavevmode\nobreak\ du\right].$ (3.19)

The reason we need an explicit form of the first derivative with respect to
$r$ is because we seek a solution that is at least $C^{1}$, i.e. smooth
solution in $r$.

Lastly, we define $\Delta\equiv\mathcal{F}(h_{2})-\mathcal{F}(h_{1})$ such
that the integro-differential equation of $\Delta$ can be derived from (3.10)
to give

$\frac{d}{du^{\prime}}\Delta=P_{2}\Delta+(P_{2}-P_{1})\mathcal{F}(h_{1})+\frac{\tilde{g}_{2}-\tilde{g}_{1}}{2}\mathcal{G}(h_{1})+(Q_{2}-Q_{1}),$
(3.20)

where $P_{1}\equiv P(h_{1})$, $P_{2}\equiv P(h_{2})$, $Q_{1}\equiv Q(h_{1})$,
and $Q_{2}\equiv Q(h_{2})$. We are going to use the maps $\mathcal{F}$ and
$\mathcal{G}$ in equations (3.14) and (3.19), respectively, to find the radius
of the open-ball where both maps map the elements into other elements inside
the same ball, and then followed by estimating differential equations for
$\Delta$, equation (3.20) to find the corresponding Lipschitz condition. Thus,
by employing Banach’s fixed-point theorem, we can show that smooth classical
solutions exist and, further, provide some conditions where the existence of
solutions might fail within this approach.

###  3.1 Local Existence Of Classical Solutions

In this subsection, we prove the existence of the unique smooth local
classical solutions to the Einstein-Skyrme system in Bondi coordinates for
some specific initial data. Following similar method in [19], we define a
function space

$\displaystyle\tilde{X}=\\{h(.,.)\in
C^{1}([0,u_{0}]\times[0,\infty))\leavevmode\nobreak\ |\leavevmode\nobreak\
\|h\|_{\tilde{X}}<\infty\\},$ (3.21)

with its norm $\|h\|_{\tilde{X}}$ defined as

$\displaystyle\|h\|_{\tilde{X}}:=\sup_{u\in[0,u_{0}]}\sup_{r\geq
0}\left\\{(1+r)^{2}|h(u,r)|+(1+r)^{3}\left|\frac{\partial h}{\partial
r}(u,r)\right|\right\\},$ (3.22)

and a larger function space $\tilde{Y}$, containing
$\tilde{X}\subset\tilde{Y}$, defined as

$\displaystyle\tilde{Y}=\\{h(.,.)\in
C^{1}([0,u_{0}]\times[0,\infty))\leavevmode\nobreak\ |\leavevmode\nobreak\
\|h\|_{\tilde{Y}}<\infty\\},$ (3.23)

with

$\displaystyle\|h\|_{\tilde{Y}}:=\sup_{u\in[0,u_{0}]}\sup_{r\geq
0}\left\\{(1+r)^{2}|h(u,r)|\right\\}.$ (3.24)

The theorem of local existence is stated as follows

######  Theorem 1.

For every smooth initial data $h_{0}(r)\in C^{1}[0,\infty)$ such that
$h_{0}(r)=O(r^{-2})$ and $\frac{\partial h_{0}(r)}{\partial r}=O(r^{-3})$ as
$r$ goes to infinity, there exist $u_{0}>0$ where unique smooth classical
solution of spherically symmetric Einstein-Skyrme system with integer
topological charge $B>0$, $h(u,r)$, exist on the interval $[0,u_{0}]$, taking
$h_{0}(r)=h(0,r)$ as its initial data. This solution behaves like
$h(u,r)=O(r^{-2})$ and $\frac{\partial h(u,r)}{\partial r}=O(r^{-3})$ as $r$
goes to infinity for every $u$ in $[0,u_{0}]$.

Proof: Suppose that our solution, $h(u,r)$ is defined in an interval
$\mathcal{U}=[0,\eta]$ that contains $[0,u_{0}]$ such that
$\eta\in\mathcal{U}$ but $\eta\notin[0,u_{0}]$. Let
$\tilde{x}=\|h\|_{\tilde{X}}$ and $\tilde{d}=\|h_{0}\|_{\tilde{X}}=\sup_{r\geq
0}\left\\{(1+r)^{2}|h(0,r)|+(1+r)^{3}\left|\frac{\partial h}{\partial
r}(0,r)\right|\right\\}$. From (3.6), (3.7), and the fact that $\frac{\partial
g}{\partial r}=\frac{g-\bar{g}}{r}$, we have the following estimate

$\displaystyle\left|\frac{g-\tilde{g}}{2r}\right|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\left|\frac{\partial g}{\partial
r}\right|+\frac{2\alpha}{r^{2}}\left|\int_{0}^{r}s^{2}g\left(2\frac{\sin^{2}\bar{h}}{s^{2}}+\frac{\sin^{4}\bar{h}}{s^{4}}\right)ds\right|$
(3.25) $\displaystyle\leq$
$\displaystyle\alpha\frac{r(\tilde{x}^{2}+\tilde{x}^{4})}{(1+r)^{3}}\leq\alpha(\tilde{x}^{2}+\tilde{x}^{4}).$

As such, integral of $P$ from $u\in[0,u_{0}]$ up to $u^{\prime}=\eta>u_{0}$
can be estimated as

$\displaystyle\int_{u}^{\eta}Pd\tilde{u}$ $\displaystyle\leq$
$\displaystyle\int_{0}^{\eta}\left|P\right|d\tilde{u}$ (3.26)
$\displaystyle\leq$
$\displaystyle\int_{0}^{\eta}\left|\alpha\frac{r(\tilde{x}^{2}+\tilde{x}^{4})}{(1+r)^{3}}+\alpha\frac{\tilde{x}^{2}+\tilde{x}^{4}}{r^{2}(1+r)^{2}}\right|du+2\frac{\tilde{x}^{2}}{r^{2}(1+r)^{2}}$
$\displaystyle\leq$ $\displaystyle 2\alpha(\tilde{x}^{2}+\tilde{x}^{4})\eta.$

Next, we deduce the following estimate for $\mathcal{F}$ and $\mathcal{G}$
from the estimate of the integral of $P$ as follow

$\displaystyle\left|\mathcal{F}(\eta,r^{\prime})\right|$ $\displaystyle\leq$
$\displaystyle\exp\left(K_{1}^{\prime}(\tilde{x}^{2}+\tilde{x}^{4})\eta\right)\left[\frac{\tilde{d}}{(1+r^{\prime})^{2}}+\int_{0}^{\eta}Qdu\right]$
(3.27) $\displaystyle\left|\mathcal{G}(\eta,r^{\prime})\right|$
$\displaystyle\leq$
$\displaystyle\exp\left(K_{1}^{\prime}(\tilde{x}^{2}+\tilde{x}^{4})\eta\right)\left[\frac{\tilde{d}}{(1+r^{\prime})^{3}}+\int_{0}^{\eta}Udu\right],$
(3.28)

with $K_{1}^{\prime}\geq 2\alpha$. Since $\tilde{g}$ is always less than one,
every point $(u,r)$ satisfies $r-r^{\prime}\geq\frac{u^{\prime}-u}{2}$ where
$(u^{\prime},r^{\prime})\in\chi$, then it is straightforward to show that both
integrals of $Q$ and $U$ up to $u^{\prime}=\eta$ in (3.27) and (3.28) can be
estimated as

$\displaystyle\left|\int_{0}^{\eta}Qdu\right|$ $\displaystyle\leq$
$\displaystyle
2\alpha\frac{(\tilde{x}^{2}+\tilde{x}^{4})\eta}{(1+r^{\prime})^{2}},$ (3.29)
$\displaystyle\left|\int_{0}^{\eta}Udu\right|$ $\displaystyle\leq$
$\displaystyle
15\alpha^{2}\frac{(\tilde{x}^{3}+\tilde{x}^{5}+\tilde{x}^{7}+\tilde{x}^{9})\eta}{(1+r^{\prime})^{3}}.$
(3.30)

As a consequence, the norm of $\mathcal{F}$ in $\tilde{X}$ satisfies the
following inequality

$\|\mathcal{F}\|_{\tilde{X}}\leq
K_{2}^{\prime}\left(2d+\tilde{x}^{2}+\tilde{x}^{3}+\tilde{x}^{4}+\tilde{x}^{5}+\tilde{x}^{7}+\tilde{x}^{9}\right)\eta\exp\left(K_{1}^{\prime}(\tilde{x}^{2}+\tilde{x}^{4})\eta\right),$
(3.31)

with $K_{2}^{\prime}\geq 15\alpha^{2}$. Now, let us take an arbitrary
$\tilde{x}_{1}>d$ and take $\eta=\eta(\tilde{x}_{1},d)$ to be the solution of

$K_{2}^{\prime}\left(2d+\tilde{x}_{1}^{2}+\tilde{x}_{1}^{3}+\tilde{x}_{1}^{4}+\tilde{x}_{1}^{5}+\tilde{x}_{1}^{7}+\tilde{x}_{1}^{9}\right)\eta\exp\left(K_{1}^{\prime}(\tilde{x}_{1}^{2}+\tilde{x}_{1}^{4})\eta\right)-\tilde{x}_{1}=0.$
(3.32)

By choosing such value444An explicit solution of equation (3.32) can be
written in principal Lambert $W$-function, $\mathcal{W}_{0}(x)$, as
$\eta=\frac{1}{K^{\prime}_{1}(\tilde{x}_{1}^{2}+\tilde{x}_{1}^{4})}\mathcal{W}_{0}\left(\frac{K^{\prime}_{1}(\tilde{x}_{1}^{3}+\tilde{x}_{1}^{5})}{K_{2}^{\prime}\left(2d+\tilde{x}_{1}^{2}+\tilde{x}_{1}^{3}+\tilde{x}_{1}^{4}+\tilde{x}_{1}^{5}+\tilde{x}_{1}^{7}+\tilde{x}_{1}^{9}\right)}\right)$,
see [27] for details. for $\eta$, we ensure that
$\|\mathcal{F}\|_{\tilde{X}}\leq\tilde{x}_{1}$. In fact, the choice of $\eta$
is not unique since it completely depends on how we estimate the norm of
$\mathcal{F}$ in $\tilde{X}$. Thus, there might be some other possible choice
for $\eta$. The conclusion of the above argument is summarized as follows

######  Lemma 1.

For every $\tilde{x}_{1}>d$, there exist $\eta(\tilde{x_{1}},d)>0$ such that
if $u_{0}<\eta$ the map $\mathcal{F}$ defined in (3.14) is contained in a
closed ball of radius $\tilde{x}_{1}$. The mentioned closed ball is a subspace
of $\tilde{X}$.

Now, let us assume that there are two different solutions,
$h_{1},h_{2}\in\tilde{X}$, for (3.10). Suppose that these solutions are
defined in the same interval $\mathcal{U}=[0,\nu]$ that contains $[0,u_{0}]$
such that $\nu\in\mathcal{U}$ but $\nu\notin[0,u_{0}]$. By assuming that
$h_{1}$ and $h_{2}$ have exactly the same initial data, the solution of (3.20)
satisfies the following inequality

$\Delta\leq\int_{0}^{\nu}\exp\left(\int_{u}^{\nu}P_{2}d\tilde{u}\right)\left[(P_{2}-P_{1})\mathcal{F}(h_{1})+\frac{\tilde{g}_{2}-\tilde{g}_{1}}{2}\mathcal{G}(h_{1})+(Q_{2}-Q_{1})\right]du$
(3.33)
$\leq\exp\left(\int_{0}^{\nu}P_{2}d\tilde{u}\right)\int_{0}^{\nu}\left|(P_{2}-P_{1})\mathcal{F}(h_{1})+\frac{\tilde{g}_{2}-\tilde{g}_{1}}{2}\mathcal{G}(h_{1})+(Q_{2}-Q_{1})\right|du$

Let us define $\tilde{y}\equiv\|h_{2}-h_{1}\|_{\tilde{Y}}$ and take
$\tilde{x}$ to be greater than
$\max\\{\|h_{1}\|_{\tilde{X}},\|h_{2}\|_{X}\\}$. The exponential factor of the
integrand on the right hand side of (3.33) satisfies a similar estimate with
the one we found on Lemma 1. Thus, we only need to find the upper-bound of the
remaining three terms. The estimate of these three terms are given as follow

$\displaystyle\int_{0}^{\nu}\left|Q_{2}-Q_{1}\right|du$ $\displaystyle\leq$
$\displaystyle
K_{3}^{\prime}\frac{(\tilde{x}^{3}+\tilde{x})y}{(1+r^{\prime})^{2}}\nu,$
(3.34)
$\displaystyle\int_{0}^{\nu}\left|(P_{2}-P_{1})\mathcal{F}(h_{1})\right|du$
$\displaystyle\leq$ $\displaystyle
K_{3}^{\prime}\exp\left(K_{1}^{\prime}(\tilde{x}^{2}+\tilde{x}^{4})\nu\right)\frac{(\tilde{x}^{3}+\tilde{x})(d+\tilde{x}^{2}+\tilde{x}^{4})y}{(1+r^{\prime})^{2}}\nu^{2},$
(3.35)
$\displaystyle\int_{0}^{\nu}\left|\frac{\tilde{g}_{2}-\tilde{g}_{1}}{2}\mathcal{G}(h_{1})\right|du$
$\displaystyle\leq$ $\displaystyle
K_{3}^{\prime}\exp\left(K_{1}^{\prime}(\tilde{x}^{2}+\tilde{x}^{4})\nu\right)\frac{(\tilde{x}^{4}+\tilde{x}^{2})(d+\tilde{x}^{3}+\tilde{x}^{5}+\tilde{x}^{7}+\tilde{x}^{9})y}{(1+r^{\prime})^{3}}\nu^{2},$

with $K_{3}^{\prime}\leq 12\alpha^{3}$. From here, we are able to deduce the
Lipschitz condition for $\Delta$ in $\tilde{Y}$, namely

$\|\Delta\|_{\tilde{Y}}\leq\tilde{\mathcal{L}}(\tilde{x})y,$ (3.37)

where we have defined

$\displaystyle\tilde{\mathcal{L}}(\tilde{x})$ $\displaystyle\equiv$
$\displaystyle
K_{3}^{\prime}\left[(\tilde{x}^{3}+\tilde{x})(d+\tilde{x}^{2}+\tilde{x}^{4})+(\tilde{x}^{4}+\tilde{x}^{2})(d+\tilde{x}^{3}+\tilde{x}^{5}+\tilde{x}^{7}+\tilde{x}^{9})+(\tilde{x}^{3}+\tilde{x})\right]\nu^{2}$
(3.38)
$\displaystyle\exp\left(K_{1}^{\prime}(\tilde{x}^{2}+\tilde{x}^{4})\nu\right).$

The function (3.38) is a monotonically increasing function and is zero for
$\tilde{x}=0$. We can always choose arbitrary value of $\tilde{x}_{1}>d$ and
take $\nu$ as a function of $\tilde{x}_{1}$ to be small enough such that
$0\leq\tilde{\mathcal{L}}(\tilde{x})<1$ for every
$\tilde{x}\in[0,\tilde{x}_{1}]$ to ensure that $\mathcal{F}$ is a contraction
mapping. Thus, from the above Lipschitz condition we can conclude that

######  Lemma 2.

For every $\tilde{x}_{1}>d$ there exist $\nu(x_{1})>0$, such that if
$u_{0}<\nu(x_{1})$ the map $\mathcal{F}$ defined in (3.14) contracts in
$\tilde{Y}$.

From Lemma 1 and Lemma 2 we deduce that unique smooth solutions of (3.10),
that decays faster than $O(r^{-2})$ as $r\rightarrow\infty$, exist in the
interval $u\in[0,u_{0}]$ where $u_{0}<\min[\eta,\nu]$. One can observe that
the existence interval, $[0,u_{0}]$, always exist for every $d$ since we can
always find either $\eta$ or $\nu$ such that $\tilde{x}_{1}>d$ and then
proceed to find the corresponding $u_{0}$.

We would also like to remark that there exists a lower bound of the norm of
initial data in $\tilde{X}$.

######  Lemma 3.

The norm of initial data, $d\equiv\|h_{0}\|_{\tilde{X}}$ is bounded from below
by,

$d\geq n\pi,$ (3.39)

where $n\in\mathbb{Z}$ is related to the topological charge, $B=\pm n$.

Proof: The proof for the lower bound shown above is straightforward. Since
$\xi(u,0)=n\pi$, then

$\displaystyle d$ $\displaystyle=$
$\displaystyle\sup_{u\in[0,u_{0}]}\sup_{r\geq
0}\left\\{(1+r)^{2}|h_{0}(r)|+(1+r)^{3}\left|\frac{\partial h_{0}(r)}{\partial
r}\right|\right\\}$ (3.40) $\displaystyle\geq$
$\displaystyle\lim_{r\rightarrow
0}\left\\{(1+r)^{2}|h_{0}(r)|+(1+r)^{3}\left|\frac{\partial h_{0}(r)}{\partial
r}\right|\right\\}\geq h_{0}(0)$ $\displaystyle=$
$\displaystyle\xi(0,0)=n\pi.$

This feature implies that $\tilde{x}_{1}>n\pi$ and implies that both $\eta$
and $\nu$ are finite because from both equations (3.32, 3.38) and the fact
that $\tilde{\mathcal{L}}(\tilde{x})<1$ for contraction mapping, we have

######  Remark 1.

$\displaystyle\eta$ $\displaystyle<$ $\displaystyle\frac{1}{4\alpha
n^{4}\pi^{4}}\mathcal{W}_{0}\left(\frac{1}{60\alpha n^{4}\pi^{4}}\right),$
(3.41) $\displaystyle\nu$ $\displaystyle<$ $\displaystyle\frac{1}{\alpha
n\pi}\mathcal{W}_{0}\left(2\sqrt{3}\alpha^{5/2}\right),$ (3.42)

where $\mathcal{W}_{0}(x)$ is the principal Lambert’s $W$-Function.

As such, we cannot extend these results to $u_{0}\rightarrow\infty$. This
means that we need a different approach in order to find global smooth
solutions and some conditions for restricting the initial data are expected.

###  3.2 Global Existence and Uniqueness Of Classical Solutions

Here, we show that a globally unique solution exists for an Einstein-Skyrme
system in Bondi coordinate with restricted initial data. To do so, by
following the methods in[22, 19] we introduce the function space $X$,

$\displaystyle X=\\{h(.,.)\in
C^{1}([0,\infty)\times[0,\infty))\leavevmode\nobreak\ |\leavevmode\nobreak\
\|h\|_{X}<\infty\\},$ (3.43)

with $\|h\|_{X}$ defined as

$\displaystyle\|h\|_{X}:=\sup_{u\geq 0}\sup_{r\geq
0}\left\\{(1+r+u)^{2}|h(u,r)|+(1+r+u)^{3}\left|\frac{\partial h}{\partial
r}(u,r)\right|\right\\}.$ (3.44)

We also define the function space

$\displaystyle X_{0}=\\{h(.)\in C^{1}([0,\infty))\leavevmode\nobreak\
|\leavevmode\nobreak\ \|h\|_{X_{0}}<\infty\\},$ (3.45)

with the norm

$\displaystyle\|h\|_{X_{0}}:=\sup_{r\geq
0}\left\\{(1+r)^{2}|h(r)|+(1+r)^{3}\left|\frac{\partial h}{\partial
r}(r)\right|\right\\}$ (3.46)

and we define the notation $d=\|h_{0}\|_{X_{0}}$. Furthermore, we define the
function space $Y$ containing $X$,

$\displaystyle Y=\\{h(.,.)\in
C^{1}([0,\infty))\times[0,\infty))\leavevmode\nobreak\ |\leavevmode\nobreak\
h(0,r)=h_{0}(r),\leavevmode\nobreak\ \|h\|_{Y}<\infty\\},$ (3.47)

with the norm

$\displaystyle\|h\|_{Y}=\sup_{u\geq 0}\sup_{r\geq
0}\left\\{(1+r+u)^{2}|h(u,r)|\right\\}.$ (3.48)

As such, the statement of the theorem is as follows:

######  Theorem 2.

Suppose we have initial data $h_{0}\in X_{0}$ where $X_{0}$ defined in (3.45).
Let $d$ be the norm of $h_{0}$, $\|h_{0}\|_{X_{0}}$, and let $X$ be a space
defined in (3.43). There exists positive $\alpha$ and $\delta>n\pi$ such that
if $n\pi\leq d<\delta$, there exists a unique global classical solution
$h(u,r)\in X$ of Einstein-Skyrme system with integer topological charge $B>0$,
possessing a nonlinear dynamical equation (3.10) with $h_{0}(r)$ as its
initial data. The decay properties of the solution are

$\displaystyle\left|h(u,r)\right|$ $\displaystyle\leq$ $\displaystyle
C(1+r+u)^{-2},$ (3.49) $\displaystyle\left|\frac{\partial h}{\partial
r}(u,r)\right|$ $\displaystyle\leq$ $\displaystyle C(1+r+u)^{-3},$ (3.50)

and the topological charge, $B$, satisfies $B=\pm n$, where $n\in\mathbb{Z}$.
The coupling constant is bounded from above by
$\alpha<\frac{58\mathcal{W}_{0}(\frac{3}{29})}{n^{2}\pi^{2}+n^{4}\pi^{4}}$.

In order to prove the theorem given above, first we need to define the map
from $X$ to the solution of (3.10), $\mathcal{F}$, such that $\mathcal{F}(h)$
solves (3.10). Thus, we need these two properties to be shown, namely

  1. 1.

$\mathcal{F}:U\rightarrow U$, with $U\subset X$.

  2. 2.

for $h_{1},h_{2}\in U\subset Y$ there exist $\lambda\in(0,1)$ such that

$\displaystyle\|\mathcal{F}(h_{2})-\mathcal{F}(h_{1})\|_{Y}\leq\lambda\|h_{2}-h_{1}\|_{Y},$
(3.51)

i.e. $\mathcal{F}$ contracts in $Y$.

As mentioned earlier, both of the conditions above lead to the existence of
unique solutions by using Banach’s fixed-point theorem. Then, we are going to
use such conditions to find suitable initial data for global solutions to
exist. Furthermore, we can also study the influence of topology on the
restrictions for initial data from the conditions above.

Firstly, we exploit the following estimates:

  * •

$\left|\bar{h}\right|\leq\frac{x}{(1+u)(1+u+r)}$ (3.52)

where $x\equiv\|h\|_{X}$, and

  * •

$\left|h(u,r)-h(u,r^{\prime})\right|\leq\frac{x}{2}\left(\frac{1}{(1+u+r^{\prime})^{2}}-\frac{1}{(1+u+r)^{2}}\right).$
(3.53)

  * •

As a consequence of both estimates above, we have

$\left|h-\bar{h}\right|\leq\frac{xr}{2(1+r+u)^{2}(1+u)}$ (3.54)

These estimates are needed to find the estimate of $g$ and $\tilde{g}$.
Similar to the space $\tilde{X}$ we use for the study of local solutions, we
also have the lower bound for the norms of space $X$ and $X_{0}$.

######  Lemma 4.

Let $x=\|h(r)\|_{X}$ and $d=\|h_{0}\|$, where $h_{0}$ is the initial data of
$h$. Both $x$ and $d$ are bounded from below,

$x\geq n\pi,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
d\geq n\pi.$ (3.55)

$n\in\mathbb{Z}$ is related to the topological charge by $B=\pm n$.

Proof:  Since the boundary condition of topological Skyrmions at the origin of
space is $\xi(u,0)=n\pi$, then $h(u,0)=n\pi$ and $h_{0}(0)=n\pi$. Thus, both
$h$ and $h_{0}$ satisfies

$\displaystyle\|h\|_{X}$ $\displaystyle\geq$ $\displaystyle\lim_{r\rightarrow
0}\left((1+r)^{2}|h(u,r)|+(1+r)^{3}\left|\frac{\partial h}{\partial
r}(u,r)\right|\right)\geq h(u,0)=n\pi,$ (3.56)
$\displaystyle\|h_{0}\|_{X_{0}}$ $\displaystyle\geq$
$\displaystyle\lim_{r\rightarrow
0}\left((1+r)^{2}|h_{0}(r)|+(1+r)^{3}\left|\frac{\partial h_{0}}{\partial
r}(r)\right|\right)\geq h_{0}(0)=n\pi.$ (3.57)

This is the end of the proof of Lemma (4).

We should expect that the lower bounds for both norm of $h$ and the norm of
$h_{0}$ above can be used to find the restriction for the initial data of
global solutions. From previous subsections, we know that in order to have
local solutions with arbitrary initial data, then the interval for the time
coordinate $u$ cannot be extended to infinity and bounded from above due to
the topology of the Skyrmion. As such, the strategy to have global solutions,
where the interval of $u$ is extended to infinity, is that we need to restrict
the norm of initial data, $d$, and we expect that the value of $d$ falls in an
interval where the lower bound is given in Lemma 4. Thus, we need to find the
upper bound of $d$ from the conditions of existence and uniqueness.

Now, consider the estimates for metric functions, ${g,\tilde{g}}$, below.

######  Proposition 1.

If $h(u,r)$ belong to the space $X$ defined in (3.43) with initial data
$h(0,r)$ belong to the space $X_{0}$ defined in (3.45), then for two functions
${g,\tilde{g}}$, given in (3.6) and (3.7), the following inequalities hold

$\displaystyle g(u,r)$ $\displaystyle\geq$
$\displaystyle\exp\left[-\alpha\frac{\|h\|_{X}^{2}+\|h\|_{X}^{4}}{58}\right],$
(3.58) $\displaystyle\tilde{g}(u,r)$ $\displaystyle\geq$
$\displaystyle\exp\left[-\alpha\frac{\|h\|_{X}^{2}+\|h\|_{X}^{4}}{58}\right]-\frac{\alpha}{6}\left(\|h\|_{X}^{2}+\|h\|_{X}^{4}\right).$
(3.59)

Furthermore, we have

$|g-\tilde{g}|\leq\alpha\frac{(\|h\|_{X}^{2}+\|h\|_{X}^{4})r^{2}}{6(1+u)^{3}(1+r+u)^{3}}$
(3.60)

Proof: At $r\rightarrow 0$ we have to impose the boundary condition
$\xi(r\rightarrow 0)=n\pi$. Thus, we can estimate $\sin\bar{h}$ as
$\sin\xi=\sin(\pi-\xi)\leq\pi-\xi\leq-r\xi_{,r}=-(h-\bar{h})$. Consider the
following estimate

$\displaystyle\int^{\infty}_{0}\frac{(h-\bar{h})^{2}}{s}\left(1+2\frac{\sin^{2}\bar{h}}{s^{2}}\right)\leavevmode\nobreak\
ds$ $\displaystyle\leq$
$\displaystyle\int^{\infty}_{0}\frac{(h-\bar{h})^{2}}{s}ds+2\int^{\infty}_{0}\frac{(h-\bar{h})^{4}}{s^{3}}ds$
(3.61) $\displaystyle\leq$ $\displaystyle\frac{x^{2}+x^{4}}{24},$

which can be used to estimate the function $g$ in (3.6) to give

$g\geq\exp\left[-\alpha\frac{x^{2}+x^{4}}{58}\right].$ (3.62)

Now, for $\tilde{g}$ we need the following estimate

$\displaystyle\int_{0}^{r}s^{2}g\left(2\frac{\sin^{2}\bar{h}}{s^{2}}+\frac{\sin^{4}\bar{h}}{s^{4}}\right)\leavevmode\nobreak\
ds$ $\displaystyle\leq$ $\displaystyle
2\int_{0}^{r}s^{2}\left(\frac{(h-\bar{h})^{2}}{s^{2}}+\frac{(h-\bar{h})^{4}}{s^{4}}\right)ds$
(3.63) $\displaystyle\leq$
$\displaystyle\frac{(x^{2}+x^{4})r^{3}}{6(1+u)^{3}(1+r+u)^{3}}.$

By using equation (3.7) with the estimate given above, we have

$\displaystyle\tilde{g}$ $\displaystyle\geq$
$\displaystyle\exp\left[-\alpha\frac{x^{2}+x^{4}}{58}\right]-\frac{\alpha}{r}\frac{(x^{2}+x^{4})r^{3}}{6(1+u)^{3}(1+r+u)^{3}}$
(3.64) $\displaystyle\geq$
$\displaystyle\exp\left[-\alpha\frac{x^{2}+x^{4}}{58}\right]-\frac{\alpha}{6}\left(x^{2}+x^{4}\right).$

Next, we estimate the following quantity

$\displaystyle\int_{r^{\prime}}^{r}\left|\frac{\partial g}{\partial
s}(u,s)\right|ds$ $\displaystyle\leq$
$\displaystyle\frac{\alpha}{2}\int_{r^{\prime}}^{r}\frac{(h-\bar{h})^{2}}{s}\left(1+2\frac{(h-\bar{h})^{2}}{s^{2}}\right)ds$
$\displaystyle\leq$
$\displaystyle\frac{\alpha}{24(1+u)^{2}}\left[x^{2}\left(\frac{1+3r^{\prime}+u}{(1+r^{\prime}+u)^{3}}-\frac{1+3r+u}{(1+r+u)^{3}}\right)\right.$
$\displaystyle+\left.x^{4}\left(\frac{1+7r^{\prime}+u}{(1+r^{\prime}+u)^{7}}-\frac{1+7r+u}{(1+r+u)^{7}}\right)\right]$

which is useful for estimate of $\left|g-\bar{g}\right|$ as follows

$\displaystyle\left|g-\bar{g}\right|$ $\displaystyle\leq$
$\displaystyle\frac{1}{r}\int_{0}^{r}\int_{r^{\prime}}^{r}\left|\frac{\partial
g}{\partial s}(u,s)\right|dsdr^{\prime}$ (3.66) $\displaystyle\leq$
$\displaystyle\alpha\frac{(x^{2}+x^{4})r^{2}}{12(1+u)^{3}(1+r+u)^{3}}.$

From equation (3.7) we know that
$g-\tilde{g}=g-\bar{g}+\frac{\alpha}{r}\int_{0}^{r}s^{2}g\left(2\frac{\sin^{2}\bar{h}}{s^{2}}+\frac{\sin^{4}\bar{h}}{s^{4}}\right)\leavevmode\nobreak\
ds$. Thus, the following inequalities hold

$\displaystyle\left|g-\tilde{g}\right|$ $\displaystyle\leq$
$\displaystyle\left|g-\bar{g}\right|+\frac{\alpha}{r}\left|\int_{0}^{r}s^{2}g\left(2\frac{\sin^{2}\bar{h}}{s^{2}}+\frac{\sin^{4}\bar{h}}{s^{4}}\right)\leavevmode\nobreak\
ds\right|$ (3.67) $\displaystyle\leq$
$\displaystyle\alpha\frac{(x^{2}+x^{4})r^{2}}{6(1+u)^{3}(1+r+u)^{3}}.$

This completes the proof of Proposition 1.

There are some implications from proposition 1. Firstly, a black hole
solution, where the lower bound of $\tilde{g}$ is less than zero, leads to an
additional condition for $\|h\|_{X}$ and $\|h\|_{X}$ is bounded from below.
However, we can always find black hole solutions for every value of $B$ since
the lower bound of $\tilde{g}$ also depends on $\alpha$, hence, the choice of
topological degree does not affect the existence of black hole solutions but
do affect the range of possible values for the effective gravitational
coupling constant, $\alpha$. Secondly, we can directly see that this bound is
connected to the flat spacetime case, $g=\tilde{g}=1$ by choosing the matter
field to be vacuum $\|h\|_{X}=0$ but this can only be done for non-topological
Skyrmions ($B=0$) or non-gravitating case ($\alpha=0$). Lastly, we would also
like to remark that both $g$ and $\tilde{g}$ converge into a single value at
spatial infinity and time infinity which implies that the well-known static
configurations (static Skyrme stars and Skyrme Blackhole) are the steady-state
limit of this dynamical model.

From here, we define a new constant

$k=\exp\left[-\alpha\frac{x^{2}+x^{4}}{58}\right]-\frac{\alpha}{6}\left(x^{2}+x^{4}\right),$
(3.68)

as a function of $x$ defined for $x\in[n\pi,x_{0})$ where $x_{0}$ is the root
of $k(x)=0$, and $k(x)$ taking values from $(0,1]$ which is deduced from the
possible values for the lower bound of $\tilde{g}$. The interval of $k$ from
the physical argument for $\tilde{g}$ leads to a restriction for the coupling
constant $\alpha$. Since we know that $x\geq n\pi$, then in order to guarantee
$k$ falls in the interval $(0,1]$, we need

$0\leq\alpha<\frac{58\mathcal{W}_{0}(\frac{3}{29})}{n^{2}\pi^{2}+n^{4}\pi^{4}}.$
(3.69)

One shall notice that the higher the topological charge means that the
interval for $\alpha$ becomes smaller. Furthermore, we shall use the constant
$k$ to simplify our estimates. The characteristic (3.8) gives the following
relation

$r=r^{\prime}+\frac{1}{2}\int_{u}^{u^{\prime}}\tilde{g}(\tilde{u},r)d\tilde{u}\geq
r^{\prime}+\frac{k}{2}(u^{\prime}-u).$ (3.70)

From the inequality given above, we can deduce that
$1+r+u\geq\frac{k}{2}(1+r^{\prime}+u^{\prime})$ because $1\geq k>0$. There is
also a useful relation for this characteristic which can be found in [22],
namely

$\int_{0}^{u^{\prime}}\frac{r^{s}}{(1+u)^{t}(1+r+u)^{q}}du\leq\frac{2^{m}}{(t+q-s-m-1)k^{m}(1+r^{\prime}+u^{\prime})^{m}}$
(3.71)

for $t+q-s-m>1$, where $t,q,s,m$ are positive integers.

Now that we have the inequalities from characteristic, we can estimate (3.14)
on the curve $\chi$. Firstly, consider the first term in (3.14). It is a
product of initial condition $h_{0}$ and the integrating factor. Because we
can estimate $h_{0}$ as

$h_{0}\leq\frac{d}{(1+r(0))^{2}}\leq\frac{4d}{k^{2}(1+r^{\prime}+u^{\prime})^{2}},$
(3.72)

where we have used the fact that $r(0)\geq r^{\prime}+\frac{k}{2}u^{\prime}$
and $d\equiv\|h_{0}\|_{X_{0}}$, then the integrating factor can be estimated
by exploiting the following relation

$\displaystyle\int_{0}^{u^{\prime}}Pdu$ $\displaystyle\leq$
$\displaystyle\int_{0}^{u^{\prime}}\left|\frac{g-\tilde{g}}{2r}\right|du+\int_{0}^{u^{\prime}}\left|\frac{\alpha}{3}gr\left(2\frac{\bar{h}^{2}}{r^{2}}+B^{2}\frac{\bar{h}^{4}}{r^{4}}\right)\right|du+\left|2\bar{h}^{2}\right|$
(3.73) $\displaystyle\leq$ $\displaystyle K_{1}(x^{2}+x^{4}),$

where satisfies $K_{1}\geq 4\alpha$, which leads us to
$\exp\left(\int_{0}^{u^{\prime}}Pdu\right)\leq\exp(K_{1}(x^{2}+x^{4}))$. Now,
for the second term, consider the following estimates

$\displaystyle\int_{0}^{u^{\prime}}Q\exp\left(\int_{u}^{u^{\prime}}Pd\tilde{u}\right)du$
$\displaystyle\leq$
$\displaystyle\exp(K_{1}(x^{2}+x^{4}))\left[\int_{0}^{u^{\prime}}\left(\left|\bar{h}\frac{g-\tilde{g}}{2r}\right|+\left|\frac{\alpha}{2}\bar{h}gr\left(2\frac{\bar{h}^{2}}{r^{2}}+\frac{\bar{h}^{4}}{r^{4}}\right)\right|\right.\right.$
(3.74)
$\displaystyle+\left.\left|g\frac{h-\bar{h}}{r}\right|\right)du+\left.2\frac{\bar{h}^{3}}{r^{2}}\right]$
$\displaystyle\leq$ $\displaystyle
K_{1}\frac{x^{3}+x^{5}}{k^{2}(1+r^{\prime}+u^{\prime})^{2}}\exp(K_{1}(x^{2}+x^{4})).$

Hence, we can conclude that $\mathcal{F}(u^{\prime},r^{\prime})$ defined in
(3.14) satisfies the following decay estimate

$\left|\mathcal{F}(u^{\prime},r^{\prime})\right|\leq
K_{1}\frac{d+x^{3}+x^{5}}{k^{2}(1+r^{\prime}+u^{\prime})^{2}}\exp(K_{1}(x^{2}+x^{4})),$
(3.75)

i.e. $\mathcal{F}$ is a bounded map whose upper bound is a decaying function
that goes like $(1+r^{\prime}+u^{\prime})^{-2}$. All smooth functions, which
decay sufficiently slower than the upper bound at
$r^{\prime}\rightarrow\infty$, falls into the family of solutions for this
case.

Next, we need to estimate the first derivative of $h$ as well. To estimate
$\mathcal{G}$ in (3.19), firstly we need to estimate the first derivative of
initial condition $h_{0}$, thus we have

$\displaystyle\frac{\partial h_{0}}{\partial
r}\leq\frac{d}{(1+r(0))^{3}}\leq\frac{8d}{k^{3}(1+r^{\prime}+u^{\prime})^{3}}.$
(3.76)

Since the first derivative of $h_{0}$ goes like $O((1+r)^{-3})$ then we should
adopt the same estimate of $\exp\left(\int_{0}^{u^{\prime}}Pdu\right)$ as the
previous case for $\mathcal{F}$. It is also necessary to find the estimate of
the first derivative of $g$ which is given by

$\displaystyle\frac{\partial g}{\partial r}$ $\displaystyle=$ $\displaystyle
g\alpha\frac{(h-\bar{h})^{2}}{2r}\left(1+2\frac{\sin^{2}\bar{h}}{r^{2}}\right)$
(3.77) $\displaystyle\leq$
$\displaystyle\alpha\frac{(x^{2}+x^{4})r}{4(1+r+u)^{4}(1+u)^{2}}.$

With the estimate of $\frac{\partial g}{\partial r}$ in hand we can proceed to
find the estimate for $\int_{0}^{u^{\prime}}Udu$ as follows

$\displaystyle\int_{0}^{u^{\prime}}Udu$ $\displaystyle\leq$
$\displaystyle\int_{0}^{u^{\prime}}\left(\left|\alpha\left(7g+r\frac{\partial
g}{\partial
r}\right)\left(\frac{\bar{h}^{2}}{r^{2}}+\frac{\bar{h}^{4}}{r^{4}}\right)\right|+\left|\frac{1}{2r}\frac{\partial
g}{\partial r}\right|\right)\left(h+\bar{h}\right)du$ (3.78)
$\displaystyle+\int_{0}^{u^{\prime}}\left(\left|\frac{g-\tilde{g}}{2r^{2}}\right|+\left|\frac{4}{r^{2}}\bar{h}^{2}\left(g+r\frac{\partial
g}{\partial
r}\right)\right|\right)\left|h-\bar{h}\right|du+10\left|\frac{\bar{h}^{4}}{r^{\prime}{}^{3}}\right|$
$\displaystyle\leq$ $\displaystyle
14\alpha^{2}\frac{x^{3}+x^{4}+x^{5}+x^{7}+x^{9}}{k^{3}(1+r^{\prime}+u^{\prime})^{3}}.$

As such, the second term of (3.19) satisfies the inequality below

$\int_{0}^{u^{\prime}}\exp\left(\int_{u}^{u^{\prime}}Pd\tilde{u}\right)Udu\leq
K_{2}\frac{x^{3}+x^{4}+x^{5}+x^{7}+x^{9}}{k^{3}(1+r^{\prime}+u^{\prime})^{3}},$
(3.79)

where, $K_{2}\geq 14\alpha^{2}$. As a conclusion,
$\mathcal{G}(u^{\prime},r^{\prime})$ defined in (3.19) is bounded from above
by

$\left|\mathcal{G}(u^{\prime},r^{\prime})\right|\leq
K_{2}\frac{d+x^{3}+x^{4}+x^{5}+x^{7}+x^{9}}{k^{3}(1+r^{\prime}+u^{\prime})^{3}}\exp\left(K_{1}(x^{2}+x^{4})\right).$
(3.80)

We conclude the estimates for $\mathcal{F}(u^{\prime},r^{\prime})$ and
$\mathcal{G}(u^{\prime},r^{\prime})$ in the following lemma

######  Lemma 5.

(Decay Properties of Einstein-Skyrme system) Suppose that $h(u,r)\in X$ is the
solution of (3.10) with initial data $h_{0}\in X_{0}$ and both $\|h\|_{X}$ and
$\|h_{0}\|_{X_{0}}$ are finite, $X$ and $X_{0}$ are defined in (3.43) and
(3.45), respectively. Then, $h(u,r)$ and $\frac{\partial h}{\partial r}(u,r)$
satisfy the following decay estimate:

$\displaystyle\left|h(u,r)\right|$ $\displaystyle\leq$ $\displaystyle
C(1+r+u)^{-2},$ (3.81) $\displaystyle\left|\frac{\partial h}{\partial
r}(u,r)\right|$ $\displaystyle\leq$ $\displaystyle C(1+r+u)^{-3}.$ (3.82)

Proof: If $x=\|h\|_{X}$ and $d=\|h_{0}\|_{X_{0}}$ are finite, then
inequalities (3.75) and (3.80) states the upper bound for $h(u,r)$ and
$\frac{\partial h}{\partial r}(u,r)$. This is the end of the proof for lemma
5.

One of the consequence of lemma 5 is the norm of $\mathcal{F}$ in $X$ is
bounded from above by

$\|\mathcal{F}\|_{X}\leq
K_{2}\frac{d+x^{3}+x^{4}+x^{5}+x^{7}+x^{9}}{k^{3}}\exp\left(K_{1}(x^{2}+x^{4})\right),$
(3.83)

which implies that finiteness of both $\|h\|_{X}$ and $\|h_{0}\|_{X_{0}}$ lead
to a bounded $\|\mathcal{F}\|_{X}$. Since the map $\mathcal{F}$ is bounded in
$X$, we proceed to show that $\mathcal{F}$ is a map from a ball to itself by
introducing the following function

$\mathcal{L}_{1}(x)\equiv\frac{xk^{3}}{K_{2}}\exp\left(-K_{1}(x^{2}+x^{4})\right)-(x^{3}+x^{4}+x^{5}+x^{7}+x^{9}).$
(3.84)

The lemma is

######  Lemma 6.

Let $X$ and $X_{0}$ are function spaces defined in (3.43) and (3.45),
respectively. Let $\alpha$ be the effective coupling constant, $x=\|h\|_{X}$
and $x_{0}$ be the root of $k(x)=0$ where $k(x)$ is defined in (3.68). If
$d=\|h_{0}\|_{X_{0}}$ as the initial data of dynamical equation (3.10)
satisfies $d<\mathcal{L}_{1}(x)$, where $\mathcal{L}_{1}(x)$ is defined in
(3.84), then there exist $x_{1}\in(n\pi,x_{0}]$ and $\alpha$ which takes
values from the interval (3.69) such that for every $x\in(n\pi,x_{1})$, we
have $\|\mathcal{F}\|_{X}<x$ with $\mathcal{F}:h\rightarrow\mathcal{F}(h)$
defined in (3.14).

Proof: It is straightforward to check that we can always choose $\alpha$ small
enough such that $\mathcal{L}_{1}(n\pi)>0$ and
$\mathcal{L}_{1}^{\prime}(n\pi)>0$. Since
$\mathcal{L}_{1}(x\rightarrow\infty)=-\infty$ then there exist a finite
$x_{1}\geq n\pi$ where $\mathcal{L}_{1}(x)$ is monotonically increasing for
$x\in(n\pi,x_{1})$. Since we can always, again, choose $\alpha$ to be small
enough such that $x_{0}>n\pi$, then we can always choose to take $x_{1}$
inside $[0,x_{0}]$. Hence, if $d<\mathcal{L}_{1}(x)$ for all
$x\in(n\pi,x_{1})$, then $\|\mathcal{F}\|_{X}<x$.

The role of lemma 6 is obvious as the first necessary condition for
$\mathcal{F}$ to have a fixed point as a contractive mapping. Suppose that we
have an open ball $\mathcal{B}(n\pi,x)=\\{h\in X|\|h\|_{X}\in(n\pi,x)\\}$. The
fact that $\|\mathcal{F}(h)\|_{X}<\|h\|_{X}$ is always true for all
$x\in(0,x_{1})$ implies that
$\mathcal{F}:\mathcal{B}(n\pi,x)\rightarrow\mathcal{B}(n\pi,x)$. Thus, the
first necessary condition is satisfied. The role of $x_{1}$ itself is to be
the maximum possible value of $x$ (i.e. the maximum radius of the ball),
hence, the contractive property of $\mathcal{F}$ must be proven for
$\|h\|_{X}\leq x_{1}$ in which
$\mathcal{F}:\mathcal{B}(n\pi,x)\rightarrow\mathcal{B}(n\pi,x)$ is guaranteed.

The condition $d<\mathcal{L}_{1}(x)$ implies that the initial condition,
$h_{0}$, is not arbitrary and it must be small enough in order to guarantee
that the necessary condition for $\mathcal{F}$ is satisfied. This came from
the fact that even though we are free to choose $K_{1}$ and $K_{2}$, both of
them are bounded from below. Thus, the highest upper bound for $d$ actually
depends on the effective gravitational coupling constant, $\alpha$, since the
lower bound of $K_{1}$ and $K_{2}$ depend on $\alpha$ and the bound for $d$
just get smaller for a higher value of coupling constant. We would like to
remark that not all values of $\alpha$ in the interval (3.69). This is coming
from the fact that the conditions from Lemma 6 and Lemma 4 give a stronger
constraint for $\alpha$ because we need $\mathcal{L}_{1}(x)>n\pi$ for
$x\in(n\pi,x_{1})$ but this constraint depends on $n$ and should be calculated
numerically for each different $n$, but it is obvious that the upper bound for
$\alpha$ corresponding to this condition is finite and non-zero.

Now, the remaining condition needed to be satisfied is that the upper bound of
$\mathcal{L}_{1}(x)$ cannot be less than $n\pi$. We can show it by proving
that there exists $\alpha$ (which should take values in the intervals
described in the previous paragraph) such that $\mathcal{L}_{1}(n\pi)>n\pi$.
Let $\alpha_{m}$ be the value of coupling constant such that
$\mathcal{L}_{1}(n\pi)=n\pi$. If $\alpha=0$ then $\mathcal{L}_{1}(n\pi)$ is
unbounded (since $K_{2}\geq 0$). Thus, for $0\leq\alpha<\alpha_{m}$,
$\mathcal{L}_{1}(n\pi)$ is strictly higher than $n\pi$, implying that there
exist $d$ which takes values between $\mathcal{L}_{1}(n\pi)$ and $n\pi$. This
completes the proof of Lemma 6.

Since the proof for the first property of the existence of unique solutions
has been established, we proceed by showing that $\mathcal{F}$ contracts in
$Y$.

######  Lemma 7.

Suppose that $h_{1},h_{2}\in X$ solves (3.10) and both share exactly the same
initial data $h_{0}\in X_{0}$, where $X$ and $X_{0}$ defined in (3.43) and
(3.45), respectively. Let $Y$ be a space defined in (3.47) and $\mathcal{F}$
be a map defined in (3.14). There exist $\lambda\in(0,1)$ such that

$\|\mathcal{F}(h_{2})-\mathcal{F}(h_{1})\|_{Y}\leq\lambda\|h_{2}-h_{1}\|_{Y}.$
(3.85)

Proof: By following the same method demonstrated in [22] and in the previous
discussion on the local existence, we define
$\Delta\equiv\mathcal{F}(h_{2})-\mathcal{F}(h_{1})$ and take $x$ to be greater
than $\max\\{\|h_{1}\|_{X},\|h_{2}\|_{X}\\}$. Then, from (3.10) we have the
integro-differential equation for $\Delta$ along the curve $\chi$ that is
given by (3.20). By assuming that $h_{1}$ and $h_{2}$ have exactly the same
initial data, the solution of (3.20) for this setup is given by

$\Delta=\int_{0}^{u^{\prime}}\exp\left(\int_{u}^{u^{\prime}}P_{2}d\tilde{u}\right)\left[(P_{2}-P_{1})\mathcal{F}(h_{1})+\frac{\tilde{g}_{2}-\tilde{g}_{1}}{2}\mathcal{G}(h_{1})+(Q_{2}-Q_{1})\right]du.$
(3.86)

Firstly, we estimate the first term,
$\int_{0}^{u^{\prime}}(P_{2}-P_{1})\mathcal{F}(h_{1})du$, as follows: the
integrand can be estimated as

$\displaystyle(P_{2}-P_{1})\mathcal{F}(h_{1})$ $\displaystyle\leq$
$\displaystyle\left[\left|\frac{g_{2}-g_{1}-\tilde{g}_{2}+\tilde{g}_{1}}{2r}\right|+\left|\frac{\alpha}{2}r\left(2\frac{\bar{h}^{2}_{2}-\bar{h}^{2}_{1}}{r^{2}}+\frac{\bar{h}^{4}_{2}-\bar{h}^{4}_{1}}{r^{4}}\right)\right|\right.$
(3.87)
$\displaystyle+\left.\frac{d}{du^{\prime}}\left|2\frac{\bar{h}^{2}_{2}-\bar{h}^{2}_{1}}{r^{2}}\right|\right]\mathcal{F}(h_{1}).$

Let $y=\|h_{2}-h_{1}\|_{Y}$, then we have

$|\bar{h}_{2}-\bar{h}_{1}|\leq\frac{y}{(1+u)(1+r+u)},$ (3.88)

which leads to

$\displaystyle|\bar{h}^{2}_{2}-\bar{h}^{2}_{1}|$ $\displaystyle\leq$
$\displaystyle\frac{2xy}{(1+u)^{2}(1+r+u)^{2}}$ (3.89)
$\displaystyle|\bar{h}^{4}_{2}-\bar{h}^{4}_{1}|$ $\displaystyle\leq$
$\displaystyle\frac{4x^{3}y}{(1+u)^{4}(1+r+u)^{4}}$ (3.90)
$\displaystyle|g_{2}-g_{1}-\tilde{g}_{2}+\tilde{g}_{1}|$ $\displaystyle\leq$
$\displaystyle\alpha\frac{2(x+x^{3})yr^{2}}{3(1+u)^{3}(1+r+u)^{3}}.$ (3.91)

Thus, we have

$\displaystyle|(P_{2}-P_{1})\mathcal{F}(h_{1})|$ $\displaystyle\leq$
$\displaystyle\left[\alpha\frac{(x+x^{3})yr}{3(1+u)^{3}(1+r+u)^{3}}+4\alpha\frac{(x+x^{3})y}{r(1+u)^{2}(1+r+u)^{2}}\right.$
$\displaystyle+\left.\frac{d}{du^{\prime}}\left(4\frac{xy}{r^{2}(1+u)^{2}(1+r+u)^{2}}\right)\right]K_{1}\frac{d+x^{3}+x^{5}}{k^{2}(1+r+u)^{2}}\exp(K_{1}(x^{2}+x^{4})).$

As such, the estimate for the first term is given by

$\displaystyle\int_{0}^{u^{\prime}}(P_{2}-P_{1})\mathcal{F}(h_{1})du$
$\displaystyle\leq$ $\displaystyle
4\alpha\frac{K_{1}}{k^{2}}(x+x^{3})(d+x^{3}+x^{5})y$ (3.93)
$\displaystyle\exp(K_{1}(x^{2}+x^{4}))\int_{0}^{u^{\prime}}\frac{du}{(1+u)^{2}(1+r+u)^{4}}$
$\displaystyle+4\frac{K_{1}}{k^{2}}\frac{x(d+x^{3}+x^{5})y}{r^{2}(1+u^{\prime})^{2}(1+r^{\prime}+u^{\prime})^{2}}\exp(K_{1}(x^{2}+x^{4}))$
$\displaystyle\leq$ $\displaystyle 16\alpha
K_{1}\frac{(x+x^{3})(d+x^{3}+x^{5})y}{k^{4}(1+r^{\prime}+u^{\prime})^{2}}\exp(K_{1}(x^{2}+x^{4})).$

Now, for the second term,
$\int_{0}^{u^{\prime}}\frac{\tilde{g}_{2}-\tilde{g}_{1}}{2}\mathcal{G}(h_{1})du$,
we need the following relation

$\displaystyle|\tilde{g}_{2}-\tilde{g}_{1}|$ $\displaystyle\leq$
$\displaystyle\left|\frac{\alpha}{r}\int_{0}^{r}s^{2}\left(2\frac{(h_{2}-\bar{h}_{2})^{2}-(h_{1}-\bar{h}_{1})^{2}}{s^{2}}+\frac{(h_{2}-\bar{h}_{2})^{4}-(h_{1}-\bar{h}_{1})^{4}}{s^{4}}\right)ds\right|$
(3.94) $\displaystyle+|\bar{g}_{2}-\bar{g}_{1}|$ $\displaystyle\leq$
$\displaystyle\frac{\alpha}{r}\int_{0}^{r}s^{2}\left(2\frac{xy}{2(1+u)^{2}(1+r+u)^{4}}+\frac{x^{3}y}{4(1+u)^{4}(1+r+u)^{8}}\right)ds$
$\displaystyle+|\bar{g}_{2}-\bar{g}_{1}|$

which is deduced from equation (3.7). This inequality requires estimate of
$|\bar{g}_{2}-\bar{g}_{1}|$ which can be deduced from (3.6). To do so,
consider the following relation

$\displaystyle|g_{2}-g_{1}|$ $\displaystyle\leq$
$\displaystyle|h_{2}-h_{1}|\left|\frac{\partial g}{\partial
h}\right|_{\max(h_{2},h_{1})}$ (3.95) $\displaystyle\leq$ $\displaystyle\alpha
y\left|\int_{r}^{\infty}s\frac{x^{2}}{8(1+u)^{2}(1+s+u)^{4}}\left(1+\frac{x^{2}}{2(1+u)^{2}(1+s+u)^{4}}\right)ds\right|$
$\displaystyle\leq$
$\displaystyle\alpha\frac{(x^{2}+x^{4})y}{16(1+u)^{2}(1+r+u)^{2}},$

that will be used to estimate $|\bar{g}_{2}-\bar{g}_{1}|$ as follows,

$\displaystyle|\bar{g}_{2}-\bar{g}_{1}|$ $\displaystyle=$
$\displaystyle\left|\frac{1}{r}\int_{0}^{r}(g_{2}-g_{1})ds\right|\leq\frac{1}{r}\int_{0}^{r}|g_{2}-g_{1}|ds$
(3.96) $\displaystyle\leq$
$\displaystyle\alpha\frac{(x^{2}+x^{4})y}{16(1+u)^{3}(1+r+u)}.$

Now that we have the estimate of $|\bar{g}_{2}-\bar{g}_{1}|$, we can proceed
to estimate the second term as follows

$\displaystyle\int_{0}^{u^{\prime}}\frac{\tilde{g}_{2}-\tilde{g}_{1}}{2}\mathcal{G}(h_{1})du$
$\displaystyle\leq$
$\displaystyle\int_{0}^{u^{\prime}}\alpha\frac{(x^{2}+x^{4})y}{32(1+u)^{3}(1+s+u)}$
$\displaystyle\left(K_{2}\frac{d+x^{3}+x^{4}+x^{5}+x^{7}+x^{9}}{k^{3}(1+r^{\prime}+u^{\prime})^{3}}\exp\left(K_{1}(x^{2}+x^{4})\right)\right)du$
$\displaystyle\leq$
$\displaystyle\frac{\alpha}{2}K_{2}\frac{(x^{2}+x^{4})(d+x^{3}+x^{4}+x^{5}+x^{7}+x^{9})}{k^{5}(1+r^{\prime}+u^{\prime})^{2}}\exp\left(K_{1}(x^{2}+x^{4})\right).$

The last term (third) to be estimated is
$\int_{0}^{u^{\prime}}\left(Q_{2}-Q_{1}\right)du$. First consider the
$|Q_{2}-Q_{1}|$ that satisfies

$\displaystyle|Q_{2}-Q_{1}|$ $\displaystyle\leq$
$\displaystyle\left|\frac{\bar{h}_{2}(g_{2}-\tilde{g}_{2})-\bar{h}_{1}(g_{1}-\tilde{g}_{1})}{2r}\right|+\left|\frac{\alpha}{2}r\left(3\frac{\bar{h}^{3}_{2}-\bar{h}^{3}_{1}}{r^{2}}+\frac{\bar{h}^{5}_{2}-\bar{h}^{5}_{1}}{r^{4}}\right)\right|$
(3.98)
$\displaystyle+\frac{d}{du^{\prime}}\left|2\frac{\bar{h}^{3}_{2}-\bar{h}^{3}_{1}}{r^{2}}\right|$
$\displaystyle\leq$ $\displaystyle
4\alpha\frac{(x^{2}+x^{4})y}{(1+u)^{3}(1+r+u)^{3}}+\frac{d}{du^{\prime}}\left|4\frac{x^{2}y}{(1+u)^{3}(1+r+u)^{5}}\right|.$

As such, the third term can be estimated as

$\displaystyle\int_{0}^{u^{\prime}}\left(Q_{2}-Q_{1}\right)du$
$\displaystyle\leq$ $\displaystyle
4\alpha(x^{2}+x^{4})y\int_{0}^{u^{\prime}}\frac{du}{(1+u)^{3}(1+r+u)^{3}}+4\frac{x^{2}y}{(1+u^{\prime})^{3}(1+r^{\prime}+u^{\prime})^{5}}$
(3.99) $\displaystyle\leq$ $\displaystyle
8\alpha\frac{(x^{2}+x^{4})y}{k^{2}(1+r^{\prime}+u^{\prime})^{2}}.$

By combining all of the three estimates of the three terms we have a decay
estimate for $\Delta$, namely

$|\Delta(u^{\prime},r^{\prime})|\leq\tilde{K}_{2}\frac{(\sigma_{1}(x)+\sigma_{2}(x)+x^{2}+x^{4})y}{k^{5}(1+r^{\prime}+u^{\prime})^{2}}\exp\left(\tilde{K}_{1}(x^{2}+x^{4})\right),$
(3.100)

where $\tilde{K}_{1}\geq 8\alpha$, $\tilde{K}_{2}\geq 7\alpha^{3}$, and the
two functions, $\sigma_{1}(x)$ and $\sigma_{2}(x)$ is given by

$\displaystyle\sigma_{1}(x)$ $\displaystyle\equiv$
$\displaystyle(x+x^{3})(d+x^{3}+x^{5})$ (3.101) $\displaystyle\sigma_{2}(x)$
$\displaystyle\equiv$
$\displaystyle(x^{2}+x^{4})(d+x^{3}+x^{4}+x^{5}+x^{7}+x^{9}).$ (3.102)

Since the decay estimate of $\Delta$ goes like
$(1+r^{\prime}+u^{\prime})^{-2}$, we can conclude that the norm of $\Delta$ in
$Y$ satisfies

$\|\Delta\|_{Y}\leq\mathcal{L}_{2}(x)y,$ (3.103)

where $\mathcal{L}_{2}(x)$ is defined as

$\mathcal{L}_{2}(x)\equiv\frac{\tilde{K}_{2}}{k^{5}}(\sigma_{1}(x)+\sigma_{2}(x)+x^{2}+x^{4})\exp\left(\tilde{K}_{1}(x^{2}+x^{4})\right).$
(3.104)

The function $\mathcal{L}_{2}(x)$ is monotonically increasing for all
$x\in[n\pi,x_{1}]$ and $\mathcal{L}_{2}(n\pi)>0$. Thus, there exist
$x_{2}\in(n\pi,x_{1}]$ and $\alpha$ which falls in the interval (3.69) such
that $\mathcal{L}_{2}(x)<1$ for all $x\in(n\pi,x_{2})$. Furthermore, the
smaller value of $\alpha$ leads to a smaller $\mathcal{L}_{2}(x)$ as well. By
taking $\lambda=\mathcal{L}_{2}(x)$ for $\|h\|_{X}<x_{2}$, we prove that
$\mathcal{F}$ contracts in $Y$. This completes our proof of the global
existence of the Einstein-Skyrme system for small initial data.

##  4 Singularity Formations

From our previous discussion, we have shown that the existence of global
smooth solutions can be guaranteed for only small initial data. Thus, there
are still large varieties of possible regular initial conditions that could
lead to singularities at a finite time. The reason why we are interested in
regular initial conditions is that we can model a gravitational collapse of a
Skyrme star since we already know that in the static limit, we have a lot of
varieties of Skyrme black holes and even a naked singularity inside a star
solution [11, 28, 12, 28].

It is known that the Einstein-Klein-Gordon system in which the scalar field
has only kinetic term does allow a singularity in finite time for regular
initial conditions which leads to the formation of naked singularities from a
scalar star [20]. This singularity turns out to be unstable, thus, it is not
physically possible to be observed and cannot be a counter-example of the
cosmic censorship conjecture [21]. Since the non-topological sector ($B=0,$ or
$A_{3}=A_{4}=0$) of the spherically symmetric Skyrme model we consider in this
work coincides with the Einstein-Klein-Gordon system, it is interesting to
study the behaviour of the topological sector ($B\neq 0$ or
$|A_{3}|=|A_{4}|=1$) and its singularity developments.

The main idea to prove that singularity might develop within this system is by
looking at the value of the Skyrmion profile function, $\xi$, at the origin,
$r=0$. Since, the Skyrme field $\phi(\vec{x})$ within this choice of ansatz,
given in (2), must be unique for every $\vec{x}$, then the value of Skyrmion
profile function cannot be arbitrary at the origin. It has been proven that in
the static limit $\xi$ must be a multiple of $\pi$ for non-gravitating cases
(flat spacetime background) [1, 2]. Furthermore, for a static Skyrme blackhole
solution where we can only determine the possible values of $\xi$ at the
blackhole horizon, it is hinted that the unique property of $\phi$ at the
origin, found in the non-gravitating case, cannot be satisfied [11, 12]. Thus,
finding such indications is sufficient to show that this Einstein-Skyrme
system develops singularities. In order to do this, we consider the self-
similar solutions that are known to produce singularity at spacetime origin
for some known scalar field theories, for example, the Einstein-Klein-Gordon
model in Bondi coordinate [20] and $SU(2)$ sigma model in flat spacetime [29].
We expect that there exist a family of self-similar solutions within the
Einstein-Skyrme system which develops singularity in finite time.

One of the main problems in analyzing the self-similar solutions of the
Einstein-Skyrme system is that the two terms in the Skyrme model Lagrangian
scales differently under global spatial scaling
$\vec{x}\rightarrow\mu\vec{x}$. In $3+1$ dimensional spacetime the quadratic
term scales like $\mu$ and the quartic term, scales like $\mu^{-1}$. Thus, the
resulting dynamical equation is not scale invariant. This fact is well
represented in (3.10) which cannot be reduced into self-similar form due to
the difference in scaling behaviour of each term.

The approach we introduce here to circumvent this problem is to consider the
singularity development of each submodel containing only one of the terms from
the dynamical equation. This can be achieved by taking $C_{2}=0$ for the
quadratic submodel, and $C_{1}=0$ for the quartic submodel. In this approach,
the resulting terms in dynamical equations of each submodel transforms
similarly under coordinate scaling transformation. Thus, we can recast them
into self-similar forms.

###  4.1 Singularities in Quadratic Submodel

The submodel we consider here is actually a sigma model possessing the first
invariant of strain tensor induced by the field
$\phi:\mathcal{M}^{1,3}\rightarrow S^{3}$ with spherically symmetric ansatz
[26]. The dynamical equation of Skyrmion field in this submodel is given by

$re^{-2\sigma}\left(\xi_{,r}\left(-r\rho_{,r}+r\sigma_{,r}-2\right)+2\left(\xi_{,u}+r\xi_{,ur}\right)e^{\sigma-\rho}-r\xi_{,rr}\right)+\sin(2\xi)=0,$
(4.1)

with Einstein’s Equations governing the dynamics of $\rho$ and $\sigma$, given
by

$\displaystyle r\left(\rho_{,r}+\sigma_{,r}\right)$ $\displaystyle=$
$\displaystyle\alpha r^{2}\xi_{,r}^{2}\leavevmode\nobreak\ ,$ (4.2)
$\displaystyle e^{2\sigma}-1-r\left(\rho_{,r}-\sigma_{,r}\right)$
$\displaystyle=$ $\displaystyle 2\alpha e^{2\sigma}\sin^{2}\xi.$ (4.3)

We can see that the system of equations is now scale invariant. To recast
these equations into a self-similar form we introduce $t=\frac{-2r}{u}$ such
that region $t>0$ lies in the past of apparent horizon [20]. It is also
advantageous to recast $\\{\rho,\sigma\\}$ into $\\{g,\tilde{g}\\}$ as defined
in (3.3,3.4) such that we can get rid of any possibility of dealing with
complex functions behind the horizon by imposing both $g$ and $\tilde{g}$ to
be at least $C^{1}$-real functions of $t$. The system of self-similar
equations is given by

$\displaystyle\left[t^{3}-t^{2}\tilde{g}\right]\xi^{\prime\prime}+\left[2t^{2}-(2t\tilde{g}+t^{2}\tilde{g}^{\prime})\right]\xi^{\prime}+g\sin\left(2\xi\right)$
$\displaystyle=$ $\displaystyle 0,$ (4.4) $\displaystyle g^{\prime}-\alpha
gt\xi^{\prime}{}^{2}$ $\displaystyle=$ $\displaystyle 0,$ (4.5)
$\displaystyle\tilde{g}^{\prime}-\frac{g-\tilde{g}}{t}+2\alpha
g\frac{\sin^{2}\xi}{t}$ $\displaystyle=$ $\displaystyle 0.$ (4.6)

From this point, we assume that the self-similar solution $\xi(t)$ is at least
a $C^{2}$-real function. Let us define the first derivative of $\xi$ to be
$\psi\equiv\xi^{\prime}$ and a new vector
$\textbf{w}\equiv\begin{bmatrix}\xi\\\ \psi\end{bmatrix}$, $\textbf{w}\in
U\subset\mathbb{R}^{2}$, such that the following lemma holds in closed
interval $\mathcal{I}=[t,t+\epsilon]$, $\epsilon>0$

######  Lemma 8.

The metric functions $g(\textbf{w},t)$ and $\tilde{g}(\textbf{w},t)$, that
satisfies equations (4.5) and (4.6), are locally Lipschitz with respect to w.

Proof: The Einstein equations (4.5) and (4.6) can be rewritten into integral
form as follows

$\displaystyle\ln g$ $\displaystyle=$ $\displaystyle\alpha\int
t^{\prime}\xi^{\prime}{}^{2}\leavevmode\nobreak\ dt^{\prime},$ (4.7)
$\displaystyle t\tilde{g}$ $\displaystyle=$ $\displaystyle\int
g\left[1-2\alpha\sin^{2}\xi\right]\leavevmode\nobreak\ dt^{\prime}.$ (4.8)

We can see that (4.7) implies $g>0$. Now, consider a solution for $g$ and a
solution for $\tilde{g}$ in the closed interval $\mathcal{I}$. Integrating
(4.7) and (4.8) in the whole domain gives us the relation

$\displaystyle|g(t+\epsilon)|_{U}$ $\displaystyle\leq$
$\displaystyle|g(t)|_{U}\exp\left[C|\alpha t\psi^{2}|\right],$ (4.9)
$\displaystyle|\tilde{g}(t+\epsilon)|_{U}$ $\displaystyle\leq$
$\displaystyle|\tilde{g}(t)|_{U}+C\left|\frac{g}{t}\left[1-2\alpha\sin^{2}\xi\right]\right|$
(4.10)

with $C>\epsilon$, implying that both $g$ and $\tilde{g}$ are bounded. Then
for $\textbf{w}_{1},\textbf{w}_{2}\in U$ we have

$\displaystyle|g(\textbf{w}_{1},t)-g(\textbf{w}_{2},t)|_{U}$
$\displaystyle\leq$ $\displaystyle\left|\exp\left[C|\alpha
t\psi_{1}^{2}|\right]-\exp\left[C|\alpha t\psi_{2}^{2}|\right]\right|,$ (4.11)
$\displaystyle|\tilde{g}(\textbf{w}_{1},t)-\tilde{g}(\textbf{w}_{2},t)|_{U}$
$\displaystyle\leq$ $\displaystyle
Cg(\textbf{w},t)\left|\frac{2\alpha}{t}\left(\sin^{2}\xi_{1}-\sin^{2}\xi_{2}\right)\right|$
(4.12)

Where $\textbf{w}\in\\{\textbf{w}_{1},\textbf{w}_{2}\\}$ which maximizes
$g(\textbf{w},t)$. Since $g$ and $\tilde{g}$ are bounded and are smooth
functions of w, then we can apply a local property

$|F(f_{1})-F(f_{2})|\leq\sup_{s\in[0,1]}\left[F^{\prime}(f_{1}+s(f_{2}-f_{1}))\right](f_{1}-f_{2}),$
(4.13)

to see that $g$ and $\tilde{g}$ are locally Lipschitz with Lipschitz constants
that are functions of $\textbf{w}\in U$,

$\displaystyle|g(\textbf{w}_{1},t)-g(\textbf{w}_{2},t)|_{U}$
$\displaystyle\leq$ $\displaystyle
C_{g}(\|\textbf{w}_{1}\|,\|\textbf{w}_{2}\|)\|\textbf{w}_{1}-\textbf{w}_{2}\|,$
(4.14)
$\displaystyle|\tilde{g}(\textbf{w}_{1},t)-\tilde{g}(\textbf{w}_{2},t)|_{U}$
$\displaystyle\leq$ $\displaystyle
C_{\tilde{g}}(\|\textbf{w}_{1}\|,\|\textbf{w}_{2}\|)\|\textbf{w}_{1}-\textbf{w}_{2}\|$
(4.15)

Both $C_{g}$ and $C_{\tilde{g}}$ are finite real functions of the norms of
$\textbf{w}_{1}$ and $\textbf{w}_{2}$.

From here, to check the existence of unique solutions, we rewrite (4.4) into

$\frac{d}{dt}\textbf{w}=\mathcal{J}_{1}(\textbf{w},t),$ (4.16)

where we have defined

$\mathcal{J}_{1}(\textbf{w},t)\equiv\begin{bmatrix}\psi\\\
J_{1}\end{bmatrix},$ (4.17)

with

$J_{1}=\left(\frac{g+\tilde{g}-2t}{t(t-\tilde{g})}-2\alpha
g\frac{\sin^{2}\xi}{t(t-\tilde{g})}\right)\psi-\left(\frac{g}{t^{2}(t-\tilde{g})}\right)\sin(2\xi).$
(4.18)

Note that the resulting system of equations is for $\xi$ that is constrained
by equations of metric function from Lemma (8). We can now state the local
property of (4.16) as follows

######  Lemma 9.

The operator $\mathcal{J}_{1}(\textbf{w},t)$ in (4.16) is locally Lipschitz
with respect to w.

Proof: From the definition of $J_{1}$ we know that for every $\textbf{w}\in U$
we can estimate

$|J_{1}|_{U}\leq\left|\frac{g+\tilde{g}}{t(t-\tilde{g})}\psi\right|+2\alpha
g\left|\frac{\sin^{2}\xi}{t(t-\tilde{g})}\psi\right|+\left|\frac{g}{t^{2}(t-\tilde{g})}\sin(2\xi)\right|.$
(4.19)

Because $\xi(t)$ belongs to the class of $C^{2}$-real functions, then it is
bounded on any closed $\mathcal{I}$. This implies that
$|\mathcal{J}_{1}(\textbf{w},t)|$ is bounded in $U$. Then, for
$\textbf{w}_{1},\textbf{w}_{2}\in U$ we have

$\displaystyle\|\mathcal{J}_{1}(\textbf{w}_{1},t)-\mathcal{J}_{1}(\textbf{w}_{2},t)\|_{U}$
$\displaystyle\leq$
$\displaystyle\left|1+\frac{g+\tilde{g}}{t(t-\tilde{g})}\right||\psi_{1}-\psi_{2}|$
$\displaystyle+\left|\frac{2\alpha
g}{t(t-\tilde{g})}\right||\psi_{1}\sin^{2}\xi_{1}-\psi_{2}\sin^{2}\xi_{2}|$
$\displaystyle+\left|\frac{g}{t^{2}(t-\tilde{g})}\right||\sin(2\xi_{1})-\sin(2\xi_{2})|,$

with $g=g(\textbf{w},t)$ and $\tilde{g}=\tilde{g}(\textbf{w},t)$ where
$\textbf{w}\in\\{\textbf{w}_{1},\textbf{w}_{2}\\}$ which maximizes
$g+\tilde{g}$. From Lemma (8) we know that the metric functions are locally
Lipschitz. Thus, by employing the property (4.13) for operator
$\mathcal{J}_{1}(\textbf{w},t)$ we have

$\|\mathcal{J}_{1}(\textbf{w}_{1},t)-\mathcal{J}_{1}(\textbf{w}_{2},t)\|_{U}\leq
C_{\mathcal{J}_{1}}(\|\textbf{w}_{1}\|,\|\textbf{w}_{2}\|)\|\textbf{w}_{1}-\textbf{w}_{2}\|.$
(4.21)

This inequality shows that $\mathcal{J}_{1}(\textbf{w},t)$ is locally
Lipschitz with a Lipschitz constant that is a function of w.

######  Remark 2.

We have two conditions that is to be satisfied in order to extend the validity
of Lemma (9) for intervals that contains $t=0$, namely

  * •

$\xi(t=0)=n\pi$, with $n\in\mathbb{Z}$, and

  * •

$\lim_{t\rightarrow 0}\psi=0$ or $\lim_{t\rightarrow 0}(g+\tilde{g})=0$.

The condition $\lim_{t\rightarrow 0}\psi=0$ gives vacuum solutions and the
other condition, $\lim_{t\rightarrow 0}(g+\tilde{g})=0$, implies changes of
sign in metric tensor components which indicate the presence of horizon near
the singularity.

The explanation for two conditions given above is straightforward. Firstly, if
the first condition is satisfied, then $|\psi(t)-\psi(0)|=|\psi(t)|\leq C|t|$
because $\xi(t)$ belongs to $C^{2}$-real functions. Then, secondly, if the
second condition holds true and the condition for $\xi(t=0)$ is satisfied,
then the first term of (4.19) is bounded, and the rest of the terms containing
$\sin\xi$ is bounded. Note that we only need one of the optional conditions to
ensure the boundedness of $|J_{1}|_{U}$. As such, if one of them is satisfied
then Lemma (9) can be extended to intervals that contains $t=0$. Since vacuum
solutions will not develop any singularities, we are now assume that the
second optional condition holds and $\psi(0)=\psi_{0}\neq 0$.

Now, consider the solution of (4.16) in $\mathcal{I}$ that is given by

$\textbf{w}(\tilde{t})=\textbf{w}(t)+\int_{t}^{\tilde{t}}\mathcal{J}_{1}(\textbf{w},t^{\prime})dt^{\prime},$
(4.22)

with $\tilde{t}\in\mathcal{I}$. By Introducing a Banach Space

$W\equiv\\{\textbf{w}\in
C(\mathcal{I},\mathbb{R}^{2}):\textbf{w}(t)=\textbf{w}_{0},\sup_{\tilde{t}\in\mathcal{I}}|\textbf{w}(\tilde{t})|\leq
L_{0}\\},$ (4.23)

equipped with norm

$\|\textbf{w}\|_{W}=\sup_{\tilde{t}\in\mathcal{I}}|\textbf{w}(\tilde{t})|,$
(4.24)

where $L_{0}$ is a positive constant, we define an operator
$\mathcal{K}:W\rightarrow W$,

$\mathcal{K}(\textbf{w}(\tilde{t}))=\textbf{w}_{0}+\int_{t}^{\tilde{t}}\mathcal{J}_{1}(\textbf{w},t^{\prime})dt^{\prime}.$
(4.25)

As such, Lemma (9) implies the uniqueness of the solution of the differential
equation (4.16).

######  Corollary 1.

The operator $\mathcal{K}$ defined in (4.25) is a map from $W$ to itself and
is a contraction mapping for $w\in W$ that is defined in a closed interval
$\mathcal{I}=[t,t+\epsilon]$, $t\in\mathbb{R}$, $\epsilon>0$, and

$\epsilon\leq\min\left[\frac{1}{C_{L_{0}}},\frac{1}{C_{L_{0}}L_{0}+\|\mathcal{J}_{1}(t)\|}\right].$
(4.26)

Thus, there exists a unique smooth solution $\textbf{w}\in W$ of differential
equation (4.16).

We can construct the maximal solution from the solutions we consider above up
to any finite $t_{m}>0$ as follows: Suppose that $\textbf{w}(t)$ be a solution
on the interval $[0,t_{m}]$ with initial values
$\textbf{w}(0)=\begin{bmatrix}n\pi\\\ \psi_{0}\end{bmatrix}$,
$n\in\mathbb{Z}$. We can repeat the arguments given above with new initial
values on $\textbf{w}(t-t_{k})$ where $0<t_{k}<t_{m}$ and then proceed to
employ the uniqueness condition to glue the resulting solutions up to
$t=t_{m}$. From here, we are ready to state the theorem on the singularity
formations of the first submodel in the Einstein-Skyrme system.

######  Theorem 3.

There exists smooth initial data for the quadratic submodel of the Einstein-
Skyrme system in the Bondi coordinates that develops singularity at the
coordinate origin in finite $u$.

Proof : In order to prove that such singularity could develop, we shall use
the self-similar solutions above as an example of initial data. The
spherically symmetric Einstein-Skyrme system’s dynamics in Bondi coordinate is
governed by three equations for three dynamical fields, namely equation (4.1),
(4.2), and (4.3). Let $[\xi_{0}(r),g_{0}(r),\tilde{g}_{0}(r)]$ be smooth
functions that are the solutions of system of equations (4.4), (4.5), and
(4.6), satisfying a set of initial values given by

$\xi_{0}(0)=0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\xi_{0}^{\prime}(0)=\psi_{0}\neq 0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ g_{0}(0)+\tilde{g}_{0}(0)=0,$ (4.27)

where $\psi_{0}$ is an arbitrary non-zero real number. Let
[$\xi(u,r),g(u,r),\tilde{g}(u,r)$] be the solutions of the system (4.1),
(4.2), and (4.3), with initial data at $u=-2$ given by

$\xi(-2,r)=\xi_{0}(r),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\partial_{u}\xi(-2,r)=\frac{r}{2}\partial_{r}\xi_{0}(r),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
g(-2,r)=g_{0}(r),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \tilde{g}(-2,r)=\tilde{g}_{0}(r).$ (4.28)

Then, in the past light-cone extended from $u=-2$ to the vertex $(u=0,r=0)$,
the solution of $\xi(u,r)$ is given by

$\xi(u,r)=\xi_{0}\left(-\frac{2r}{u}\right),$ (4.29)

hence satisfies

$\partial_{r}\xi(u,0)=-\frac{2}{u}\partial_{r}\xi_{0}(0).$ (4.30)

Because $\partial_{r}\xi_{0}(0)=\psi_{0}\neq 0$ then
$\partial_{r}\xi(u,0)\rightarrow\infty$ as $u\rightarrow 0$. From Einstein’s
equation, we can see that the singular point for $\xi$ is also a singular
point for both $g$ and $\tilde{g}$, resulting in a singularity for the metric
functions as well. We would like to remark that the singularity from the self-
similar solution demonstrated above is hidden behind a horizon (see Remark 2),
in contrast with the one found from a similar setup for the Einstein-Klein-
Gordon system where the singularity is naked (does not located inside the
horizon) [20, 21]. Interestingly, the solutions keep the topological charge to
be $B=\pm n$ during the process of singularity formation because
$\lim_{r\rightarrow 0}\xi=n\pi$ for all $u\in[-2,0)$. The value of the
topological charge might change after the singularity is formed, $u=0$, but
the solutions should behave differently and the governing equations (4.1-4.3)
cannot be used anymore due to the existence of horizon. This might be related
to the fact that the Skyrme black holes usually possess a fractional
topological charge [5, 6, 10].

###  4.2 Remarks on The Quartic Submodel

In this subsection, we are going to discuss the singularity formation for the
second submodel. As mentioned before, the different scaling behaviour of the
quartic (Skyrme) term with the rest of the terms in the field equations raises
a problem when we try to introduce the same self-similar form from the
previous case for the quadratic submodel. To see this explicitly, consider the
following expression of Einstein equations for the quartic submodel

$\displaystyle r(\rho_{,r}+\sigma_{,r})$ $\displaystyle=$ $\displaystyle
2\alpha\xi_{,r}^{2}\sin^{2}\xi,$ (4.31) $\displaystyle
e^{2\sigma}-1-r(\rho_{,r}-\sigma_{,r})$ $\displaystyle=$ $\displaystyle\alpha
e^{2\sigma}\frac{\sin^{4}\xi}{r^{2}}.$ (4.32)

We can see that, if we impose the same assumption that $\sin\xi$ is invariant
under rescaling of $r$ then the right-hand and left-hand side of both
equations above scales differently, but this is not the case for the Skyrme
field equation that is given by

$\displaystyle-2(\cos\xi)_{,ur}+g\sin^{2}\xi\frac{\cos\xi}{r^{2}}+\left[\tilde{g}_{,r}(\cos\xi)_{,r}+\tilde{g}(\cos\xi)_{,rr}\right]=0.$
(4.33)

In order to keep the same form of solution as the previous submodel, here we
can only consider the limit $\alpha=0$. Thus, the Einstein equations after
substitution of $g$ and $\tilde{g}$, and then followed by assumption that both
$g$ and $\tilde{g}$ are functions of $t=\frac{-2r}{u}$, are now given by

$g^{\prime}=0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{g}^{\prime}=\frac{g-\tilde{g}}{t}.$ (4.34)

The first implication is that $g$ is a constant that should be chosen as 1
since the metric components are flat at the asymptotic boundary. Then, the
second implication of the Einstein equation shown above is that $\tilde{g}$
takes a solution that blows up at $t=0$, namely

$\tilde{g}=1-\frac{l}{t}.$ (4.35)

with $l$ arbitrary integration constant. As such, $r=0$ with arbitrary $u$ is
singular points of the metric functions.

Now, the substitution of this form of the metric function to the field
equation of this submodel gives

$\displaystyle-\left(v^{\prime}+tv^{\prime\prime}\right)+(1-v^{2})\frac{v}{t^{2}}+\left[\frac{l}{t^{2}}v^{\prime}+\left(1-\frac{l}{t}\right)v^{\prime\prime}\right]=0.$
(4.36)

where we have defined a new function $v(t)\equiv\cos(\xi(t))$ such that $v$
takes values in $[-1,1]$. We can recast the second-order differential equation
of $v$ into two first-order equation by introducing
$\textbf{V}=\begin{bmatrix}v\\\ z\end{bmatrix}$ with $z=v^{\prime}$, such that
the resulting system f differential equation is given by

$\frac{d}{dt}\textbf{V}=\mathcal{J}_{2}(\textbf{V},t),$ (4.37)

where we have defined

$\mathcal{J}_{2}(\textbf{w},t)\equiv\begin{bmatrix}v\\\ J_{2}\end{bmatrix},$
(4.38)

with

$J_{2}=\left(\frac{t^{2}-l}{t(t-l-t^{2})}\right)z-\left(\frac{1}{t(t-l-t^{2})}\right)\left(1-v^{2}\right)v.$
(4.39)

Because $l$ is arbitrary, we have three possible cases to be considered. The
first one is $l>1/4$ where the possibility of $J_{2}$ become unbounded is
located only at $t=0$. The second one is $l=0$ where $J_{2}$ possibly become
unbounded at $t=0$ and $t=1$. The last one is $0<l\leq 1/4$ where we have
three possible singular points for $J_{2}$, implying that we might need a
strong condition for V in order to make $J_{2}$ bounded for every possible
interval.

The resulting submodel does not belong to the Einstein-Skyrme system anymore
since we get rid of the back-reaction between gravity and the Skyrme field by
choosing $\alpha=0$ and further details of this submodel are going to be
addressed in future works.

##  5 Conclusions and Outlook

We have considered the spherically symmetric Einstein-Skyrme system with
ansatz that depends on the radial coordinate, $r$, and retarded time, $u$. The
resulting system of dynamical equations consists of an integro-differential
equation for the Skyrme field profile function constrained by two equations
from the Einstein field equations governing the profile of metric functions.
Firstly, It is shown that unique local classical solutions exist with bounded
time intervals that depend on the norm of the initial data, shown in Theorem
1. The corresponding solutions cannot be extended into global solutions
because the time interval cannot be extended to infinity due to the upper
bound which comes from the topology of the Skyrmion. Thus the proof of the
existence of unique global solutions requires a different strategy.

Secondly, we show that in order for the global unique solutions to exist, the
initial data and and the coupling constant, $\alpha$, cannot be arbitrary. The
corresponding solutions and their first derivative with respect to $r$ are
bounded above by decaying functions stated in Theorem 2. Furthermore, we have
shown that there exist some possible configurations of initial data within the
Einstein-Skyrme system that produce singularity at the light cone vertex. As
expected, this singular solution does not belong to the family of global
unique solutions from Theorem 2.

These results open up the possibility of studying the dynamical stability of
the Einstein-Skyrme system, provided the value of the norm of initial data and
the coupling constant are inside the allowed intervals in order to make sure
that the classical solutions exist. Unfortunately, if we consider the more
general case of unrestricted initial data, then the existence of global
classical solutions is not guaranteed, as implied by Theorem 1 and Theorem 3.
However, it is actually possible to circumvent this problem by introducing the
concept of a generalized solution for the Einstein-Klein-Gordon system
proposed in [19]. Some modifications to the concept of generalized solutions
mentioned above are expected in order to apply it to the Einstein-Skyrme
system since we have internal symmetries and topology from the Skyrme fields.
Thus, we are going to address these problems in the future.

## Acknowledgements

The work in this paper is supported by GTA Research Group ITB, Riset ITB 2022,
and PDUPT Kemenristek-ITB 2022. A. N. A. would like to acknowledge the support
from the ICTP through the Associates Programme (2018-2023). B. E. G. would
also acknowledge the support from the ICTP through the Associates Programme
(2017-2022) and BRIN Visiting Researcher Programme 2023. E. S. F. also would
like to acknowledge the support from BRIN through the Research Assistant
Programme 2022.

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