∎

e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>
11institutetext: Department of Physics, Eastern Mediterranean University, G.
Magusa, north Cyprus, Mersin 10, Turkey 22institutetext: Institute of
Theoretical Physics, Faculty of Mathematics and Physics, Charles University in
Prague, V Holešovičkách 2, 180 00 Prague 8, Czech Republic

# Resolution of curvature singularities from quantum mechanical and loop
perspective

T. Tahamtane1,addr1  O. Svíteke2,addr2

(Received: date / Accepted: date)

###### Abstract

We analyze the persistence of curvature singularities when analyzed using
quantum theory. First, quantum test particles obeying the Klein-Gordon and
Chandrasekhar-Dirac equation are used to probe the classical timelike naked
singularity. We show that the classical singularity is felt even by our
quantum probes. Next, we use loop quantization to resolve singularity hidden
beneath the horizon. The singularity is resolved in this case.

###### Keywords:

Singularity resolution global monopole loop quantization

##  1 Introduction

One of the important predictions of the Einstein’s theory of general
relativity is the formation of spacetime singularities. In classical general
relativity, singularities are defined as points in which the evolution of
timelike or null geodesics is not defined after a finite proper time.
According to the classification of the classical singularities devised by
Ellis and Schmidt 01 , scalar curvature singularities are the most strongest
one in the sense that the spacetime posses incomplete geodesics ending in them
and all the physical quantities such as the gravitational field (scalars
formed from curvature tensor), energy density and tidal forces diverge at the
singular point.

But such divergence of physical quantities signify the breakdown of predictive
power of classical general relativity. If these singularities are covered by
horizon (as supposed by Cosmic Censorship Conjecture) then at least the
physically most relevant region of spacetime is under control. Naked
singularities (those not covered by horizon), on the other hand, provide an
observer with causal access to the region of diverging quantities and should
be avoided. However, even singularities covered by the horizon can be accessed
by an infalling observer and, more importantly, we would like to have a theory
that lacks divergences, at least effectively.

The natural direction for resolving the problem of singularities in classical
theory is investigating their persistence in quantum picture. Although we do
not have a final quantum theory of gravity we still have several tools for
analyzing quantum singularities. The first approach relies on examining
properties of quantum particle wave functions on the background represented by
the studied geometry. This is a frequently used technique based on well
understood properties of operators on a Hilbert space. To move further, one
might proceed to using quantum fields and possibly even the backreaction of
background geometry using semiclassical Einstein equations with suitably
regularized stress energy tensor. Finally, one can apply quantization of the
geometry itself. The last approach is in principle the most precise but relies
on the selected quantization method and we have no generally accepted one in
case of gravity.

Quantum singularities were studied for different specific situations (and
using also generalizations), mainly using the first approach 6 ; 7 ; 8 ; 9 ;
10 ; 11 ; 12 ; 13 ; 140 ; 14 ; 15 ; 16 ; 17 . Recently, singularities in f(R)
gravity were investigated in the presence of linear electromagnetic field 170
.

We will apply two of the above mentioned approaches for analysis of
singularity in case of the general metric of global monopole 1 , which is
determined by two parameters - one characterizing the "Schwarzschild-type
mass" and the other one the deficit of solid angle. The singularity is
generally covered by single horizon but the class of metrics also contains, as
a special case, a naked singularity which is analyzed from quantum mechanical
point of view using the technique of Horowitz and Marolf 4  (who continued the
pioneering work of Wald 3 ). This method for analyzing timelike singularities
is based on investigation of self-adjoint extensions of the evolution operator
associated with the given wave equation. If it is unique the spacetime is
deemed quantum mechanically non-singular. The analysis is carried out for
relativistic quantum particle wave equations on a fixed background.
Specifically, we review the previous results for Klein-Gordon equation and
show the calculation using Newman-Penrose formalism for the Dirac equation,
both in the case of pure global monopole with naked singularity for which the
method was developed.

But as already mentioned, the most reliable method when trying to investigate
the possible removal of the singularities from geometry is quantum gravity.
Here we have selected loop quantization method inspired by 22 ; 23 ; 24 ,
where the spacetime beneath the horizon (in the non-naked subclass) is
isometric to the Kantowski-Sachs cosmology. Then one can apply methods from
Loop Quantum Cosmology (LQC), that are based on loop quantization on the
restricted configuration space. In this way, the results for resolution of
initial cosmological singularity are translated to statements about the
singularity at the origin $r=0$.

##  2 The General Metric for Global monopole

It is well known that different types of non-standard topological objects may
have been formed during initial Universe evolution, such as domain walls,
cosmic strings and monopoles 1 ; 2 . The basic idea is that these topological
defects have formed as a result of a breakdown of local or global gauge
symmetries. The simplest model that gives rise to global monopole is described
by the Lagrangian

$L=\frac{1}{2}\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{a},$ (1)

where $\phi^{a}$ is a triplet of scalar fields, $a=1,2,3.$ The model has a
global $O\left(3\right)$ symmetry, which is spontaneously broken to
$U\left(1\right)$. The field configuration describing the monopole is

$\phi^{a}=\eta\frac{x^{a}}{r}$

where $x^{a}x^{a}=r^{2}$. We assume that underlying geometry is general static
spherically symmetric described by the line element

$ds^{2}=-B\left(r\right)dt^{2}+\frac{dr^{2}}{A\left(r\right)}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (2)

with the usual relation between the spherical coordinates, $r,\theta,\phi$ and
the Cartesian coordinates $x^{a}$. The Lagrangian for the above given field
configuration simplifies in the following way

$L=\frac{1}{2}\left(\partial_{\theta}\phi^{a}\partial^{\theta}\phi^{a}+\partial_{\phi}\phi^{a}\partial^{\phi}\phi^{a}\right)=\frac{\eta^{2}}{r^{2}},$
(3)

and the diagonal energy momentum tensor is given by these components

$T_{t}^{t}=T_{r}^{r}=-\frac{\eta^{2}}{r^{2}},\text{ \ \
}T_{\theta}^{\theta}=T_{\phi}^{\phi}=0.\text{\ }$ (4)

The general solution of the Einstein equations with this $T_{\mu}^{\nu}$ is

$B=A^{-1}=1-8\pi G\eta^{2}-\frac{2GM}{r}$ (5)

where $M$ is a constant of integration. The metric describes a black hole of
mass $M$, carrying a global monopole charge characterized by $\eta$. Such a
black hole can be formed if a global monopole is swallowed by an ordinary
black hole 1 .

The Kretschmann scalar which indicates the formation of curvature singularity
is given by

$\mathcal{K}=\frac{48M^{2}G^{2}}{r^{6}}+\frac{128M\pi
G^{2}\eta^{2}}{r^{5}}+\frac{256\pi^{2}G^{2}\eta^{4}}{r^{4}}.$ (6)

It is obvious that $r=0$ is a typical central curvature singularity (scalar
curvature singularity according to above mentioned classification) and the
dominant contribution comes from term corresponding to black hole mass $M$. If
$M>0$ the singularity is evidently spacelike and covered by a single horizon.

##  3 Global monopole and its singularity

If we assume that the mass term is negligible on the astrophysical scale or
vanishing, we will have

$ds^{2}=-\left(1-8\pi G\eta^{2}\right)dt^{2}+\frac{dr^{2}}{\left(1-8\pi
G\eta^{2}\right)}+r^{2}d\Omega^{2},$ (7)

For simplicity we choose $\alpha^{2}=1-8\pi G\eta^{2}$ and by rescaling $r$
and $t$ variables, we can rewrite the monopole metric as

$ds^{2}=-dt^{2}+dr^{2}+\alpha^{2}r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right),$ (8)

If we calculate the Kretschmann scalar,

$\mathcal{K}=4\frac{\left(\alpha^{2}-1\right)^{2}}{r^{4}\alpha^{4}}.$

still there is a weaker singularity at $r=0$. From the metric (7) one can
immediately see that the singularity is timelike. This time, because our
simplified metric does not have the horizon the singularity is naked.

##  4 Naked Singularity

As mentioned in the Introduction naked singularity poses a serious problems
and its resolution would be desirable. In this section, the occurrence of
naked singularities in global monopole will be analyzed from quantum
mechanical point of view. In probing the singularity, quantum test particles
obeying the Klein-Gordon and Dirac equations are used. The reason for using
two different types of fields is to clarify whether the classical singularity
is sensitive to spin of the fields.

According to Horowitz and Marolf (HM) 4 , the singular character of the
spacetime is defined as the ambiguity in the evolution of the wave functions.
That is to say, the singular character is determined based on the number of
self-adjoint extensions of the evolution operator to the entire Hilbert space.
If the extension is unique, it is said that the spacetime is quantum
mechanically regular. The brief review of the method follows:

Consider a static spacetime $\left(\mathcal{M},g_{\mu\nu}\right)$ with a
timelike Killing vector field $\xi^{\mu}$. Let $t$ denote the Killing
parameter and $\Sigma$ denote a static slice. The Klein-Gordon equation in
this space is

$\left(\nabla^{\mu}\nabla_{\mu}-M^{2}\right)\psi=0.$ (9)

This equation can be written in the form

$\frac{\partial^{2}\psi}{\partial
t^{2}}=\sqrt{f}D^{i}\left(\sqrt{f}D_{i}\psi\right)-fM^{2}\psi=-A\psi,$ (10)

in which $f=-\xi^{\mu}\xi_{\mu}$ and $D_{i}$ is the spatial covariant
derivative on $\Sigma$. We assume that the Hilbert space
$\mathcal{H}=L^{2}\left(\Sigma,\mu\right)$ is the space of square integrable
functions on $\Sigma$ with appropriate measure $\mu$. Initially the operator
$A$ is defined on smooth functions with compact support
$C_{0}^{\infty}(\Sigma)$. Since the operator $A$ is real, positive and
symmetric its self-adjoint extensions always exist. If it has a unique
extension $A_{E},$ then $A$ is called essentially self-adjoint 18 ; 19 ; 20 .
Accordingly, the Klein-Gordon equation for a free particle satisfies

$i\frac{d\psi}{dt}=\sqrt{A_{E}}\psi,$ (11)

with the solution

$\psi\left(t\right)=\exp\left[-it\sqrt{A_{E}}\right]\psi\left(0\right).$ (12)

If $A$ is not essentially self-adjoint, the future time evolution of the wave
function (12) is ambiguous. Then, HM criterion defines the spacetime as
quantum mechanically singular. However, if there is only a single self-adjoint
extension, the operator $A$ is said to be essentially self-adjoint and the
quantum evolution described by equation (12) is uniquely determined by the
initial conditions. According to the HM criterion, this spacetime is said to
be quantum mechanically non-singular. In order to determine the number of
self-adjoint extensions, the concept of deficiency indices is used. The
deficiency subspaces $N_{\pm}$ are defined by ( see Ref.5  for a detailed
mathematical background),

$\displaystyle N_{+}$ $\displaystyle=\\{\psi\in D(A^{\ast}),\text{\ \
}A^{\ast}\psi=Z_{+}\psi,\text{\ \ }ImZ_{+}>0\\}$ $\displaystyle\text{\ \ with
dimension }n_{+}$ (13) $\displaystyle N_{-}$ $\displaystyle=\\{\psi\in
D(A^{\ast}),\text{ \ \ }A^{\ast}\psi=Z_{-}\psi,\text{\ \ }ImZ_{-}<0\\}$
$\displaystyle\text{\ \ with dimension }n_{-}$

The dimensions $\left(\text{ }n_{+},n_{-}\right)$ are the deficiency indices
of the operator $A$. The indices $n_{+}(n_{-})$ are completely independent of
the choice of $Z_{+}(Z_{-})$ depending only on whether $Z$ lies in the upper
(lower) half complex plane. Generally one takes $Z_{+}=i\lambda$ and
$Z_{-}=-i\lambda$ , where $\lambda$ is an arbitrary positive constant
necessary for dimensional reasons. The determination of deficiency indices
then reduces to counting the number of solutions of $A^{\ast}\psi=Z\psi$ ;
(for $\lambda=1$),

$A^{\ast}\psi\pm i\psi=0$ (14)

that belong to the Hilbert space $\mathcal{H}$. If there are no square
integrable solutions ( i.e. $n_{+}=n_{-}=0)$, the operator $A$ possesses a
unique self-adjoint extension and it is essentially self-adjoint.
Consequently, a sufficient condition for the operator $A$ to be essentially
self-adjoint is to find only solutions satisfying Eq. (14) that do not belong
to the Hilbert space.

###  4.1 Klein - Gordon Fields

The Klein-Gordon equation for a massless scalar particle is given by,

$\square\psi=g^{-1/2}\partial_{\mu}\left[g^{1/2}g^{\mu\nu}\partial_{\nu}\right]\psi=M^{2}\psi.$
(15)

For the metric (8), the Klein-Gordon equation becomes,

$\frac{\partial^{2}\psi}{\partial
t^{2}}=-\left\\{\frac{\partial^{2}\psi}{\partial
r^{2}}+\frac{1}{r^{2}\alpha^{2}}\frac{\partial^{2}\psi}{\partial\theta^{2}}+\frac{1}{r^{2}\alpha^{2}\sin^{2}\theta}\frac{\partial^{2}\psi}{\partial\varphi^{2}}+\right.$
(16)
$\left.\qquad\qquad+\frac{\cos\theta}{r^{2}\alpha^{2}\sin\theta}\frac{\partial\psi}{\partial\theta}+\frac{2}{r}\frac{\partial\psi}{\partial
r}\right\\}.$

In analogy with the equation (10), the spatial operator $A$ is

$\emph{A}=\left\\{\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r^{2}\alpha^{2}}\frac{\partial^{2}}{\partial\theta^{2}}+\frac{1}{r^{2}\alpha^{2}\sin^{2}\theta}\frac{\partial^{2}}{\partial\varphi^{2}}\right.+$
(17)
$\left.\qquad\qquad+\frac{\cos\theta}{r^{2}\alpha^{2}\sin\theta}\frac{\partial}{\partial\theta}+\frac{2}{r}\frac{\partial}{\partial
r}\right\\}.$

and the equation to be solved is $\left(\emph{A}^{\ast}\pm i\right)\psi=0$.
Using separation of variables,
$\psi=R\left(r\right)Y_{l}^{m}\left(\theta,\varphi\right)$, we get the radial
portion of equation (14) as,

$\frac{d^{2}R\left(r\right)}{dr^{2}}+\frac{2}{r}\frac{dR\left(r\right)}{dr}+\left(\frac{-l\left(l+1\right)}{r^{2}\alpha^{2}}\pm
i\right)R\left(r\right)=0.$ (18)

The square integrability of the above solution is checked by calculating the
squared norm of the above solution in which the function space on each $t=$
constant hypersurface $\Sigma$ is defined as
$\mathcal{H=}L^{2}\left(\Sigma,\mu\right)$ where $\mu$ is the measure given by
the spatial metric volume element.

We easily recover the results showed in 12 : The spacetime of global monopole
remains singular in the view of relativistic quantum mechanics: the future of
a given initial wave packet obeying the Klein-Gordon equation is not generally
well determined, similarly to the future of a classical particle which reaches
the classical singularity at $r=0$.

###  4.2 Dirac Fields

The Newman-Penrose formalism will be used here to analyze massless Dirac
particle propagating in the space of global monopole. The signature of the
metric (8) is changed to $-2$ in order to use the Dirac equation in Newman-
Penrose formalism. Thus, the metric is given by,

$ds^{2}=dt^{2}-dr^{2}-r^{2}\alpha^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right).$ (19)

The Chandrasekhar-Dirac (CD) 140  equations in Newman-Penrose formalism are
given by

$\displaystyle\left(D+\epsilon-\rho\right)F_{1}+\left(\bar{\delta}+\pi-\alpha\right)F_{2}$
$\displaystyle=$ $\displaystyle 0,$ (20)
$\displaystyle\left(\nabla+\mu-\gamma\right)F_{2}+\left(\delta+\beta-\tau\right)F_{1}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\left(D+\bar{\epsilon}-\bar{\rho}\right)G_{2}-\left(\delta+\bar{\pi}-\bar{\alpha}\right)G_{1}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\left(\nabla+\bar{\mu}-\bar{\gamma}\right)G_{1}-\left(\bar{\delta}+\bar{\beta}-\bar{\tau}\right)G_{2}$
$\displaystyle=$ $\displaystyle 0,$

where $F_{1},F_{2},G_{1}$ and $G_{2}$ are the components of the wave function,
$\epsilon,\rho,\pi,\alpha,\mu,\gamma,\beta$ and $\tau$ are the spin
coefficients to be found and the "bar" denotes complex conjugation. The null
tetrad vectors for the metric (19) are defined by

$\displaystyle l^{a}$ $\displaystyle=$ $\displaystyle\left(1,1,0,0\right),$
(21) $\displaystyle n^{a}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2},-\frac{1}{2},0,0\right),$ $\displaystyle
m^{a}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(0,0,\frac{1}{\alpha
r},\frac{i}{r\alpha\sin\theta}\right).$

The directional derivatives in the Dirac equation are defined by
$D=l^{a}\partial_{a},\nabla=n^{a}\partial_{a}$ and $\delta=m^{a}\partial_{a}.$
We define operators in the following way

$\displaystyle\mathbf{D}_{0}$ $\displaystyle=$ $\displaystyle D$
$\displaystyle\mathbf{D}_{0}^{\dagger}$ $\displaystyle=$
$\displaystyle-2\nabla$ (22) $\displaystyle\mathbf{L}_{0}^{\dagger}$
$\displaystyle=$ $\displaystyle\sqrt{2}r\text{ }\alpha\delta\text{ and
}\mathbf{L}_{1}^{\dagger}=\mathbf{L}_{0}^{\dagger}+\frac{\cot\theta}{2}$
$\displaystyle\mathbf{L}_{0}$ $\displaystyle=$
$\displaystyle\sqrt{2}r\alpha\text{ }\bar{\delta}\text{ and
}\mathbf{L}_{1}=\mathbf{L}_{0}+\frac{\cot\theta}{2}$

The nonzero spin coefficients are,

$\mu=-\frac{1}{2r},\rho=-\frac{1}{r},\beta=-\mathbf{\alpha}=\frac{1}{2\sqrt{2}}\frac{\cot\theta}{r\alpha}.$
(23)

Substituting nonzero spin coefficients and the definitions of the operators
given above into the CD equations leads to

$\displaystyle\left(\mathbf{D}_{0}+\frac{1}{r}\right)F_{1}+\frac{1}{r\alpha\sqrt{2}}\mathbf{L}_{1}F_{2}=0,$
$\displaystyle-\frac{1}{2}\left(\mathbf{D}_{0}^{\dagger}+\frac{1}{r}\right)F_{2}+\frac{1}{r\alpha\sqrt{2}}\mathbf{L}_{1}^{\dagger}F_{1}=0,$
$\displaystyle\left(\mathbf{D}_{0}+\frac{1}{r}\right)G_{2}-\frac{1}{r\alpha\sqrt{2}}\mathbf{L}_{1}^{\dagger}G_{1}=0,$
$\displaystyle\frac{1}{2}\left(\mathbf{D}_{0}^{\dagger}+\frac{1}{r}\right)G_{1}+\frac{1}{r\alpha\sqrt{2}}\mathbf{L}_{1}G_{2}=0.$
(24)

For the solution of the CD equations, we assume separable solution in the form
of

$\displaystyle F_{1}$ $\displaystyle=$ $\displaystyle
f_{1}(r)Y_{1}(\theta)e^{i\left(kt+m\varphi\right)},$ $\displaystyle F_{2}$
$\displaystyle=$ $\displaystyle
f_{2}(r)Y_{2}(\theta)e^{i\left(kt+m\varphi\right)},$ $\displaystyle G_{1}$
$\displaystyle=$ $\displaystyle
g_{1}(r)Y_{3}(\theta)e^{i\left(kt+m\varphi\right)},$ $\displaystyle G_{2}$
$\displaystyle=$ $\displaystyle
g_{2}(r)Y_{4}(\theta)e^{i\left(kt+m\varphi\right)}.$ (25)

Here $\left\\{f_{1},f_{2},g_{1},g_{2}\right\\}$ and
$\left\\{Y_{1},Y_{2},Y_{3},Y_{4}\right\\}$ are functions of $r$ and $\theta$
respectively, $m$ is the azimuthal quantum number and $k$ is the frequency of
the Dirac spinor, which is assumed to be positive and real. By substituting
(25) in (24) we will see that with these assumptions

$\displaystyle\text{\ }f_{1}(r)$ $\displaystyle=$ $\displaystyle
g_{2}(r)\text{ \ \ \ \ and \ \ \ }f_{2}(r)=g_{1}(r)\text{\ \ },$ (26)
$\displaystyle Y_{1}(\theta)$ $\displaystyle=$ $\displaystyle
Y_{3}(\theta)\text{ \ \ \ \ and \ \ \ }Y_{2}(\theta)=Y_{4}(\theta)$ (27)

Dirac equation reduces to two equations. The radial part of the Dirac
equations become

$\displaystyle\left(\mathbf{D}_{0}+\frac{1}{r}\right)f_{1}\left(r\right)=\frac{\lambda}{r\alpha\sqrt{2}}f_{2}\left(r\right),$
(28)
$\displaystyle\frac{1}{2}\left(\mathbf{D}_{0}^{\dagger}+\frac{1}{r}\right)f_{2}\left(r\right)=\frac{\lambda}{r\alpha\sqrt{2}}f_{1}\left(r\right).$

where $\lambda$ comes from separation of variables. We further assume that

$\displaystyle f_{1}\left(r\right)$ $\displaystyle=$
$\displaystyle\frac{\Psi_{1}\left(r\right)}{r},$ $\displaystyle
f_{2}\left(r\right)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}\Psi_{2}\left(r\right)}{r},$

then equation (28) transforms into,

$\displaystyle\mathbf{D}_{0}\Psi_{1}=\frac{\lambda^{{}^{\prime}}}{r}\Psi_{2},$
(29)
$\displaystyle\mathbf{D}_{0}^{\dagger}\Psi_{2}=\frac{\lambda^{{}^{\prime}}}{r}\Psi_{1}.$

where $\lambda^{{}^{\prime}}=\frac{\lambda}{\alpha},$ so we will have

$\displaystyle\left(\frac{d}{dr}+ik\right)\Psi_{1}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{{}^{\prime}}}{r}\Psi_{2},$ (30)
$\displaystyle\left(\frac{d}{dr}-ik\right)\Psi_{2}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{{}^{\prime}}}{r}\Psi_{1},$

In order to write the above equation in a more compact form we combine the
solutions in the following way,

$\displaystyle Z_{+}$ $\displaystyle=$ $\displaystyle\Psi_{1}+\Psi_{2},$
$\displaystyle Z_{-}$ $\displaystyle=$ $\displaystyle\Psi_{2}-\Psi_{1}.$

After doing some calculations we end up with a pair of one-dimensional
Schrödinger-like wave equations with effective potentials,

$\displaystyle\left(\frac{d^{2}}{dr^{2}}+k^{2}\right)Z_{\pm}=V_{\pm}Z_{\pm},$
(31) $\displaystyle
V_{\pm}=\frac{\lambda^{{}^{\prime}2}}{r^{2}}\mp\frac{\lambda^{{}^{\prime}}}{r^{2}}.$
(32)

In analogy with the equation (10), the spatial operator $A$ for the massless
case is

$A=-\frac{d^{2}}{dr^{2}}+V_{\pm},$

so we have

$\left(\frac{d^{2}}{dr^{2}}-\left[\frac{\lambda^{{}^{\prime}2}}{r^{2}}\mp\frac{\lambda^{{}^{\prime}}}{r^{2}}\right]\mp
i\right)Z_{\pm}=0.$ (33)

The solutions of the above equations are expressible using Bessel functions of
the first and second kind in the following way

$\displaystyle Z_{+}$ $\displaystyle=$ $\displaystyle
C_{1}\sqrt{r}J\left(\lambda^{{}^{\prime}}-\frac{1}{2},\frac{r}{\sqrt{2}}\left(1-i\right)\right)+$
$\displaystyle
C_{2}\sqrt{r}Y\left(\lambda^{{}^{\prime}}-\frac{1}{2},\frac{r}{\sqrt{2}}\left(1-i\right)\right),$
$\displaystyle Z_{-}$ $\displaystyle=$ $\displaystyle
C_{1}^{{}^{\prime}}\sqrt{r}J\left(\lambda^{{}^{\prime}}+\frac{1}{2},\frac{r}{\sqrt{2}}\left(1+i\right)\right)+$
(34) $\displaystyle
C_{2}^{{}^{\prime}}\sqrt{r}Y\left(\lambda^{{}^{\prime}}+\frac{1}{2},\frac{r}{\sqrt{2}}\left(1+i\right)\right).$

Using the asymptotic formulas for Bessel functions when $r\rightarrow\infty$
($Y(\kappa,z)\approx z^{-1/2}\sin(z-\kappa\pi/2-\pi/4)$ and
$J(\kappa,z)\approx z^{-1/2}\cos(z-\kappa\pi/2-\pi/4)$) and noting the complex
argument in both solutions one can find a combination of constants
$C_{1},C_{2}$ or $C_{1}^{\prime},C_{2}^{\prime}$ which is square integrable
near infinity. (But, it is also possible to choose the constants differently
so that both solutions are not square integrable!).

When $r\rightarrow 0$ the approximate expressions for Bessel functions
($Y(\kappa,z)\approx z^{-\kappa}$ for $\kappa\neq 0$, $Y(0,z)\approx\ln(z/2)$
and $J(\kappa,z)\approx z^{\kappa}$) imply that for $C_{2}=0$ and
$C_{2}^{\prime}=0$ we have square integrable solution near zero. (Here again
if we suppose $C_{1}=0$ and $C_{1}^{\prime}=0$, for $\kappa\geq 3/2$, the
solutions are not square integrable!. One could restrict an analysis to only
certain wave modes and purposely choose the modes to be quantum regular).

But since we have a solution of equations valid on the whole domain (not just
asymptotic forms of equations) we can match the behaviour at zero and
infinity. Based on the results we can have solution square integrable over the
whole domain and therefore our deficiency indices are nonzero. The operator is
not essentially self-adjoint and the spacetime is quantum mechanically
singular.

##  5 Quantum Gravity

Now we are going to investigate the singularity of general global monopole
using techniques from loop quantization in the manner of 23 . Consider
equation (2), for $r<\frac{2GM}{1-8\pi G\eta^{2}}$. This metric describes
spacetime inside the horizon of a black hole. The coordinate $r$ is timelike
and the coordinate $t$ is spatial there; for convenience we rename them as
$r\equiv T$ and $t\equiv r$ with $T\in[0,\frac{2GM}{1-8\pi G\eta^{2}}]$ and
$r\in[-\infty,+\infty]$ and the metric becomes

$ds^{2}=-\left(\alpha^{2}-\frac{2GM}{T}\right)dr^{2}+\frac{dT^{2}}{\left(\alpha^{2}-\frac{2GM}{T}\right)}+T^{2}d\Omega^{2},$
(35)

we eliminate the coefficient of $dT^{2}$ by defining a new temporal variable
$\tau$ via

$d\tau=\frac{dT}{\sqrt{\frac{2GM}{T}-\alpha^{2}}}.$ (36)

Accordingly, the metric becomes

$ds^{2}=-d\tau^{2}+\left(\frac{2GM}{T}-\alpha^{2}\right)dr^{2}+T^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right).$ (37)

We introduce two functions
$a^{2}\left(\tau\right)\equiv\frac{2Gm}{T}-\alpha^{2}$ and
$b^{2}\left(\tau\right)\equiv T^{2}\left(\tau\right)$ and redefine $\tau\equiv
t$. The metric becomes

$ds^{2}=-dt^{2}+a^{2}\left(t\right)dr^{2}+b^{2}\left(t\right)\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right),$ (38)

this metric describes a homogeneous, anisotropic Kanto-wski-Sachs cosmological
mmodel with spatial section having topology $\mathbf{R\times S}^{2}$. From
this observation comes the motivation to use LQC approach. In our case
$a\left(t\right)$ is a function of $b\left(t\right)$.

###  5.1 Classical observables

The corresponding action for gravity minimally coupled with scalar field can
be written in the form

$S=\frac{1}{16\pi G}\int dtd^{3}xNh^{1/2}\left[K_{ij}K^{ij}-K^{2}+\right.$
$\left.\qquad\qquad\qquad+^{\left(3\right)}R-\frac{16\pi
G\eta^{2}}{b^{2}}\right],$ (39)

by considering the metric (38), the action becomes

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{-1}{8\pi G}\int
dt\int_{0}^{R}dr\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta
ab^{2}\times$ (40)
$\displaystyle\times\left[\frac{\dot{b}^{2}}{b^{2}}+\frac{2\dot{a}\dot{b}}{ab}-\frac{\alpha^{2}}{b^{2}}\right]$

by using the relation between $a$ and $b$, we will be able to write the action
in terms of a single function

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{R\alpha^{2}}{2G}\int
dt\sqrt{\frac{b}{2GM}}\left(1-\frac{\alpha^{2}b}{2GM}\right)^{-1/2}\times$
(41)
$\displaystyle\times\left[\dot{b}^{2}+\frac{2GM}{b}\left(1-\frac{\alpha^{2}b}{2GM}\right)\right].$

Now, we will compute the Hamiltonian (Hamiltonian constraint). The momentum
associated to the chosen configuration variable is

$p_{b}=\frac{R\alpha^{2}\dot{b}}{G}\sqrt{\frac{b}{2GM}}\left(1-\frac{\alpha^{2}b}{2GM}\right)^{-1/2},$
(42)

and therefore we obtain

$\displaystyle H$ $\displaystyle=$ $\displaystyle p_{b}\dot{b}-L$
$\displaystyle=$
$\displaystyle\sqrt{\frac{2GM}{b}}\sqrt{1-\frac{\alpha^{2}b}{2GM}}\left[\frac{Gp_{b}^{2}}{2R\alpha^{2}}-\frac{R\alpha^{2}}{2G}\right].$

Now, we calculate the Hamiltonian constraint in terms of $\dot{b}$

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{R\alpha^{2}}{2G}\sqrt{\frac{b}{2GM}}\left(1-\frac{\alpha^{2}b}{2GM}\right)^{-1/2}\times$
(44)
$\displaystyle\times\left[\dot{b}^{2}-\frac{2GM}{b}\left(1-\frac{\alpha^{2}b}{2GM}\right)\right]=0,$

and immediately get the following solution

$\dot{b}^{2}=\frac{2GM}{b}-\alpha^{2},$ (45)

which is exactly the equation (36). When the horizon radius,
$r_{h}=\frac{2GM}{\alpha^{2}}$, is much larger than the scale on which we are
probing the singularity, we can write

$1-\frac{\alpha^{2}b}{2GM}\sim 1$

so the Hamiltonian would be

$H=\sqrt{\frac{2GM}{b}}\left[\frac{GP_{b}^{2}}{2R\alpha^{2}}-\frac{R\alpha^{2}}{2G}\right].$
(46)

The volume

$\displaystyle V$ $\displaystyle=$ $\displaystyle\int drd\theta
d\phi\sqrt{h}=4\pi Rab^{2}$ (47) $\displaystyle=$ $\displaystyle 4\pi
Rb^{3/2}\sqrt{2GM}\sqrt{1-\frac{\alpha^{2}b}{2GM}}$

simplifies when using the above approximation and we obtain

$\displaystyle V$ $\displaystyle=$ $\displaystyle l_{0}b^{3/2}$ $\displaystyle
l_{0}$ $\displaystyle=$ $\displaystyle 4\pi R\sqrt{2GM}$ (48)

The canonical pair is given by $b\equiv x$ and $p_{b}$, with Poisson bracket
$\left\\{x,p_{b}\right\\}=1$.

For isotropic models, only holonomies evaluated in isotropic connections
$A_{a}^{i}=\tilde{c}\delta_{a}^{i}$ appear. Along straight lines in the
direction of translation symmetries $X_{I}^{a}=\left(\partial/\partial
X^{I}\right)^{a}$, holonomies $\exp\left(\int
X_{I}^{a}A_{a}^{i}\tau_{i}\right)$ in the fundamental representation of
$SU\left(2\right)$ have matrix elements of the form $\exp\left(i\mu c\right)$,
where $\mu$ depends on the length of the curve used. Here, it turns out to be
useful to introduce $c:=V_{0}^{1/3}\tilde{c}$ defined in terms of the
coordinate size $V_{0}$ of the region used to define the isotropic phase space
21 .

Using this motivation we introduce the following function which will be used
instead of the momentum (from now on we leave out the subscript $b$ for
momentum associated with this observable) 23

$U_{\gamma}\left(p\right)\equiv\exp\left(8\pi G\frac{i\gamma}{L}p\right)$ (49)

where $\gamma$ is a real parameter and $L$ fixes the length scale. The
parameter $\gamma$ determines the separation of momentum points in the phase
space.

The pair $\left(x,U_{\gamma}\left(p\right)\right)$ has the following Poisson
bracket algebra

$\left\\{x,U_{\gamma}\left(p\right)\right\\}=8\pi
G\frac{i\gamma}{L}U_{\gamma}\left(p\right)$ (50)

A straightforward calculation gives

$\displaystyle U_{\gamma}^{-1}\left\\{V^{n},U_{\gamma}\right\\}$
$\displaystyle=$ $\displaystyle
l_{0}^{n}U_{\gamma}^{-1}\left\\{\left|x\right|^{3n/2},U_{\gamma}\right\\}$
$\displaystyle=$ $\displaystyle i8\pi
Gl_{0}^{n}\frac{\gamma}{L}\frac{3n}{2}sgn\left(x\right)\left|x\right|^{3n/2-1}$

We are concerned with the quantity $\frac{1}{\left|x\right|}$ which can serve
as an indicator for singularity presence because classically it diverges for
$\left|x\right|\rightarrow 0$ thus producing singularity. From this moment we
choose $n=1/3$

$\frac{sgn\left(x\right)}{\sqrt{\left|x\right|}}=-\frac{2Li}{8\pi
Gl_{0}^{1/3}\gamma}U_{\gamma}^{-1}\left\\{V^{1/3},U_{\gamma}\right\\}.$ (52)

###  5.2 Quantization

We will use the basis of Hillbert space introduced in 22 ; 23 , which is
formed by eigenstates of $\hat{x}$. This implies the existence of a self-
adjoint operator $\hat{x}$, acting on the basis states according to

$\hat{x}\left|\mathbf{\mu}\right\rangle=L\mu\left|\mathbf{\mu}\right\rangle$
(53)

Next, we want to promote the classical momentum function
$U_{\gamma}=e^{\left(8\pi G\frac{i\gamma}{L}p\right)}$ to operator. We can do
so by defining the action of $\hat{U}_{\gamma}$ on the basis states with the
help of the definition equation (53) and using commutation relation based on
the Poisson bracket between $x$ and $U_{\gamma}$ we obtain

$\hat{U}_{\gamma}\left|\mathbf{\mu}\right\rangle=\left|\mathbf{\mu-\gamma}\right\rangle,\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ }\left[\hat{x},\hat{U}_{\gamma}\right]=-\gamma
L\hat{U}_{\gamma}.$ (54)

Using canonical quantization of Poison bracket $\left[,\right]\rightarrow
i\hbar\left\\{,\right\\}$, and using equation (51) we get a relation for the
length scale

$L=\sqrt{8\pi}l_{p}$ (55)

###  5.3 Volume operator and disappearance of the singularity

In the vicinity of the singularity we assume the approximate equation (48).
Then the volume operator acts in the following way on the basis states

$\hat{V}\left|\mathbf{\mu}\right\rangle=l_{0}\left|x\right|^{3/2}\left|\mathbf{\mu}\right\rangle=l_{0}\left|L\mu\right|^{3/2}\left|\mathbf{\mu}\right\rangle$
(56)

Using the equation (52) and promoting the Poisson brackets to commutators,
while setting $\gamma=1$, we find

$\frac{\widehat{1}}{\left|x\right|}=\frac{1}{2\pi
l_{p}^{2}l_{0}^{2/3}}\left(\hat{U}_{\gamma}^{-1}\left[\hat{V}^{1/3},\hat{U}_{\gamma}\right]\right)^{2}.$
(57)

On the basis states this operator acts in the following way

$\displaystyle\hat{U}_{\gamma}^{-1}\left[\hat{V}^{1/3},\hat{U}_{\gamma}\right]\left|\mathbf{\mu}\right\rangle$
$\displaystyle=$
$\displaystyle\left(\hat{U}_{\gamma}^{-1}\hat{V}^{1/3}\hat{U}_{\gamma}-\hat{U}_{\gamma}^{-1}\hat{U}_{\gamma}\hat{V}^{1/3}\right)\left|\mathbf{\mu}\right\rangle$
(58) $\displaystyle=$ $\displaystyle
l_{0}^{1/3}l_{p}^{1/2}\left(\sqrt{\mu-1}-\sqrt{\mu}\right)\left|\mathbf{\mu}\right\rangle$

so finally we get

$\frac{\widehat{1}}{\left|x\right|}\left|\mathbf{\mu}\right\rangle=\sqrt{\frac{2}{\pi
l_{p}^{2}}}\left(\sqrt{\mu-1}-\sqrt{\mu}\right)^{2}\left|\mathbf{\mu}\right\rangle.$
(59)

We can see that the spectrum is bounded from above and so the singularity is
resolved in the quantum theory (the theory gives finite predictions for
observables related to singularity). In fact, the eigenvalue of operator
$\frac{\widehat{1}}{\left|x\right|}$ corresponding to the state
$\left|\mathbf{0}\right\rangle$ which probes the classical singularity is
equal to $\sqrt{\frac{2}{\pi l_{p}^{2}}}$, which is the highest eigenvalue of
the spectrum. Specifically, the operator corresponding to the curvature
invariant

$\displaystyle\mathcal{~{}R}_{\mu\nu\rho\sigma}\mathcal{R}^{\mu\nu\rho\sigma}$
$\displaystyle=$ $\displaystyle\frac{48M^{2}G^{2}}{r^{6}}+\frac{128M\pi
G^{2}\eta^{2}}{r^{5}}+\frac{256G^{2}\pi^{2}\eta^{4}}{r^{4}}$
$\displaystyle\equiv\frac{48M^{2}G^{2}}{b\left(t\right)^{6}}$ $\displaystyle+$
$\displaystyle\frac{128M\pi
G^{2}\eta^{2}}{b\left(t\right)^{5}}+\frac{256G^{2}\pi^{2}\eta^{4}}{b\left(t\right)^{4}}$
(60)

is then automatically finite in quantum mechanics. Promoting it to operator
and evaluating on $\left|\mathbf{0}\right\rangle$ we get

$\displaystyle\mathcal{~{}}\widehat{\mathcal{R}_{\mu\nu\rho\sigma}\mathcal{R}^{\mu\nu\rho\sigma}}\left|\mathbf{0}\right\rangle$
$\displaystyle=\left(\widehat{\frac{48M^{2}G^{2}}{\left|x\right|^{6}}}+\widehat{\frac{128M\pi
G^{2}\eta^{2}}{\left|x\right|^{5}}}+\widehat{\frac{256\pi^{2}G^{2}\eta^{4}}{\left|x\right|^{4}}}\right)\left|\mathbf{0}\right\rangle$
$\displaystyle=\left(\textstyle{\frac{384M^{2}G^{2}}{\pi^{3}l_{p}^{6}}+\sqrt{\frac{2}{\pi^{5}}}\frac{512M\pi
G^{2}\eta^{2}}{l_{p}^{5}}+\frac{1024\pi^{2}G^{2}\eta^{4}}{\pi^{2}l_{p}^{4}}}\right)\left|\mathbf{0}\right\rangle$
(61)

On the other hand, when $\left|\mu\right|\rightarrow\infty$ the eigenvalue of
$\frac{\widehat{1}}{\left|x\right|}$ goes to zero which is natural behaviour
for large $\left|x\right|$.

Also, it is possible to show that the quantum Hamiltonian constraint gives a
discrete difference equation for the coefficients of the physical states.

##  6 Conclusion

We have seen that we have not been successful in removing the naked
singularity by using relativistic quantum mechanics (for both Klein-Gordon and
Dirac equations). On the other hand we have shown that the curvature
singularity of general global monopole is resolved when the geometry is
quantized using loop techniques. Unfortunately, one cannot directly compare
the results because the loop quantization relied on radial coordinate being
timelike beneath the horizon which is not the case for naked singularity of
pure monopole. But still, this might be an indication that the first method is
not reliable for determining the fate of singularities in quantum theory and
one should rather focus on quantization of the geometry itself. But even the
approach using loop quantization that relied on restricted class of geometries
should not be trusted completely. One should allow, e.g., for deviations from
spherical symmetry to be completely sure about the fate of singularities.

###### Acknowledgements.

T.T. gratefully acknowledges the hospitality of Institute of Theoretical
Physics (Charles University in Prague) during her stay. O. S. was supported by
grant GAČR 202/09/0772.

##  Appendix A Geometric quantities

The spatial metric is

$h_{ij}=\left(a^{2}\left(t\right),b^{2}\left(t\right),b^{2}\left(t\right)\sin^{2}\theta\right),$
(62)

The extrinsic curvature is $K_{ij}=-\frac{1}{2}\frac{\partial h_{ij}}{\partial
t}$, and so

$\displaystyle K_{ij}=-\left(a\dot{a},b\dot{b},b\dot{b}\sin^{2}\theta\right),$
$\displaystyle
K=K_{ij}h^{ij}=-\left(\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}\right)$ (63)
$\displaystyle
K_{ij}K^{ij}=\left(\frac{\dot{a}^{2}}{a^{2}}+2\frac{\dot{b}^{2}}{b^{2}}\right)\text{\
\ \ \ \ \ \ }$ $\displaystyle
K_{ij}K^{ij}-K^{2}=-2\left(\frac{\dot{b}^{2}}{b^{2}}+2\frac{\dot{a}\dot{b}}{ab}\right)$
(64)

The Ricci curvature for the space section is

${}^{\left(3\right)}R=\frac{2}{b^{2}}.$ (65)

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