# Quantum system ascribed to the Oppenheimer-Snyder
model of massive star

Andrzej Góźdź<EMAIL_ADDRESS>Institute of Physics, Maria Curie-
Skłodowska University, pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland
Jan J. Ostrowski<EMAIL_ADDRESS>Department of Fundamental
Research, National Centre for Nuclear Research, Pasteura 7, 02-093 Warszawa,
Poland  Aleksandra Pȩdrak<EMAIL_ADDRESS>Department of
Fundamental Research, National Centre for Nuclear Research, Pasteura 7, 02-093
Warszawa, Poland  Włodzimierz Piechocki<EMAIL_ADDRESS>Department of Fundamental Research, National Centre for Nuclear Research,
Pasteura 7, 02-093 Warszawa, Poland

(August 28, 2024)

###### Abstract

We quantize the Oppenheimer-Snyder model of black hole using the integral
quantization method. We treat spatial and temporal coordinates on the same
footing both at classical and quantum levels. Our quantization resolves or
smears the singularities of the classical curvature invariants. Quantum
trajectories with bounces can replace singular classical ones. The considered
quantum black hole may have finite bouncing time. As a byproduct, we obtain
the resolution of the gravitational singularity of the Schwarzschild black
hole at quantum level.

###### Contents

  1. I Introduction
  2. II Spherically symmetric spacetime
    1. II.1 The perfect fluid case
    2. II.2 The dust case
    3. II.3 Singularities
      1. II.3.1 Shell crossing singularity
      2. II.3.2 Gravitational singularities
      3. II.3.3 Apparent horizon
    4. II.4 Oppenheimer-Snyder black hole
  3. III Quantum Oppenheimer-Snyder black hole
    1. III.1 Integral quantization method
    2. III.2 Quantum dynamics
    3. III.3 Fate of classical singularities
  4. IV Quantum Schwarzschild black hole
  5. V Conclusions
  6. A General analytical solution
  7. B Coordinate transformations
  8. C The state spaces

##  I Introduction

Cosmological models can be used to describe black holes after imposing the
condition that involves dealing with isolated objects. In the case of
spherically symmetric objects, this issue may be reduced to the problem of
matching the Schwarzschild spacetime with finite region of specific spacetime
[1, 2]. This idea was recently used to obtain the Oppenheimer-Snyder (OS) and
Lemaître-Tolman-Bondi (LTB) models of isolated objects within one formalism
(see [3] and references therein). The former model concerns a spherical cloud
of homogeneous dust (pressureless matter), whereas the latter one deals with a
spherical but inhomogeneous cloud of dust. However, the merging condition
leads sometimes to complicated equations defining variables, which cannot be
resolved analytically but only numerically, creating additional difficulties
in analyses (e.g., see [3]).

Different strategy of obtaining a description of an isolated astrophysical
object was proposed recently for LTB model [5, 6, 4, 7]. A metric of both
interior and exterior regions of isolated body is expressed in one coordinate
system, which avoids the need for imposing the matching conditions across the
interface between matter and vacuum regions provided that certain functions
are continuous across the boundary. We apply this approach in the present
paper.

In this article we present the quantum system ascribed to the OS model of
collapsing pressureless dust star. For this purpose we use the so-called
integral quantization (IQ) method applied quite recently to the quantization
of the Schwarzschild spacetime [8] and a thin matter shell in vacuum [9].

In this article, we disregard the black hole evaporation by the Hawking
radiation. Thus, the global mass of considered star is conserved during its
evolution.

Following the idea presented in [8], we quantize not only spatial but also
temporal coordinates. Rationale for such an approach is the covariance of
general relativity with respect to the transformations of these coordinates.
Treating temporal and spatial coordinates on the same footing at quantum level
has enabled the construction of a consistent quantum theory.

One of the main issues addressed in the quantization of black holes is the
singularity avoidance. It was widely discussed in the literature. In the case
of the OS black hole, various approaches motivated by the loop quantum gravity
have led to the resolution of the singularity of the classical model (see,
e.g., [10, 11, 12, 13, 14, 15, 16] and references therein). There exist many
investigations done in the context of quantum geometrodynamics. See, for
instance, [17] and references therein. On the other hand, there are few
results obtained within the IQ method based on coherent states [18, 19]. The
latter method is applicable even in the cases the canonical quantization has
methodological problems [19]. We extend this discussion in the conclusion
section.

The paper is organized as follows: In Sec. II we recall the formalism of
spherically symmetric spacetime specialized to black holes. The singularities
and horizon issues are exhibited. The case of the dust black hole is made
specific. Sec. III is devoted to the quantization of the OS model. It includes
recalling an essence of the IQ quantization and presents the quantum dynamics
of considered black hole. In particular, we discuss the fate of the classical
singularities at quantum level. We reduce the results to the case of the
quantum Schwarzschild black hole in Sec. IV. We conclude in Sec. V. App. A
recalls general solution for the dynamics of the LTB spacetime. App. B
specifies the transformation between two coordinate systems applied in our
paper. The two state spaces used in the affine quantization are discussed in
App. C.

In the following we choose $\;G=c=1=\hbar\;$ except where otherwise noted.

##  II Spherically symmetric spacetime

###  II.1 The perfect fluid case

We begin our considerations with recalling the metric and field equations of
the LTB model of black hole presented in the article [5], and describing a
general spherically symmetric perfect fluid.

The metric of the collapsing massive star of a spherically symmetric spacetime
in the coordinates $(t,r,\theta,\phi)$ (see, e.g., [21]) and considered in [5]
reads

$\mathrm{d}s^{2}=-\alpha^{2}\mathrm{d}t^{2}+\frac{(\alpha\sqrt{2M/r+E}\,\mathrm{d}t+\mathrm{d}r)^{2}}{1+E}+r^{2}\mathrm{d}\Omega^{2}\,,$
(1)

where $(t,r)\in\mathbb{R}\times\mathbb{R}_{+}$, with
$\mathbb{R}_{+}:=\\{r\in\mathbb{R}\leavevmode\nobreak\ |\leavevmode\nobreak\
r>0\\}$, and where $\alpha(t,r)>0$ is the lapse function (specific to the
$3+1$ decomposition of the spacetime metric), whereas
$\mathrm{d}\Omega^{2}:=\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}$.
The so-called energy function $E(t,r)$, arbitrary up to the constraint $E>-1$,
is a measure of the energy of a shell at a radius $r$. The so-called mass
function $M(t,r)$, at some radius $r$, is defined to be

$M(t,r)=4\pi\int_{0}^{r}\rho(t,\sigma)\sigma^{2}\;d\sigma\,.$ (2)

The field equations have the form [5]

$\frac{\partial M}{\partial
t}-\alpha\,\sqrt{\frac{2M}{r}+E}\;\;\Big{(}\frac{\partial M}{\partial r}+4\pi
Pr^{2}\Big{)}=0\,,$ (3) $\frac{\partial E}{\partial
t}-\alpha\,\sqrt{\frac{2M}{r}+E}\;\;\Big{(}\frac{\partial E}{\partial
r}+2\,\frac{1+E}{\rho+P}\;\frac{\partial P}{\partial r}\Big{)}=0\,,$ (4)
$\frac{1}{\alpha}\;\frac{\partial\alpha}{\partial
r}+\frac{1}{\rho+P}\;\frac{\partial P}{\partial r}=0\,,$ (5) $f(\rho,P)=0\,.$
(6)

The energy-momentum tensor for a perfect fluid reads

$T^{\mu\nu}=(\rho+P)\,n^{\mu}n^{\nu}+Pg^{\mu\nu}\,,$ (7)

where $P$ is the pressure, $\rho\neq-P$ is the energy density, $n^{\mu}$ is
the vector field tangent to the fluid, and where (6) is an equation of state.

In the OS collapse model of black hole, we have two regions: a “star region”
defined by non-vanishing $\rho$ and a vacuum region with vanishing energy
density. In this configuration, the system is asymptotically flat and $M$
presents the global mass/energy of the system. This energy is conserved during
the evolution as a static spherically symmetric black hole does not radiate
gravitational waves.

Eqs. (3)–(6) require the specification of initial and boundary conditions to
be well defined. Additionally, one should impose the boundary conditions at
the interface between the matter that fills the interior of the ball of matter
and the vacuum exterior. For these important issues we recommend Sec. III of
Ref. [5].

The metric (1) is expressed in terms of a generalization of the Painlevé-
Gullstrand (PG) coordinates [22, 23]. In this choice of coordinates, the
radial coordinate $r$ is always spacelike. We call the above formalism, the
spherically symmetric spacetime model in the PG coordinates.

###  II.2 The dust case

In what follows, we consider the dust case. It can be introduced by the
conditions

$P=0\;\;,\;\;\alpha=1\;,$ (8)

implemented into Eqs. (1)–(5). As the result, the metric and field equations
become

$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=$
$\displaystyle-\mathrm{d}t^{2}+\frac{(\sqrt{2M/r+E}\,\mathrm{d}t+\mathrm{d}r)^{2}}{1+E}+r^{2}\mathrm{d}\Omega^{2}\,,$
(9) $\displaystyle\frac{\partial M}{\partial t}$ $\displaystyle-$
$\displaystyle\sqrt{\frac{2M}{r}+E}\;\;\frac{\partial M}{\partial r}=0\,,$
(10) $\displaystyle\frac{\partial E}{\partial t}$ $\displaystyle-$
$\displaystyle\sqrt{\frac{2M}{r}+E}\;\;\frac{\partial E}{\partial r}=0\,,$
(11)

where $M$ is defined by Eq. (2), and where $t$ is the time measured by an in-
falling geodesic observer.

The system (9)–(11) can describe the interior and exterior regions of the LTB
dust star in the single PG coordinate patch. Given the initial density profile
with the density vanishing at some finite radius, the latter region can be
shown to be diffeomorphic to the Schwarzschild spacetime [4]. The so-called
marginally bound model [24], considered in this article, is the case with
$E=0$ (see, App. A). The classical dynamics of this model is defined, due to
(10), by the equation

$\frac{\partial M}{\partial t}-\sqrt{\frac{2M}{r}}\;\;\frac{\partial
M}{\partial r}=0\,.$ (12)

This equation has a general implicit solution (as presented in [5]):

$M=\mathcal{F}\left(\frac{2}{3}r^{3/2}+t\sqrt{M}\right)\;,$ (13)

where $\mathcal{F}$ is a function of integration coming from solving the
equation with the method of characteristics. In what follows we present a
simpler but explicit, separable analytical solution to Eq. (12), which is a
particular case satisfying Eq. (13) (see, App. A for a general solution in
$(t,R)$ coordinates).

Making the assumption $M(t,r)=M_{1}(t)M_{2}(r)$, splits (12) into the two
equations:

$\displaystyle\frac{\partial M_{1}}{\partial t}-\lambda
M_{1}^{\frac{3}{2}}=0\,,$ (14) $\displaystyle\frac{\partial M_{2}}{\partial
r}-\lambda\sqrt{\frac{M_{2}r}{2}}=0\,.$ (15)

Direct integration of the equation (14) gives the solution

$\sqrt{M_{1}}=\frac{-2}{\lambda t+C_{1}}\ ,$ (16)

where $\lambda$ is the separation parameter and $C_{1}$ an arbitrary constant.

Similarly, integration of the equation (15) gives the following solution

$\sqrt{M_{2}}=\frac{\lambda}{3\sqrt{2}}r^{\frac{3}{2}}+C_{2}\ ,$ (17)

where $C_{2}$ is an arbitrary constant.

Combining both factors $M_{1}$ and $M_{2}$, the full solution of the dust
equation can be written as

$\sqrt{M(t,r)}=\sqrt{M_{1}(t)}\sqrt{M_{2}(r)}=\left(\frac{-2}{\lambda
t+C_{1}}\right)\left(\frac{\lambda}{3\sqrt{2}}r^{\frac{3}{2}}+C_{2}\right)\,,$
(18)

so that

$M(t,r)=\left(\sqrt{M_{1}(t)}\sqrt{M_{2}(r)}\right)^{2}=\left\\{\left(\frac{-2}{\lambda
t+C_{1}}\right)\left(\frac{\lambda}{3\sqrt{2}}r^{\frac{3}{2}}+C_{2}\right)\right\\}^{2}\,.$
(19)

Formally, the derivation of the solution (19) requires the restrictions of the
domains of the functions $\sqrt{M_{1}(t,r)}$ and $\sqrt{M_{2}(t,r)}$, which
also implies the restriction for the domain of $M(t,r)$. However, since the
function (19) fulfills the condition of mass positivity for arbitrary $(t,r)$,
we take the mass (19) to be defined on the full domain
$(t,r)\in\mathbb{R}\times\mathbb{R}_{+}$, except the singularities at
$t_{c}=-C_{1}/\lambda$. This result can be mathematically well defined by
taking the definition of the complex root in the solutions to Eqs. (14) and
(15).

###  II.3 Singularities

In the synchronous comoving coordinates $(\tau,R,\theta,\phi)$, the line
element and field equations read (see, e.g., [25])

$\mathrm{d}s^{2}=-\mathrm{d}\tau^{2}+\frac{\left(\frac{\partial r}{\partial
R}\right)^{2}}{1+2E}\mathrm{d}R^{2}+r^{2}\mathrm{d}\Omega^{2}\;,$ (20)
$\displaystyle\left(\frac{\partial r}{\partial\tau}\right)^{2}$
$\displaystyle=$ $\displaystyle\frac{2M}{r}+E\;,$ (21) $\displaystyle
4\pi\rho$ $\displaystyle=$ $\displaystyle\frac{\partial M}{\partial
R}\left(r^{2}\frac{\partial r}{\partial R}\right)^{-1}\;.$ (22)

Eq. (21) describes the dynamics of the shells. Taking a square root of this
equation requires choosing a sign, which represents expanding or collapsing
shells for negative or positive sign, respectively. The second equation (22)
defines the density, which can become singular under two conditions leading to
two different types of singularities. The $\partial r/\partial R=0$ is usually
referred to as a shell crossing and may be viewed as a weak singularity,
inherent to the lack of pressure in the matter model. The second one however,
i.e. $r=0$, is a strong singularity and cannot be avoided for the collapsing
scenarios. For the relationship between the line elements (9) and (20)
expressed in two different coordinate systems, see App B.

####  II.3.1 Shell crossing singularity

Let us find an explicit form of the condition $\partial r/\partial R=0$ for
the separable solution (19). We rewrite it as

$M(t,r)=\left(Ar^{\frac{3}{2}}+B\right)^{2}\;,$ (23)

where

$A=\frac{-2\lambda}{\left(\lambda
t+C_{1}\right)3\sqrt{2}}\;\;,\;\;B=\frac{-2C_{2}}{\lambda t+C_{1}}\;.$ (24)

Let us take some instance of time $t=0$ so that

$M(t_{i},r_{i})=\left(\left(\frac{-2}{C_{1}}\right)\left(\frac{\lambda}{3\sqrt{2}}r_{i}^{\frac{3}{2}}+C_{2}\right)\right)^{2}=\left(A_{i}r_{i}^{\frac{3}{2}}+B_{i}\right)^{2}\;.$
(25)

We can use the freedom to choose our radial coordinate and set $R=r_{i}$,
where $r_{i}$ is the areal radius at $t=0$. We then have

$M(0,R)=\left(A_{i}R^{\frac{3}{2}}+B_{i}\right)^{2}\;.$ (26)

We also get from (23)

$r=\left(\frac{\sqrt{M}-B}{A}\right)^{\frac{2}{3}}\;.$ (27)

Differentiating the above equations leads to:

$\frac{\partial r}{\partial
R}=\frac{1}{3A}r^{-\frac{1}{2}}M^{-\frac{1}{2}}\frac{\partial M}{\partial
R}\;,$ (28)

and

$\frac{\partial M}{\partial R}=3\sqrt{M}A_{i}R^{\frac{1}{2}}\;.$ (29)

From the two equations above it is evident that the shell-crossing cannot
occur as for our setup we have

$\frac{\partial r}{\partial R}=\frac{\lambda t+C_{1}}{3\sqrt{2}\,C_{1}}\neq
0\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{for}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ t\neq t_{c}=-C_{1}/\lambda\;.$ (30)

####  II.3.2 Gravitational singularities

The existence of gravitational singularity is usually signalized by the blow-
up of the curvature scalars. The Kretschmann scalar can be decomposed in the
following way:

$K=R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}=C_{\mu\nu\lambda\sigma}C^{\mu\nu\lambda\sigma}+2R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}{\mathcal{R}}^{2}\;,$
(31)

where $R_{\mu\nu\lambda\sigma}$, $C_{\mu\nu\lambda\sigma}$, $R_{\mu\nu}$ and
${\mathcal{R}}$ are the Riemann, Weyl and Ricci tensors and the scalar
curvature. Term by term, we have:

$\displaystyle C_{\mu\nu\lambda\sigma}C^{\mu\nu\lambda\sigma}$
$\displaystyle=48\left(\frac{1}{3}\frac{\partial M}{\partial
R}\left(\frac{\partial r}{\partial R}\right)^{-1}r^{-2}-Mr^{-3}\right)^{2}\;,$
(32) $\displaystyle R_{\mu\nu}R^{\mu\nu}$
$\displaystyle=64\pi^{2}\rho^{2}=\left(2\frac{\partial M}{\partial
R}\left(\frac{\partial r}{\partial R}\right)^{-1}r^{-2}\right)^{2}\;,$ (33)
$\displaystyle{\mathcal{R}}$ $\displaystyle=8\pi\rho=2\frac{\partial
M}{\partial R}\left(\frac{\partial r}{\partial R}\right)^{-1}r^{-2}\;.$ (34)

For the separable solution (19), with $C_{2}=0$ corresponding to the OS model
(see, subsection II.4), we get:

$\displaystyle C_{\mu\nu\lambda\sigma}C^{\mu\nu\lambda\sigma}$
$\displaystyle=48\left(AM^{\frac{1}{2}}r^{-\frac{3}{2}}-Mr^{-3}\right)^{2}=0\;,$
(35) $\displaystyle R_{\mu\nu}R^{\mu\nu}$
$\displaystyle=36A^{2}Mr^{-3}=576\,\lambda^{4}\,\epsilon^{-4}\;,$ (36)
$\displaystyle{\mathcal{R}}$
$\displaystyle=6AM^{\frac{1}{2}}r^{-\frac{3}{2}}=24\,\lambda^{2}\,\epsilon^{-2}\;,$
(37) $\displaystyle K$
$\displaystyle=108A^{2}Mr^{-3}-96AM^{\frac{3}{2}}r^{-\frac{9}{2}}+48M^{2}r^{-6}=\frac{240}{81}\,\lambda^{4}\,\epsilon^{-4}\;,$
(38)

where $\epsilon:=\lambda t+C_{1}$. The solution (19) implies that for
$\epsilon\rightarrow 0$ we have the gravitational singularity.

####  II.3.3 Apparent horizon

Very often, in the case of dynamical space-times it is not possible to
establish the event horizon (EH), since it requires a knowledge of the whole
space-time, including null and spatial infinities. One of the ways to replace
the notion of event horizon by something in principle determinable locally is
to resort to the outermost marginally outer trapped surface called the
apparent horizon (AH). Although its definition is dependent on the foliation
of space-time, it is still one of the most convenient ways to define the black
hole, especially in the numerical general relativity. In the spherically
symmetric configuration, the AH is a sphere residing on the hyper-surface.
Condition for this sphere to be an outer trapped surface requires calculating
the divergence of the outgoing (from the surface) null vectors and demanding
it to be zero. The radial null vector associated with the LTB metric (for
$E=0$) in PG coordinates (see e.g. [4] for derivation) reads:

$k^{\mu}=\left(1,1-\sqrt{\frac{2M}{R}},0,0\right)\;.$ (39)

The condition for the divergence of (39) to be zero, $k^{\mu}_{\;\;;\mu}=0$,
on a given surface is equivalent to requiring that for this surface we have

$r(t)=2M(t,r(t))\;.$ (40)

It is worth noting that for the surfaces located in the vacuum part of the
solution this condition is equivalent with the condition for the EH.

###  II.4 Oppenheimer-Snyder black hole

In the following we will focus on the interior region of the solution.

For $C_{2}=0$, we have from (19) the formula

$M(t,r)=\frac{2}{9}\,\frac{\lambda^{2}\,r^{3}}{(\lambda t+C_{1})^{2}}\,,$ (41)

which resolved with respect to $r$ gives

$r=\Big{(}\frac{9}{2}\Big{)}^{1/3}M^{1/3}\Big{(}\frac{\lambda
t+C_{1}}{\lambda}\Big{)}^{2/3}\,.$ (42)

Eq. (42) shows that specifying $M$ and $t$ leads to corresponding $r$. Thus,
inserting into (42) $M=M(0,R)=M_{0}$ leads to

$r=r_{0}(t,R)=:f(R)g(t)\,.$ (43)

The covariant condition for spherically symmetric dust solution of Einstein
equations to be equivalent to the Friedmann–Lemaître–Robertson–Walker (FLRW)
model is vanishing of the shear, acceleration and rotation (see, e.g., [25]).
The only non-trivial quantity in our case is the shear $\sigma$, which reads

$\sigma=\frac{1}{3}\left(\frac{1}{r}\frac{\partial r}{\partial
t}-\left(\frac{\partial r}{\partial
R}\right)^{-1}\frac{\partial^{2}r}{\partial t\partial R}\right)\;.$ (44)

Taking $r=f(R)g(t)$ gives $\sigma=0$ and hence $C_{2}=0$ leads to the FLRW
model and after imposing the Schwarzschild solution in the outer region, to
the OS black hole. This can be achieved by prescribing the initial density
profile, e.g., with the use of the Heaviside step function. Such setup
guaranties the Schwarzschild solution in the exterior region throughout the
evolution.

The equation defining the horizon can be written, due to (40), in the form

$r_{0}(t_{h},R)=2M_{0}\,,$ (45)

where $t_{h}$ denotes the time at which the most outer shell reaches the
horizon. Resolving (45) with respect to $t_{h}$ leads to the formula
$t_{h}=t_{c}\mp\frac{4}{3}M_{0}$.

Since we consider the collapsing star, the comoving observer passes the
horizon before approaching the gravitational singularity. At the singularity
the classical dynamics breaks down. Thus, there is no classical evolution for
$t\geq t_{c}$ so that we choose

$t_{h}=t_{c}-\frac{4}{3}M_{0}\,.$ (46)

##  III Quantum Oppenheimer-Snyder black hole

In the standard approach to quantization of classical dynamics defined by the
Hamiltonian $H$, that is a generator of the dynamics, one maps $H$ onto an
operator $\hat{H}$ defined in some Hilbert space $\mathcal{H}$. If $\hat{H}$
is self-adjoint in $\mathcal{H}$, it can be used to define quantum dynamics in
the form of the Schrödinger equation defined in $\mathcal{H}$.

In the case considered in this paper, we do not quantize Hamilton’s dynamics,
but we use the solution to the classical dynamics to construct three quantum
observables generated by their classical counterparts: the time operator
$\hat{t}$, the radius operator $\hat{r}$ and the total mass operator
$\hat{M}$. To this purpose we use the classical dynamics similarly as in the
cosmological case considered in [32]. The classical dynamics is defined by the
equations of motion obtained by inserting the metric (9) into the Einstein
equations. In what follows we consider the marginally bound model so that the
dynamics is defined by Eq. (12). The equations of motion, with suitable set
of boundary condition, carry the same information as the general solution to
these equations. In our case as Eq. (19) presents the solution to Eq. (12)
so that the solution $M(t,r)$ includes the dynamics of this gravitational
system.

###  III.1 Integral quantization method

In what follows we apply the affine coherent states (ACS) quantization (see,
[8, 29, 30, 31, 32] and references therein) to the quantization of the
classical dynamics (12) corresponding to the marginally bound case. The
extension to the general dust case (10)–(11) and perfect fluid case can be
done by analogy.

We begin with introducing the extended configuration space $T$ for our system.
It is defined as follows [8]

$T=\\{(t,r)\in\mathbb{R}\times\mathbb{R}_{+}\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{R}_{+}=(0,+\infty)\,,$ (47)

where $t$ and $r$ are the time and radial coordinates, respectively, which
occur in the line element (9).

Since the configuration space is a half plane, it can be identified with the
affine group $\textrm{Aff}(\mathbb{R})=:G$ for which the multiplication rule
is given by

$(t^{\prime},r^{\prime})\cdot(t,r)=(r^{\prime}t+t^{\prime},r^{\prime}r),$ (48)

with the unity $(0,1)$ and the inverse

$(t,r)^{-1}=\left(-\frac{t}{r},\frac{1}{r}\right).$ (49)

The affine group has two, nontrivial, inequivalent irreducible unitary
representations. Both are realized in the Hilbert space
$\mathcal{H}=L^{2}(\mathbb{R}_{+},d\nu(x))$, where $d\nu(x)=dx/x$ is the
invariant measure on the multiplicative group $(\mathbb{R}_{+},\cdot)$. In
what follows we choose the one defined by

$U(t,r)\psi(x)=e^{itx}\psi(rx)\,.$ (50)

The integration over the affine group reads

$\int_{G}d\mu(t,r):=\frac{1}{2\pi}\int_{-\infty}^{\infty}dt\int_{0}^{\infty}dr/r^{2}\,,$
(51)

where the measure $d\mu(t,r)$ is left invariant.

Fixing the normalized vector $|\Phi_{0}\rangle\in
L^{2}(\mathbb{R}_{+},d\nu(x))$, called the fiducial vector, one can define a
continuous family of affine coherent states $|t,r\rangle\in
L^{2}(\mathbb{R}_{+},d\nu(x))$ as follows

$|t,r\rangle=U(t,r)|\Phi_{0}\rangle.$ (52)

The fiducial vector can be taken to be any vector of $\mathcal{H}$ that
satisfies certain conditions to be specified during the quantization process.
It is a sort of free “parameter” of ACS quantization. First of all, it should
be normalized so that we should have

$\langle\Phi_{0}|\Phi_{0}\rangle:=\int_{0}^{\infty}d\nu(x)\langle\Phi_{0}|x\rangle\langle
x|\Phi_{0}\rangle=\int_{0}^{\infty}d\nu(x)|\Phi_{0}(x)|^{2}=1\,,$ (53)

where we have used the formula [30]

$\int_{0}^{\infty}d\nu(x)|x\rangle\langle x|=\hat{1\kern-4.75pt1}\,,$ (54)

which applies to $\mathcal{H}$.

The space of coherent states is highly entangled in the sense that we have

$\langle t,r|t^{\prime},r^{\prime}\rangle\neq 0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{if}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ t\neq t^{\prime}\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{or}\leavevmode\nobreak\ \leavevmode\nobreak\ r\neq
r^{\prime}\,,$ (55) $\langle t,r|t,r\rangle=1\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{if}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\langle\Phi_{0}|\Phi_{0}\rangle=1\,.$ (56)

The irreducibility of the representation, used to define the coherent states
(52), enables making use of Schur’s lemma, which leads to the resolution of
the unity $\hat{1\kern-4.75pt1}$ in $L^{2}(\mathbb{R}_{+},d\nu(x))$:

$\int_{G}d\mu(t,r)|t,r\rangle\langle
t,r|=A_{\Phi_{0}}\;\hat{1\kern-4.75pt1}\;,$ (57)

where

$A_{\Phi_{0}}:=\int_{0}^{\infty}\frac{dp}{p^{2}}|\Phi_{0}(p)|^{2}<\infty\,.$
(58)

Making use of the resolution of the unity (57), we define the quantization of
a classical observable $f:T\rightarrow\mathbb{R}$ as follows

$f\longrightarrow\hat{f}:=\frac{1}{A_{\Phi_{0}}}\int_{\textrm{G}}d\mu(t,r)|t,r\rangle
f(t,r)\langle t,r|\,,$ (59)

where $\hat{f}:\mathcal{H}\rightarrow\mathcal{H}$ is the corresponding quantum
observable.

The mapping (59) is covariant in the sense that one has

$\displaystyle
U(\xi_{0})\hat{f}U^{\dagger}(\xi_{0})=\frac{1}{A_{\Phi_{0}}}\int_{\textrm{G}}d\mu(\xi)U(\xi_{0})|\xi\rangle
f(\xi)\langle\xi|U^{\dagger}(\xi_{0})$
$\displaystyle=\frac{1}{A_{\Phi_{0}}}\int_{\textrm{G}}d\mu(\xi)|\xi\rangle
f(\xi_{0}^{-1}\cdot\xi)\langle\xi|\,,$ (60)

where
$\xi_{0}^{-1}\cdot\xi=(t_{0},r_{0})^{-1}\cdot(t,r)=(\frac{t-t_{0}}{r_{0}},\frac{r}{r_{0}})$,
with $\xi:=(t,r)$. It means, no point in the configuration space $T$ is
privileged.

Eq. (59) defines a linear mapping and the observable $\hat{f}$ is a symmetric
operator by the construction. It results from the Schwartz inequality that
$|\langle t,r|\Psi\rangle|\leq\|\Psi\|$ for every vector $\Psi\in\mathcal{H}$
and every $(t,r)$. This implies that the sesquilinear form corresponding to
the operator $\hat{f}$ fulfill the following inequalities

$\displaystyle|\langle\Psi_{2}|\hat{f}|\Psi_{1}\rangle|\leq\frac{1}{A_{\phi}}\int_{\textrm{G}}d\mu(t,r)|f(t,r)||\langle
t,r|\Psi_{2}\rangle^{\star}||\langle t,r|\Psi_{1}\rangle|$
$\displaystyle\leq\left[\frac{1}{A_{\phi}}\int_{\textrm{G}}d\mu(t,r)|f(t,r)|\right]\|\Psi_{2}\|\|\Psi_{1}\|\,.$
(61)

The second inequality proves that for every finite value of the integral in
the square bracket, i.e., $\int_{\textrm{G}}d\mu(t,r)|f(t,r)|<\infty$, the
operator $\hat{f}$ is bounded so that it is a self-adjoint operator.

###  III.2 Quantum dynamics

We propose to apply the procedure of mapping a classical dynamics onto a
quantum dynamics considered in [32]. In that method, the time variable of the
classical level is mapped onto an operator acting at the quantum level so that
it is no longer an evolution parameter of the quantum level, but a quantum
observable like other quantum observables. In this way, the time is considered
on the same footing as the spatial coordinates.

In our classical model we have three observables: the time-like $t$, the
space-like $r$, and the mass $M(t,r)$. The maximum $M(t,r)$ equal to $M_{0}$,
represents the total energy of this system which has to be a conserved
quantity. The imposition of the energy conservation constraint onto the mass
operator $\hat{M}$, calculated in the appropriate Hilbert space, leads to a
quantum dynamics.

Classical observables should be related to the corresponding quantum
observables by their expectation values. This is the leading idea we use to
find quantum states of our system in a Hilbert space. In our case, these three
classical basic observables are mapped onto the corresponding operators
$\hat{t}$, $\hat{r}$, and $\hat{M}$ by using Eq. (59). Since $M$ is of a
fundamental importance, as it represents the global energy of considered
gravitational system, we treat the corresponding operator $\hat{M}$ as the
main quantum observable. We use the expectation value of $\hat{M}$ to
determine the family of vector states $\psi_{\eta}(t,r)$ in the Hilbert space
$\mathcal{K}=L^{2}(G,d\mu(t,r))$, where $\eta=(\eta_{1},\eta_{2})$ are the
parameters specifying the quantum evolution.

Consistently, we require the states $|\psi_{\eta}\rangle$ to satisfy the
constraint

$\langle\psi_{\eta}|\hat{M}|\psi_{\eta}\rangle=M_{0}\,,$ (62)

where $M_{0}$ is the mass of considered dust star defined by the density
$\rho$ of the star with the radius $r_{0}$. For the OS star with the density

$\rho(t,r)=\left\\{\begin{array}[]{lll}\frac{\lambda^{2}}{6\pi}\frac{1}{(\lambda
t+C_{1})^{2}}&\mbox{for}&r\leq r_{0}(t)\\\
0&\mbox{for}&r>r_{0}(t)\end{array}\right.\;,$ (63)

the corresponding mass defined by Eq. (2) reads

$M(t,r)=\left\\{\begin{array}[]{lll}\frac{2\lambda^{2}}{9}\frac{r^{3}}{(\lambda
t+C_{1})^{2}}&\mbox{for}&r\leq r_{0}(t)\\\
M_{0}&\mbox{for}&r>r_{0}(t)\end{array}\right.\;,$ (64)

so that the relationship between $M_{0}$ and $r_{0}$ is

$r_{0}(t)=\Big{(}\frac{9M_{0}}{2\lambda^{2}}\Big{)}^{1/3}(\lambda
t+C_{1})^{2/3}\,.$ (65)

Due to (64), the expression for the mass operator can be rewritten as

$\hat{M}=\frac{1}{2\pi
A_{\phi}}\int_{-\infty}^{\infty}dt\,\left\\{\int_{0}^{r_{0}(t)}\frac{dr}{r^{2}}|t,r\rangle\left[\frac{2\lambda^{2}}{9}\frac{r^{3}}{(\lambda
t+C_{1})^{2}}\right]\langle
t,r|+M_{0}\int_{r_{0}(t)}^{\infty}\frac{dr}{r^{2}}|t,r\rangle\langle
t,r|\right\\}\,.$ (66)

This operator fulfill the condition (III.1)

$|\langle\Psi_{2}|\hat{M}|\Psi_{1}\rangle|\leq
M_{0}\|\Psi_{2}\|\|\Psi_{1}\|\,,$ (67)

which implies that $\hat{M}$ is a bounded operator.

Using the resolution of the unity (57), the mass operator can be rewritten as

$\hat{M}=M_{0}\hat{1\kern-4.75pt1}+\frac{1}{2\pi
A_{\phi}}\int_{-\infty}^{\infty}dt\,\int_{0}^{r_{0}(t)}\frac{dr}{r^{2}}|t,r\rangle\left[\frac{2\lambda^{2}}{9}\frac{r^{3}}{(\lambda
t+C_{1})^{2}}-M_{0}\right]\langle t,r|\,.$ (68)

This form of the mass operator shows that the calculated mass is independent
of the shape of the wave function $\Psi(t,r):=\langle t,r|\Psi\rangle$ outside
of the star, i.e., for $r\geq r_{0}(t)$. This also implies that all states
$|\Psi_{M_{0}}\rangle\in\mathcal{H}$ satisfying the condition $\langle
t,r|\Psi_{M_{0}}\rangle=0$ for all $r\leq r_{0}(t)$ are eigenstates of the
mass operator which belong to the eigenvalue $M_{0}$

$\hat{M}|\Psi_{M_{0}}\rangle=M_{0}|\Psi_{M_{0}}\rangle\,,$ (69)

i.e., $M_{0}$ is the upper bound of the mass spectrum. On the other hand, the
mass function is non-negative. The condition $M(t,r)\geq 0$ determines the
lower bound of the mass operator spectrum to be equal to zero.

According to the interpretation of the state spaces of our quantum system
described in App. (C), the corresponding scalar product $\langle
t,r|\Psi_{M_{0}}\rangle$ should be interpreted as the probability amplitude of
finding our OS star at the time $t$ in a form of a spherical three dimensional
object with the radius $r$. The constructed mass operator $\hat{M}$ (see
(66)), measures the full mass of the OS star bounded by the classical radius
$r_{0}(t)$. This feature and the fact that $M_{0}$ is the maximum of the mass
operator, imply that all the amplitudes $\langle t,r|\Psi\rangle$ have to be
equal to zero inside the star, for every acceptable physical state
$|\Psi\rangle=|\Psi_{M_{0}}\rangle\in\mathcal{H}$.

The above suggests that the mass operator (66) has also eigenstates
corresponding to masses smaller than $M_{0}$. It is clear that all the states
$|\Psi\rangle\in\mathcal{H}$ for which the amplitudes $\langle
t,r|\Psi\rangle$ are not identically equal to zero inside of the star
correspond to smaller masses than $M_{0}$. In the case of an isolated OS star,
such states do not occur because of the conservation of the total energy.
However, some perturbations of the eigenstates (69) can lead to the mass
smaller than $M_{0}$ and possibly a new dynamics of the OS star.

Such mass variation may come, for example, from the physical process of the
Hawking radiation. However, some analyses suggest (see, e.g., [20]) that this
dissipative process has minor effect on deeply quantum regime inside black
hole (but should be implemented into analysis for completeness). Our formalism
gives some possibility of a description of such a process. However,
mathematical description requires the preparation of a kind of transition
operator which transforms a state of given mass to a state of lower mass. This
interesting feature requires further analysis and will be consider elsewhere.

In what follows, we consider the quantum states corresponding to the total
mass $M_{0}$, defined to be wave packets localized within a given time and
space intervals, and having additionally a “hole” including the singularity at
$t=t_{c}$. Qualitatively, each packet represents a wave function with compact
support. The wave function without a hole can be obtained in the limit
$\epsilon\to 0^{+}$. This type of wave packet can be written as

$\psi_{\eta_{1},\eta_{2}}(t,r)=\frac{1}{\sqrt{N}}\,r\,\chi_{\left(r(t,\eta_{1}),r(t,\eta_{1}+\delta_{1})\right)}(r)\,\chi_{\left(\eta_{2}-\frac{\delta_{2}}{2},\eta_{2}+\frac{\delta_{2}}{2}\right)}(t)\,\left(1-\chi_{\left(t_{c}-\frac{\epsilon}{2},t_{c}+\frac{\epsilon}{2}\right)}(t)\right)\,,$
(70)

where $\delta_{1},\delta_{2}>0$, $\eta_{1}\in\langle 0,\infty)$,
$\eta_{2}\in(-\infty,\infty)$. In addition we assume the condition
$0<\epsilon<\delta_{2}$, which defines the wave functions with the width
larger than the hole around the singularity. It means, we consider a subset of
wave functions which totally cover the hole. They are equal to zero within the
hole and are different from zero outside the hole. In this way they connect
both regions separated by the singularity.

In what follows we use the following auxiliary functions: The characteristic
function $\chi_{(a,b)}(x)$ of the set $(a,b)$ defined as

$\chi_{(a,b)}(x)=\left\\{\begin{array}[]{lll}1&\mbox{for}&x\in(a,b)\\\
0&\mbox{for}&x\not\in(a,b)\end{array}\right.\;,$ (71)

and the function $r(x,y)$ defined to be

$r(x,y)=\sqrt[3]{\frac{9}{2}}y^{\frac{1}{3}}(x-t_{c})^{\frac{2}{3}}\;,\qquad
t_{c}=-\frac{C_{1}}{\lambda}\,,$ (72)

which represents the radius of the shell with the mass $y$ at the time $x$.

Next, we define two new functions in terms of integrals

$\mathcal{G}_{\alpha}(a,b)=\int_{a}^{b}dx\,x^{\alpha-1}=\frac{1}{\alpha}\left(b^{\alpha}-a^{\alpha}\right)\,,\quad\alpha\neq
0\,,$ (73)

and

$\displaystyle\mathcal{F}_{\alpha}(\eta_{2})=\int_{\eta_{2}-\frac{\delta_{2}}{2}}^{\eta_{2}+\frac{\delta_{2}}{2}}dx\,(x-t_{c})^{\alpha-1}\left(1-\chi_{\left(t_{c}-\frac{\epsilon}{2},\;t_{c}+\frac{\epsilon}{2}\right)}(x)\right)$
(74)
$\displaystyle=\left\\{\begin{array}[]{lll}\mathcal{G}_{\alpha}\left(\eta_{2}^{-},\eta_{2}^{+}\right)&\mbox{for}&\eta_{2}\leq
t_{c}-\frac{\epsilon+\delta_{2}}{2}\\\
\mathcal{G}_{\alpha}\left(\eta_{2}^{-},-\frac{\epsilon}{2}\right)&\mbox{for}&t_{c}-\frac{\epsilon+\delta_{2}}{2}<\eta_{2}\leq
t_{c}-\frac{\delta_{2}-\epsilon}{2}\\\
\mathcal{G}_{\alpha}\left(\eta_{2}^{-},-\frac{\epsilon}{2}\right)+\mathcal{G}_{\alpha}\left(\frac{\epsilon}{2},\eta_{2}^{+}\right)&\mbox{for}&t_{c}-\frac{\delta_{2}-\epsilon}{2}<\eta_{2}\leq
t_{c}+\frac{\delta_{2}-\epsilon}{2}\\\
\mathcal{G}_{\alpha}\left(\frac{\epsilon}{2},\eta_{2}^{+}\right)&\mbox{for}&t_{c}+\frac{\delta_{2}-\epsilon}{2}<\eta_{2}\leq
t_{c}+\frac{\delta_{2}+\epsilon}{2}\\\
\mathcal{G}_{\alpha}\left(\eta_{2}^{-},\eta_{2}^{+}\right)&\mbox{for}&\eta_{2}>t_{c}+\frac{\delta_{2}+\epsilon}{2}\end{array}\right.$
(80)

where

$\eta_{2}^{-}:=\eta_{2}-\frac{\delta_{2}}{2}-t_{c},\qquad\eta_{2}^{+}:=\eta_{2}+\frac{\delta_{2}}{2}-t_{c}\leavevmode\nobreak\
\,.\\\ $ (81)

It is worth to notice that we have

$\mathcal{F}_{\alpha}(\eta_{2})\xrightarrow[]{\epsilon\rightarrow
0}\left\\{\begin{array}[]{lll}\mathcal{G}_{\alpha}(\eta_{2}^{-},\eta_{2}^{+})&\mbox{for}&\alpha>0\\\
\infty&\mbox{for}&\alpha<0\,.\end{array}\right.$ (82)

The normalization constant of the wave packets (70) reads

$N=\frac{1}{2\pi
A_{\phi}}\frac{1}{3}\,\sqrt[3]{\frac{9}{2}}\;\mathcal{G}_{\frac{1}{3}}(\eta_{1},\eta_{1}+\delta_{1})\;\mathcal{F}_{\frac{5}{3}}(\eta_{2})\,.$
(83)

The expectation value of the operator $\hat{t}$ is the following

$\displaystyle
t_{Q}(\eta_{1},\eta_{2}):=\langle\psi_{\eta_{1},\eta_{2}}|\hat{t}|\psi_{\eta_{1},\eta_{2}}\rangle$
$\displaystyle=\frac{1}{2\pi
A_{\phi}}\frac{1}{N}\int_{\eta_{2}-\frac{\delta_{2}}{2}}^{\eta_{2}+\frac{\delta_{2}}{2}}dt\,t\left(1-\chi_{\left(t_{c}-\frac{\epsilon}{2},t_{c}+\frac{\epsilon}{2}\right)}(t)\right)\int_{r(t,\eta_{1})}^{r(t,\eta_{1}+\delta_{1})}dr$
$\displaystyle=\frac{\mathcal{F}_{\frac{8}{3}}(\eta_{2})}{\mathcal{F}_{\frac{5}{3}}(\eta_{2})}+t_{c}\,.$
(84)

The expectation value of the operator $\hat{r}$ is found to be

$\displaystyle
r_{Q}(\eta_{1},\eta_{2}):=\langle\psi_{\eta_{,}\eta_{2}}|\hat{r}|\psi_{\eta_{1},\eta_{2}}\rangle$
$\displaystyle=\frac{1}{2\pi
A_{\phi}}\frac{1}{N}\int_{\eta_{2}-\frac{\delta_{2}}{2}}^{\eta_{2}+\frac{\delta_{2}}{2}}dt\left(1-\chi_{\left(t_{c}-\frac{\epsilon}{2},t_{c}+\frac{\epsilon}{2}\right)}(t)\right)\,\int_{r(t,\eta_{1})}^{r(t,\eta_{1}+\delta_{1})}dr\,r$
$\displaystyle=\sqrt[3]{\frac{9}{2}}\;\frac{\mathcal{G}_{\frac{2}{3}}(\eta_{1},\eta_{1}+\delta_{1})}{\mathcal{G}_{\frac{1}{3}}(\eta_{1},\eta_{1}+\delta_{1})}\;\frac{\mathcal{F}_{\frac{7}{3}}(\eta_{2})}{\mathcal{F}_{\frac{5}{3}}(\eta_{2})}\,.$
(85)

The expectation value of the mass operator $\hat{M}$ reads

$\langle\psi_{\eta_{1},\eta_{2}}|\hat{M}|\psi_{\eta_{1},\eta_{2}}\rangle=M_{0}\,,$
(86)

where $\eta_{1}\geq M_{0}$.

For very small values of $\delta_{2}$ and $\epsilon$ the functions
$t_{Q}(\eta_{1},\eta_{2})$ and $r_{Q}(\eta_{1},\eta_{2})$ can be approximated
as follows

$\displaystyle t_{Q}(\eta_{1},\eta_{2})\approx\eta_{2}\,,$ (87)
${}_{Q}(\eta_{1},\eta_{2})\approx\sqrt[3]{\frac{9}{2}}\eta_{1}^{\frac{1}{3}}(\eta_{2}-t_{c})^{\frac{2}{3}}\,.$
(88)

Comparing these formulae with their classical counterparts, we get
$\eta_{1}=M_{0}$, and we see that the parameter $\eta_{2}$ corresponds to the
classical time (but only to some extent).

With fixed $\eta_{1}$, the quantum radius of the OS star
$r_{Q}(\eta_{1},\eta_{2})$ is found to achieve its minimum at $\eta_{2}=t_{c}$
and reads

$r_{Q}(\eta_{1},t_{c})=\sqrt[3]{\frac{9}{2}}\;\frac{\mathcal{G}_{\frac{2}{3}}(\eta_{1},\eta_{1}+\delta_{1})}{\mathcal{G}_{\frac{1}{3}}(\eta_{1},\eta_{1}+\delta_{1})}\;\frac{5}{7}\left(\frac{1}{2}\right)^{\frac{2}{3}}\frac{\delta_{2}^{\frac{7}{3}}-\epsilon^{\frac{7}{3}}}{\delta_{2}^{\frac{5}{3}}-\epsilon^{\frac{5}{3}}}\,.$
(89)

The radius $r_{Q}(\eta_{1},t_{c})$ represents the bouncing radius of the OS
star. The corresponding quantum time $t_{Q}(\eta_{1},t_{c})=t_{c}$, as
$\mathcal{F}_{\frac{8}{3}}(t_{c})=0$ in (84), so that it is equal to the time
at which the classical singularity is achieved. After the bounce the quantum
star expands (see Figure 1). This phenomenon does not disappear even for the
states without the whole, i.e., for $\epsilon=0$.

Figure 1: The $r_{Q}(\eta_{1},\eta_{2})$ function (see Eq. (85)) for
$\eta_{2}\in[-10,10]$, with $\eta_{1}=2=M_{0}$ (blue curve), $\eta_{1}=5$
(orange curve), $\eta_{1}=20$ (green curve), where $\delta_{1}=\delta_{2}=1$,
$\epsilon=10^{-5}$, $t_{c}=5$.

Another interesting aspect of the evolution of the OS star is its horizon,
which the collapsing star reaches, corresponding to the classical time $t_{h}$
given by the formula (40). To estimate the parameters $\eta_{2}$ , while
$\eta_{1}=M_{0}$, let us impose the condition

$t_{Q}(M_{0},\eta_{2})=t_{h}=t_{c}-\frac{4}{3}M_{0}.$ (90)

For small $\delta_{1},\delta_{2}$ and $\epsilon$, the approximate solution of
Eq. (90) is found to be $\eta_{2}=t_{c}-\frac{4}{3}M_{0}$. In this
approximation the horizon radius $r_{Q}(M_{0},\eta_{2})\approx 2M_{0}$, i.e.,
it reproduces the classical condition for the horizon.

The vector state in the form of the wave packet (70) is well defined for any
$t$, including the neighborhood of the singularity at $t_{c}$. Thus, the
quantum dynamics in terms of expectation values (84) and (85) is well defined
before and after the singularity. See, for instance, the case of a very small
$\delta_{2}$ and $\epsilon$ presented by (87) and (88). In particular, the
equation

$\langle\psi_{\eta_{1},\eta_{2}}|\hat{r}|\psi_{\eta_{1},\eta_{2}}\rangle=2\;\langle\psi_{\eta_{1},\eta_{2}}|\hat{M}|\psi_{\eta_{1},\eta_{2}}\rangle\,,$
(91)

for $\eta_{1}=M_{0}$, is formally well defined. This equation corresponds to
the classical equation (40) that determines the horizon (46), but is defined
at the quantum level. Due to (85) and (86), Eq. (91) can be rewritten as

$\sqrt[3]{\frac{9}{2}}\;\frac{\mathcal{G}_{\frac{2}{3}}(M_{0},M_{0}+\delta_{1})}{\mathcal{G}_{\frac{1}{3}}(M_{0},M_{0}+\delta_{1})}\;\frac{\mathcal{F}_{\frac{7}{3}}(\eta_{2})}{\mathcal{F}_{\frac{5}{3}}(\eta_{2})}=2M_{0}\,.$
(92)

Let us solve numerically Eq. (92). We take $t_{c}=1$ and $M_{0}=\eta_{1}=1$;
the state parameters $\delta_{1}=0.01$, $\delta_{2}=0.2$, and $\epsilon=0.1$.
For such values of the state parameters Eq. (91) has two solutions for
$\eta_{2}$, which we denote as $\eta_{2h}=1\mp 1.32877\approx
t_{c}\mp\frac{4}{3}M_{0}$. The second solution, $\eta_{2h}=1+1.32877$, that
follows the bounce does not have as clear physical meaning as the one
preceding the bounce. It will be further discussed in Sec. V.

These numerical results are not changed qualitatively for different choice of
parameters and also for the states without a hole, i.e., for $\epsilon=0$.

###  III.3 Fate of classical singularities

In what follows we address the issue of the classical singularities of the
Oppenheimer-Snyder black hole at the quantum level.

Let us introduce the following function

$\mathcal{A}(\alpha,\beta,\gamma)=\frac{1}{(t-t_{c})^{\alpha}}M(t,r)^{\beta}r^{-\gamma}\,.$
(93)

Our curvature invariants (36)-(38) can be expressed in terms of $\mathcal{A}$
in the following way:

$\displaystyle R_{\mu\nu}R^{\mu\nu}=8\,\mathcal{A}(2,1,3)\,,$ (94)
$\displaystyle\mathcal{R}=(-2\sqrt{2})\,\mathcal{A}\left(1,\frac{1}{2},\frac{3}{2}\right)\,,$
(95) $\displaystyle
K=48\,\mathcal{A}(2,1,3)+32\sqrt{2}\,\mathcal{A}\left(2,\frac{3}{2},\frac{9}{2}\right)+48\,\mathcal{A}(0,2,6)\,.$
(96)

The expectation value of the operator $\hat{\mathcal{A}}(\alpha,\beta,\gamma)$
is found to be:

$\displaystyle\langle\psi_{\eta_{1},\eta_{2}}|\hat{\mathcal{A}}(\alpha,\beta,\gamma)|\psi_{\eta_{1},\eta_{2}}\rangle$
$\displaystyle=\frac{1}{2\pi
A_{\psi}}\,\frac{1}{N}\int_{\eta_{2}-\frac{\delta_{2}}{2}}^{\eta_{2}+\frac{\delta_{2}}{2}}dt\left(1-\chi_{\left(t_{c}-\frac{\epsilon}{2},t_{c}+\frac{\epsilon}{2}\right)}(t)\right)\,\int_{r(t,\eta_{1})}^{r(t,\eta_{1}+\delta_{1})}dr\frac{M(t,r)^{\beta}}{(t-t_{c})^{\alpha}r^{\gamma}}$
$\displaystyle=\left(\frac{2}{9}\right)^{\frac{\gamma}{3}}\frac{\mathcal{F}_{\frac{5}{3}-\frac{2}{3}\gamma-\alpha}(\eta_{2})}{\mathcal{F}_{\frac{5}{3}}(\eta_{2})}\cdot$
(97)
$\displaystyle\cdot\left\\{\begin{array}[]{lll}\frac{\mathcal{G}_{\beta-\frac{\gamma}{3}+\frac{1}{3}}(\eta_{1},\eta_{1}+\delta_{1})}{\mathcal{G}_{\frac{1}{3}}(\eta_{1},\eta_{1}+\delta_{1})}&\mbox{for}&\eta_{1}<M_{0}-\delta_{1}\\\
\frac{\mathcal{G}_{\beta-\frac{\gamma}{3}+\frac{1}{3}}(\eta_{1},M_{0})+M_{0}^{\beta}\mathcal{G}_{\frac{1}{3}-\frac{\gamma}{3}}(M_{0},\eta_{1}+\delta_{1})}{\mathcal{G}_{\frac{1}{3}}(\eta_{1},\eta_{1}+\delta_{1})}&\mbox{for}&M_{0}-\delta_{1}\leq\eta_{1}<M_{0}\,.\\\
M_{0}^{\beta}\frac{\mathcal{G}_{\frac{1}{3}-\frac{\gamma}{3}}(\eta_{1},\eta_{1}+\delta_{1})}{\mathcal{G}_{\frac{1}{3}}(\eta_{1},\eta_{1}+\delta_{1})}&\mbox{for}&\eta_{1}\geq
M_{0}\end{array}\right.$ (100)

Is it possible for the expectation value of the operator $\hat{\mathcal{A}}$
to be infinite? Let us examine this issue:

Since the numerators in Eq.  (III.3) are defined in terms of the functions
$\mathcal{G}_{\alpha}(a,b)$, the infinity can be achieved if and only if one
has division by zero. In the denominators we get two kinds of functions which
potentially can be equal to zero. The first one is
$\mathcal{G}_{\frac{1}{3}}(a,b)=0$ for $a=b$. This condition cannot be
satisfied for $\delta_{1}>0$. The second one is the function
$\mathcal{F}_{\frac{5}{3}}(\eta_{2})$ which is a combination of the functions
$\mathcal{G}$. Making use of the definition (74) and the fact that
$\mathcal{G}_{\frac{5}{3}}(a,b)=0$ is true only for $a=b$, the only case of
division by zero is when $\eta_{2}^{-}\leq-\frac{\epsilon}{2}$ and
$\frac{\epsilon}{2}\leq\eta_{2}^{+}$. However, in the case
$\delta_{2}>\epsilon$ the two boundary conditions
$\eta_{2}^{-}=-\frac{\epsilon}{2}$ and $\frac{\epsilon}{2}=\eta_{2}^{+}$ can
not be fulfilled simultaneously. Therefore, the function
$\mathcal{F}_{\frac{5}{3}}(\eta_{2})$ is not equal to zero for any $\eta_{2}$.
We conclude that the expectation value of the operator $\hat{\mathcal{A}}$ for
the states $\psi_{\eta_{1}\eta_{2}}$ is not singular. Therefore, the
expectation values of the operators corresponding to the invariants (94)-(96)
nowhere diverge.

On the other hand, due to the equation (82), it is easily seen that we have

$\langle\psi_{\eta_{1},\eta_{2}}|\hat{\mathcal{A}}|\psi_{\eta_{1},\eta_{2}}\rangle\xrightarrow[]{\epsilon\rightarrow
0}\infty\,,$ (101)

so that the expectation values of the operators corresponding to (94)–(96)
diverge.

It is instructive to examine the variance of this operator in the case
$\epsilon\rightarrow 0^{+}$. The variance of the symmetric operator
$\hat{\mathcal{A}}$ in the quantum state $|\psi\rangle\in\mathcal{K}$ with
compact support, for $\epsilon>0$, is the following [8, 33]

$\mathrm{var}(\hat{\mathcal{A}},\psi):=\langle\psi|\big{(}\hat{\mathcal{A}}-\langle\hat{\mathcal{A}}\rangle_{\psi}\hat{1\kern-4.75pt1}\big{)}^{2}\psi\rangle=\langle\big{(}\hat{\mathcal{A}}-\langle\hat{\mathcal{A}}\rangle_{\psi}\hat{1\kern-4.75pt1}\big{)}\psi|\big{(}\hat{\mathcal{A}}-\langle\hat{\mathcal{A}}\rangle_{\psi}\hat{1\kern-4.75pt1}\big{)}\psi\rangle=\|\big{(}\hat{\mathcal{A}}-\langle\hat{\mathcal{A}}\rangle_{\psi}\hat{1\kern-4.75pt1}\big{)}\psi\|^{2}\,,$
(102)

where
$\langle\hat{\mathcal{A}}\rangle_{\psi}:=\langle\psi|\hat{\mathcal{A}}\psi\rangle$.
Thus, $\mathrm{var}(\hat{\mathcal{A}},\psi)>0$ for any non-zero vector of the
Hilbert space $\mathcal{K}$ if and only if
$\hat{\mathcal{A}}|\psi\rangle\neq\langle\hat{\mathcal{A}}\rangle_{\psi}\hat{1\kern-4.75pt1}\psi\rangle$.
The latter means that $|\psi\rangle$ is not an eigenstate of
$\hat{\mathcal{A}}$, which is the case for the wave packet (70) for any
$\epsilon$, and in particular for $\epsilon=0$ due to (101). Thus, the
singularity in this case does occur, but it is smeared. Consequently, the
expectation values of the operators corresponding to the invariants (94)–(96)
are singular, but these singularities occur with non-zero fluctuations.

##  IV Quantum Schwarzschild black hole

Recently, we have quantized the Schwarzschild spacetime with the negative mass
parameter, $M<0$, to address the case with a naked singularity [8, 36]. Here
we consider the case with a covered singularity $M>0$. The case $M=0$
corresponds to the Minkowski spacetime.

The Schwarzschild black hole (SBH) is a spherically symmetric, static,
solution of the Einstein field equations, parameterized by the constant $M>0$,
having the horizon at $r=2M$, and the gravitational singularity at $r=0$. It
is an exact spacetime geometry of a non-rotating “point particle” with the
mass parameter $M$ devoid of an “internal” dynamics [37].

Let us identify the SBH within our formalism. To this end, let us consider the
line element defined by (9) and (10) with $E=0$. It is clear that constant $M$
satisfies Eq. (12). To see that it corresponds to the SBH, we argue as
follows. Evaluating the Einstein tensor with so obtained metric leads to zero,
which means that it corresponds to the vacuum case. The Birkhoff theorem
implies that this spherically symmetric vacuum solution must be diffeomorphic
to the Schwarzschild metric (for more details see [4]).

It results from (31)–(46) that the only non vanishing invariant is the
Kretschmann scalar $K=48M^{2}r^{-6}$. In our recent paper [8] we have shown
that quantization smears the singularity indicated by the Kretschmann scalar
avoiding its localization in the configuration space. Here we can obtain quite
similar result using the quantum state (70) with $\epsilon=0$. Applying the
reasoning following Eq. (101) leads to the result

$\mathrm{var}(\hat{K};\psi_{\eta_{1},\eta_{2}})>0,$ (103)

where $\hat{K}$ is the quantum operator corresponding to the Kretschmann
scalar $K$. Thus, the singularity in this case does occur, but it is smeared.

However, making use of the quantum state (70) with $\epsilon>0$, we obtain the
result (III.3) specialized to the case of the Kretschmann operator. Therefore,
we conclude that the expectation value of the Kretschmann operator is finite,
which resolves the classical singularity problem.

##  V Conclusions

We have shown in the paper that the evolution of the considered OS dust star
model consists of three phases: classical collapse towards the gravitational
singularity, quantum evolution, and classical expansion away from the
singularity. The quantum evolution consists of quantum collapse, strongly
quantum regime, and quantum expansion. The quantum collapse and expansion are
described by the evolution of the expectation value of the position operator.
The strong quantum regime may present regular quantum bounce or smeared
singularity.

We have obtained the above scenario by making use of the vector state in the
form of the wave packet with compact support parameterized by two real
parameters $\eta_{1}$ and $\eta_{2}$. Thus, these vector states represent a
wide class of quantum states in considered Hilbert space
$\mathcal{K}^{\prime}$ (see App. C). Therefore, our resolution of the
classical gravitational singularity of the OS model is not generic, but
possible within considered quantum system ascribed to that classical one.

Separate interpretation needs the issue of the horizons of considered isolated
gravitational system. As the system is spherically symmetric, it possesses the
horizon that is formed during the first classical phase of its evolution if
the dust density is high enough (indirectly due to the Birkhoff theorem). The
second expression for the horizon, derived from Eq. (91), following the bounce
does not have so clear meaning. We suggest, it may describe the disappearance
of the first horizon created before the bounce. In such case, the bouncing
time of considered black hole can be estimated by $\frac{8}{3}M_{0}$. The
quantum evolution may describe the so called black-to-white hole transition
(see, e.g., [14] and references therein).

We present the quantum Schwarzschild spacetime for the positive mass
parameter, $M>0$, promised to be carried out in our recent paper [8]
concerning the case $M<0$. That paper concerns the naked singularity. Here we
are concerned with the covered singularity, i.e. the black hole case. The
resolution of the classical singularity is done in the way quite similar as
for the OS spacetime owing to the fact that the former is a special case of
the latter.

Our quantum description of the collapsing dust star corresponds to the flat
sector of the dust collapse presented in [19] (while described in comoving
coordinates). However, we apply different parametrization of the affine group.
It is worth to mention that different parameterizations may lead to unitarily
inequivalent quantum theories [30]. In our parametrization the expectation
values of elementary observables calculated in coherent states coincide with
corresponding classical observables. Therefore, we use the quantum states in
the form of wave packets. They enable obtaining the quantum bounce of the
space operator. All the states parameterized by the parameters like deltas’and
epsilon in Eq. (70) satisfy required physical condition and improve
flexibility of the model. In principle, such parameters can be determined
either by observational data or more realistic model of matter field. Another
difference is that we have analysed the evolution of the curvature invariants.
Obtained expectation values of corresponding quantum operators are either
singular but smeared, or regular.

In both approaches, ours and that of [19], the bouncing time of considered
quantum black hole is numerically similar. In both papers, the issue of
possible Hawking radiation is relegated to future papers.

In this paper we have used an idea of time to be a quantum observable,
similarly to the space position quantum observable. A proposal of treating
time on the same footing as other quantum observables is recently published as
the Projection Evolution approach (PEv) in [38]. In the present paper we do
not apply the full PEv formalism which allows, in principle, to get detailed
dynamics of the system by construction of the so called evolution operators.
Instead, we build a family of possible, but not all, evolution scenarios of
our OS star by constructing a family of states with required physical
properties. One of the most important feature of our wave packets is that in a
limit of very narrow functions $\psi_{\eta_{1},\eta_{2}}$ the expectation
values of $\hat{t}$,$\leavevmode\nobreak\ \hat{r}$ and $\hat{M}$ operators
reproduce the classical motion of the OS star, far from singularity. This
constraint allows to choose some possible evolution paths and calculate them
as relations among expectation values of the fundamental observables in this
model, i.e., the time, the position and the global mass.

Our next paper will concern the quantum system ascribed to the LTB model of
collapsing star. Within this model one can consider the inclusion of realistic
features of matter field such as pressure and inhomogeneity in density
distribution, and reasonable equation of state. Considering the cases with
covered singularity and naked singularity separately, will enable making
deeper insight into the issue of the quantum bounce.

###### Acknowledgements.

We would like to thank Claus Kiefer, Jerzy Lewandowski and Edward Wilson-Ewing
for helpful discussions. JJO acknowledges the support of the National Science
Centre (NCN, Poland) under the Sonata-15 research grant
UMO-2019/35/D/ST9/00342.

##  Appendix A General analytical solution

The equations of (21)–(22) have analytical solutions depending on the sign of
$E$ function. They can be written in the parametric form:

$\displaystyle E<0$
$\displaystyle,\;\;r=-\frac{M}{E}\left(1-\cos\eta\right),\;\;\eta-\sin\eta=\frac{\left(-E\right)^{3/2}}{M}\left(t-t_{B}(R)\right),$
$\displaystyle E=0$
$\displaystyle,\;\;r=\left(\frac{9}{2}M\left(t-t_{B}(R)\right)^{2}\right)^{1/3},$
$\displaystyle E>0$
$\displaystyle,\;\;r=\frac{M}{E}\left(\cosh\eta-1\right),\;\;\sinh\eta-\eta=\frac{\left(E\right)^{3/2}}{M}\left(t-t_{B}(R)\right),$

where $\eta$ is an auxiliary parameter and $t_{B}(R)$ is the integration
function. The cases $E>0$, $E<0$ or $E=0$ correspond to unbounded, bounded,
and marginally bounded models, respectively (see, e.g. [25] for more details).

##  Appendix B Coordinate transformations

In order for this article to be self-contained we present the transformations
between different coordinate systems used throughout the text (see, e.g.
[Gieseletal]). Let us recall that we denote the generalised Painlevé-
Gullstrand coordinates (GPG) as $(t,r,\theta,\phi)$, and co-moving synchronous
coordinates as $(\tau,R,\theta,\phi)$. We can derive the proper coordinate
transformation by direct inspection of the line elements. The LTB metric in
co-moving synchronous coordinates reads

$\mathrm{d}s^{2}=-\mathrm{d}\tau^{2}+\frac{\left(\frac{\partial r}{\partial
R}\right)^{2}}{1+E}\mathrm{d}R^{2}+r^{2}\mathrm{d}\Omega^{2}\;.$ (105)

The GPG coordinates are defined via $t=\tau$ and $r=r(R,\tau)$. Since

$\mathrm{d}r=\frac{\partial r}{\partial R}\mathrm{d}R+\frac{\partial
r}{\partial\tau}\mathrm{d}\tau\;,$ (106)

we get by direct substitution

$\mathrm{d}s^{2}=-\mathrm{d}\tau^{2}+\frac{\left(\mathrm{d}r-\frac{\partial
r}{\partial\tau}\mathrm{d}\tau\right)^{2}}{1+E}+r^{2}\mathrm{d}\Omega^{2}\;.$
(107)

Using (21) (with the negative sign of the root) we arrive at

$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+\frac{(\sqrt{2M/r+E}\,\mathrm{d}t+\mathrm{d}r)^{2}}{1+E}+r^{2}\mathrm{d}\Omega^{2}\,,$
(108)

where we re-inserted the $t$ coordinate. It is worth noticing that the field
equations (10)–(11) and (21)–(22) are equivalent (multiply both sides of (10)
by $\left(\partial M/\partial r\right)^{-1}$.

##  Appendix C The state spaces

Within the affine quantization method, we deal with two Hilbert spaces. The
first one $\mathcal{H}=L^{2}(\mathbb{R}_{+},d\nu(x))$ is the carrier space of
the unitary representation $U(t,r)$ of the affine group
$\textrm{Aff}(\mathbb{R})$. The second Hilbert space
$\mathcal{K}=L^{2}(\textrm{Aff}(\mathbb{R}),d\mu(t,r))$, of square integrable
functions on the affine group $\textrm{Aff}(\mathbb{R})$ with the scalar
product

$(\Phi_{2}|\Phi_{1}):=\frac{1}{A_{\phi}}\int_{\textrm{Aff}(\mathbb{R})}d\mu(t,r)\,,\Phi_{2}(t,r)^{\star}\Phi_{1}(t,r)$
(109)

provides a physical interpretation of the vectors $\Psi\in\mathcal{H}$ using
the mapping $\iota_{G}:\mathcal{H}\to\mathcal{K}$ defined as:

$\iota_{G}(\Psi):=\langle t,r|\Psi\rangle\,,$ (110)

where $\langle t,r|\Psi\rangle\in\mathcal{K}$ is a probability amplitude of
localization of the state $\Psi\in\mathcal{H}$ in the coherent state
$|t,r\rangle$. The coherent states, in turn, are quantum counterparts of the
classical configuration space points, where

$\kappa:T\ni(t,r)\to|t,r\rangle\in\mathcal{H}$ (111)

is a one-to-one mapping between classical and quantum configuration spaces.

The mapping (110) is an unitary transformation of the Hilbert space
$\mathcal{H}$ into a subspace $\mathcal{K}^{\prime}$ of the Hilbert space
$\mathcal{K}$, i.e., it is not a one-to-one transformation between the spaces
$\mathcal{H}$ an $\mathcal{K}$. Using standard decomposition of any vector
$|\Psi\rangle\in\mathcal{H}$ into the coherent states $|t,r\rangle$, one gets
the equality of the scalar products in both spaces $\mathcal{H}$ and
$\mathcal{K}$

$\langle\Psi_{2}|\Psi_{1}\rangle=(\iota_{G}(\Psi_{2})|\iota_{G}(\Psi_{1}))$
(112)

for all $\Psi_{2},\Psi_{1}\in\mathcal{H}$.

For further considerations let us introduce a projection operator
$\hat{\mathcal{N}}$ in the state space $\mathcal{K}$ defined as

$\hat{\mathcal{N}}\Phi(t,r):=\frac{1}{A_{\Phi}}\int_{\textrm{Aff}(\mathbb{R})}d\mu(t^{\prime},r^{\prime})\langle
t,r|t^{\prime},r^{\prime}\rangle\Phi(t^{\prime},r^{\prime})\,.$ (113)

A direct calculation shows that for every $\Psi\in\mathcal{H}$ the amplitude
$\langle t,r|\Psi\rangle$ belongs to the subspace
$\hat{\mathcal{N}}\mathcal{K}$, i.e.,

$\hat{\mathcal{N}}\,\langle t,r|\Psi\rangle=\langle t,r|\Psi\rangle\,.$ (114)

On the other hand, every $\Psi\in\mathcal{H}$ can be expanded into the
coherent states

$\Psi(x)=\frac{1}{A_{\phi}}\int_{\textrm{Aff}(\mathbb{R})}d\mu(t,r)\langle
x|t,r\rangle\langle t,r|\Psi\rangle$ (115)

what implies that Eq. (115) defines an inverse transformation to the unitary
mapping (110) between the spaces $\mathcal{H}$ and $\mathcal{K}^{\prime}$. The
subspace $\mathcal{K}^{\prime}\subset\mathcal{K}$ is spanned by the functions
$\langle t,r|\Psi\rangle$, which symbolically can be written as

$\mathcal{K}^{\prime}=\mathrm{Hilbert}\\{\iota_{G}(\Psi):\Psi\in\mathcal{H}\\}\,.$
(116)

Summarizing, we have the following hierarchy of Hilbert spaces:
$\mathcal{K}^{\prime}\subset\hat{\mathcal{N}}\mathcal{K}\subset\mathcal{K}$
and the spaces $\mathcal{H}$ and $\mathcal{K}^{\prime}$ are unitarily
isomorphic.

These properties allow to use either the carrier Hilbert space $\mathcal{H}$
or the Hilbert space $\mathcal{K}^{\prime}$ to perform calculations, depending
on context, while keeping their appropriate physical interpretations.

In the case of the spherical symmetry, $(t,r)\in T$ denotes the time $t$ and
the two dimensional sphere with the radius $r$. That point is represented by
the coherent state $|t,r\rangle\in\mathcal{H}$ for which the expectation
values of the time and the radius operators are $\langle
t,r|\hat{t}|t,r\rangle=t$ and $\langle t,r|\hat{r}|t,r\rangle=r$,
respectively. The scalar product $\langle t,r|\Psi\rangle$, where
$|\Psi\rangle\in\mathcal{H}$, is the probability amplitude of the localization
of the system at the time $t$ in a form of a three dimensional spherical
object with the radius $r$.

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